% Copyright 1990 - 1995 by AT&T Bell Laboratories. % Permission to use, copy, modify, and distribute this software % and its documentation for any purpose and without fee is hereby % granted, provided that the above copyright notice appear in all % copies and that both that the copyright notice and this % permission notice and warranty disclaimer appear in supporting % documentation, and that the names of AT&T Bell Laboratories or % any of its entities not be used in advertising or publicity % pertaining to distribution of the software without specific, % written prior permission. % AT&T disclaims all warranties with regard to this software, % including all implied warranties of merchantability and fitness. % In no event shall AT&T be liable for any special, indirect or % consequential damages or any damages whatsoever resulting from % loss of use, data or profits, whether in an action of contract, % negligence or other tortious action, arising out of or in % connection with the use or performance of this software. % Much of this program was copied with permission from MF.web Version 1.9 % It interprets a language very similar to D.E. Knuth's METAFONT, but with % changes designed to make it more suitable for PostScript output. % TeX is a trademark of the American Mathematical Society. % METAFONT is a trademark of Addison-Wesley Publishing Company. % PostScript is a trademark of Adobe Systems Incorporated. % Here is TeX material that gets inserted after \input webmac \def\hang{\hangindent 3em\noindent\ignorespaces} \def\textindent#1{\hangindent2.5em\noindent\hbox to2.5em{\hss#1 }\ignorespaces} \def\PASCAL{Pascal} \def\ps{PostScript} \def\ph{\hbox{Pascal-H}} \def\psqrt#1{\sqrt{\mathstrut#1}} \def\k{_{k+1}} \def\pct!{{\char`\%}} % percent sign in ordinary text \font\tenlogo=logo10 % font used for the METAFONT logo \font\logos=logosl10 \def\MF{{\tenlogo META}\-{\tenlogo FONT}} \def\MP{{\tenlogo META}\-{\tenlogo POST}} \def\<#1>{$\langle#1\rangle$} \def\section{\mathhexbox278} \let\swap=\leftrightarrow \def\round{\mathop{\rm round}\nolimits} \mathchardef\vb="026A % synonym for `\|' \def\(#1){} % this is used to make section names sort themselves better \def\9#1{} % this is used for sort keys in the index via @@:sort key}{entry@@> \outer\def\N#1. \[#2]#3.{\MN#1.\vfil\eject % begin starred section \def\rhead{PART #2:\uppercase{#3}} % define running headline \message{*\modno} % progress report \edef\next{\write\cont{\Z{\?#2]#3}{\modno}{\the\pageno}}}\next \ifon\startsection{\bf\ignorespaces#3.\quad}\ignorespaces} \let\?=\relax % we want to be able to \write a \? \def\title{MetaPost} \def\topofcontents{\hsize 5.5in \vglue -30pt plus 1fil minus 1.5in \def\?##1]{\hbox to 1in{\hfil##1.\ }} } \def\botofcontents{\vskip 0pt plus 1fil minus 1.5in} \pageno=3 \def\glob{13} % this should be the section number of "" \def\gglob{20, 26} % this should be the next two sections of "" @* \[1] Introduction. This is \MP, a graphics-language processor based on D. E. Knuth's \MF. The \PASCAL\ program that follows defines a standard version @:PASCAL}{\PASCAL@> of \MP\ that is designed to be highly portable so that identical output will be obtainable on a great variety of computers. The main purpose of the following program is to explain the algorithms of \MP\ as clearly as possible. As a result, the program will not necessarily be very efficient when a particular \PASCAL\ compiler has translated it into a particular machine language. However, the program has been written so that it can be tuned to run efficiently in a wide variety of operating environments by making comparatively few changes. Such flexibility is possible because the documentation that follows is written in the \.{WEB} language, which is at a higher level than \PASCAL; the preprocessing step that converts \.{WEB} to \PASCAL\ is able to introduce most of the necessary refinements. Semi-automatic translation to other languages is also feasible, because the program below does not make extensive use of features that are peculiar to \PASCAL. A large piece of software like \MP\ has inherent complexity that cannot be reduced below a certain level of difficulty, although each individual part is fairly simple by itself. The \.{WEB} language is intended to make the algorithms as readable as possible, by reflecting the way the individual program pieces fit together and by providing the cross-references that connect different parts. Detailed comments about what is going on, and about why things were done in certain ways, have been liberally sprinkled throughout the program. These comments explain features of the implementation, but they rarely attempt to explain the \MP\ language itself, since the reader is supposed to be familiar with {\sl The {\logos METAFONT\/}book} as well as the manual @.WEB@> @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> {\sl A User's Manual for MetaPost}, Computing Science Technical Report 162, AT\AM T Bell Laboratories. @ The present implementation is a preliminary version, but the possibilities for new features are limited by the desire to remain as nearly compatible with \MF\ as possible. On the other hand, the \.{WEB} description can be extended without changing the core of the program, and it has been designed so that such extensions are not extremely difficult to make. The |banner| string defined here should be changed whenever \MP\ undergoes any modifications, so that it will be clear which version of \MP\ might be the guilty party when a problem arises. @^extensions to \MP@> @^system dependencies@> @d banner=='This is MetaPost, Version 0.641' {printed when \MP\ starts} @ Different \PASCAL s have slightly different conventions, and the present @!@:PASCAL H}{\ph@> program is expressed in a version of \PASCAL\ that D. E. Knuth used for \MF. Constructions that apply to this particular compiler, which we shall call \ph, should help the reader see how to make an appropriate interface for other systems if necessary. (\ph\ is Charles Hedrick's modification of a compiler @^Hedrick, Charles Locke@> for the DECsystem-10 that was originally developed at the University of Hamburg; cf.\ {\sl SOFTWARE---Practice \AM\ Experience \bf6} (1976), 29--42. The \MP\ program below is intended to be adaptable, without extensive changes, to most other versions of \PASCAL\ and commonly used \PASCAL-to-C translators, so it does not fully @!@:C@> use the admirable features of \ph. Indeed, a conscious effort has been made here to avoid using several idiosyncratic features of standard \PASCAL\ itself, so that most of the code can be translated mechanically into other high-level languages. For example, the `\&{with}' and `\\{new}' features are not used, nor are pointer types, set types, or enumerated scalar types; there are no `\&{var}' parameters, except in the case of files; there are no tag fields on variant records; there are no |real| variables; no procedures are declared local to other procedures.) The portions of this program that involve system-dependent code, where changes might be necessary because of differences between \PASCAL\ compilers and/or differences between operating systems, can be identified by looking at the sections whose numbers are listed under `system dependencies' in the index. Furthermore, the index entries for `dirty \PASCAL' list all places where the restrictions of \PASCAL\ have not been followed perfectly, for one reason or another. @!@^system dependencies@> @!@^dirty \PASCAL@> @ The program begins with a normal \PASCAL\ program heading, whose components will be filled in later, using the conventions of \.{WEB}. @.WEB@> For example, the portion of the program called `\X\glob:Global variables\X' below will be replaced by a sequence of variable declarations that starts in $\section\glob$ of this documentation. In this way, we are able to define each individual global variable when we are prepared to understand what it means; we do not have to define all of the globals at once. Cross references in $\section\glob$, where it says ``See also sections \gglob, \dots,'' also make it possible to look at the set of all global variables, if desired. Similar remarks apply to the other portions of the program heading. Actually the heading shown here is not quite normal: The |program| line does not mention any |output| file, because \ph\ would ask the \MP\ user to specify a file name if |output| were specified here. @^system dependencies@> @d mtype==t@&y@&p@&e {this is a \.{WEB} coding trick:} @f mtype==type {`\&{mtype}' will be equivalent to `\&{type}'} @f type==true {but `|type|' will not be treated as a reserved word} @p @t\4@>@@/ program MP; {all file names are defined dynamically} label @@/ const @@/ mtype @@/ var @@/ @# procedure initialize; {this procedure gets things started properly} var @@/ begin @@/ end;@# @t\4@>@@/ @t\4@>@@/ @ The overall \MP\ program begins with the heading just shown, after which comes a bunch of procedure declarations and function declarations. Finally we will get to the main program, which begins with the comment `|start_here|'. If you want to skip down to the main program now, you can look up `|start_here|' in the index. But the author suggests that the best way to understand this program is to follow pretty much the order of \MP's components as they appear in the \.{WEB} description you are now reading, since the present ordering is intended to combine the advantages of the ``bottom up'' and ``top down'' approaches to the problem of understanding a somewhat complicated system. @ Three labels must be declared in the main program, so we give them symbolic names. @d start_of_MP=1 {go here when \MP's variables are initialized} @d end_of_MP=9998 {go here to close files and terminate gracefully} @d final_end=9999 {this label marks the ending of the program} @= start_of_MP@t\hskip-2pt@>, end_of_MP@t\hskip-2pt@>,@,final_end; {key control points} @ Some of the code below is intended to be used only when diagnosing the strange behavior that sometimes occurs when \MP\ is being installed or when system wizards are fooling around with \MP\ without quite knowing what they are doing. Such code will not normally be compiled; it is delimited by the codewords `$|debug|\ldots|gubed|$', with apologies to people who wish to preserve the purity of English. Similarly, there is some conditional code delimited by `$|stat|\ldots|tats|$' that is intended for use when statistics are to be kept about \MP's memory usage. @^debugging@> @d debug==@{ {change this to `$\\{debug}\equiv\null$' when debugging} @d gubed==@t@>@} {change this to `$\\{gubed}\equiv\null$' when debugging} @f debug==begin @f gubed==end @# @d stat==@{ {change this to `$\\{stat}\equiv\null$' when gathering usage statistics} @d tats==@t@>@} {change this to `$\\{tats}\equiv\null$' when gathering usage statistics} @f stat==begin @f tats==end @ This program has two important variations: (1) There is a long and slow version called \.{INIMP}, which does the extra calculations needed to @.INIMP@> initialize \MP's internal tables; and (2)~there is a shorter and faster production version, which cuts the initialization to a bare minimum. Parts of the program that are needed in (1) but not in (2) are delimited by the codewords `$|init|\ldots|tini|$'. @d init== {change this to `$\\{init}\equiv\.{@@\{}$' in the production version} @d tini== {change this to `$\\{tini}\equiv\.{@@\}}$' in the production version} @f init==begin @f tini==end @ If the first character of a \PASCAL\ comment is a dollar sign, \ph\ treats the comment as a list of ``compiler directives'' that will affect the translation of this program into machine language. The directives shown below specify full checking and inclusion of the \PASCAL\ debugger when \MP\ is being debugged, but they cause range checking and other redundant code to be eliminated when the production system is being generated. Arithmetic overflow will be detected in all cases. @^system dependencies@> @^Overflow in arithmetic@> @= @{@&$C-,A+,D-@} {no range check, catch arithmetic overflow, no debug overhead} @!debug @{@&$C+,D+@}@+ gubed {but turn everything on when debugging} @ This \MP\ implementation conforms to the rules of the {\sl Pascal User @:PASCAL}{\PASCAL@> @^system dependencies@> Manual} published by Jensen and Wirth in 1975, except where system-dependent @^Wirth, Niklaus@> @^Jensen, Kathleen@> code is necessary to make a useful system program, and except in another respect where such conformity would unnecessarily obscure the meaning and clutter up the code: We assume that |case| statements may include a default case that applies if no matching label is found. Thus, we shall use constructions like $$\vbox{\halign{\ignorespaces#\hfil\cr |case x of|\cr 1: $\langle\,$code for $x=1\,\rangle$;\cr 3: $\langle\,$code for $x=3\,\rangle$;\cr |othercases| $\langle\,$code for |x<>1| and |x<>3|$\,\rangle$\cr |endcases|\cr}}$$ since most \PASCAL\ compilers have plugged this hole in the language by incorporating some sort of default mechanism. For example, the \ph\ compiler allows `|others|:' as a default label, and other \PASCAL s allow syntaxes like `\&{else}' or `\&{otherwise}' or `\\{otherwise}:', etc. The definitions of |othercases| and |endcases| should be changed to agree with local conventions. Note that no semicolon appears before |endcases| in this program, so the definition of |endcases| should include a semicolon if the compiler wants one. (Of course, if no default mechanism is available, the |case| statements of \MP\ will have to be laboriously extended by listing all remaining cases. People who are stuck with such \PASCAL s have, in fact, done this, successfully but not happily!) @d othercases == others: {default for cases not listed explicitly} @d endcases == @+end {follows the default case in an extended |case| statement} @f othercases == else @f endcases == end @ The following parameters can be changed at compile time to extend or reduce \MP's capacity. They may have different values in \.{INIMP} and in production versions of \MP. @.INIMP@> @^system dependencies@> @= @!mem_max=30000; {greatest index in \MP's internal |mem| array; must be strictly less than |max_halfword|; must be equal to |mem_top| in \.{INIMP}, otherwise |>=mem_top|} @!max_internal=100; {maximum number of internal quantities} @!buf_size=500; {maximum number of characters simultaneously present in current lines of open files; must not exceed |max_halfword|} @!error_line=72; {width of context lines on terminal error messages} @!half_error_line=42; {width of first lines of contexts in terminal error messages; should be between 30 and |error_line-15|} @!max_print_line=79; {width of longest text lines output; should be at least 60} @!emergency_line_length=255; {\ps\ output lines can be this long in unusual circumstances} @!stack_size=30; {maximum number of simultaneous input sources} @!max_read_files=4; {maximum number of simultaneously open \&{readfrom} files} @!max_strings=2500; {maximum number of strings; must not exceed |max_halfword|} @!string_vacancies=9000; {the minimum number of characters that should be available for the user's identifier names and strings, after \MP's own error messages are stored} @!strings_vacant=1000; {the minimum number of strings that should be available} @!pool_size=32000; {maximum number of characters in strings, including all error messages and help texts, and the names of all identifiers; must exceed |string_vacancies| by the total length of \MP's own strings, which is currently about 22000} @!font_max=50; {maximum font number for included text fonts} @!font_mem_size=10000; {number of words for \.{TFM} information for text fonts} @!file_name_size=40; {file names shouldn't be longer than this} @!pool_name='MPlib:MP.POOL '; {string of length |file_name_size|; tells where the string pool appears} @.MPlib@> @!ps_tab_name='MPlib:PSFONTS.MAP '; {string of length |file_name_size|; locates font name translation table} @!path_size=300; {maximum number of knots between breakpoints of a path} @!bistack_size=785; {size of stack for bisection algorithms; should probably be left at this value} @!header_size=100; {maximum number of \.{TFM} header words, times~4} @!lig_table_size=5000; {maximum number of ligature/kern steps, must be at least 255 and at most 32510} @!max_kerns=500; {maximum number of distinct kern amounts} @!max_font_dimen=50; {maximum number of \&{fontdimen} parameters} @ Like the preceding parameters, the following quantities can be changed at compile time to extend or reduce \MP's capacity. But if they are changed, it is necessary to rerun the initialization program \.{INIMP} @.INIMP@> to generate new tables for the production \MP\ program. One can't simply make helter-skelter changes to the following constants, since certain rather complex initialization numbers are computed from them. They are defined here using \.{WEB} macros, instead of being put into \PASCAL's |const| list, in order to emphasize this distinction. @d mem_min=0 {smallest index in the |mem| array, must not be less than |min_halfword|} @d mem_top==30000 {largest index in the |mem| array dumped by \.{INIMP}; must be substantially larger than |mem_min| and not greater than |mem_max|} @d hash_size=2100 {maximum number of symbolic tokens, must be less than |max_halfword-3*param_size|} @d hash_prime=1777 {a prime number equal to about 85\pct! of |hash_size|} @d max_in_open=6 {maximum number of input files and error insertions that can be going on simultaneously} @d param_size=150 {maximum number of simultaneous macro parameters} @d max_write_files=4 {maximum number of simultaneously open \&{write} files} @^system dependencies@> @ In case somebody has inadvertently made bad settings of the ``constants,'' \MP\ checks them using a global variable called |bad|. This is the first of many sections of \MP\ where global variables are defined. @= @!bad:integer; {is some ``constant'' wrong?} @ Later on we will say `\ignorespaces|if mem_max>=max_halfword then bad:=10|', or something similar. (We can't do that until |max_halfword| has been defined.) @= bad:=0; if (half_error_line<30)or(half_error_line>error_line-15) then bad:=1; if max_print_line<60 then bad:=2; if emergency_line_lengthmem_top then bad:=4; if hash_prime>hash_size then bad:=5; if header_size mod 4 <> 0 then bad:=6; if(lig_table_size<255)or(lig_table_size>32510)then bad:=7; @ Labels are given symbolic names by the following definitions, so that occasional |goto| statements will be meaningful. We insert the label `|exit|:' just before the `\ignorespaces|end|\unskip' of a procedure in which we have used the `|return|' statement defined below; the label `|restart|' is occasionally used at the very beginning of a procedure; and the label `|reswitch|' is occasionally used just prior to a |case| statement in which some cases change the conditions and we wish to branch to the newly applicable case. Loops that are set up with the |loop| construction defined below are commonly exited by going to `|done|' or to `|found|' or to `|not_found|', and they are sometimes repeated by going to `|continue|'. If two or more parts of a subroutine start differently but end up the same, the shared code may be gathered together at `|common_ending|'. Incidentally, this program never declares a label that isn't actually used, because some fussy \PASCAL\ compilers will complain about redundant labels. @d exit=10 {go here to leave a procedure} @d restart=20 {go here to start a procedure again} @d reswitch=21 {go here to start a case statement again} @d continue=22 {go here to resume a loop} @d done=30 {go here to exit a loop} @d done1=31 {like |done|, when there is more than one loop} @d done2=32 {for exiting the second loop in a long block} @d done3=33 {for exiting the third loop in a very long block} @d done4=34 {for exiting the fourth loop in an extremely long block} @d done5=35 {for exiting the fifth loop in an immense block} @d done6=36 {for exiting the sixth loop in a block} @d found=40 {go here when you've found it} @d found1=41 {like |found|, when there's more than one per routine} @d found2=42 {like |found|, when there's more than two per routine} @d not_found=45 {go here when you've found nothing} @d common_ending=50 {go here when you want to merge with another branch} @ Here are some macros for common programming idioms. @d incr(#) == #:=#+1 {increase a variable by unity} @d decr(#) == #:=#-1 {decrease a variable by unity} @d negate(#) == #:=-# {change the sign of a variable} @d double(#) == #:=#+# {multiply a variable by two} @d loop == @+ while true do@+ {repeat over and over until a |goto| happens} @f loop == xclause {\.{WEB}'s |xclause| acts like `\ignorespaces|while true do|\unskip'} @d do_nothing == {empty statement} @d return == goto exit {terminate a procedure call} @f return == nil {\.{WEB} will henceforth say |return| instead of \\{return}} @* \[2] The character set. In order to make \MP\ readily portable to a wide variety of computers, all of its input text is converted to an internal eight-bit code that includes standard ASCII, the ``American Standard Code for Information Interchange.'' This conversion is done immediately when each character is read in. Conversely, characters are converted from ASCII to the user's external representation just before they are output to a text file. @^ASCII code@> Such an internal code is relevant to users of \MP\ only with respect to the \&{char} and \&{ASCII} operations, and the comparison of strings. @ Characters of text that have been converted to \MP's internal form are said to be of type |ASCII_code|, which is a subrange of the integers. @= @!ASCII_code=0..255; {eight-bit numbers} @ The original \PASCAL\ compiler was designed in the late 60s, when six-bit character sets were common, so it did not make provision for lowercase letters. Nowadays, of course, we need to deal with both capital and small letters in a convenient way, especially in a program for font design; so the present specification of \MP\ has been written under the assumption that the \PASCAL\ compiler and run-time system permit the use of text files with more than 64 distinguishable characters. More precisely, we assume that the character set contains at least the letters and symbols associated with ASCII codes @'40 through @'176; all of these characters are now available on most computer terminals. Since we are dealing with more characters than were present in the first \PASCAL\ compilers, we have to decide what to call the associated data type. Some \PASCAL s use the original name |char| for the characters in text files, even though there now are more than 64 such characters, while other \PASCAL s consider |char| to be a 64-element subrange of a larger data type that has some other name. In order to accommodate this difference, we shall use the name |text_char| to stand for the data type of the characters that are converted to and from |ASCII_code| when they are input and output. We shall also assume that |text_char| consists of the elements |chr(first_text_char)| through |chr(last_text_char)|, inclusive. The following definitions should be adjusted if necessary. @^system dependencies@> @d text_char == char {the data type of characters in text files} @d first_text_char=0 {ordinal number of the smallest element of |text_char|} @d last_text_char=255 {ordinal number of the largest element of |text_char|} @= @!i:integer; @ The \MP\ processor converts between ASCII code and the user's external character set by means of arrays |xord| and |xchr| that are analogous to \PASCAL's |ord| and |chr| functions. @= @!xord: array [text_char] of ASCII_code; {specifies conversion of input characters} @!xchr: array [ASCII_code] of text_char; {specifies conversion of output characters} @ Since we are assuming that our \PASCAL\ system is able to read and write the visible characters of standard ASCII (although not necessarily using the ASCII codes to represent them), the following assignment statements initialize the standard part of the |xchr| array properly, without needing any system-dependent changes. On the other hand, it is possible to implement \MP\ with less complete character sets, and in such cases it will be necessary to change something here. @^system dependencies@> @= xchr[@'40]:=' '; xchr[@'41]:='!'; xchr[@'42]:='"'; xchr[@'43]:='#'; xchr[@'44]:='$'; xchr[@'45]:='%'; xchr[@'46]:='&'; xchr[@'47]:='''';@/ xchr[@'50]:='('; xchr[@'51]:=')'; xchr[@'52]:='*'; xchr[@'53]:='+'; xchr[@'54]:=','; xchr[@'55]:='-'; xchr[@'56]:='.'; xchr[@'57]:='/';@/ xchr[@'60]:='0'; xchr[@'61]:='1'; xchr[@'62]:='2'; xchr[@'63]:='3'; xchr[@'64]:='4'; xchr[@'65]:='5'; xchr[@'66]:='6'; xchr[@'67]:='7';@/ xchr[@'70]:='8'; xchr[@'71]:='9'; xchr[@'72]:=':'; xchr[@'73]:=';'; xchr[@'74]:='<'; xchr[@'75]:='='; xchr[@'76]:='>'; xchr[@'77]:='?';@/ xchr[@'100]:='@@'; xchr[@'101]:='A'; xchr[@'102]:='B'; xchr[@'103]:='C'; xchr[@'104]:='D'; xchr[@'105]:='E'; xchr[@'106]:='F'; xchr[@'107]:='G';@/ xchr[@'110]:='H'; xchr[@'111]:='I'; xchr[@'112]:='J'; xchr[@'113]:='K'; xchr[@'114]:='L'; xchr[@'115]:='M'; xchr[@'116]:='N'; xchr[@'117]:='O';@/ xchr[@'120]:='P'; xchr[@'121]:='Q'; xchr[@'122]:='R'; xchr[@'123]:='S'; xchr[@'124]:='T'; xchr[@'125]:='U'; xchr[@'126]:='V'; xchr[@'127]:='W';@/ xchr[@'130]:='X'; xchr[@'131]:='Y'; xchr[@'132]:='Z'; xchr[@'133]:='['; xchr[@'134]:='\'; xchr[@'135]:=']'; xchr[@'136]:='^'; xchr[@'137]:='_';@/ xchr[@'140]:='`'; xchr[@'141]:='a'; xchr[@'142]:='b'; xchr[@'143]:='c'; xchr[@'144]:='d'; xchr[@'145]:='e'; xchr[@'146]:='f'; xchr[@'147]:='g';@/ xchr[@'150]:='h'; xchr[@'151]:='i'; xchr[@'152]:='j'; xchr[@'153]:='k'; xchr[@'154]:='l'; xchr[@'155]:='m'; xchr[@'156]:='n'; xchr[@'157]:='o';@/ xchr[@'160]:='p'; xchr[@'161]:='q'; xchr[@'162]:='r'; xchr[@'163]:='s'; xchr[@'164]:='t'; xchr[@'165]:='u'; xchr[@'166]:='v'; xchr[@'167]:='w';@/ xchr[@'170]:='x'; xchr[@'171]:='y'; xchr[@'172]:='z'; xchr[@'173]:='{'; xchr[@'174]:='|'; xchr[@'175]:='}'; xchr[@'176]:='~';@/ @ The ASCII code is ``standard'' only to a certain extent, since many computer installations have found it advantageous to have ready access to more than 94 printing characters. If \MP\ is being used on a garden-variety \PASCAL\ for which only standard ASCII codes will appear in the input and output files, it doesn't really matter what codes are specified in |xchr[0..@'37]|, but the safest policy is to blank everything out by using the code shown below. However, other settings of |xchr| will make \MP\ more friendly on computers that have an extended character set, so that users can type things like `\.^^Z' instead of `\.{<>}'. People with extended character sets can assign codes arbitrarily, giving an |xchr| equivalent to whatever characters the users of \MP\ are allowed to have in their input files. Appropriate changes to \MP's |char_class| table should then be made. (Unlike \TeX, each installation of \MP\ has a fixed assignment of category codes, called the |char_class|.) Such changes make portability of programs more difficult, so they should be introduced cautiously if at all. @^character set dependencies@> @^system dependencies@> @= for i:=0 to @'37 do xchr[i]:=' '; for i:=@'177 to @'377 do xchr[i]:=' '; @ The following system-independent code makes the |xord| array contain a suitable inverse to the information in |xchr|. Note that if |xchr[i]=xchr[j]| where |i= for i:=first_text_char to last_text_char do xord[chr(i)]:=@'177; for i:=@'200 to @'377 do xord[xchr[i]]:=i; for i:=0 to @'176 do xord[xchr[i]]:=i; @* \[3] Input and output. The bane of portability is the fact that different operating systems treat input and output quite differently, perhaps because computer scientists have not given sufficient attention to this problem. People have felt somehow that input and output are not part of ``real'' programming. Well, it is true that some kinds of programming are more fun than others. With existing input/output conventions being so diverse and so messy, the only sources of joy in such parts of the code are the rare occasions when one can find a way to make the program a little less bad than it might have been. We have two choices, either to attack I/O now and get it over with, or to postpone I/O until near the end. Neither prospect is very attractive, so let's get it over with. The basic operations we need to do are (1)~inputting and outputting of text, to or from a file or the user's terminal; (2)~inputting and outputting of eight-bit bytes, to or from a file; (3)~instructing the operating system to initiate (``open'') or to terminate (``close'') input or output from a specified file; (4)~testing whether the end of an input file has been reached; (5)~display of bits on the user's screen. The bit-display operation will be discussed in a later section; we shall deal here only with more traditional kinds of I/O. \MP\ needs to deal with two kinds of files. We shall use the term |alpha_file| for a file that contains textual data, and the term |byte_file| for a file that contains eight-bit binary information. These two types turn out to be the same on many computers, but sometimes there is a significant distinction, so we shall be careful to distinguish between them. Standard protocols for transferring such files from computer to computer, via high-speed networks, are now becoming available to more and more communities of users. The program actually makes use also of a third kind of file, called a |word_file|, when dumping and reloading mem information for its own initialization. We shall define a word file later; but it will be possible for us to specify simple operations on word files before they are defined. @= @!eight_bits=0..255; {unsigned one-byte quantity} @!alpha_file=packed file of text_char; {files that contain textual data} @!byte_file=packed file of eight_bits; {files that contain binary data} @ Most of what we need to do with respect to input and output can be handled by the I/O facilities that are standard in \PASCAL, i.e., the routines called |get|, |put|, |eof|, and so on. But standard \PASCAL\ does not allow file variables to be associated with file names that are determined at run time, so it cannot be used to implement \MP; some sort of extension to \PASCAL's ordinary |reset| and |rewrite| is crucial for our purposes. We shall assume that |name_of_file| is a variable of an appropriate type such that the \PASCAL\ run-time system being used to implement \MP\ can open a file whose external name is specified by |name_of_file|. @^system dependencies@> @= @!name_of_file:packed array[1..file_name_size] of char;@;@/ {on some systems this may be a \&{record} variable} @!name_length:0..file_name_size;@/{this many characters are actually relevant in |name_of_file| (the rest are blank)} @ The \ph\ compiler with which the original version of \MF\ was prepared extends the rules of \PASCAL\ in a very convenient way. To open file~|f|, we can write $$\vbox{\halign{#\hfil\qquad&#\hfil\cr |reset(f,@t\\{name}@>,'/O')|&for input;\cr |rewrite(f,@t\\{name}@>,'/O')|&for output.\cr}}$$ The `\\{name}' parameter, which is of type `\ignorespaces|packed array[@t\<\\{any}>@>] of text_char|', stands for the name of the external file that is being opened for input or output. Blank spaces that might appear in \\{name} are ignored. The `\.{/O}' parameter tells the operating system not to issue its own error messages if something goes wrong. If a file of the specified name cannot be found, or if such a file cannot be opened for some other reason (e.g., someone may already be trying to write the same file), we will have |@!erstat(f)<>0| after an unsuccessful |reset| or |rewrite|. This allows \MP\ to undertake appropriate corrective action. @:PASCAL H}{\ph@> @^system dependencies@> \MP's file-opening procedures return |false| if no file identified by |name_of_file| could be opened. @d reset_OK(#)==erstat(#)=0 @d rewrite_OK(#)==erstat(#)=0 @p function a_open_in(var @!f:alpha_file):boolean; {open a text file for input} begin reset(f,name_of_file,'/O'); a_open_in:=reset_OK(f); end; @# function a_open_out(var @!f:alpha_file):boolean; {open a text file for output} begin rewrite(f,name_of_file,'/O'); a_open_out:=rewrite_OK(f); end; @# function b_open_in(var @!f:byte_file):boolean; {open a binary file for input} begin reset(f,name_of_file,'/O'); b_open_in:=reset_OK(f); end; @# function b_open_out(var @!f:byte_file):boolean; {open a binary file for output} begin rewrite(f,name_of_file,'/O'); b_open_out:=rewrite_OK(f); end; @# function w_open_in(var @!f:word_file):boolean; {open a word file for input} begin reset(f,name_of_file,'/O'); w_open_in:=reset_OK(f); end; @# function w_open_out(var @!f:word_file):boolean; {open a word file for output} begin rewrite(f,name_of_file,'/O'); w_open_out:=rewrite_OK(f); end; @ Files can be closed with the \ph\ routine `|close(f)|', which @^system dependencies@> should be used when all input or output with respect to |f| has been completed. This makes |f| available to be opened again, if desired; and if |f| was used for output, the |close| operation makes the corresponding external file appear on the user's area, ready to be read. @p procedure a_close(var @!f:alpha_file); {close a text file} begin close(f); end; @# procedure b_close(var @!f:byte_file); {close a binary file} begin close(f); end; @# procedure w_close(var @!f:word_file); {close a word file} begin close(f); end; @ Binary input and output are done with \PASCAL's ordinary |get| and |put| procedures, so we don't have to make any other special arrangements for binary~I/O. Text output is also easy to do with standard \PASCAL\ routines. The treatment of text input is more difficult, however, because of the necessary translation to |ASCII_code| values. \MP's conventions should be efficient, and they should blend nicely with the user's operating environment. @ Input from text files is read one line at a time, using a routine called |input_ln|. This function is defined in terms of global variables called |buffer|, |first|, and |last| that will be described in detail later; for now, it suffices for us to know that |buffer| is an array of |ASCII_code| values, and that |first| and |last| are indices into this array representing the beginning and ending of a line of text. @= @!buffer:array[0..buf_size] of ASCII_code; {lines of characters being read} @!first:0..buf_size; {the first unused position in |buffer|} @!last:0..buf_size; {end of the line just input to |buffer|} @!max_buf_stack:0..buf_size; {largest index used in |buffer|} @ The |input_ln| function brings the next line of input from the specified field into available positions of the buffer array and returns the value |true|, unless the file has already been entirely read, in which case it returns |false| and sets |last:=first|. In general, the |ASCII_code| numbers that represent the next line of the file are input into |buffer[first]|, |buffer[first+1]|, \dots, |buffer[last-1]|; and the global variable |last| is set equal to |first| plus the length of the line. Trailing blanks are removed from the line; thus, either |last=first| (in which case the line was entirely blank) or |buffer[last-1]<>" "|. @^inner loop@> An overflow error is given, however, if the normal actions of |input_ln| would make |last>=buf_size|; this is done so that other parts of \MP\ can safely look at the contents of |buffer[last+1]| without overstepping the bounds of the |buffer| array. Upon entry to |input_ln|, the condition |first=max_buf_stack then begin max_buf_stack:=last+1; if max_buf_stack=buf_size then @; end; buffer[last]:=xord[f^]; get(f); incr(last); if buffer[last-1]<>" " then last_nonblank:=last; end; last:=last_nonblank; input_ln:=true; end; end; @ The user's terminal acts essentially like other files of text, except that it is used both for input and for output. When the terminal is considered an input file, the file variable is called |term_in|, and when it is considered an output file the file variable is |term_out|. @^system dependencies@> @= @!term_in:alpha_file; {the terminal as an input file} @!term_out:alpha_file; {the terminal as an output file} @ Here is how to open the terminal files in \ph. The `\.{/I}' switch suppresses the first |get|. @^system dependencies@> @d t_open_in==reset(term_in,'TTY:','/O/I') {open the terminal for text input} @d t_open_out==rewrite(term_out,'TTY:','/O') {open the terminal for text output} @ Sometimes it is necessary to synchronize the input/output mixture that happens on the user's terminal, and three system-dependent procedures are used for this purpose. The first of these, |update_terminal|, is called when we want to make sure that everything we have output to the terminal so far has actually left the computer's internal buffers and been sent. The second, |clear_terminal|, is called when we wish to cancel any input that the user may have typed ahead (since we are about to issue an unexpected error message). The third, |wake_up_terminal|, is supposed to revive the terminal if the user has disabled it by some instruction to the operating system. The following macros show how these operations can be specified in \ph: @^system dependencies@> @d update_terminal == break(term_out) {empty the terminal output buffer} @d clear_terminal == break_in(term_in,true) {clear the terminal input buffer} @d wake_up_terminal == do_nothing {cancel the user's cancellation of output} @ We need a special routine to read the first line of \MP\ input from the user's terminal. This line is different because it is read before we have opened the transcript file; there is sort of a ``chicken and egg'' problem here. If the user types `\.{input cmr10}' on the first line, or if some macro invoked by that line does such an \.{input}, the transcript file will be named `\.{cmr10.log}'; but if no \.{input} commands are performed during the first line of terminal input, the transcript file will acquire its default name `\.{mpout.log}'. (The transcript file will not contain error messages generated by the first line before the first \.{input} command.) The first line is even more special if we are lucky enough to have an operating system that treats \MP\ differently from a run-of-the-mill \PASCAL\ object program. It's nice to let the user start running a \MP\ job by typing a command line like `\.{MP cmr10}'; in such a case, \MP\ will operate as if the first line of input were `\.{cmr10}', i.e., the first line will consist of the remainder of the command line, after the part that invoked \MP. The first line is special also because it may be read before \MP\ has input a mem file. In such cases, normal error messages cannot yet be given. The following code uses concepts that will be explained later. @= if mem_ident=0 then begin write_ln(term_out,'Buffer size exceeded!'); goto final_end; @.Buffer size exceeded@> end else begin cur_input.loc_field:=first; cur_input.limit_field:=last-1; overflow("buffer size",buf_size); @:MetaPost capacity exceeded buffer size}{\quad buffer size@> end @ Different systems have different ways to get started. But regardless of what conventions are adopted, the routine that initializes the terminal should satisfy the following specifications: \yskip\textindent{1)}It should open file |term_in| for input from the terminal. (The file |term_out| will already be open for output to the terminal.) \textindent{2)}If the user has given a command line, this line should be considered the first line of terminal input. Otherwise the user should be prompted with `\.{**}', and the first line of input should be whatever is typed in response. \textindent{3)}The first line of input, which might or might not be a command line, should appear in locations |first| to |last-1| of the |buffer| array. \textindent{4)}The global variable |loc| should be set so that the character to be read next by \MP\ is in |buffer[loc]|. This character should not be blank, and we should have |loc @p function init_terminal:boolean; {gets the terminal input started} label exit; begin t_open_in; loop@+begin wake_up_terminal; write(term_out,'**'); update_terminal; @.**@> if not input_ln(term_in,true) then {this shouldn't happen} begin write_ln(term_out); write(term_out,'! End of file on the terminal... why?'); @.End of file on the terminal@> init_terminal:=false; return; end; loc:=first; while (loc which converts single-character strings into the ASCII code number of the single character involved, while it converts other strings into integers and builds a string pool file. Thus, when the string constant \.{"."} appears in the program below, \.{WEB} converts it into the integer 46, which is the ASCII code for a period, while \.{WEB} will convert a string like \.{"hello"} into some integer greater than~255. String number 46 will presumably be the single character `\..'\thinspace; but some ASCII codes have no standard visible representation, and \MP\ may need to be able to print an arbitrary ASCII character, so the first 256 strings are used to specify exactly what should be printed for each of the 256 possibilities. Elements of the |str_pool| array must be ASCII codes that can actually be printed; i.e., they must have an |xchr| equivalent in the local character set. (This restriction applies only to preloaded strings, not to those generated dynamically by the user.) Some \PASCAL\ compilers won't pack integers into a single byte unless the integers lie in the range |-128..127|. To accommodate such systems we access the string pool via macros that can easily be redefined. When accessing character dimensions for the \&{infont} operator, an explicit offset is used to convert from |pool_ASCII_code| to |ASCII_code|. @d si(#) == # {convert from |ASCII_code| to |pool_ASCII_code|} @d so(#) == # {convert from |pool_ASCII_code| to |ASCII_code|} @d min_pool_ASCII=0 {added to an |ASCII_code| to make a |pool_ASCII_code|} @= @!pool_pointer = 0..pool_size; {for variables that point into |str_pool|} @!str_number = 0..max_strings; {for variables that point into |str_start|} @!pool_ASCII_code = 0..255; {elements of |str_pool| array} @ @= @!str_pool:packed array[pool_pointer] of pool_ASCII_code; {the characters} @!str_start : array[str_number] of pool_pointer; {the starting pointers} @!next_str : array[str_number] of str_number; {for linking strings in order} @!pool_ptr : pool_pointer; {first unused position in |str_pool|} @!str_ptr : str_number; {number of the current string being created} @!init_pool_ptr : pool_pointer; {the starting value of |pool_ptr|} @!init_str_use : str_number; {the initial number of strings in use} @!max_pool_ptr : pool_pointer; {the maximum so far of |pool_ptr|} @!max_str_ptr : str_number; {the maximum so far of |str_ptr|} @ Except for |strs_used_up|, the following string statistics are only maintained when code between |stat| $\ldots$ |tats| delimiters is not commented out: @= @!strs_used_up:integer; {strings in use or unused but not reclaimed} @!pool_in_use:integer; {total number of cells of |str_pool| actually in use} @!strs_in_use:integer; {total number of strings actually in use} @!max_pl_used:integer; {maximum |pool_in_use| so far} @!max_strs_used:integer; {maximum |strs_in_use| so far} @ Several of the elementary string operations are performed using \.{WEB} macros instead of \PASCAL\ procedures, because many of the operations are done quite frequently and we want to avoid the overhead of procedure calls. For example, here is a simple macro that computes the length of a string. @.WEB@> @d str_stop(#)==str_start[next_str[#]] {one cell past the end of string number \#} @d length(#)==(str_stop(#)-str_start[#]) {the number of characters in string \#} @ The length of the current string is called |cur_length|. If we decide that the current string is not needed, |flush_cur_string| resets |pool_ptr| so that |cur_length| becomes zero. @d cur_length == (pool_ptr - str_start[str_ptr]) @d flush_cur_string == pool_ptr:=str_start[str_ptr] @ Strings are created by appending character codes to |str_pool|. The |append_char| macro, defined here, does not check to see if the value of |pool_ptr| has gotten too high; this test is supposed to be made before |append_char| is used. To test if there is room to append |l| more characters to |str_pool|, we shall write |str_room(l)|, which tries to make sure there is enough room by compacting the string pool if necessary. If this does not work, |do_compaction| aborts \MP\ and gives an apologetic error message. @d append_char(#) == {put |ASCII_code| \# at the end of |str_pool|} begin str_pool[pool_ptr]:=si(#); incr(pool_ptr); end @d str_room(#) == {make sure that the pool hasn't overflowed} begin if pool_ptr+# > max_pool_ptr then if pool_ptr+# > pool_size then do_compaction(#) else max_pool_ptr:=pool_ptr+#; end @ The following routine is similar to |str_room(1)| but it uses the argument |pool_size| to prevent |do_compaction| from aborting when string space is exhausted. @= procedure unit_str_room; begin if pool_ptr>=pool_size then do_compaction(pool_size); if pool_ptr>=max_pool_ptr then max_pool_ptr:=pool_ptr+1; end; @ \MP's string expressions are implemented in a brute-force way: Every new string or substring that is needed is simply copied into the string pool. Space is eventually reclaimed by a procedure called |do_compaction| with the aid of a simple system system of reference counts. @^reference counts@> The number of references to string number |s| will be |str_ref[s]|. The special value |str_ref[s]=max_str_ref=127| is used to denote an unknown positive number of references; such strings will never be recycled. If a string is ever referred to more than 126 times, simultaneously, we put it in this category. Hence a single byte suffices to store each |str_ref|. @d max_str_ref=127 {``infinite'' number of references} @d add_str_ref(#)==begin if str_ref[#]= @!str_ref:array[str_number] of 0..max_str_ref; @ Here's what we do when a string reference disappears: @d delete_str_ref(#)== begin if str_ref[#]1 then decr(str_ref[#])@+else flush_string(#); end @= procedure flush_string(@!s:str_number); begin stat pool_in_use:=pool_in_use-length(s); decr(strs_in_use); tats@; if next_str[s]<>str_ptr then str_ref[s]:=0 else begin str_ptr:=s; decr(strs_used_up); end; pool_ptr:=str_start[str_ptr]; end; @ Once a sequence of characters has been appended to |str_pool|, it officially becomes a string when the function |make_string| is called. This function returns the identification number of the new string as its value. When getting the next unused string number from the linked list, we pretend that $$ \hbox{|max_str_ptr+1|, |max_str_ptr+2|, $\ldots$, |max_strings|} $$ are linked sequentially even though the |next_str| entries have not been initialized yet. We never allow |str_ptr| to reach |max_strings|; |do_compaction| is responsible for making sure of this. @p @t\4@>@@; @t\4@>@@; function make_string : str_number; {current string enters the pool} label restart; var @!s:str_number; {the new string} begin restart: s:=str_ptr; str_ptr:=next_str[s]; if str_ptr>max_str_ptr then if str_ptr=max_strings then begin str_ptr:=s; do_compaction(0); goto restart; end else begin debug if strs_used_up<>max_str_ptr then confusion("s");@+gubed@/ @:this can't happen s}{\quad \.s@> max_str_ptr:=str_ptr; next_str[str_ptr]:=max_str_ptr+1; end; str_ref[s]:=1; str_start[str_ptr]:=pool_ptr; incr(strs_used_up); stat incr(strs_in_use); pool_in_use:=pool_in_use+length(s); if pool_in_use>max_pl_used then max_pl_used:=pool_in_use; if strs_in_use>max_strs_used then max_strs_used:=strs_in_use; tats@; make_string:=s; end; @ On rare occasions, we might decide after calling |make_string| that some characters should be removed from the end of the last string and transferred to the beginning of a string under construction. This basically a matter of resetting |str_start[str_ptr]|. It is not practical to ensure that the new value for this pointer is in range, so this procedure should be used carefully. @p procedure chop_last_string(@!p:pool_pointer); begin stat pool_in_use:=pool_in_use-(str_start[str_ptr]-p); @+tats; str_start[str_ptr]:=p; end; @ The most interesting string operation is string pool compaction. The idea is to recover unused space in the |str_pool| array by recopying the strings to close the gaps created when some strings become unused. All string numbers~$k$ where |str_ref[k]=0| are to be linked into the list of free string numbers after |str_ptr|. If this fails to free enough pool space we issue an |overflow| error unless |needed=pool_size|. Calling |do_compaction| with |needed=pool_size| supresses all overflow tests. The compaction process starts with |last_fixed_str| because all lower numbered strings are permanently allocated with |max_str_ref| in their |str_ref| entries. @= @!last_fixed_str:str_number; {last permanently allocated string} @!fixed_str_use:str_number; {number of permanently allocated strings} @ @= procedure do_compaction(@!needed:pool_pointer); label done; var @!str_use:str_number; {a count of strings in use} @!r,@!s,@!t:str_number; {strings being manipulated} @!p,@!q:pool_pointer; {destination and source for copying string characters} begin @; r:=last_fixed_str; s:=next_str[r]; p:=str_start[s]; while s<>str_ptr do begin while str_ref[s]=0 do @; r:=s; s:=next_str[s]; incr(str_use); @; end; done: @; if needed; stat @; tats@; strs_used_up:=str_use; end; @ @= t:=next_str[last_fixed_str]; while (str_ref[t]=max_str_ref)and(t<>str_ptr) do begin incr(fixed_str_use); last_fixed_str:=t; t:=next_str[t]; end; str_use:=fixed_str_use @ Because of the way |flush_string| has been written, it should never be necessary to |goto done| here. The extra line of code seems worthwhile to preserve the generality of |do_compaction|. @= begin t:=s; s:=next_str[s]; next_str[r]:=s; next_str[t]:=next_str[str_ptr]; next_str[str_ptr]:=t; if s=str_ptr then goto done; end @ The string currently starts at |str_start[r]| and ends just before |str_start[s]|. We don't change |str_start[s]| because it might be needed to locate the next string. @= q:=str_start[r]; str_start[r]:=p; while q= q:=str_start[str_ptr]; str_start[str_ptr]:=p; while q= begin if str_use>=max_strings-1 then begin str_overflowed:=true; overflow("number of strings", max_strings-1-init_str_use); @:MetaPost capacity exceeded number of strings}{\quad number of strings@> end; if pool_ptr+needed>max_pool_ptr then if pool_ptr+needed>pool_size then begin str_overflowed:=true; overflow("pool size", pool_size-init_pool_ptr); @:MetaPost capacity exceeded pool size}{\quad pool size@> end else max_pool_ptr:=pool_ptr+needed; end @ Routines that can be called after string overflow need a way of checking whether it is safe to use |str_room|, |make_string|, or |do_compaction|. @= @!str_overflowed:boolean; {is \MP\ aborting due to pool size of number of strings?} @ @= if (str_start[str_ptr]<>pool_in_use)or(str_use<>strs_in_use) then confusion("string"); @:this can't happen string}{\quad string@> incr(pact_count); pact_chars:=pact_chars+pool_ptr-str_stop(last_fixed_str); pact_strs:=pact_strs+str_use-fixed_str_use; debug s:=str_ptr; t:=str_use; while s<=max_str_ptr do begin if t>max_str_ptr then confusion(""""); incr(t); s:=next_str[s]; end; if t<=max_str_ptr then confusion(""""); gubed @ A few more global variables are needed to keep track of statistics when |stat| $\ldots$ |tats| blocks are not commented out. @= @!pact_count:integer; {number of string pool compactions so far} @!pact_chars:integer; {total number of characters moved during compactions} @!pact_strs:integer; {total number of strings moved during compactions} @ @= pact_count:=0; pact_chars:=0; pact_strs:=0@; @ The following subroutine compares string |s| with another string of the same length that appears in |buffer| starting at position |k|; the result is |true| if and only if the strings are equal. @p function str_eq_buf(@!s:str_number;@!k:integer):boolean; {test equality of strings} label not_found; {loop exit} var @!j: pool_pointer; {running index} @!result: boolean; {result of comparison} begin j:=str_start[s]; while jbuffer[k] then begin result:=false; goto not_found; end; incr(j); incr(k); end; result:=true; not_found: str_eq_buf:=result; end; @ Here is a similar routine, but it compares two strings in the string pool, and it does not assume that they have the same length. If the first string is lexicographically greater than, less than, or equal to the second, the result is respectively positive, negative, or zero. @p function str_vs_str(@!s,@!t:str_number):integer; {test equality of strings} label exit; var @!j,@!k: pool_pointer; {running indices} @!ls,@!lt:integer; {lengths} @!l:integer; {length remaining to test} begin ls:=length(s); lt:=length(t); if ls<=lt then l:=ls@+else l:=lt; j:=str_start[s]; k:=str_start[t]; while l>0 do begin if str_pool[j]<>str_pool[k] then begin str_vs_str:=str_pool[j]-str_pool[k]; return; end; incr(j); incr(k); decr(l); end; str_vs_str:=ls-lt; exit:end; @ The initial values of |str_pool|, |str_start|, |pool_ptr|, and |str_ptr| are computed by the \.{INIMP} program, based in part on the information that \.{WEB} has output while processing \MP. @.INIMP@> @^string pool@> @p @!init function get_strings_started:boolean; {initializes the string pool, but returns |false| if something goes wrong} label done,exit; var @!k,@!l:0..255; {small indices or counters} @!m,@!n:text_char; {characters input from |pool_file|} @!g:str_number; {garbage} @!a:integer; {accumulator for check sum} @!c:boolean; {check sum has been checked} begin pool_ptr:=0; str_ptr:=0; max_pool_ptr:=0; max_str_ptr:=0; str_start[0]:=0; next_str[0]:=1; str_overflowed:=false; stat pool_in_use:=0; strs_in_use:=0; max_pl_used:=0; max_strs_used:=0; @; tats@; strs_used_up:=0; @; @; last_fixed_str:=str_ptr-1; fixed_str_use:=str_ptr; exit:end; tini @ The first 256 strings will consist of a single character only. @= for k:=0 to 255 do begin append_char(k); g:=make_string; str_ref[g]:=max_str_ref; end; @ The first 128 strings will contain 95 standard ASCII characters, and the other 33 characters will be printed in three-symbol form like `\.{\^\^A}' unless a system-dependent change is made here. Installations that have an extended character set, where for example |xchr[@'32]=@t\.{\'^^Z\'}@>|, would like string @'32 to be printed as the single character @'32 instead of the three characters @'136, @'136, @'132 (\.{\^\^Z}). On the other hand, even people with an extended character set will want to represent string @'15 by \.{\^\^M}, since @'15 is ASCII's ``carriage return'' code; the idea is to produce visible strings instead of tabs or line-feeds or carriage-returns or bell-rings or characters that are treated anomalously in text files. Unprintable characters of codes 128--255 are, similarly, rendered \.{\^\^80}--\.{\^\^ff}. The boolean expression defined here should be |true| unless \MP\ internal code number~|k| corresponds to a non-troublesome visible symbol in the local character set. If character |k| cannot be printed, and |k<@'200|, then character |k+@'100| or |k-@'100| must be printable; moreover, ASCII codes |[@'60..@'71, @'141..@'146]| must be printable. @^character set dependencies@> @^system dependencies@> @= (k<" ")or(k>"~") @ When the \.{WEB} system program called \.{TANGLE} processes the \.{MP.WEB} description that you are now reading, it outputs the \PASCAL\ program \.{MP.PAS} and also a string pool file called \.{MP.POOL}. The \.{INIMP} @.WEB@>@.INIMP@> program reads the latter file, where each string appears as a two-digit decimal length followed by the string itself, and the information is recorded in \MP's string memory. @= @!init @!pool_file:alpha_file; {the string-pool file output by \.{TANGLE}} tini @ @d bad_pool(#)==begin wake_up_terminal; write_ln(term_out,#); a_close(pool_file); get_strings_started:=false; return; end @= name_of_file:=pool_name; {we needn't set |name_length|} if a_open_in(pool_file) then begin c:=false; repeat @; until c; a_close(pool_file); get_strings_started:=true; end else bad_pool('! I can''t read MP.POOL.') @.I can't read MP.POOL@> @ @= begin if eof(pool_file) then bad_pool('! MP.POOL has no check sum.'); @.MP.POOL has no check sum@> read(pool_file,m,n); {read two digits of string length} if m='*' then @ else begin if (xord[m]<"0")or(xord[m]>"9")or@| (xord[n]<"0")or(xord[n]>"9") then bad_pool('! MP.POOL line doesn''t begin with two digits.'); @.MP.POOL line doesn't...@> l:=xord[m]*10+xord[n]-"0"*11; {compute the length} if pool_ptr+l+string_vacancies>pool_size then bad_pool('! You have to increase POOLSIZE.'); @.You have to increase POOLSIZE@> if str_ptr+strings_vacant>=max_strings then bad_pool('! You have to increase MAXSTRINGS.'); @.You have to increase MAXSTRINGS@> for k:=1 to l do begin if eoln(pool_file) then m:=' '@+else read(pool_file,m); append_char(xord[m]); end; read_ln(pool_file); g:=make_string; str_ref[g]:=max_str_ref; end; end @ The \.{WEB} operation \.{@@\$} denotes the value that should be at the end of this \.{MP.POOL} file; any other value means that the wrong pool file has been loaded. @^check sum@> @= begin a:=0; k:=1; loop@+ begin if (xord[n]<"0")or(xord[n]>"9") then bad_pool('! MP.POOL check sum doesn''t have nine digits.'); @.MP.POOL check sum...@> a:=10*a+xord[n]-"0"; if k=9 then goto done; incr(k); read(pool_file,n); end; done: if a<>@$ then bad_pool('! MP.POOL doesn''t match; TANGLE me again.'); @.MP.POOL doesn't match@> c:=true; end @* \[5] On-line and off-line printing. Messages that are sent to a user's terminal and to the transcript-log file are produced by several `|print|' procedures. These procedures will direct their output to a variety of places, based on the setting of the global variable |selector|, which has the following possible values: \yskip \hang |term_and_log|, the normal setting, prints on the terminal and on the transcript file. \hang |log_only|, prints only on the transcript file. \hang |term_only|, prints only on the terminal. \hang |no_print|, doesn't print at all. This is used only in rare cases before the transcript file is open. \hang |ps_file_only| prints only on the \ps\ output file. \hang |pseudo|, puts output into a cyclic buffer that is used by the |show_context| routine; when we get to that routine we shall discuss the reasoning behind this curious mode. \hang |new_string|, appends the output to the current string in the string pool. \hang |0..max_write_files-1| prints on one of the files used for the \&{write} @:write_}{\&{write} primitive@> command. \yskip \noindent The symbolic names `|term_and_log|', etc., have been assigned numeric codes that satisfy the convenient relations |no_print+1=term_only|, |no_print+2=log_only|, |term_only+2=log_only+1=term_and_log|. These relations are not used when |selector| could be |pseudo|, |new_string|, or |ps_file_only|. We need not check for unprintable characters when |selector= @!log_file : alpha_file; {transcript of \MP\ session} @!ps_file: alpha_file; {the generic font output goes here} @!selector : 0..max_selector; {where to print a message} @!dig : array[0..22] of 0..15; {digits in a number being output} @!tally : integer; {the number of characters recently printed} @!term_offset : 0..max_print_line; {the number of characters on the current terminal line} @!file_offset : 0..max_print_line; {the number of characters on the current file line} @!ps_offset : integer; {the number of characters on the current \ps\ file line} @!trick_buf:array[0..error_line] of ASCII_code; {circular buffer for pseudoprinting} @!trick_count: integer; {threshold for pseudoprinting, explained later} @!first_count: integer; {another variable for pseudoprinting} @ @= selector:=term_only; tally:=0; term_offset:=0; file_offset:=0; ps_offset:=0; @ Macro abbreviations for output to the terminal and to the log file are defined here for convenience. Some systems need special conventions for terminal output, and it is possible to adhere to those conventions by changing |wterm|, |wterm_ln|, and |wterm_cr| here. @^system dependencies@> @d wterm(#)==write(term_out,#) @d wterm_ln(#)==write_ln(term_out,#) @d wterm_cr==write_ln(term_out) @d wlog(#)==write(log_file,#) @d wlog_ln(#)==write_ln(log_file,#) @d wlog_cr==write_ln(log_file) @d wps(#)==write(ps_file,#) @d wps_ln(#)==write_ln(ps_file,#) @d wps_cr==write_ln(ps_file) @ To end a line of text output, we call |print_ln|. Cases |0..max_write_files| use an array |wr_file| that will be declared later. @= procedure print_ln; {prints an end-of-line} begin case selector of term_and_log: begin wterm_cr; wlog_cr; term_offset:=0; file_offset:=0; end; log_only: begin wlog_cr; file_offset:=0; end; term_only: begin wterm_cr; term_offset:=0; end; ps_file_only: begin wps_cr; ps_offset:=0; end; no_print,pseudo,new_string: do_nothing; othercases write_ln(wr_file[selector]) endcases; end; {note that |tally| is not affected} @ The |print_visible_char| procedure sends one character to the desired destination, using the |xchr| array to map it into an external character compatible with |input_ln|. (It assumes that it is always called with a visible ASCII character.) All printing comes through |print_ln| or |print_char|, which ultimately calls |print_visible_char|, hence these routines are the ones that limit lines to at most |max_print_line| characters. But we must make an exception for the \ps\ output file since it is not safe to cut up lines arbitrarily in \ps. Procedure |unit_str_room| needs to be declared |forward| here because it calls |do_compaction| and |do_compaction| can call the error routines. Actually, |unit_str_room| avoids |overflow| errors but it can call |confusion|. @= procedure@?unit_str_room; forward;@t\2@>@/ procedure print_visible_char(@!s:ASCII_code); {prints a single character} label done; begin case selector of term_and_log: begin wterm(xchr[s]); wlog(xchr[s]); incr(term_offset); incr(file_offset); if term_offset=max_print_line then begin wterm_cr; term_offset:=0; end; if file_offset=max_print_line then begin wlog_cr; file_offset:=0; end; end; log_only: begin wlog(xchr[s]); incr(file_offset); if file_offset=max_print_line then print_ln; end; term_only: begin wterm(xchr[s]); incr(term_offset); if term_offset=max_print_line then print_ln; end; ps_file_only: begin wps(xchr[s]); incr(ps_offset); end; no_print: do_nothing; pseudo: if tally=max_pool_ptr then begin unit_str_room; if pool_ptr>=pool_size then goto done; {drop characters if string space is full} end; append_char(s); end; othercases write(wr_file[selector],xchr[s]) endcases; done:incr(tally); end; @ The |print_char| procedure sends one character to the desired destination. File names and string expressions might contain |ASCII_code| values that can't be printed using |print_visible_char|. These characters will be printed in three- or four-symbol form like `\.{\^\^A}' or `\.{\^\^e4}'. (This procedure assumes that it is safe to bypass all checks for unprintable characters when |selector| is in the range |0..max_write_files-1| or when |selector=ps_file_only|. In the former case the user might want to write unprintable characters, and in the latter case the \ps\ printing routines check their arguments themselves before calling |print_char| or |print|.) @d print_lc_hex(#)==l:=#; if l<10 then print_visible_char(l+"0")@+else print_visible_char(l-10+"a") @= procedure print_char(@!k:ASCII_code); {prints a single character} var l:0..255; {small index or counter} begin if selector then begin print_visible_char("^"); print_visible_char("^"); if k<@'100 then print_visible_char(k+@'100) else if k<@'200 then print_visible_char(k-@'100) else begin print_lc_hex(k div 16); print_lc_hex(k mod 16); end; end else print_visible_char(k); end; @ An entire string is output by calling |print|. Note that if we are outputting the single standard ASCII character \.c, we could call |print("c")|, since |"c"=99| is the number of a single-character string, as explained above. But |print_char("c")| is quicker, so \MP\ goes directly to the |print_char| routine when it knows that this is safe. (The present implementation assumes that it is always safe to print a visible ASCII character.) @^system dependencies@> @= procedure print(@!s:integer); {prints string |s|} var @!j:pool_pointer; {current character code position} begin if (s<0)or(s>max_str_ptr) then s:="???"; {this can't happen} @.???@> j:=str_start[s]; while j= update_terminal; @ The procedure |print_nl| is like |print|, but it makes sure that the string appears at the beginning of a new line. @= procedure print_nl(@!s:str_number); {prints string |s| at beginning of line} begin case selector of term_and_log: if (term_offset>0)or(file_offset>0) then print_ln; log_only: if file_offset>0 then print_ln; term_only: if term_offset>0 then print_ln; ps_file_only: if ps_offset>0 then print_ln; no_print,pseudo,new_string: do_nothing; end; {there are no other cases} print(s); end; @ An array of digits in the range |0..9| is printed by |print_the_digs|. @= procedure print_the_digs(@!k:eight_bits); {prints |dig[k-1]|$\,\ldots\,$|dig[0]|} begin while k>0 do begin decr(k); print_char("0"+dig[k]); end; end; @ The following procedure, which prints out the decimal representation of a given integer |n|, has been written carefully so that it works properly if |n=0| or if |(-n)| would cause overflow. It does not apply |mod| or |div| to negative arguments, since such operations are not implemented consistently by all \PASCAL\ compilers. @= procedure print_int(@!n:integer); {prints an integer in decimal form} var k:0..23; {index to current digit; we assume that $|n|<10^{23}$} @!m:integer; {used to negate |n| in possibly dangerous cases} begin k:=0; if n<0 then begin print_char("-"); if n>-100000000 then negate(n) else begin m:=-1-n; n:=m div 10; m:=(m mod 10)+1; k:=1; if m<10 then dig[0]:=m else begin dig[0]:=0; incr(n); end; end; end; repeat dig[k]:=n mod 10; n:=n div 10; incr(k); until n=0; print_the_digs(k); end; @ \MP\ also makes use of a trivial procedure to print two digits. The following subroutine is usually called with a parameter in the range |0<=n<=99|. @p procedure print_dd(@!n:integer); {prints two least significant digits} begin n:=abs(n) mod 100; print_char("0"+(n div 10)); print_char("0"+(n mod 10)); end; @ Here is a procedure that asks the user to type a line of input, assuming that the |selector| setting is either |term_only| or |term_and_log|. The input is placed into locations |first| through |last-1| of the |buffer| array, and echoed on the transcript file if appropriate. This procedure is never called when |interaction term_offset:=0; {the user's line ended with \<\rm return>} decr(selector); {prepare to echo the input} if last<>first then for k:=first to last-1 do print(buffer[k]); print_ln; buffer[last]:="%"; incr(selector); {restore previous status} end; @* \[6] Reporting errors. When something anomalous is detected, \MP\ typically does something like this: $$\vbox{\halign{#\hfil\cr |print_err("Something anomalous has been detected");|\cr |help3("This is the first line of my offer to help.")|\cr |("This is the second line. I'm trying to")|\cr |("explain the best way for you to proceed.");|\cr |error;|\cr}}$$ A two-line help message would be given using |help2|, etc.; these informal helps should use simple vocabulary that complements the words used in the official error message that was printed. (Outside the U.S.A., the help messages should preferably be translated into the local vernacular. Each line of help is at most 60 characters long, in the present implementation, so that |max_print_line| will not be exceeded.) The |print_err| procedure supplies a `\.!' before the official message, and makes sure that the terminal is awake if a stop is going to occur. The |error| procedure supplies a `\..' after the official message, then it shows the location of the error; and if |interaction=error_stop_mode|, it also enters into a dialog with the user, during which time the help message may be printed. @^system dependencies@> @ The global variable |interaction| has four settings, representing increasing amounts of user interaction: @d batch_mode=0 {omits all stops and omits terminal output} @d nonstop_mode=1 {omits all stops} @d scroll_mode=2 {omits error stops} @d error_stop_mode=3 {stops at every opportunity to interact} @d print_err(#)==begin if interaction=error_stop_mode then wake_up_terminal; print_nl("! "); print(#); @.!\relax@> end @= @!interaction:batch_mode..error_stop_mode; {current level of interaction} @ @=interaction:=error_stop_mode; @ \MP\ is careful not to call |error| when the print |selector| setting might be unusual. The only possible values of |selector| at the time of error messages are \yskip\hang|no_print| (when |interaction=batch_mode| and |log_file| not yet open); \hang|term_only| (when |interaction>batch_mode| and |log_file| not yet open); \hang|log_only| (when |interaction=batch_mode| and |log_file| is open); \hang|term_and_log| (when |interaction>batch_mode| and |log_file| is open). @= if interaction=batch_mode then selector:=no_print@+else selector:=term_only @ A global variable |deletions_allowed| is set |false| if the |get_next| routine is active when |error| is called; this ensures that |get_next| will never be called recursively. @^recursion@> The global variable |history| records the worst level of error that has been detected. It has four possible values: |spotless|, |warning_issued|, |error_message_issued|, and |fatal_error_stop|. Another global variable, |error_count|, is increased by one when an |error| occurs without an interactive dialog, and it is reset to zero at the end of every statement. If |error_count| reaches 100, \MP\ decides that there is no point in continuing further. @d spotless=0 {|history| value when nothing has been amiss yet} @d warning_issued=1 {|history| value when |begin_diagnostic| has been called} @d error_message_issued=2 {|history| value when |error| has been called} @d fatal_error_stop=3 {|history| value when termination was premature} @= @!deletions_allowed:boolean; {is it safe for |error| to call |get_next|?} @!history:spotless..fatal_error_stop; {has the source input been clean so far?} @!error_count:-1..100; {the number of scrolled errors since the last statement ended} @ The value of |history| is initially |fatal_error_stop|, but it will be changed to |spotless| if \MP\ survives the initialization process. @= deletions_allowed:=true; error_count:=0; {|history| is initialized elsewhere} @ Since errors can be detected almost anywhere in \MP, we want to declare the error procedures near the beginning of the program. But the error procedures in turn use some other procedures, which need to be declared |forward| before we get to |error| itself. It is possible for |error| to be called recursively if some error arises when |get_next| is being used to delete a token, and/or if some fatal error occurs while \MP\ is trying to fix a non-fatal one. But such recursion @^recursion@> is never more than two levels deep. @= procedure@?normalize_selector; forward;@t\2@>@/ procedure@?get_next; forward;@t\2@>@/ procedure@?term_input; forward;@t\2@>@/ procedure@?show_context; forward;@t\2@>@/ procedure@?begin_file_reading; forward;@t\2@>@/ procedure@?open_log_file; forward;@t\2@>@/ procedure@?close_files_and_terminate; forward;@t\2@>@/ procedure@?clear_for_error_prompt; forward;@t\2@>@/ @t\4\hskip-\fontdimen2\font@>@;@+@!debug@+procedure@?debug_help; forward;@;@+gubed@;@/ @t\4@>@ @ Individual lines of help are recorded in the array |help_line|, which contains entries in positions |0..(help_ptr-1)|. They should be printed in reverse order, i.e., with |help_line[0]| appearing last. @d hlp1(#)==help_line[0]:=#;@+end @d hlp2(#)==help_line[1]:=#; hlp1 @d hlp3(#)==help_line[2]:=#; hlp2 @d hlp4(#)==help_line[3]:=#; hlp3 @d hlp5(#)==help_line[4]:=#; hlp4 @d hlp6(#)==help_line[5]:=#; hlp5 @d help0==help_ptr:=0 {sometimes there might be no help} @d help1==@+begin help_ptr:=1; hlp1 {use this with one help line} @d help2==@+begin help_ptr:=2; hlp2 {use this with two help lines} @d help3==@+begin help_ptr:=3; hlp3 {use this with three help lines} @d help4==@+begin help_ptr:=4; hlp4 {use this with four help lines} @d help5==@+begin help_ptr:=5; hlp5 {use this with five help lines} @d help6==@+begin help_ptr:=6; hlp6 {use this with six help lines} @= @!help_line:array[0..5] of str_number; {helps for the next |error|} @!help_ptr:0..6; {the number of help lines present} @!use_err_help:boolean; {should the |err_help| string be shown?} @!err_help:str_number; {a string set up by \&{errhelp}} @ @= help_ptr:=0; use_err_help:=false; err_help:=0; @ The |jump_out| procedure just cuts across all active procedure levels and goes to |end_of_MP|. This is the only nonlocal |@!goto| statement in the whole program. It is used when there is no recovery from a particular error. Some \PASCAL\ compilers do not implement non-local |goto| statements. @^system dependencies@> In such cases the body of |jump_out| should simply be `|close_files_and_terminate|;\thinspace' followed by a call on some system procedure that quietly terminates the program. @= procedure jump_out; begin goto end_of_MP; end; @ Here now is the general |error| routine. @= procedure error; {completes the job of error reporting} label continue,exit; var @!c:ASCII_code; {what the user types} @!s1,@!s2,@!s3:integer; {used to save global variables when deleting tokens} @!j:pool_pointer; {character position being printed} begin if history; incr(error_count); if error_count=100 then begin print_nl("(That makes 100 errors; please try again.)"); @.That makes 100 errors...@> history:=fatal_error_stop; jump_out; end; @; exit:end; @ @= loop@+begin continue: clear_for_error_prompt; prompt_input("? "); @.?\relax@> if last=first then return; c:=buffer[first]; if c>="a" then c:=c+"A"-"a"; {convert to uppercase} @; end @ It is desirable to provide an `\.E' option here that gives the user an easy way to return from \MP\ to the system editor, with the offending line ready to be edited. But such an extension requires some system wizardry, so the present implementation simply types out the name of the file that should be edited and the relevant line number. @^system dependencies@> There is a secret `\.D' option available when the debugging routines haven't been commented~out. @^debugging@> @= case c of "0","1","2","3","4","5","6","7","8","9": if deletions_allowed then @; @t\4\4@>@;@+@!debug "D":begin debug_help;goto continue;@+end;@+gubed@/ "E": if file_ptr>0 then begin print_nl("You want to edit file "); @.You want to edit file x@> print(input_stack[file_ptr].name_field); print(" at line "); print_int(true_line);@/ interaction:=scroll_mode; jump_out; end; "H": @; "I":@; "Q","R","S":@; "X":begin interaction:=scroll_mode; jump_out; end; othercases do_nothing endcases;@/ @ @ @= begin print("Type to proceed, S to scroll future error messages,");@/ @.Type to proceed...@> print_nl("R to run without stopping, Q to run quietly,");@/ print_nl("I to insert something, "); if file_ptr>0 then print("E to edit your file,"); if deletions_allowed then print_nl("1 or ... or 9 to ignore the next 1 to 9 tokens of input,"); print_nl("H for help, X to quit."); end @ Here the author of \MP\ apologizes for making use of the numerical relation between |"Q"|, |"R"|, |"S"|, and the desired interaction settings |batch_mode|, |nonstop_mode|, |scroll_mode|. @^Knuth, Donald Ervin@> @= begin error_count:=0; interaction:=batch_mode+c-"Q"; print("OK, entering "); case c of "Q":begin print("batchmode"); decr(selector); end; "R":print("nonstopmode"); "S":print("scrollmode"); end; {there are no other cases} print("..."); print_ln; update_terminal; return; end @ When the following code is executed, |buffer[(first+1)..(last-1)]| may contain the material inserted by the user; otherwise another prompt will be given. In order to understand this part of the program fully, you need to be familiar with \MP's input stacks. @= begin begin_file_reading; {enter a new syntactic level for terminal input} if last>first+1 then begin loc:=first+1; buffer[first]:=" "; end else begin prompt_input("insert>"); loc:=first; @.insert>@> end; first:=last+1; cur_input.limit_field:=last; return; end @ We allow deletion of up to 99 tokens at a time. @= begin s1:=cur_cmd; s2:=cur_mod; s3:=cur_sym; OK_to_interrupt:=false; if (last>first+1) and (buffer[first+1]>="0")and(buffer[first+1]<="9") then c:=c*10+buffer[first+1]-"0"*11 else c:=c-"0"; while c>0 do begin get_next; {one-level recursive call of |error| is possible} @; decr(c); end; cur_cmd:=s1; cur_mod:=s2; cur_sym:=s3; OK_to_interrupt:=true; help2("I have just deleted some text, as you asked.")@/ ("You can now delete more, or insert, or whatever."); show_context; goto continue; end @ @= begin if use_err_help then begin @; use_err_help:=false; end else begin if help_ptr=0 then help2("Sorry, I don't know how to help in this situation.")@/ @t\kern1em@>("Maybe you should try asking a human?"); repeat decr(help_ptr); print(help_line[help_ptr]); print_ln; until help_ptr=0; end; help4("Sorry, I already gave what help I could...")@/ ("Maybe you should try asking a human?")@/ ("An error might have occurred before I noticed any problems.")@/ ("``If all else fails, read the instructions.''");@/ goto continue; end @ @= j:=str_start[err_help]; while jsi("%") then print(so(str_pool[j])) else if j+1=str_stop(err_help) then print_ln else if str_pool[j+1]<>si("%") then print_ln else begin incr(j); print_char("%"); end; incr(j); end @ @= if interaction>batch_mode then decr(selector); {avoid terminal output} if use_err_help then begin print_nl(""); @; end else while help_ptr>0 do begin decr(help_ptr); print_nl(help_line[help_ptr]); end; print_ln; if interaction>batch_mode then incr(selector); {re-enable terminal output} print_ln @ In anomalous cases, the print selector might be in an unknown state; the following subroutine is called to fix things just enough to keep running a bit longer. @p procedure normalize_selector; begin if log_opened then selector:=term_and_log else selector:=term_only; if job_name=0 then open_log_file; if interaction=batch_mode then decr(selector); end; @ The following procedure prints \MP's last words before dying. @d succumb==begin if interaction=error_stop_mode then interaction:=scroll_mode; {no more interaction} if log_opened then error; @!debug if interaction>batch_mode then debug_help;@;@+gubed@;@/ history:=fatal_error_stop; jump_out; {irrecoverable error} end @= procedure fatal_error(@!s:str_number); {prints |s|, and that's it} begin normalize_selector;@/ print_err("Emergency stop"); help1(s); succumb; @.Emergency stop@> end; @ Here is the most dreaded error message. @= procedure overflow(@!s:str_number;@!n:integer); {stop due to finiteness} begin normalize_selector; print_err("MetaPost capacity exceeded, sorry ["); @.MetaPost capacity exceeded ...@> print(s); print_char("="); print_int(n); print_char("]"); help2("If you really absolutely need more capacity,")@/ ("you can ask a wizard to enlarge me."); succumb; end; @ The program might sometime run completely amok, at which point there is no choice but to stop. If no previous error has been detected, that's bad news; a message is printed that is really intended for the \MP\ maintenance person instead of the user (unless the user has been particularly diabolical). The index entries for `this can't happen' may help to pinpoint the problem. @^dry rot@> @= procedure confusion(@!s:str_number); {consistency check violated; |s| tells where} begin normalize_selector; if history help1("I'm broken. Please show this to someone who can fix can fix"); end else begin print_err("I can't go on meeting you like this"); @.I can't go on...@> help2("One of your faux pas seems to have wounded me deeply...")@/ ("in fact, I'm barely conscious. Please fix it and try again."); end; succumb; end; @ Users occasionally want to interrupt \MP\ while it's running. If the \PASCAL\ runtime system allows this, one can implement a routine that sets the global variable |interrupt| to some nonzero value when such an interrupt is signaled. Otherwise there is probably at least a way to make |interrupt| nonzero using the \PASCAL\ debugger. @^system dependencies@> @^debugging@> @d check_interrupt==begin if interrupt<>0 then pause_for_instructions; end @= @!interrupt:integer; {should \MP\ pause for instructions?} @!OK_to_interrupt:boolean; {should interrupts be observed?} @ @= interrupt:=0; OK_to_interrupt:=true; @ When an interrupt has been detected, the program goes into its highest interaction level and lets the user have the full flexibility of the |error| routine. \MP\ checks for interrupts only at times when it is safe to do this. @p procedure pause_for_instructions; begin if OK_to_interrupt then begin interaction:=error_stop_mode; if (selector=log_only)or(selector=no_print) then incr(selector); print_err("Interruption"); @.Interruption@> help3("You rang?")@/ ("Try to insert some instructions for me (e.g.,`I show x'),")@/ ("unless you just want to quit by typing `X'."); deletions_allowed:=false; error; deletions_allowed:=true; interrupt:=0; end; end; @ Many of \MP's error messages state that a missing token has been inserted behind the scenes. We can save string space and program space by putting this common code into a subroutine. @p procedure missing_err(@!s:str_number); begin print_err("Missing `"); print(s); print("' has been inserted"); @.Missing...inserted@> end; @* \[7] Arithmetic with scaled numbers. The principal computations performed by \MP\ are done entirely in terms of integers less than $2^{31}$ in magnitude; thus, the arithmetic specified in this program can be carried out in exactly the same way on a wide variety of computers, including some small ones. @^small computers@> But \PASCAL\ does not define the @!|div| operation in the case of negative dividends; for example, the result of |(-2*n-1) div 2| is |-(n+1)| on some computers and |-n| on others. There are two principal types of arithmetic: ``translation-preserving,'' in which the identity |(a+q*b)div b=(a div b)+q| is valid; and ``negation-preserving,'' in which |(-a)div b=-(a div b)|. This leads to two \MP s, which can produce different results, although the differences should be negligible when the language is being used properly. The \TeX\ processor has been defined carefully so that both varieties of arithmetic will produce identical output, but it would be too inefficient to constrain \MP\ in a similar way. @d el_gordo == @'17777777777 {$2^{31}-1$, the largest value that \MP\ likes} @ One of \MP's most common operations is the calculation of $\lfloor{a+b\over2}\rfloor$, the midpoint of two given integers |a| and~|b|. The only decent way to do this in \PASCAL\ is to write `|(a+b) div 2|'; but on most machines it is far more efficient to calculate `|(a+b)| right shifted one bit'. Therefore the midpoint operation will always be denoted by `|half(a+b)|' in this program. If \MP\ is being implemented with languages that permit binary shifting, the |half| macro should be changed to make this operation as efficient as possible. Since some languages have shift operators that can only be trusted to work on positive numbers, there is also a macro |halfp| that is used only when the quantity being halved is known to be positive or zero. @d half(#)==(#) div 2 @d halfp(#)==(#) div 2 @ A single computation might use several subroutine calls, and it is desirable to avoid producing multiple error messages in case of arithmetic overflow. So the routines below set the global variable |arith_error| to |true| instead of reporting errors directly to the user. @= @!arith_error:boolean; {has arithmetic overflow occurred recently?} @ @= arith_error:=false; @ At crucial points the program will say |check_arith|, to test if an arithmetic error has been detected. @d check_arith==begin if arith_error then clear_arith;@+end @p procedure clear_arith; begin print_err("Arithmetic overflow"); @.Arithmetic overflow@> help4("Uh, oh. A little while ago one of the quantities that I was")@/ ("computing got too large, so I'm afraid your answers will be")@/ ("somewhat askew. You'll probably have to adopt different")@/ ("tactics next time. But I shall try to carry on anyway."); error; arith_error:=false; end; @ Addition is not always checked to make sure that it doesn't overflow, but in places where overflow isn't too unlikely the |slow_add| routine is used. @p function slow_add(@!x,@!y:integer):integer; begin if x>=0 then if y<=el_gordo-x then slow_add:=x+y else begin arith_error:=true; slow_add:=el_gordo; end else if -y<=el_gordo+x then slow_add:=x+y else begin arith_error:=true; slow_add:=-el_gordo; end; end; @ Fixed-point arithmetic is done on {\sl scaled integers\/} that are multiples of $2^{-16}$. In other words, a binary point is assumed to be sixteen bit positions from the right end of a binary computer word. @d quarter_unit == @'40000 {$2^{14}$, represents 0.250000} @d half_unit == @'100000 {$2^{15}$, represents 0.50000} @d three_quarter_unit == @'140000 {$3\cdot2^{14}$, represents 0.75000} @d unity == @'200000 {$2^{16}$, represents 1.00000} @d two == @'400000 {$2^{17}$, represents 2.00000} @d three == @'600000 {$2^{17}+2^{16}$, represents 3.00000} @= @!scaled = integer; {this type is used for scaled integers} @!small_number=0..63; {this type is self-explanatory} @ The following function is used to create a scaled integer from a given decimal fraction $(.d_0d_1\ldots d_{k-1})$, where |0<=k<=17|. The digit $d_i$ is given in |dig[i]|, and the calculation produces a correctly rounded result. @p function round_decimals(@!k:small_number) : scaled; {converts a decimal fraction} var @!a:integer; {the accumulator} begin a:=0; while k>0 do begin decr(k); a:=(a+dig[k]*two) div 10; end; round_decimals:=halfp(a+1); end; @ Conversely, here is a procedure analogous to |print_int|. If the output of this procedure is subsequently read by \MP\ and converted by the |round_decimals| routine above, it turns out that the original value will be reproduced exactly. A decimal point is printed only if the value is not an integer. If there is more than one way to print the result with the optimum number of digits following the decimal point, the closest possible value is given. The invariant relation in the \&{repeat} loop is that a sequence of decimal digits yet to be printed will yield the original number if and only if they form a fraction~$f$ in the range $s-\delta\L10\cdot2^{16}f= procedure print_scaled(@!s:scaled); {prints scaled real, rounded to five digits} var @!delta:scaled; {amount of allowable inaccuracy} begin if s<0 then begin print_char("-"); negate(s); {print the sign, if negative} end; print_int(s div unity); {print the integer part} s:=10*(s mod unity)+5; if s<>5 then begin delta:=10; print_char("."); repeat if delta>unity then s:=s+@'100000-(delta div 2); {round the final digit} print_char("0"+(s div unity)); s:=10*(s mod unity); delta:=delta*10; until s<=delta; end; end; @ We often want to print two scaled quantities in parentheses, separated by a comma. @= procedure print_two(@!x,@!y:scaled); {prints `|(x,y)|'} begin print_char("("); print_scaled(x); print_char(","); print_scaled(y); print_char(")"); end; @ The |scaled| quantities in \MP\ programs are generally supposed to be less than $2^{12}$ in absolute value, so \MP\ does much of its internal arithmetic with 28~significant bits of precision. A |fraction| denotes a scaled integer whose binary point is assumed to be 28 bit positions from the right. @d fraction_half==@'1000000000 {$2^{27}$, represents 0.50000000} @d fraction_one==@'2000000000 {$2^{28}$, represents 1.00000000} @d fraction_two==@'4000000000 {$2^{29}$, represents 2.00000000} @d fraction_three==@'6000000000 {$3\cdot2^{28}$, represents 3.00000000} @d fraction_four==@'10000000000 {$2^{30}$, represents 4.00000000} @= @!fraction=integer; {this type is used for scaled fractions} @ In fact, the two sorts of scaling discussed above aren't quite sufficient; \MP\ has yet another, used internally to keep track of angles in units of $2^{-20}$ degrees. @d forty_five_deg==@'264000000 {$45\cdot2^{20}$, represents $45^\circ$} @d ninety_deg==@'550000000 {$90\cdot2^{20}$, represents $90^\circ$} @d one_eighty_deg==@'1320000000 {$180\cdot2^{20}$, represents $180^\circ$} @d three_sixty_deg==@'2640000000 {$360\cdot2^{20}$, represents $360^\circ$} @= @!angle=integer; {this type is used for scaled angles} @ The |make_fraction| routine produces the |fraction| equivalent of |p/q|, given integers |p| and~|q|; it computes the integer $f=\lfloor2^{28}p/q+{1\over2}\rfloor$, when $p$ and $q$ are positive. If |p| and |q| are both of the same scaled type |t|, the ``type relation'' |make_fraction(t,t)=fraction| is valid; and it's also possible to use the subroutine ``backwards,'' using the relation |make_fraction(t,fraction)=t| between scaled types. If the result would have magnitude $2^{31}$ or more, |make_fraction| sets |arith_error:=true|. Most of \MP's internal computations have been designed to avoid this sort of error. If this subroutine were programmed in assembly language on a typical machine, we could simply compute |(@t$2^{28}$@>*p)div q|, since a double-precision product can often be input to a fixed-point division instruction. But when we are restricted to \PASCAL\ arithmetic it is necessary either to resort to multiple-precision maneuvering or to use a simple but slow iteration. The multiple-precision technique would be about three times faster than the code adopted here, but it would be comparatively long and tricky, involving about sixteen additional multiplications and divisions. This operation is part of \MP's ``inner loop''; indeed, it will consume nearly 10\pct! of the running time (exclusive of input and output) if the code below is left unchanged. A machine-dependent recoding will therefore make \MP\ run faster. The present implementation is highly portable, but slow; it avoids multiplication and division except in the initial stage. System wizards should be careful to replace it with a routine that is guaranteed to produce identical results in all cases. @^system dependencies@> As noted below, a few more routines should also be replaced by machine-dependent code, for efficiency. But when a procedure is not part of the ``inner loop,'' such changes aren't advisable; simplicity and robustness are preferable to trickery, unless the cost is too high. @^inner loop@> @p function make_fraction(@!p,@!q:integer):fraction; var @!f:integer; {the fraction bits, with a leading 1 bit} @!n:integer; {the integer part of $\vert p/q\vert$} @!negative:boolean; {should the result be negated?} @!be_careful:integer; {disables certain compiler optimizations} begin if p>=0 then negative:=false else begin negate(p); negative:=true; end; if q<=0 then begin debug if q=0 then confusion("/");@;@+gubed@;@/ @:this can't happen /}{\quad \./@> negate(q); negative:=not negative; end; n:=p div q; p:=p mod q; if n>=8 then begin arith_error:=true; if negative then make_fraction:=-el_gordo@+else make_fraction:=el_gordo; end else begin n:=(n-1)*fraction_one; @; if negative then make_fraction:=-(f+n)@+else make_fraction:=f+n; end; end; @ The |repeat| loop here preserves the following invariant relations between |f|, |p|, and~|q|: (i)~|0<=p @= f:=1; repeat be_careful:=p-q; p:=be_careful+p; if p>=0 then f:=f+f+1 else begin double(f); p:=p+q; end; until f>=fraction_one; be_careful:=p-q; if be_careful+p>=0 then incr(f) @ The dual of |make_fraction| is |take_fraction|, which multiplies a given integer~|q| by a fraction~|f|. When the operands are positive, it computes $p=\lfloor qf/2^{28}+{1\over2}\rfloor$, a symmetric function of |q| and~|f|. This routine is even more ``inner loopy'' than |make_fraction|; the present implementation consumes almost 20\pct! of \MP's computation time during typical jobs, so a machine-language substitute is advisable. @^inner loop@> @^system dependencies@> @p function take_fraction(@!q:integer;@!f:fraction):integer; var @!p:integer; {the fraction so far} @!negative:boolean; {should the result be negated?} @!n:integer; {additional multiple of $q$} @!be_careful:integer; {disables certain compiler optimizations} begin @=0| and |q>0|@>; if f; be_careful:=n-el_gordo; if be_careful+p>0 then begin arith_error:=true; n:=el_gordo-p; end; if negative then take_fraction:=-(n+p) else take_fraction:=n+p; end; @ @=0| and |q>0|@>= if f>=0 then negative:=false else begin negate(f); negative:=true; end; if q<0 then begin negate(q); negative:=not negative; end; @ The invariant relations in this case are (i)~$\lfloor(qf+p)/2^k\rfloor =\lfloor qf_0/2^{28}+{1\over2}\rfloor$, where $k$ is an integer and $f_0$ is the original value of~$f$; (ii)~$2^k\L f<2^{k+1}$. @^inner loop@> @= p:=fraction_half; {that's $2^{27}$; the invariants hold now with $k=28$} if q @p function take_scaled(@!q:integer;@!f:scaled):integer; var @!p:integer; {the fraction so far} @!negative:boolean; {should the result be negated?} @!n:integer; {additional multiple of $q$} @!be_careful:integer; {disables certain compiler optimizations} begin @=0| and |q>0|@>; if f; be_careful:=n-el_gordo; if be_careful+p>0 then begin arith_error:=true; n:=el_gordo-p; end; if negative then take_scaled:=-(n+p) else take_scaled:=n+p; end; @ @= p:=half_unit; {that's $2^{15}$; the invariants hold now with $k=16$} @^inner loop@> if q=0 then negative:=false else begin negate(p); negative:=true; end; if q<=0 then begin debug if q=0 then confusion("/");@+gubed@;@/ @:this can't happen /}{\quad \./@> negate(q); negative:=not negative; end; n:=p div q; p:=p mod q; if n>=@'100000 then begin arith_error:=true; if negative then make_scaled:=-el_gordo@+else make_scaled:=el_gordo; end else begin n:=(n-1)*unity; @; if negative then make_scaled:=-(f+n)@+else make_scaled:=f+n; end; end; @ @= f:=1; repeat be_careful:=p-q; p:=be_careful+p; if p>=0 then f:=f+f+1 else begin double(f); p:=p+q; end; until f>=unity; be_careful:=p-q; if be_careful+p>=0 then incr(f) @ Here is a typical example of how the routines above can be used. It computes the function $${1\over3\tau}f(\theta,\phi)= {\tau^{-1}\bigl(2+\sqrt2\,(\sin\theta-{1\over16}\sin\phi) (\sin\phi-{1\over16}\sin\theta)(\cos\theta-\cos\phi)\bigr)\over 3\,\bigl(1+{1\over2}(\sqrt5-1)\cos\theta+{1\over2}(3-\sqrt5\,)\cos\phi\bigr)},$$ where $\tau$ is a |scaled| ``tension'' parameter. This is \MP's magic fudge factor for placing the first control point of a curve that starts at an angle $\theta$ and ends at an angle $\phi$ from the straight path. (Actually, if the stated quantity exceeds 4, \MP\ reduces it to~4.) The trigonometric quantity to be multiplied by $\sqrt2$ is less than $\sqrt2$. (It's a sum of eight terms whose absolute values can be bounded using relations such as $\sin\theta\cos\theta\L{1\over2}$.) Thus the numerator is positive; and since the tension $\tau$ is constrained to be at least $3\over4$, the numerator is less than $16\over3$. The denominator is nonnegative and at most~6. Hence the fixed-point calculations below are guaranteed to stay within the bounds of a 32-bit computer word. The angles $\theta$ and $\phi$ are given implicitly in terms of |fraction| arguments |st|, |ct|, |sf|, and |cf|, representing $\sin\theta$, $\cos\theta$, $\sin\phi$, and $\cos\phi$, respectively. @p function velocity(@!st,@!ct,@!sf,@!cf:fraction;@!t:scaled):fraction; var @!acc,@!num,@!denom:integer; {registers for intermediate calculations} begin acc:=take_fraction(st-(sf div 16), sf-(st div 16)); acc:=take_fraction(acc,ct-cf); num:=fraction_two+take_fraction(acc,379625062); {$2^{28}\sqrt2\approx379625062.497$} denom:=fraction_three+take_fraction(ct,497706707)+take_fraction(cf,307599661); {$3\cdot2^{27}\cdot(\sqrt5-1)\approx497706706.78$ and $3\cdot2^{27}\cdot(3-\sqrt5\,)\approx307599661.22$} if t<>unity then num:=make_scaled(num,t); {|make_scaled(fraction,scaled)=fraction|} if num div 4>=denom then velocity:=fraction_four else velocity:=make_fraction(num,denom); end; @ The following somewhat different subroutine tests rigorously if $ab$ is greater than, equal to, or less than~$cd$, given integers $(a,b,c,d)$. In most cases a quick decision is reached. The result is $+1$, 0, or~$-1$ in the three respective cases. @d return_sign(#)==begin ab_vs_cd:=#; return; end @p function ab_vs_cd(@!a,b,c,d:integer):integer; label exit; var @!q,@!r:integer; {temporary registers} begin @=0|, |b,d>0|@>; loop@+ begin q := a div d; r := c div b; if q<>r then if q>r then return_sign(1)@+else return_sign(-1); q := a mod d; r := c mod b; if r=0 then if q=0 then return_sign(0)@+else return_sign(1); if q=0 then return_sign(-1); a:=b; b:=q; c:=d; d:=r; end; {now |a>d>0| and |c>b>0|} exit:end; @ @= if a<0 then begin negate(a); negate(b); end; if c<0 then begin negate(c); negate(d); end; if d<=0 then begin if b>=0 then if ((a=0)or(b=0))and((c=0)or(d=0)) then return_sign(0) else return_sign(1); if d=0 then if a=0 then return_sign(0)@+else return_sign(-1); q:=a; a:=c; c:=q; q:=-b; b:=-d; d:=q; end else if b<=0 then begin if b<0 then if a>0 then return_sign(-1); if c=0 then return_sign(0) else return_sign(-1); end @ We conclude this set of elementary routines with some simple rounding and truncation operations that are coded in a machine-independent fashion. The routines are slightly complicated because we want them to work without overflow whenever $-2^{31}\L x<2^{31}$. @p function floor_scaled(@!x:scaled):scaled; {$2^{16}\lfloor x/2^{16}\rfloor$} var @!be_careful:integer; {temporary register} begin if x>=0 then floor_scaled:=x-(x mod unity) else begin be_careful:=x+1; floor_scaled:=x+((-be_careful) mod unity)+1-unity; end; end; @# function round_unscaled(@!x:scaled):integer; {$\lfloor x/2^{16}+.5\rfloor$} var @!be_careful:integer; {temporary register} begin if x>=half_unit then round_unscaled:=1+((x-half_unit) div unity) else if x>=-half_unit then round_unscaled:=0 else begin be_careful:=x+1; round_unscaled:=-(1+((-be_careful-half_unit) div unity)); end; end; @# function round_fraction(@!x:fraction):scaled; {$\lfloor x/2^{12}+.5\rfloor$} var @!be_careful:integer; {temporary register} begin if x>=2048 then round_fraction:=1+((x-2048) div 4096) else if x>=-2048 then round_fraction:=0 else begin be_careful:=x+1; round_fraction:=-(1+((-be_careful-2048) div 4096)); end; end; @* \[8] Algebraic and transcendental functions. \MP\ computes all of the necessary special functions from scratch, without relying on |real| arithmetic or system subroutines for sines, cosines, etc. @ To get the square root of a |scaled| number |x|, we want to calculate $s=\lfloor 2^8\!\sqrt x +{1\over2}\rfloor$. If $x>0$, this is the unique integer such that $2^{16}x-s\L s^2<2^{16}x+s$. The following subroutine determines $s$ by an iterative method that maintains the invariant relations $x=2^{46-2k}x_0\bmod 2^{30}$, $0 else begin k:=23; q:=2; while x|\unskip} begin decr(k); x:=x+x+x+x; end; if x; until k=0; square_rt:=halfp(q); end; end; @ @= begin if x<0 then begin print_err("Square root of "); @.Square root...replaced by 0@> print_scaled(x); print(" has been replaced by 0"); help2("Since I don't take square roots of negative numbers,")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; end; square_rt:=0; end @ @= double(x); double(y); if x>=fraction_four then {note that |fraction_four=@t$2^{30}$@>|} begin x:=x-fraction_four; incr(y); end; double(x); y:=y+y-q; double(q); if x>=fraction_four then begin x:=x-fraction_four; incr(y); end; if y>q then begin y:=y-q; q:=q+2; end else if y<=0 then begin q:=q-2; y:=y+q; end; decr(k) @ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by an elegant iterative scheme due to Cleve Moler and Donald Morrison [{\sl IBM Journal @^Moler, Cleve Barry@> @^Morrison, Donald Ross@> of Research and Development\/ \bf27} (1983), 577--581]. It modifies |a| and~|b| in such a way that their Pythagorean sum remains invariant, while the smaller argument decreases. @p function pyth_add(@!a,@!b:integer):integer; label done; var @!r:fraction; {register used to transform |a| and |b|} @!big:boolean; {is the result dangerously near $2^{31}$?} begin a:=abs(a); b:=abs(b); if a0 then begin if a; if big then if a= loop@+ begin r:=make_fraction(b,a); r:=take_fraction(r,r); {now $r\approx b^2/a^2$} if r=0 then goto done; r:=make_fraction(r,fraction_four+r); a:=a+take_fraction(a+a,r); b:=take_fraction(b,r); end; done: @ Here is a similar algorithm for $\psqrt{a^2-b^2}$. It converges slowly when $b$ is near $a$, but otherwise it works fine. @p function pyth_sub(@!a,@!b:integer):integer; label done; var @!r:fraction; {register used to transform |a| and |b|} @!big:boolean; {is the input dangerously near $2^{31}$?} begin a:=abs(a); b:=abs(b); if a<=b then @ else begin if a; if big then a:=a+a; end; pyth_sub:=a; end; @ @= loop@+ begin r:=make_fraction(b,a); r:=take_fraction(r,r); {now $r\approx b^2/a^2$} if r=0 then goto done; r:=make_fraction(r,fraction_four-r); a:=a-take_fraction(a+a,r); b:=take_fraction(b,r); end; done: @ @= begin if a help2("Since I don't take square roots of negative numbers,")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; end; a:=0; end @ The subroutines for logarithm and exponential involve two tables. The first is simple: |two_to_the[k]| equals $2^k$. The second involves a bit more calculation, which the author claims to have done correctly: |spec_log[k]| is $2^{27}$ times $\ln\bigl(1/(1-2^{-k})\bigr)= 2^{-k}+{1\over2}2^{-2k}+{1\over3}2^{-3k}+\cdots\,$, rounded to the nearest integer. @= @!two_to_the:array[0..30] of integer; {powers of two} @!spec_log:array[1..28] of integer; {special logarithms} @ @= @!k:integer; {all-purpose loop index} @ @= two_to_the[0]:=1; for k:=1 to 30 do two_to_the[k]:=2*two_to_the[k-1]; spec_log[1]:=93032640; spec_log[2]:=38612034; spec_log[3]:=17922280; spec_log[4]:=8662214; spec_log[5]:=4261238; spec_log[6]:=2113709; spec_log[7]:=1052693; spec_log[8]:=525315; spec_log[9]:=262400; spec_log[10]:=131136; spec_log[11]:=65552; spec_log[12]:=32772; spec_log[13]:=16385; for k:=14 to 27 do spec_log[k]:=two_to_the[27-k]; spec_log[28]:=1; @ Here is the routine that calculates $2^8$ times the natural logarithm of a |scaled| quantity; it is an integer approximation to $2^{24}\ln(x/2^{16})$, when |x| is a given positive integer. The method is based on exercise 1.2.2--25 in {\sl The Art of Computer Programming\/}: During the main iteration we have $1\L 2^{-30}x<1/(1-2^{1-k})$, and the logarithm of $2^{30}x$ remains to be added to an accumulator register called~$y$. Three auxiliary bits of accuracy are retained in~$y$ during the calculation, and sixteen auxiliary bits to extend |y| are kept in~|z| during the initial argument reduction. (We add $100\cdot2^{16}=6553600$ to~|z| and subtract 100 from~|y| so that |z| will not become negative; also, the actual amount subtracted from~|y| is~96, not~100, because we want to add~4 for rounding before the final division by~8.) @p function m_log(@!x:scaled):scaled; var @!y,@!z:integer; {auxiliary registers} @!k:integer; {iteration counter} begin if x<=0 then @ else begin y:=1302456956+4-100; {$14\times2^{27}\ln2\approx1302456956.421063$} z:=27595+6553600; {and $2^{16}\times .421063\approx 27595$} while xfraction_four+4 do @; m_log:=y div 8; end; end; @ @= begin z:=((x-1) div two_to_the[k])+1; {$z=\lceil x/2^k\rceil$} while x= begin print_err("Logarithm of "); @.Logarithm...replaced by 0@> print_scaled(x); print(" has been replaced by 0"); help2("Since I don't take logs of non-positive numbers,")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; m_log:=0; end @ Conversely, the exponential routine calculates $\exp(x/2^8)$, when |x| is |scaled|. The result is an integer approximation to $2^{16}\exp(x/2^{24})$, when |x| is regarded as an integer. @p function m_exp(@!x:scaled):scaled; var @!k:small_number; {loop control index} @!y,@!z:integer; {auxiliary registers} begin if x>174436200 then {$2^{24}\ln((2^{31}-1)/2^{16})\approx 174436199.51$} begin arith_error:=true; m_exp:=el_gordo; end else if x<-197694359 then m_exp:=0 {$2^{24}\ln(2^{-1}/2^{16})\approx-197694359.45$} else begin if x<=0 then begin z:=-8*x; y:=@'4000000; {$y=2^{20}$} end else begin if x<=127919879 then z:=1023359037-8*x {$2^{27}\ln((2^{31}-1)/2^{20})\approx 1023359037.125$} else z:=8*(174436200-x); {|z| is always nonnegative} y:=el_gordo; end; @; if x<=127919879 then m_exp:=(y+8) div 16@+else m_exp:=y; end; end; @ The idea here is that subtracting |spec_log[k]| from |z| corresponds to multiplying |y| by $1-2^{-k}$. A subtle point (which had to be checked) was that if $x=127919879$, the value of~|y| will decrease so that |y+8| doesn't overflow. In fact, $z$ will be 5 in this case, and |y| will decrease by~64 when |k=25| and by~16 when |k=27|. @= k:=1; while z>0 do begin while z>=spec_log[k] do begin z:=z-spec_log[k]; y:=y-1-((y-two_to_the[k-1]) div two_to_the[k]); end; incr(k); end @ The trigonometric subroutines use an auxiliary table such that |spec_atan[k]| contains an approximation to the |angle| whose tangent is~$1/2^k$. @= @!spec_atan:array[1..26] of angle; {$\arctan2^{-k}$ times $2^{20}\cdot180/\pi$} @ @= spec_atan[1]:=27855475; spec_atan[2]:=14718068; spec_atan[3]:=7471121; spec_atan[4]:=3750058; spec_atan[5]:=1876857; spec_atan[6]:=938658; spec_atan[7]:=469357; spec_atan[8]:=234682; spec_atan[9]:=117342; spec_atan[10]:=58671; spec_atan[11]:=29335; spec_atan[12]:=14668; spec_atan[13]:=7334; spec_atan[14]:=3667; spec_atan[15]:=1833; spec_atan[16]:=917; spec_atan[17]:=458; spec_atan[18]:=229; spec_atan[19]:=115; spec_atan[20]:=57; spec_atan[21]:=29; spec_atan[22]:=14; spec_atan[23]:=7; spec_atan[24]:=4; spec_atan[25]:=2; spec_atan[26]:=1; @ Given integers |x| and |y|, not both zero, the |n_arg| function returns the |angle| whose tangent points in the direction $(x,y)$. This subroutine first determines the correct octant, then solves the problem for |0<=y<=x|, then converts the result appropriately to return an answer in the range |-one_eighty_deg<=@t$\theta$@><=one_eighty_deg|. (The answer is |+one_eighty_deg| if |y=0| and |x<0|, but an answer of |-one_eighty_deg| is possible if, for example, |y=-1| and $x=-2^{30}$.) The octants are represented in a ``Gray code,'' since that turns out to be computationally simplest. @d negate_x=1 @d negate_y=2 @d switch_x_and_y=4 @d first_octant=1 @d second_octant=first_octant+switch_x_and_y @d third_octant=first_octant+switch_x_and_y+negate_x @d fourth_octant=first_octant+negate_x @d fifth_octant=first_octant+negate_x+negate_y @d sixth_octant=first_octant+switch_x_and_y+negate_x+negate_y @d seventh_octant=first_octant+switch_x_and_y+negate_y @d eighth_octant=first_octant+negate_y @p function n_arg(@!x,@!y:integer):angle; var @!z:angle; {auxiliary register} @!t:integer; {temporary storage} @!k:small_number; {loop counter} @!octant:first_octant..sixth_octant; {octant code} begin if x>=0 then octant:=first_octant else begin negate(x); octant:=first_octant+negate_x; end; if y<0 then begin negate(y); octant:=octant+negate_y; end; if x else begin @; @; end; end; @ @= begin print_err("angle(0,0) is taken as zero"); @.angle(0,0)...zero@> help2("The `angle' between two identical points is undefined.")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); error; n_arg:=0; end @ @= case octant of first_octant:n_arg:=z; second_octant:n_arg:=ninety_deg-z; third_octant:n_arg:=ninety_deg+z; fourth_octant:n_arg:=one_eighty_deg-z; fifth_octant:n_arg:=z-one_eighty_deg; sixth_octant:n_arg:=-z-ninety_deg; seventh_octant:n_arg:=z-ninety_deg; eighth_octant:n_arg:=-z; end {there are no other cases} @ At this point we have |x>=y>=0|, and |x>0|. The numbers are scaled up or down until $2^{28}\L x<2^{29}$, so that accurate fixed-point calculations will be made. @= while x>=fraction_two do begin x:=halfp(x); y:=halfp(y); end; z:=0; if y>0 then begin while x; end @ During the calculations of this section, variables |x| and~|y| represent actual coordinates $(x,2^{-k}y)$. We will maintain the condition |x>=y|, so that the tangent will be at most $2^{-k}$. If $x<2y$, the tangent is greater than $2^{-k-1}$. The transformation $(a,b)\mapsto(a+b\tan\phi,b-a\tan\phi)$ replaces $(a,b)$ by coordinates whose angle has decreased by~$\phi$; in the special case $a=x$, $b=2^{-k}y$, and $\tan\phi=2^{-k-1}$, this operation reduces to the particularly simple iteration shown here. [Cf.~John E. Meggitt, @^Meggitt, John E.@> {\sl IBM Journal of Research and Development\/ \bf6} (1962), 210--226.] The initial value of |x| will be multiplied by at most $(1+{1\over2})(1+{1\over8})(1+{1\over32})\cdots\approx 1.7584$; hence there is no chance of integer overflow. @= k:=0; repeat double(y); incr(k); if y>x then begin z:=z+spec_atan[k]; t:=x; x:=x+(y div two_to_the[k+k]); y:=y-t; end; until k=15; repeat double(y); incr(k); if y>x then begin z:=z+spec_atan[k]; y:=y-x; end; until k=26 @ Conversely, the |n_sin_cos| routine takes an |angle| and produces the sine and cosine of that angle. The results of this routine are stored in global integer variables |n_sin| and |n_cos|. @= @!n_sin,@!n_cos:fraction; {results computed by |n_sin_cos|} @ Given an integer |z| that is $2^{20}$ times an angle $\theta$ in degrees, the purpose of |n_sin_cos(z)| is to set |x=@t$r\cos\theta$@>| and |y=@t$r\sin\theta$@>| (approximately), for some rather large number~|r|. The maximum of |x| and |y| will be between $2^{28}$ and $2^{30}$, so that there will be hardly any loss of accuracy. Then |x| and~|y| are divided by~|r|. @p procedure n_sin_cos(@!z:angle); {computes a multiple of the sine and cosine} var @!k:small_number; {loop control variable} @!q:0..7; {specifies the quadrant} @!r:fraction; {magnitude of |(x,y)|} @!x,@!y,@!t:integer; {temporary registers} begin while z<0 do z:=z+three_sixty_deg; z:=z mod three_sixty_deg; {now |0<=z; @; r:=pyth_add(x,y); n_cos:=make_fraction(x,r); n_sin:=make_fraction(y,r); end; @ In this case the octants are numbered sequentially. @= case q of 0:do_nothing; 1:begin t:=x; x:=y; y:=t; end; 2:begin t:=x; x:=-y; y:=t; end; 3:negate(x); 4:begin negate(x); negate(y); end; 5:begin t:=x; x:=-y; y:=-t; end; 6:begin t:=x; x:=y; y:=-t; end; 7:negate(y); end {there are no other cases} @ The main iteration of |n_sin_cos| is similar to that of |n_arg| but applied in reverse. The values of |spec_atan[k]| decrease slowly enough that this loop is guaranteed to terminate before the (nonexistent) value |spec_atan[27]| would be required. @= k:=1; while z>0 do begin if z>=spec_atan[k] then begin z:=z-spec_atan[k]; t:=x;@/ x:=t+y div two_to_the[k]; y:=y-t div two_to_the[k]; end; incr(k); end; if y<0 then y:=0 {this precaution may never be needed} @ And now let's complete our collection of numeric utility routines by considering random number generation. \MP\ generates pseudo-random numbers with the additive scheme recommended in Section 3.6 of {\sl The Art of Computer Programming}; however, the results are random fractions between 0 and |fraction_one-1|, inclusive. There's an auxiliary array |randoms| that contains 55 pseudo-random fractions. Using the recurrence $x_n=(x_{n-55}-x_{n-31})\bmod 2^{28}$, we generate batches of 55 new $x_n$'s at a time by calling |new_randoms|. The global variable |j_random| tells which element has most recently been consumed. @= @!randoms:array[0..54] of fraction; {the last 55 random values generated} @!j_random:0..54; {the number of unused |randoms|} @ To consume a random fraction, the program below will say `|next_random|' and then it will fetch |randoms[j_random]|. @d next_random==if j_random=0 then new_randoms else decr(j_random) @p procedure new_randoms; var @!k:0..54; {index into |randoms|} @!x:fraction; {accumulator} begin for k:=0 to 23 do begin x:=randoms[k]-randoms[k+31]; if x<0 then x:=x+fraction_one; randoms[k]:=x; end; for k:=24 to 54 do begin x:=randoms[k]-randoms[k-24]; if x<0 then x:=x+fraction_one; randoms[k]:=x; end; j_random:=54; end; @ To initialize the |randoms| table, we call the following routine. @p procedure init_randoms(@!seed:scaled); var @!j,@!jj,@!k:fraction; {more or less random integers} @!i:0..54; {index into |randoms|} begin j:=abs(seed); while j>=fraction_one do j:=halfp(j); k:=1; for i:=0 to 54 do begin jj:=k; k:=j-k; j:=jj; if k<0 then k:=k+fraction_one; randoms[(i*21)mod 55]:=j; end; new_randoms; new_randoms; new_randoms; {``warm up'' the array} end; @ To produce a uniform random number in the range |0<=u=u>x| or |0=u=x|, given a |scaled| value~|x|, we proceed as shown here. Note that the call of |take_fraction| will produce the values 0 and~|x| with about half the probability that it will produce any other particular values between 0 and~|x|, because it rounds its answers. @p function unif_rand(@!x:scaled):scaled; var @!y:scaled; {trial value} begin next_random; y:=take_fraction(abs(x),randoms[j_random]); if y=abs(x) then unif_rand:=0 else if x>0 then unif_rand:=y else unif_rand:=-y; end; @ Finally, a normal deviate with mean zero and unit standard deviation can readily be obtained with the ratio method (Algorithm 3.4.1R in {\sl The Art of Computer Programming\/}). @p function norm_rand:scaled; var @!x,@!u,@!l:integer; {what the book would call $2^{16}X$, $2^{28}U$, and $-2^{24}\ln U$} begin repeat repeat next_random; x:=take_fraction(112429,randoms[j_random]-fraction_half); {$2^{16}\sqrt{8/e}\approx 112428.82793$} next_random; u:=randoms[j_random]; until abs(x)=0; norm_rand:=x; end; @* \[9] Packed data. In order to make efficient use of storage space, \MP\ bases its major data structures on a |memory_word|, which contains either a (signed) integer, possibly scaled, or a small number of fields that are one half or one quarter of the size used for storing integers. If |x| is a variable of type |memory_word|, it contains up to four fields that can be referred to as follows: $$\vbox{\halign{\hfil#&#\hfil&#\hfil\cr |x|&.|int|&(an |integer|)\cr |x|&.|sc|\qquad&(a |scaled| integer)\cr |x.hh.lh|, |x.hh|&.|rh|&(two halfword fields)\cr |x.hh.b0|, |x.hh.b1|, |x.hh|&.|rh|&(two quarterword fields, one halfword field)\cr |x.qqqq.b0|, |x.qqqq.b1|, |x.qqqq|&.|b2|, |x.qqqq.b3|\hskip-100pt &\qquad\qquad\qquad(four quarterword fields)\cr}}$$ This is somewhat cumbersome to write, and not very readable either, but macros will be used to make the notation shorter and more transparent. The \PASCAL\ code below gives a formal definition of |memory_word| and its subsidiary types, using packed variant records. \MP\ makes no assumptions about the relative positions of the fields within a word. Since we are assuming 32-bit integers, a halfword must contain at least 16 bits, and a quarterword must contain at least 8 bits. @^system dependencies@> But it doesn't hurt to have more bits; for example, with enough 36-bit words you might be able to have |mem_max| as large as 262142. N.B.: Valuable memory space will be dreadfully wasted unless \MP\ is compiled by a \PASCAL\ that packs all of the |memory_word| variants into the space of a single integer. Some \PASCAL\ compilers will pack an integer whose subrange is `|0..255|' into an eight-bit field, but others insist on allocating space for an additional sign bit; on such systems you can get 256 values into a quarterword only if the subrange is `|-128..127|'. The present implementation tries to accommodate as many variations as possible, so it makes few assumptions. If integers having the subrange `|min_quarterword..max_quarterword|' can be packed into a quarterword, and if integers having the subrange `|min_halfword..max_halfword|' can be packed into a halfword, everything should work satisfactorily. It is usually most efficient to have |min_quarterword=min_halfword=0|, so one should try to achieve this unless it causes a severe problem. The values defined here are recommended for most 32-bit computers. @d min_quarterword=0 {smallest allowable value in a |quarterword|} @d max_quarterword=255 {largest allowable value in a |quarterword|} @d min_halfword==0 {smallest allowable value in a |halfword|} @d max_halfword==65535 {largest allowable value in a |halfword|} @ Here are the inequalities that the quarterword and halfword values must satisfy (or rather, the inequalities that they mustn't satisfy): @= init if mem_max<>mem_top then bad:=8;@+tini@;@/ if mem_max0)or(max_quarterword<127) then bad:=9; if (min_halfword>0)or(max_halfword<32767) then bad:=10; if (min_quarterwordmax_halfword) then bad:=11; if (mem_min=max_halfword) then bad:=12; if max_strings>max_halfword then bad:=13; if buf_size>max_halfword then bad:=14; if font_max>max_halfword then bad:=15; if (max_quarterword-min_quarterword<255)or@| (max_halfword-min_halfword<65535) then bad:=16; @ The operation of subtracting |min_halfword| occurs rather frequently in \MP, so it is convenient to abbreviate this operation by using the macro |ho| defined here. \MP\ will run faster with respect to compilers that don't optimize the expression `|x-0|', if this macro is simplified in the obvious way when |min_halfword=0|. Similarly, |qi| and |qo| are used for input to and output from quarterwords. @^system dependencies@> @d ho(#)==#-min_halfword {to take a sixteen-bit item from a halfword} @d qo(#)==#-min_quarterword {to read eight bits from a quarterword} @d qi(#)==#+min_quarterword {to store eight bits in a quarterword} @ The reader should study the following definitions closely: @^system dependencies@> @d sc==int {|scaled| data is equivalent to |integer|} @= @!quarterword = min_quarterword..max_quarterword; {1/4 of a word} @!halfword=min_halfword..max_halfword; {1/2 of a word} @!two_choices = 1..2; {used when there are two variants in a record} @!three_choices = 1..3; {used when there are three variants in a record} @!two_halves = packed record@;@/ @!rh:halfword; case two_choices of 1: (@!lh:halfword); 2: (@!b0:quarterword; @!b1:quarterword); end; @!four_quarters = packed record@;@/ @!b0:quarterword; @!b1:quarterword; @!b2:quarterword; @!b3:quarterword; end; @!memory_word = record@;@/ case three_choices of 1: (@!int:integer); 2: (@!hh:two_halves); 3: (@!qqqq:four_quarters); end; @!word_file = file of memory_word; @ When debugging, we may want to print a |memory_word| without knowing what type it is; so we print it in all modes. @^dirty \PASCAL@>@^debugging@> @p @!debug procedure print_word(@!w:memory_word); {prints |w| in all ways} begin print_int(w.int); print_char(" ");@/ print_scaled(w.sc); print_char(" "); print_scaled(w.sc div @'10000); print_ln;@/ print_int(w.hh.lh); print_char("="); print_int(w.hh.b0); print_char(":"); print_int(w.hh.b1); print_char(";"); print_int(w.hh.rh); print_char(" ");@/ print_int(w.qqqq.b0); print_char(":"); print_int(w.qqqq.b1); print_char(":"); print_int(w.qqqq.b2); print_char(":"); print_int(w.qqqq.b3); end; gubed @* \[10] Dynamic memory allocation. The \MP\ system does nearly all of its own memory allocation, so that it can readily be transported into environments that do not have automatic facilities for strings, garbage collection, etc., and so that it can be in control of what error messages the user receives. The dynamic storage requirements of \MP\ are handled by providing a large array |mem| in which consecutive blocks of words are used as nodes by the \MP\ routines. Pointer variables are indices into this array, or into another array called |eqtb| that will be explained later. A pointer variable might also be a special flag that lies outside the bounds of |mem|, so we allow pointers to assume any |halfword| value. The minimum memory index represents a null pointer. @d pointer==halfword {a flag or a location in |mem| or |eqtb|} @d null==mem_min {the null pointer} @ The |mem| array is divided into two regions that are allocated separately, but the dividing line between these two regions is not fixed; they grow together until finding their ``natural'' size in a particular job. Locations less than or equal to |lo_mem_max| are used for storing variable-length records consisting of two or more words each. This region is maintained using an algorithm similar to the one described in exercise 2.5--19 of {\sl The Art of Computer Programming}. However, no size field appears in the allocated nodes; the program is responsible for knowing the relevant size when a node is freed. Locations greater than or equal to |hi_mem_min| are used for storing one-word records; a conventional \.{AVAIL} stack is used for allocation in this region. Locations of |mem| between |mem_min| and |mem_top| may be dumped as part of preloaded format files, by the \.{INIMP} preprocessor. @.INIMP@> Production versions of \MP\ may extend the memory at the top end in order to provide more space; these locations, between |mem_top| and |mem_max|, are always used for single-word nodes. The key pointers that govern |mem| allocation have a prescribed order: $$\hbox{|null=mem_min= @!mem : array[mem_min..mem_max] of memory_word; {the big dynamic storage area} @!lo_mem_max : pointer; {the largest location of variable-size memory in use} @!hi_mem_min : pointer; {the smallest location of one-word memory in use} @ Users who wish to study the memory requirements of particular applications can can use optional special features that keep track of current and maximum memory usage. When code between the delimiters |@!stat| $\ldots$ |tats| is not ``commented out,'' \MP\ will run a bit slower but it will report these statistics when |tracing_stats| is positive. @= @!var_used, @!dyn_used : integer; {how much memory is in use} @ Let's consider the one-word memory region first, since it's the simplest. The pointer variable |mem_end| holds the highest-numbered location of |mem| that has ever been used. The free locations of |mem| that occur between |hi_mem_min| and |mem_end|, inclusive, are of type |two_halves|, and we write |info(p)| and |link(p)| for the |lh| and |rh| fields of |mem[p]| when it is of this type. The single-word free locations form a linked list $$|avail|,\;\hbox{|link(avail)|},\;\hbox{|link(link(avail))|},\;\ldots$$ terminated by |null|. @d link(#) == mem[#].hh.rh {the |link| field of a memory word} @d info(#) == mem[#].hh.lh {the |info| field of a memory word} @= @!avail : pointer; {head of the list of available one-word nodes} @!mem_end : pointer; {the last one-word node used in |mem|} @ If one-word memory is exhausted, it might mean that the user has forgotten a token like `\&{enddef}' or `\&{endfor}'. We will define some procedures later that try to help pinpoint the trouble. @p @t\4@>@@; @t\4@>@ @ The function |get_avail| returns a pointer to a new one-word node whose |link| field is null. However, \MP\ will halt if there is no more room left. @^inner loop@> @p function get_avail : pointer; {single-word node allocation} var @!p:pointer; {the new node being got} begin p:=avail; {get top location in the |avail| stack} if p<>null then avail:=link(avail) {and pop it off} else if mem_end end; end; link(p):=null; {provide an oft-desired initialization of the new node} @!stat incr(dyn_used);@+tats@;{maintain statistics} get_avail:=p; end; @ Conversely, a one-word node is recycled by calling |free_avail|. @d free_avail(#)== {single-word node liberation} begin link(#):=avail; avail:=#; @!stat decr(dyn_used);@+tats@/ end @ There's also a |fast_get_avail| routine, which saves the procedure-call overhead at the expense of extra programming. This macro is used in the places that would otherwise account for the most calls of |get_avail|. @^inner loop@> @d fast_get_avail(#)==@t@>@;@/ begin #:=avail; {avoid |get_avail| if possible, to save time} if #=null then #:=get_avail else begin avail:=link(#); link(#):=null; @!stat incr(dyn_used);@+tats@/ end; end @ The available-space list that keeps track of the variable-size portion of |mem| is a nonempty, doubly-linked circular list of empty nodes, pointed to by the roving pointer |rover|. Each empty node has size 2 or more; the first word contains the special value |max_halfword| in its |link| field and the size in its |info| field; the second word contains the two pointers for double linking. Each nonempty node also has size 2 or more. Its first word is of type |two_halves|\kern-1pt, and its |link| field is never equal to |max_halfword|. Otherwise there is complete flexibility with respect to the contents of its other fields and its other words. (We require |mem_max= @!rover : pointer; {points to some node in the list of empties} @ A call to |get_node| with argument |s| returns a pointer to a new node of size~|s|, which must be 2~or more. The |link| field of the first word of this new node is set to null. An overflow stop occurs if no suitable space exists. If |get_node| is called with $s=2^{30}$, it simply merges adjacent free areas and returns the value |max_halfword|. @p function get_node(@!s:integer):pointer; {variable-size node allocation} label found,exit,restart; var @!p:pointer; {the node currently under inspection} @!q:pointer; {the node physically after node |p|} @!r:integer; {the newly allocated node, or a candidate for this honor} @!t,@!tt:integer; {temporary registers} @^inner loop@> begin restart: p:=rover; {start at some free node in the ring} repeat @; p:=rlink(p); {move to the next node in the ring} until p=rover; {repeat until the whole list has been traversed} if s=@'10000000000 then begin get_node:=max_halfword; return; end; if lo_mem_max+2; overflow("main memory size",mem_max+1-mem_min); {sorry, nothing satisfactory is left} @:MetaPost capacity exceeded main memory size}{\quad main memory size@> found: link(r):=null; {this node is now nonempty} @!stat var_used:=var_used+s; {maintain usage statistics} tats@;@/ get_node:=r; exit:end; @ The lower part of |mem| grows by 1000 words at a time, unless we are very close to going under. When it grows, we simply link a new node into the available-space list. This method of controlled growth helps to keep the |mem| usage consecutive when \MP\ is implemented on ``virtual memory'' systems. @^virtual memory@> @= begin if hi_mem_min-lo_mem_max>=1998 then t:=lo_mem_max+1000 else t:=lo_mem_max+1+(hi_mem_min-lo_mem_max) div 2; {|lo_mem_max+2<=tmem_min+max_halfword then t:=mem_min+max_halfword; p:=llink(rover); q:=lo_mem_max; rlink(p):=q; llink(rover):=q;@/ rlink(q):=rover; llink(q):=p; link(q):=empty_flag; node_size(q):=t-lo_mem_max;@/ lo_mem_max:=t; link(lo_mem_max):=null; info(lo_mem_max):=null; rover:=q; goto restart; end @ @= q:=p+node_size(p); {find the physical successor} while is_empty(q) do {merge node |p| with node |q|} begin t:=rlink(q); tt:=llink(q); @^inner loop@> if q=rover then rover:=t; llink(t):=tt; rlink(tt):=t;@/ q:=q+node_size(q); end; r:=q-s; if r>p+1 then @; if r=p then if rlink(p)<>p then @; node_size(p):=q-p {reset the size in case it grew} @ @= begin node_size(p):=r-p; {store the remaining size} rover:=p; {start searching here next time} goto found; end @ Here we delete node |p| from the ring, and let |rover| rove around. @= begin rover:=rlink(p); t:=llink(p); llink(rover):=t; rlink(t):=rover; goto found; end @ Conversely, when some variable-size node |p| of size |s| is no longer needed, the operation |free_node(p,s)| will make its words available, by inserting |p| as a new empty node just before where |rover| now points. @p procedure free_node(@!p:pointer; @!s:halfword); {variable-size node liberation} var @!q:pointer; {|llink(rover)|} begin node_size(p):=s; link(p):=empty_flag; @^inner loop@> q:=llink(rover); llink(p):=q; rlink(p):=rover; {set both links} llink(rover):=p; rlink(q):=p; {insert |p| into the ring} @!stat var_used:=var_used-s;@+tats@;{maintain statistics} end; @ Just before \.{INIMP} writes out the memory, it sorts the doubly linked available space list. The list is probably very short at such times, so a simple insertion sort is used. The smallest available location will be pointed to by |rover|, the next-smallest by |rlink(rover)|, etc. @p @!init procedure sort_avail; {sorts the available variable-size nodes by location} var @!p,@!q,@!r: pointer; {indices into |mem|} @!old_rover:pointer; {initial |rover| setting} begin p:=get_node(@'10000000000); {merge adjacent free areas} p:=rlink(rover); rlink(rover):=max_halfword; old_rover:=rover; while p<>old_rover do @; p:=rover; while rlink(p)<>max_halfword do begin llink(rlink(p)):=p; p:=rlink(p); end; rlink(p):=rover; llink(rover):=p; end; tini @ The following |while| loop is guaranteed to terminate, since the list that starts at |rover| ends with |max_halfword| during the sorting procedure. @= if p= @^data structure assumptions@> rover:=lo_mem_stat_max+1; {initialize the dynamic memory} link(rover):=empty_flag; node_size(rover):=1000; {which is a 1000-word available node} llink(rover):=rover; rlink(rover):=rover;@/ lo_mem_max:=rover+1000; link(lo_mem_max):=null; info(lo_mem_max):=null;@/ for k:=hi_mem_stat_min to mem_top do mem[k]:=mem[lo_mem_max]; {clear list heads} avail:=null; mem_end:=mem_top; hi_mem_min:=hi_mem_stat_min; {initialize the one-word memory} var_used:=lo_mem_stat_max+1-mem_min; dyn_used:=mem_top+1-(hi_mem_stat_min); {initialize statistics} @; @ The procedure |flush_list(p)| frees an entire linked list of one-word nodes that starts at a given position, until coming to |sentinel| or a pointer that is not in the one-word region. Another procedure, |flush_node_list|, frees an entire linked list of one-word and two-word nodes, until coming to a |null| pointer. @^inner loop@> @p procedure flush_list(@!p:pointer); {makes list of single-word nodes available} label done; var @!q,@!r:pointer; {list traversers} begin if p>=hi_mem_min then if p<>sentinel then begin r:=p; repeat q:=r; r:=link(r); @!stat decr(dyn_used);@+tats@/ if rnull do begin q:=p; p:=link(p); if q are debugging.) @= @!debug @!free: packed array [mem_min..mem_max] of boolean; {free cells} @t\hskip1em@>@!was_free: packed array [mem_min..mem_max] of boolean; {previously free cells} @t\hskip1em@>@!was_mem_end,@!was_lo_max,@!was_hi_min: pointer; {previous |mem_end|, |lo_mem_max|,and |hi_mem_min|} @t\hskip1em@>@!panicking:boolean; {do we want to check memory constantly?} gubed @ @= @!debug was_mem_end:=mem_min; {indicate that everything was previously free} was_lo_max:=mem_min; was_hi_min:=mem_max; panicking:=false; gubed @ Procedure |check_mem| makes sure that the available space lists of |mem| are well formed, and it optionally prints out all locations that are reserved now but were free the last time this procedure was called. @p @!debug procedure check_mem(@!print_locs : boolean); label done1,done2,done3; {loop exits} var @!p,@!q,@!r:pointer; {current locations of interest in |mem|} @!clobbered:boolean; {is something amiss?} begin for p:=mem_min to lo_mem_max do free[p]:=false; {you can probably do this faster} for p:=hi_mem_min to mem_end do free[p]:=false; {ditto} @; @; @; @; if print_locs then @; for p:=mem_min to lo_mem_max do was_free[p]:=free[p]; for p:=hi_mem_min to mem_end do was_free[p]:=free[p]; {|was_free:=free| might be faster} was_mem_end:=mem_end; was_lo_max:=lo_mem_max; was_hi_min:=hi_mem_min; end; gubed @ @= p:=avail; q:=null; clobbered:=false; while p<>null do begin if (p>mem_end)or(p print_int(q); goto done1; end; free[p]:=true; q:=p; p:=link(q); end; done1: @ @= p:=rover; q:=null; clobbered:=false; repeat if (p>=lo_mem_max)or(p=lo_mem_max)or(rlink(p)lo_mem_max)or@| (llink(rlink(p))<>p) then clobbered:=true; if clobbered then begin print_nl("Double-AVAIL list clobbered at "); @.Double-AVAIL list clobbered...@> print_int(q); goto done2; end; for q:=p to p+node_size(p)-1 do {mark all locations free} begin if free[q] then begin print_nl("Doubly free location at "); @.Doubly free location...@> print_int(q); goto done2; end; free[q]:=true; end; q:=p; p:=rlink(p); until p=rover; done2: @ @= p:=mem_min; while p<=lo_mem_max do {node |p| should not be empty} begin if is_empty(p) then begin print_nl("Bad flag at "); print_int(p); @.Bad flag...@> end; while (p<=lo_mem_max) and not free[p] do incr(p); while (p<=lo_mem_max) and free[p] do incr(p); end @ @= begin @; print_nl("New busy locs:"); @.New busy locs@> for p:=mem_min to lo_mem_max do if not free[p] and ((p>was_lo_max) or was_free[p]) then @; for p:=hi_mem_min to mem_end do if not free[p] and ((pwas_mem_end) or was_free[p]) then @; @; end @ There might be many new busy locations so we are careful to print contiguous blocks compactly. During this operation |q| is the last new busy location and |r| is the start of the block containing |q|. @= begin if p>q+1 then begin if q>r then begin print(".."); print_int(q); end; print_char(" "); print_int(p); r:=p; end; q:=p; end @ @= q:=mem_max; r:=mem_max @ @= if q>r then begin print(".."); print_int(q); end @ The |search_mem| procedure attempts to answer the question ``Who points to node~|p|?'' In doing so, it fetches |link| and |info| fields of |mem| that might not be of type |two_halves|. Strictly speaking, this is @^dirty \PASCAL@> undefined in \PASCAL, and it can lead to ``false drops'' (words that seem to point to |p| purely by coincidence). But for debugging purposes, we want to rule out the places that do {\sl not\/} point to |p|, so a few false drops are tolerable. @p @!debug procedure search_mem(@!p:pointer); {look for pointers to |p|} var @!q:integer; {current position being searched} begin for q:=mem_min to lo_mem_max do begin if link(q)=p then begin print_nl("LINK("); print_int(q); print_char(")"); end; if info(q)=p then begin print_nl("INFO("); print_int(q); print_char(")"); end; end; for q:=hi_mem_min to mem_end do begin if link(q)=p then begin print_nl("LINK("); print_int(q); print_char(")"); end; if info(q)=p then begin print_nl("INFO("); print_int(q); print_char(")"); end; end; @; end; gubed @* \[12] The command codes. Before we can go much further, we need to define symbolic names for the internal code numbers that represent the various commands obeyed by \MP. These codes are somewhat arbitrary, but not completely so. For example, some codes have been made adjacent so that |case| statements in the program need not consider cases that are widely spaced, or so that |case| statements can be replaced by |if| statements. A command can begin an expression if and only if its code lies between |min_primary_command| and |max_primary_command|, inclusive. The first token of a statement that doesn't begin with an expression has a command code between |min_command| and |max_statement_command|, inclusive. Anything less than |min_command| is eliminated during macro expansions, and anything no more than |max_pre_command| is eliminated when expanding \TeX\ material. Ranges such as |min_secondary_command..max_secondary_command| are used when parsing expressions, but the relative ordering within such a range is generally not critical. The ordering of the highest-numbered commands (|commacomma @d semicolon=82 {the operator `\.;', must be |comma+1|} @d end_group=83 {end a group (\&{endgroup}), must be |semicolon+1|} @d stop=84 {end a job (\&{end}, \&{dump}), must be |end_group+1|} @d max_command_code=stop @d outer_tag=max_command_code+1 {protection code added to command code} @= @!command_code=1..max_command_code; @ Variables and capsules in \MP\ have a variety of ``types,'' distinguished by the code numbers defined here. These numbers are also not completely arbitrary. Things that get expanded must have types |>independent|; a type remaining after expansion is numeric if and only if its code number is at least |numeric_type|; objects containing numeric parts must have types between |transform_type| and |pair_type|; all other types must be smaller than |transform_type|; and among the types that are not unknown or vacuous, the smallest two must be |boolean_type| and |string_type| in that order. @d undefined=0 {no type has been declared} @d unknown_tag=1 {this constant is added to certain type codes below} @d vacuous=1 {no expression was present} @d boolean_type=2 {\&{boolean} with a known value} @d unknown_boolean=boolean_type+unknown_tag @d string_type=4 {\&{string} with a known value} @d unknown_string=string_type+unknown_tag @d pen_type=6 {\&{pen} with a known value} @d unknown_pen=pen_type+unknown_tag @d path_type=8 {\&{path} with a known value} @d unknown_path=path_type+unknown_tag @d picture_type=10 {\&{picture} with a known value} @d unknown_picture=picture_type+unknown_tag @d transform_type=12 {\&{transform} variable or capsule} @d color_type=13 {\&{color} variable or capsule} @d pair_type=14 {\&{pair} variable or capsule} @d numeric_type=15 {variable that has been declared \&{numeric} but not used} @d known=16 {\&{numeric} with a known value} @d dependent=17 {a linear combination with |fraction| coefficients} @d proto_dependent=18 {a linear combination with |scaled| coefficients} @d independent=19 {\&{numeric} with unknown value} @d token_list=20 {variable name or suffix argument or text argument} @d structured=21 {variable with subscripts and attributes} @d unsuffixed_macro=22 {variable defined with \&{vardef} but no \.{\AT!\#}} @d suffixed_macro=23 {variable defined with \&{vardef} and \.{\AT!\#}} @# @d unknown_types==unknown_boolean,unknown_string, unknown_pen,unknown_picture,unknown_path @= procedure print_type(@!t:small_number); begin case t of vacuous:print("vacuous"); boolean_type:print("boolean"); unknown_boolean:print("unknown boolean"); string_type:print("string"); unknown_string:print("unknown string"); pen_type:print("pen"); unknown_pen:print("unknown pen"); path_type:print("path"); unknown_path:print("unknown path"); picture_type:print("picture"); unknown_picture:print("unknown picture"); transform_type:print("transform"); color_type:print("color"); pair_type:print("pair"); known:print("known numeric"); dependent:print("dependent"); proto_dependent:print("proto-dependent"); numeric_type:print("numeric"); independent:print("independent"); token_list:print("token list"); structured:print("structured"); unsuffixed_macro:print("unsuffixed macro"); suffixed_macro:print("suffixed macro"); othercases print("undefined") endcases; end; @ Values inside \MP\ are stored in two-word nodes that have a |name_type| as well as a |type|. The possibilities for |name_type| are defined here; they will be explained in more detail later. @d root=0 {|name_type| at the top level of a variable} @d saved_root=1 {same, when the variable has been saved} @d structured_root=2 {|name_type| where a |structured| branch occurs} @d subscr=3 {|name_type| in a subscript node} @d attr=4 {|name_type| in an attribute node} @d x_part_sector=5 {|name_type| in the \&{xpart} of a node} @d y_part_sector=6 {|name_type| in the \&{ypart} of a node} @d xx_part_sector=7 {|name_type| in the \&{xxpart} of a node} @d xy_part_sector=8 {|name_type| in the \&{xypart} of a node} @d yx_part_sector=9 {|name_type| in the \&{yxpart} of a node} @d yy_part_sector=10 {|name_type| in the \&{yypart} of a node} @d red_part_sector=11 {|name_type| in the \&{redpart} of a node} @d green_part_sector=12 {|name_type| in the \&{greenpart} of a node} @d blue_part_sector=13 {|name_type| in the \&{bluepart} of a node} @d capsule=14 {|name_type| in stashed-away subexpressions} @d token=15 {|name_type| in a numeric token or string token} @ Primitive operations that produce values have a secondary identification code in addition to their command code; it's something like genera and species. For example, `\.*' has the command code |primary_binary|, and its secondary identification is |times|. The secondary codes start at 30 so that they don't overlap with the type codes; some type codes (e.g., |string_type|) are used as operators as well as type identifications. The relative values are not critical, except for |true_code..false_code|, |or_op..and_op|, and |filled_op..bounded_op|. The restrictions are that |and_op-false_code=or_op-true_code|, that the ordering of |x_part...blue_part| must match that of |x_part_sector..blue_part_sector|, and the ordering of |filled_op..bounded_op| must match that of the code values they test for. @d true_code=30 {operation code for \.{true}} @d false_code=31 {operation code for \.{false}} @d null_picture_code=32 {operation code for \.{nullpicture}} @d null_pen_code=33 {operation code for \.{nullpen}} @d job_name_op=34 {operation code for \.{jobname}} @d read_string_op=35 {operation code for \.{readstring}} @d pen_circle=36 {operation code for \.{pencircle}} @d normal_deviate=37 {operation code for \.{normaldeviate}} @d read_from_op=38 {operation code for \.{readfrom}} @d close_from_op=39 {operation code for \.{closefrom}} @d odd_op=40 {operation code for \.{odd}} @d known_op=41 {operation code for \.{known}} @d unknown_op=42 {operation code for \.{unknown}} @d not_op=43 {operation code for \.{not}} @d decimal=44 {operation code for \.{decimal}} @d reverse=45 {operation code for \.{reverse}} @d make_path_op=46 {operation code for \.{makepath}} @d make_pen_op=47 {operation code for \.{makepen}} @d oct_op=48 {operation code for \.{oct}} @d hex_op=49 {operation code for \.{hex}} @d ASCII_op=50 {operation code for \.{ASCII}} @d char_op=51 {operation code for \.{char}} @d length_op=52 {operation code for \.{length}} @d turning_op=53 {operation code for \.{turningnumber}} @d x_part=54 {operation code for \.{xpart}} @d y_part=55 {operation code for \.{ypart}} @d xx_part=56 {operation code for \.{xxpart}} @d xy_part=57 {operation code for \.{xypart}} @d yx_part=58 {operation code for \.{yxpart}} @d yy_part=59 {operation code for \.{yypart}} @d red_part=60 {operation code for \.{redpart}} @d green_part=61 {operation code for \.{greenpart}} @d blue_part=62 {operation code for \.{bluepart}} @d font_part=63 {operation code for \.{fontpart}} @d text_part=64 {operation code for \.{textpart}} @d path_part=65 {operation code for \.{pathpart}} @d pen_part=66 {operation code for \.{penpart}} @d dash_part=67 {operation code for \.{dashpart}} @d sqrt_op=68 {operation code for \.{sqrt}} @d m_exp_op=69 {operation code for \.{mexp}} @d m_log_op=70 {operation code for \.{mlog}} @d sin_d_op=71 {operation code for \.{sind}} @d cos_d_op=72 {operation code for \.{cosd}} @d floor_op=73 {operation code for \.{floor}} @d uniform_deviate=74 {operation code for \.{uniformdeviate}} @d char_exists_op=75 {operation code for \.{charexists}} @d font_size=76 {operation code for \.{fontsize}} @d ll_corner_op=77 {operation code for \.{llcorner}} @d lr_corner_op=78 {operation code for \.{lrcorner}} @d ul_corner_op=79 {operation code for \.{ulcorner}} @d ur_corner_op=80 {operation code for \.{urcorner}} @d arc_length=81 {operation code for \.{arclength}} @d angle_op=82 {operation code for \.{angle}} @d cycle_op=83 {operation code for \.{cycle}} @d filled_op=84 {operation code for \.{filled}} @d stroked_op=85 {operation code for \.{stroked}} @d textual_op=86 {operation code for \.{textual}} @d clipped_op=87 {operation code for \.{clipped}} @d bounded_op=88 {operation code for \.{bounded}} @d plus=89 {operation code for \.+} @d minus=90 {operation code for \.-} @d times=91 {operation code for \.*} @d over=92 {operation code for \./} @d pythag_add=93 {operation code for \.{++}} @d pythag_sub=94 {operation code for \.{+-+}} @d or_op=95 {operation code for \.{or}} @d and_op=96 {operation code for \.{and}} @d less_than=97 {operation code for \.<} @d less_or_equal=98 {operation code for \.{<=}} @d greater_than=99 {operation code for \.>} @d greater_or_equal=100 {operation code for \.{>=}} @d equal_to=101 {operation code for \.=} @d unequal_to=102 {operation code for \.{<>}} @d concatenate=103 {operation code for \.\&} @d rotated_by=104 {operation code for \.{rotated}} @d slanted_by=105 {operation code for \.{slanted}} @d scaled_by=106 {operation code for \.{scaled}} @d shifted_by=107 {operation code for \.{shifted}} @d transformed_by=108 {operation code for \.{transformed}} @d x_scaled=109 {operation code for \.{xscaled}} @d y_scaled=110 {operation code for \.{yscaled}} @d z_scaled=111 {operation code for \.{zscaled}} @d in_font=112 {operation code for \.{infont}} @d intersect=113 {operation code for \.{intersectiontimes}} @d double_dot=114 {operation code for improper \.{..}} @d substring_of=115 {operation code for \.{substring}} @d min_of=substring_of @d subpath_of=116 {operation code for \.{subpath}} @d direction_time_of=117 {operation code for \.{directiontime}} @d point_of=118 {operation code for \.{point}} @d precontrol_of=119 {operation code for \.{precontrol}} @d postcontrol_of=120 {operation code for \.{postcontrol}} @d pen_offset_of=121 {operation code for \.{penoffset}} @d arc_time_of=122 {operation code for \.{arctime}} @p procedure print_op(@!c:quarterword); begin if c<=numeric_type then print_type(c) else case c of true_code:print("true"); false_code:print("false"); null_picture_code:print("nullpicture"); null_pen_code:print("nullpen"); job_name_op:print("jobname"); read_string_op:print("readstring"); pen_circle:print("pencircle"); normal_deviate:print("normaldeviate"); read_from_op:print("readfrom"); close_from_op:print("closefrom"); odd_op:print("odd"); known_op:print("known"); unknown_op:print("unknown"); not_op:print("not"); decimal:print("decimal"); reverse:print("reverse"); make_path_op:print("makepath"); make_pen_op:print("makepen"); oct_op:print("oct"); hex_op:print("hex"); ASCII_op:print("ASCII"); char_op:print("char"); length_op:print("length"); turning_op:print("turningnumber"); x_part:print("xpart"); y_part:print("ypart"); xx_part:print("xxpart"); xy_part:print("xypart"); yx_part:print("yxpart"); yy_part:print("yypart"); red_part:print("redpart"); green_part:print("greenpart"); blue_part:print("bluepart"); font_part:print("fontpart"); text_part:print("textpart"); path_part:print("pathpart"); pen_part:print("penpart"); dash_part:print("dashpart"); sqrt_op:print("sqrt"); m_exp_op:print("mexp"); m_log_op:print("mlog"); sin_d_op:print("sind"); cos_d_op:print("cosd"); floor_op:print("floor"); uniform_deviate:print("uniformdeviate"); char_exists_op:print("charexists"); font_size:print("fontsize"); ll_corner_op:print("llcorner"); lr_corner_op:print("lrcorner"); ul_corner_op:print("ulcorner"); ur_corner_op:print("urcorner"); arc_length:print("arclength"); angle_op:print("angle"); cycle_op:print("cycle"); filled_op:print("filled"); stroked_op:print("stroked"); textual_op:print("textual"); clipped_op:print("clipped"); bounded_op:print("bounded"); plus:print_char("+"); minus:print_char("-"); times:print_char("*"); over:print_char("/"); pythag_add:print("++"); pythag_sub:print("+-+"); or_op:print("or"); and_op:print("and"); less_than:print_char("<"); less_or_equal:print("<="); greater_than:print_char(">"); greater_or_equal:print(">="); equal_to:print_char("="); unequal_to:print("<>"); concatenate:print("&"); rotated_by:print("rotated"); slanted_by:print("slanted"); scaled_by:print("scaled"); shifted_by:print("shifted"); transformed_by:print("transformed"); x_scaled:print("xscaled"); y_scaled:print("yscaled"); z_scaled:print("zscaled"); in_font:print("infont"); intersect:print("intersectiontimes"); substring_of:print("substring"); subpath_of:print("subpath"); direction_time_of:print("directiontime"); point_of:print("point"); precontrol_of:print("precontrol"); postcontrol_of:print("postcontrol"); pen_offset_of:print("penoffset"); arc_time_of:print("arctime"); othercases print("..") endcases; end; @ \MP\ also has a bunch of internal parameters that a user might want to fuss with. Every such parameter has an identifying code number, defined here. @d tracing_titles=1 {show titles online when they appear} @d tracing_equations=2 {show each variable when it becomes known} @d tracing_capsules=3 {show capsules too} @d tracing_choices=4 {show the control points chosen for paths} @d tracing_specs=5 {show path subdivision prior to filling with polygonal a pen} @d tracing_commands=6 {show commands and operations before they are performed} @d tracing_restores=7 {show when a variable or internal is restored} @d tracing_macros=8 {show macros before they are expanded} @d tracing_output=9 {show digitized edges as they are output} @d tracing_stats=10 {show memory usage at end of job} @d tracing_lost_chars=11 {show characters that aren't \&{infont}} @d tracing_online=12 {show long diagnostics on terminal and in the log file} @d year=13 {the current year (e.g., 1984)} @d month=14 {the current month (e.g, 3 $\equiv$ March)} @d day=15 {the current day of the month} @d time=16 {the number of minutes past midnight when this job started} @d char_code=17 {the number of the next character to be output} @d char_ext=18 {the extension code of the next character to be output} @d char_wd=19 {the width of the next character to be output} @d char_ht=20 {the height of the next character to be output} @d char_dp=21 {the depth of the next character to be output} @d char_ic=22 {the italic correction of the next character to be output} @d design_size=23 {the unit of measure used for |char_wd..char_ic|, in points} @d pausing=24 {positive to display lines on the terminal before they are read} @d showstopping=25 {positive to stop after each \&{show} command} @d fontmaking=26 {positive if font metric output is to be produced} @d linejoin=27 {as in \ps: 0 for mitered, 1 for round, 2 for beveled} @d linecap=28 {as in \ps: 0 for butt, 1 for round, 2 for square} @d miterlimit=29 {controls miter length as in \ps} @d warning_check=30 {controls error message when variable value is large} @d boundary_char=31 {the right boundary character for ligatures} @d prologues=32 {positive to output conforming PostScript using built-in fonts} @d true_corners=33 {positive to make \&{llcorner} etc. ignore \&{setbounds}} @d max_given_internal=33 @= @!internal:array[1..max_internal] of scaled; {the values of internal quantities} @!int_name:array[1..max_internal] of str_number; {their names} @!int_ptr:max_given_internal..max_internal; {the maximum internal quantity defined so far} @ @= for k:=1 to max_given_internal do internal[k]:=0; int_ptr:=max_given_internal; @ The symbolic names for internal quantities are put into \MP's hash table by using a routine called |primitive|, which will be defined later. Let us enter them now, so that we don't have to list all those names again anywhere else. @= primitive("tracingtitles",internal_quantity,tracing_titles);@/ @!@:tracingtitles_}{\&{tracingtitles} primitive@> primitive("tracingequations",internal_quantity,tracing_equations);@/ @!@:tracing_equations_}{\&{tracingequations} primitive@> primitive("tracingcapsules",internal_quantity,tracing_capsules);@/ @!@:tracing_capsules_}{\&{tracingcapsules} primitive@> primitive("tracingchoices",internal_quantity,tracing_choices);@/ @!@:tracing_choices_}{\&{tracingchoices} primitive@> primitive("tracingspecs",internal_quantity,tracing_specs);@/ @!@:tracing_specs_}{\&{tracingspecs} primitive@> primitive("tracingcommands",internal_quantity,tracing_commands);@/ @!@:tracing_commands_}{\&{tracingcommands} primitive@> primitive("tracingrestores",internal_quantity,tracing_restores);@/ @!@:tracing_restores_}{\&{tracingrestores} primitive@> primitive("tracingmacros",internal_quantity,tracing_macros);@/ @!@:tracing_macros_}{\&{tracingmacros} primitive@> primitive("tracingoutput",internal_quantity,tracing_output);@/ @!@:tracing_output_}{\&{tracingoutput} primitive@> primitive("tracingstats",internal_quantity,tracing_stats);@/ @!@:tracing_stats_}{\&{tracingstats} primitive@> primitive("tracinglostchars",internal_quantity,tracing_lost_chars);@/ @!@:tracing_lost_chars_}{\&{tracinglostchars} primitive@> primitive("tracingonline",internal_quantity,tracing_online);@/ @!@:tracing_online_}{\&{tracingonline} primitive@> primitive("year",internal_quantity,year);@/ @!@:year_}{\&{year} primitive@> primitive("month",internal_quantity,month);@/ @!@:month_}{\&{month} primitive@> primitive("day",internal_quantity,day);@/ @!@:day_}{\&{day} primitive@> primitive("time",internal_quantity,time);@/ @!@:time_}{\&{time} primitive@> primitive("charcode",internal_quantity,char_code);@/ @!@:char_code_}{\&{charcode} primitive@> primitive("charext",internal_quantity,char_ext);@/ @!@:char_ext_}{\&{charext} primitive@> primitive("charwd",internal_quantity,char_wd);@/ @!@:char_wd_}{\&{charwd} primitive@> primitive("charht",internal_quantity,char_ht);@/ @!@:char_ht_}{\&{charht} primitive@> primitive("chardp",internal_quantity,char_dp);@/ @!@:char_dp_}{\&{chardp} primitive@> primitive("charic",internal_quantity,char_ic);@/ @!@:char_ic_}{\&{charic} primitive@> primitive("designsize",internal_quantity,design_size);@/ @!@:design_size_}{\&{designsize} primitive@> primitive("pausing",internal_quantity,pausing);@/ @!@:pausing_}{\&{pausing} primitive@> primitive("showstopping",internal_quantity,showstopping);@/ @!@:showstopping_}{\&{showstopping} primitive@> primitive("fontmaking",internal_quantity,fontmaking);@/ @!@:fontmaking_}{\&{fontmaking} primitive@> primitive("linejoin",internal_quantity,linejoin);@/ @!@:linejoin_}{\&{linejoin} primitive@> primitive("linecap",internal_quantity,linecap);@/ @!@:linecap_}{\&{linecap} primitive@> primitive("miterlimit",internal_quantity,miterlimit);@/ @!@:miterlimit_}{\&{miterlimit} primitive@> primitive("warningcheck",internal_quantity,warning_check);@/ @!@:warning_check_}{\&{warningcheck} primitive@> primitive("boundarychar",internal_quantity,boundary_char);@/ @!@:boundary_char_}{\&{boundarychar} primitive@> primitive("prologues",internal_quantity,prologues);@/ @!@:prologues_}{\&{prologues} primitive@> primitive("truecorners",internal_quantity,true_corners);@/ @!@:true_corners_}{\&{truecorners} primitive@> @ Well, we do have to list the names one more time, for use in symbolic printouts. @= int_name[tracing_titles]:="tracingtitles"; int_name[tracing_equations]:="tracingequations"; int_name[tracing_capsules]:="tracingcapsules"; int_name[tracing_choices]:="tracingchoices"; int_name[tracing_specs]:="tracingspecs"; int_name[tracing_commands]:="tracingcommands"; int_name[tracing_restores]:="tracingrestores"; int_name[tracing_macros]:="tracingmacros"; int_name[tracing_output]:="tracingoutput"; int_name[tracing_stats]:="tracingstats"; int_name[tracing_lost_chars]:="tracinglostchars"; int_name[tracing_online]:="tracingonline"; int_name[year]:="year"; int_name[month]:="month"; int_name[day]:="day"; int_name[time]:="time"; int_name[char_code]:="charcode"; int_name[char_ext]:="charext"; int_name[char_wd]:="charwd"; int_name[char_ht]:="charht"; int_name[char_dp]:="chardp"; int_name[char_ic]:="charic"; int_name[design_size]:="designsize"; int_name[pausing]:="pausing"; int_name[showstopping]:="showstopping"; int_name[fontmaking]:="fontmaking"; int_name[linejoin]:="linejoin"; int_name[linecap]:="linecap"; int_name[miterlimit]:="miterlimit"; int_name[warning_check]:="warningcheck"; int_name[boundary_char]:="boundarychar"; int_name[prologues]:="prologues"; int_name[true_corners]:="truecorners"; @ The following procedure, which is called just before \MP\ initializes its input and output, establishes the initial values of the date and time. @^system dependencies@> Since standard \PASCAL\ cannot provide such information, something special is needed. The program here simply specifies July 4, 1776, at noon; but users probably want a better approximation to the truth. Note that the values are |scaled| integers. Hence \MP\ can no longer be used after the year 32767. @p procedure fix_date_and_time; begin internal[time]:=12*60*unity; {minutes since midnight} internal[day]:=4*unity; {fourth day of the month} internal[month]:=7*unity; {seventh month of the year} internal[year]:=1776*unity; {Anno Domini} end; @ \MP\ is occasionally supposed to print diagnostic information that goes only into the transcript file, unless |tracing_online| is positive. Now that we have defined |tracing_online| we can define two routines that adjust the destination of print commands: @= @@; procedure begin_diagnostic; {prepare to do some tracing} begin old_setting:=selector; if selector=ps_file_only then selector:=non_ps_setting; if(internal[tracing_online]<=0)and(selector=term_and_log) then begin decr(selector); if history=spotless then history:=warning_issued; end; end; @# procedure end_diagnostic(@!blank_line:boolean); {restore proper conditions after tracing} begin print_nl(""); if blank_line then print_ln; selector:=old_setting; end; @ The global variable |non_ps_setting| is initialized when it is time to print on |ps_file|. @= @!old_setting,@!non_ps_setting:0..max_selector; @ We will occasionally use |begin_diagnostic| in connection with line-number printing, as follows. (The parameter |s| is typically |"Path"| or |"Cycle spec"|, etc.) @= procedure print_diagnostic(@!s,@!t:str_number;@!nuline:boolean); begin begin_diagnostic; if nuline then print_nl(s)@+else print(s); print(" at line "); print_int(true_line); print(t); print_char(":"); end; @ The 256 |ASCII_code| characters are grouped into classes by means of the |char_class| table. Individual class numbers have no semantic or syntactic significance, except in a few instances defined here. There's also |max_class|, which can be used as a basis for additional class numbers in nonstandard extensions of \MP. @d digit_class=0 {the class number of \.{0123456789}} @d period_class=1 {the class number of `\..'} @d space_class=2 {the class number of spaces and nonstandard characters} @d percent_class=3 {the class number of `\.\%'} @d string_class=4 {the class number of `\."'} @d right_paren_class=8 {the class number of `\.)'} @d isolated_classes==5,6,7,8 {characters that make length-one tokens only} @d letter_class=9 {letters and the underline character} @d left_bracket_class=17 {`\.['} @d right_bracket_class=18 {`\.]'} @d invalid_class=20 {bad character in the input} @d max_class=20 {the largest class number} @= @!char_class:array[ASCII_code] of 0..max_class; {the class numbers} @ If changes are made to accommodate non-ASCII character sets, they should follow the guidelines in Appendix~C of {\sl The {\logos METAFONT\/}book}. @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> @^system dependencies@> @= for k:="0" to "9" do char_class[k]:=digit_class; char_class["."]:=period_class; char_class[" "]:=space_class; char_class["%"]:=percent_class; char_class[""""]:=string_class;@/ char_class[","]:=5; char_class[";"]:=6; char_class["("]:=7; char_class[")"]:=right_paren_class; for k:="A" to "Z" do char_class[k]:=letter_class; for k:="a" to "z" do char_class[k]:=letter_class; char_class["_"]:=letter_class;@/ char_class["<"]:=10; char_class["="]:=10; char_class[">"]:=10; char_class[":"]:=10; char_class["|"]:=10;@/ char_class["`"]:=11; char_class["'"]:=11;@/ char_class["+"]:=12; char_class["-"]:=12;@/ char_class["/"]:=13; char_class["*"]:=13; char_class["\"]:=13;@/ char_class["!"]:=14; char_class["?"]:=14;@/ char_class["#"]:=15; char_class["&"]:=15; char_class["@@"]:=15; char_class["$"]:=15;@/ char_class["^"]:=16; char_class["~"]:=16;@/ char_class["["]:=left_bracket_class; char_class["]"]:=right_bracket_class;@/ char_class["{"]:=19; char_class["}"]:=19;@/ for k:=0 to " "-1 do char_class[k]:=invalid_class; for k:=127 to 255 do char_class[k]:=invalid_class; @* \[13] The hash table. Symbolic tokens are stored and retrieved by means of a fairly standard hash table algorithm called the method of ``coalescing lists'' (cf.\ Algorithm 6.4C in {\sl The Art of Computer Programming\/}). Once a symbolic token enters the table, it is never removed. The actual sequence of characters forming a symbolic token is stored in the |str_pool| array together with all the other strings. An auxiliary array |hash| consists of items with two halfword fields per word. The first of these, called |next(p)|, points to the next identifier belonging to the same coalesced list as the identifier corresponding to~|p|; and the other, called |text(p)|, points to the |str_start| entry for |p|'s identifier. If position~|p| of the hash table is empty, we have |text(p)=0|; if position |p| is either empty or the end of a coalesced hash list, we have |next(p)=0|. An auxiliary pointer variable called |hash_used| is maintained in such a way that all locations |p>=hash_used| are nonempty. The global variable |st_count| tells how many symbolic tokens have been defined, if statistics are being kept. The first 256 locations of |hash| are reserved for symbols of length one. There's a parallel array called |eqtb| that contains the current equivalent values of each symbolic token. The entries of this array consist of two halfwords called |eq_type| (a command code) and |equiv| (a secondary piece of information that qualifies the |eq_type|). @d next(#) == hash[#].lh {link for coalesced lists} @d text(#) == hash[#].rh {string number for symbolic token name} @d eq_type(#) == eqtb[#].lh {the current ``meaning'' of a symbolic token} @d equiv(#) == eqtb[#].rh {parametric part of a token's meaning} @d hash_base=257 {hashing actually starts here} @d hash_is_full == (hash_used=hash_base) {are all positions occupied?} @= @!hash_used:pointer; {allocation pointer for |hash|} @!st_count:integer; {total number of known identifiers} @ Certain entries in the hash table are ``frozen'' and not redefinable, since they are used in error recovery. @d hash_top==hash_base+hash_size {the first location of the frozen area} @d frozen_inaccessible==hash_top {|hash| location to protect the frozen area} @d frozen_repeat_loop==hash_top+1 {|hash| location of a loop-repeat token} @d frozen_right_delimiter==hash_top+2 {|hash| location of a permanent `\.)'} @d frozen_left_bracket==hash_top+3 {|hash| location of a permanent `\.['} @d frozen_slash==hash_top+4 {|hash| location of a permanent `\./'} @d frozen_colon==hash_top+5 {|hash| location of a permanent `\.:'} @d frozen_semicolon==hash_top+6 {|hash| location of a permanent `\.;'} @d frozen_end_for==hash_top+7 {|hash| location of a permanent \&{endfor}} @d frozen_end_def==hash_top+8 {|hash| location of a permanent \&{enddef}} @d frozen_fi==hash_top+9 {|hash| location of a permanent \&{fi}} @d frozen_end_group==hash_top+10 {|hash| location of a permanent `\.{endgroup}'} @d frozen_etex==hash_top+11 {|hash| location of a permanent \&{etex}} @d frozen_mpx_break==hash_top+12 {|hash| location of a permanent \&{mpxbreak}} @d frozen_bad_vardef==hash_top+13 {|hash| location of `\.{a bad variable}'} @d frozen_undefined==hash_top+14 {|hash| location that never gets defined} @d hash_end==hash_top+14 {the actual size of the |hash| and |eqtb| arrays} @= @!hash: array[1..hash_end] of two_halves; {the hash table} @!eqtb: array[1..hash_end] of two_halves; {the equivalents} @ @= next(1):=0; text(1):=0; eq_type(1):=tag_token; equiv(1):=null; for k:=2 to hash_end do begin hash[k]:=hash[1]; eqtb[k]:=eqtb[1]; end; @ @= hash_used:=frozen_inaccessible; {nothing is used} st_count:=0;@/ text(frozen_bad_vardef):="a bad variable"; text(frozen_etex):="etex"; text(frozen_mpx_break):="mpxbreak"; text(frozen_fi):="fi"; text(frozen_end_group):="endgroup"; text(frozen_end_def):="enddef"; text(frozen_end_for):="endfor";@/ text(frozen_semicolon):=";"; text(frozen_colon):=":"; text(frozen_slash):="/"; text(frozen_left_bracket):="["; text(frozen_right_delimiter):=")";@/ text(frozen_inaccessible):=" INACCESSIBLE";@/ eq_type(frozen_right_delimiter):=right_delimiter; @ @= if hash_end+max_internal>max_halfword then bad:=17; @ Here is the subroutine that searches the hash table for an identifier that matches a given string of length~|l| appearing in |buffer[j.. (j+l-1)]|. If the identifier is not found, it is inserted; hence it will always be found, and the corresponding hash table address will be returned. @p function id_lookup(@!j,@!l:integer):pointer; {search the hash table} label found; {go here when you've found it} var @!h:integer; {hash code} @!p:pointer; {index in |hash| array} @!k:pointer; {index in |buffer| array} begin if l=1 then @; @; p:=h+hash_base; {we start searching here; note that |0<=h0 then if length(text(p))=l then if str_eq_buf(text(p),j) then goto found; if next(p)=0 then @; p:=next(p); end; found: id_lookup:=p; end; @ @= begin p:=buffer[j]+1; text(p):=p-1; goto found; end @ @= begin if text(p)>0 then begin repeat if hash_is_full then overflow("hash size",hash_size); @:MetaPost capacity exceeded hash size}{\quad hash size@> decr(hash_used); until text(hash_used)=0; {search for an empty location in |hash|} next(p):=hash_used; p:=hash_used; end; str_room(l); for k:=j to j+l-1 do append_char(buffer[k]); text(p):=make_string; str_ref[text(p)]:=max_str_ref; @!stat incr(st_count);@+tats@;@/ goto found; end @ The value of |hash_prime| should be roughly 85\pct! of |hash_size|, and it should be a prime number. The theory of hashing tells us to expect fewer than two table probes, on the average, when the search is successful. [See J.~S. Vitter, {\sl Journal of the ACM\/ \bf30} (1983), 231--258.] @^Vitter, Jeffrey Scott@> @= h:=buffer[j]; for k:=j+1 to j+l-1 do begin h:=h+h+buffer[k]; while h>=hash_prime do h:=h-hash_prime; end @ @= for q:=1 to hash_end do begin if equiv(q)=p then begin print_nl("EQUIV("); print_int(q); print_char(")"); end; end @ We need to put \MP's ``primitive'' symbolic tokens into the hash table, together with their command code (which will be the |eq_type|) and an operand (which will be the |equiv|). The |primitive| procedure does this, in a way that no \MP\ user can. The global value |cur_sym| contains the new |eqtb| pointer after |primitive| has acted. @p @!init procedure primitive(@!s:str_number;@!c:halfword;@!o:halfword); var @!k:pool_pointer; {index into |str_pool|} @!j:small_number; {index into |buffer|} @!l:small_number; {length of the string} begin k:=str_start[s]; l:=str_stop(s)-k; {we will move |s| into the (empty) |buffer|} for j:=0 to l-1 do buffer[j]:=so(str_pool[k+j]); cur_sym:=id_lookup(0,l);@/ if s>=256 then {we don't want to have the string twice} begin flush_string(text(cur_sym)); text(cur_sym):=s; end; eq_type(cur_sym):=c; equiv(cur_sym):=o; end; tini @ Many of \MP's primitives need no |equiv|, since they are identifiable by their |eq_type| alone. These primitives are loaded into the hash table as follows: @= primitive("..",path_join,0);@/ @!@:.._}{\.{..} primitive@> primitive("[",left_bracket,0); eqtb[frozen_left_bracket]:=eqtb[cur_sym];@/ @!@:[ }{\.{[} primitive@> primitive("]",right_bracket,0);@/ @!@:] }{\.{]} primitive@> primitive("}",right_brace,0);@/ @!@:]]}{\.{\char`\}} primitive@> primitive("{",left_brace,0);@/ @!@:][}{\.{\char`\{} primitive@> primitive(":",colon,0); eqtb[frozen_colon]:=eqtb[cur_sym];@/ @!@:: }{\.{:} primitive@> primitive("::",double_colon,0);@/ @!@::: }{\.{::} primitive@> primitive("||:",bchar_label,0);@/ @!@:::: }{\.{\char'174\char'174:} primitive@> primitive(":=",assignment,0);@/ @!@::=_}{\.{:=} primitive@> primitive(",",comma,0);@/ @!@:, }{\., primitive@> primitive(";",semicolon,0); eqtb[frozen_semicolon]:=eqtb[cur_sym];@/ @!@:; }{\.; primitive@> primitive("\",relax,0);@/ @!@:]]\\}{\.{\char`\\} primitive@> @# primitive("addto",add_to_command,0);@/ @!@:add_to_}{\&{addto} primitive@> primitive("atleast",at_least,0);@/ @!@:at_least_}{\&{atleast} primitive@> primitive("begingroup",begin_group,0); bg_loc:=cur_sym;@/ @!@:begin_group_}{\&{begingroup} primitive@> primitive("controls",controls,0);@/ @!@:controls_}{\&{controls} primitive@> primitive("curl",curl_command,0);@/ @!@:curl_}{\&{curl} primitive@> primitive("delimiters",delimiters,0);@/ @!@:delimiters_}{\&{delimiters} primitive@> primitive("endgroup",end_group,0); eqtb[frozen_end_group]:=eqtb[cur_sym]; eg_loc:=cur_sym;@/ @!@:endgroup_}{\&{endgroup} primitive@> primitive("everyjob",every_job_command,0);@/ @!@:every_job_}{\&{everyjob} primitive@> primitive("exitif",exit_test,0);@/ @!@:exit_if_}{\&{exitif} primitive@> primitive("expandafter",expand_after,0);@/ @!@:expand_after_}{\&{expandafter} primitive@> primitive("interim",interim_command,0);@/ @!@:interim_}{\&{interim} primitive@> primitive("let",let_command,0);@/ @!@:let_}{\&{let} primitive@> primitive("newinternal",new_internal,0);@/ @!@:new_internal_}{\&{newinternal} primitive@> primitive("of",of_token,0);@/ @!@:of_}{\&{of} primitive@> primitive("randomseed",random_seed,0);@/ @!@:random_seed_}{\&{randomseed} primitive@> primitive("save",save_command,0);@/ @!@:save_}{\&{save} primitive@> primitive("scantokens",scan_tokens,0);@/ @!@:scan_tokens_}{\&{scantokens} primitive@> primitive("shipout",ship_out_command,0);@/ @!@:ship_out_}{\&{shipout} primitive@> primitive("skipto",skip_to,0);@/ @!@:skip_to_}{\&{skipto} primitive@> primitive("special",special_command,0); @!@:special}{\&{special} primitive@> primitive("step",step_token,0);@/ @!@:step_}{\&{step} primitive@> primitive("str",str_op,0);@/ @!@:str_}{\&{str} primitive@> primitive("tension",tension,0);@/ @!@:tension_}{\&{tension} primitive@> primitive("to",to_token,0);@/ @!@:to_}{\&{to} primitive@> primitive("until",until_token,0);@/ @!@:until_}{\&{until} primitive@> primitive("within",within_token,0);@/ @!@:within_}{\&{within} primitive@> primitive("write",write_command,0);@/ @!@:write_}{\&{write} primitive@> @ Each primitive has a corresponding inverse, so that it is possible to display the cryptic numeric contents of |eqtb| in symbolic form. Every call of |primitive| in this program is therefore accompanied by some straightforward code that forms part of the |print_cmd_mod| routine explained below. @= add_to_command:print("addto"); assignment:print(":="); at_least:print("atleast"); bchar_label:print("||:"); begin_group:print("begingroup"); colon:print(":"); comma:print(","); controls:print("controls"); curl_command:print("curl"); delimiters:print("delimiters"); double_colon:print("::"); end_group:print("endgroup"); every_job_command:print("everyjob"); exit_test:print("exitif"); expand_after:print("expandafter"); interim_command:print("interim"); left_brace:print("{"); left_bracket:print("["); let_command:print("let"); new_internal:print("newinternal"); of_token:print("of"); path_join:print(".."); random_seed:print("randomseed"); relax:print_char("\"); right_brace:print("}"); right_bracket:print("]"); save_command:print("save"); scan_tokens:print("scantokens"); semicolon:print(";"); ship_out_command:print("shipout"); skip_to:print("skipto"); special_command: print("special"); step_token:print("step"); str_op:print("str"); tension:print("tension"); to_token:print("to"); until_token:print("until"); within_token:print("within"); write_command:print("write"); @ We will deal with the other primitives later, at some point in the program where their |eq_type| and |equiv| values are more meaningful. For example, the primitives for macro definitions will be loaded when we consider the routines that define macros. It is easy to find where each particular primitive was treated by looking in the index at the end; for example, the section where |"def"| entered |eqtb| is listed under `\&{def} primitive'. @* \[14] Token lists. A \MP\ token is either symbolic or numeric or a string, or it denotes a macro parameter or capsule; so there are five corresponding ways to encode it @^token@> internally: (1)~A symbolic token whose hash code is~|p| is represented by the number |p|, in the |info| field of a single-word node in~|mem|. (2)~A numeric token whose |scaled| value is~|v| is represented in a two-word node of~|mem|; the |type| field is |known|, the |name_type| field is |token|, and the |value| field holds~|v|. The fact that this token appears in a two-word node rather than a one-word node is, of course, clear from the node address. (3)~A string token is also represented in a two-word node; the |type| field is |string_type|, the |name_type| field is |token|, and the |value| field holds the corresponding |str_number|. (4)~Capsules have |name_type=capsule|, and their |type| and |value| fields represent arbitrary values (in ways to be explained later). (5)~Macro parameters are like symbolic tokens in that they appear in |info| fields of one-word nodes. The $k$th parameter is represented by |expr_base+k| if it is of type \&{expr}, or by |suffix_base+k| if it is of type \&{suffix}, or by |text_base+k| if it is of type \&{text}. (Here |0<=k= if text_base+param_size>max_halfword then bad:=18; @ We have set aside a two word node beginning at |null| so that we can have |value(null)=0|. We will make use of this coincidence later. @= link(null):=null; value(null):=0; @ A numeric token is created by the following trivial routine. @p function new_num_tok(@!v:scaled):pointer; var @!p:pointer; {the new node} begin p:=get_node(token_node_size); value(p):=v; type(p):=known; name_type(p):=token; new_num_tok:=p; end; @ A token list is a singly linked list of nodes in |mem|, where each node contains a token and a link. Here's a subroutine that gets rid of a token list when it is no longer needed. @p procedure@?token_recycle; forward;@t\2@>@;@/ procedure flush_token_list(@!p:pointer); var @!q:pointer; {the node being recycled} begin while p<>null do begin q:=p; p:=link(p); if q>=hi_mem_min then free_avail(q) else begin case type(q) of vacuous,boolean_type,known:do_nothing; string_type:delete_str_ref(value(q)); unknown_types,pen_type,path_type,picture_type,pair_type,color_type, transform_type,dependent,proto_dependent,independent: begin g_pointer:=q; token_recycle; end; othercases confusion("token") @:this can't happen token}{\quad token@> endcases;@/ free_node(q,token_node_size); end; end; end; @ The procedure |show_token_list|, which prints a symbolic form of the token list that starts at a given node |p|, illustrates these conventions. The token list being displayed should not begin with a reference count. However, the procedure is intended to be fairly robust, so that if the memory links are awry or if |p| is not really a pointer to a token list, almost nothing catastrophic can happen. An additional parameter |q| is also given; this parameter is either null or it points to a node in the token list where a certain magic computation takes place that will be explained later. (Basically, |q| is non-null when we are printing the two-line context information at the time of an error message; |q| marks the place corresponding to where the second line should begin.) The generation will stop, and `\.{\char`\ ETC.}' will be printed, if the length of printing exceeds a given limit~|l|; the length of printing upon entry is assumed to be a given amount called |null_tally|. (Note that |show_token_list| sometimes uses itself recursively to print variable names within a capsule.) @^recursion@> Unusual entries are printed in the form of all-caps tokens preceded by a space, e.g., `\.{\char`\ BAD}'. @= procedure@?print_capsule; forward; @t\2@>@;@/ procedure show_token_list(@!p,@!q:integer;@!l,@!null_tally:integer); label exit; var @!class,@!c:small_number; {the |char_class| of previous and new tokens} @!r,@!v:integer; {temporary registers} begin class:=percent_class; tally:=null_tally; while (p<>null) and (tally; @; class:=c; p:=link(p); end; if p<>null then print(" ETC."); @.ETC@> exit: end; @ @= c:=letter_class; {the default} if (pmem_end) then begin print(" CLOBBERED"); return; @.CLOBBERED@> end; if p else begin r:=info(p); if r>=expr_base then @ else if r<1 then if r=0 then @ else print(" IMPOSSIBLE") @.IMPOSSIBLE@> else begin r:=text(r); if (r<0)or(r>max_str_ptr) then print(" NONEXISTENT") @.NONEXISTENT@> else @; end; end @ @= if name_type(p)=token then if type(p)=known then @ else if type(p)<>string_type then print(" BAD") @.BAD@> else begin print_char(""""); print(value(p)); print_char(""""); c:=string_class; end else if (name_type(p)<>capsule)or(type(p)independent) then print(" BAD") else begin g_pointer:=p; print_capsule; c:=right_paren_class; end @ @= begin if class=digit_class then print_char(" "); v:=value(p); if v<0 then begin if class=left_bracket_class then print_char(" "); print_char("["); print_scaled(v); print_char("]"); c:=right_bracket_class; end else begin print_scaled(v); c:=digit_class; end; end @ Strictly speaking, a genuine token will never have |info(p)=0|. But we will see later (in the |print_variable_name| routine) that it is convenient to let |info(p)=0| stand for `\.{[]}'. @= begin if class=left_bracket_class then print_char(" "); print("[]"); c:=right_bracket_class; end @ @= begin if r end else if r end else begin print("(TEXT"); r:=r-(text_base); @.TEXT@> end; print_int(r); print_char(")"); c:=right_paren_class; end @ @= begin c:=char_class[so(str_pool[str_start[r]])]; if c=class then case c of letter_class:print_char("."); isolated_classes:do_nothing; othercases print_char(" ") endcases; print(r); end @ The following procedures have been declared |forward| with no parameters, because the author dislikes \PASCAL's convention about |forward| procedures with parameters. It was necessary to do something, because |show_token_list| is recursive (although the recursion is limited to one level), and because |flush_token_list| is syntactically (but not semantically) recursive. @^recursion@> @= procedure print_capsule; begin print_char("("); print_exp(g_pointer,0); print_char(")"); end; @# procedure token_recycle; begin recycle_value(g_pointer); end; @ @= @!g_pointer:pointer; {(global) parameter to the |forward| procedures} @ Macro definitions are kept in \MP's memory in the form of token lists that have a few extra one-word nodes at the beginning. The first node contains a reference count that is used to tell when the list is no longer needed. To emphasize the fact that a reference count is present, we shall refer to the |info| field of this special node as the |ref_count| field. @^reference counts@> The next node or nodes after the reference count serve to describe the formal parameters. They either contain a code word that specifies all of the parameters, or they contain zero or more parameter tokens followed by the code `|general_macro|'. @d ref_count==info {reference count preceding a macro definition or picture header} @d add_mac_ref(#)==incr(ref_count(#)) {make a new reference to a macro list} @d general_macro=0 {preface to a macro defined with a parameter list} @d primary_macro=1 {preface to a macro with a \&{primary} parameter} @d secondary_macro=2 {preface to a macro with a \&{secondary} parameter} @d tertiary_macro=3 {preface to a macro with a \&{tertiary} parameter} @d expr_macro=4 {preface to a macro with an undelimited \&{expr} parameter} @d of_macro=5 {preface to a macro with undelimited `\&{expr} |x| \&{of}~|y|' parameters} @d suffix_macro=6 {preface to a macro with an undelimited \&{suffix} parameter} @d text_macro=7 {preface to a macro with an undelimited \&{text} parameter} @p procedure delete_mac_ref(@!p:pointer); {|p| points to the reference count of a macro list that is losing one reference} begin if ref_count(p)=null then flush_token_list(p) else decr(ref_count(p)); end; @ The following subroutine displays a macro, given a pointer to its reference count. @p @t\4@>@@; procedure show_macro(@!p:pointer;@!q,@!l:integer); label exit; var @!r:pointer; {temporary storage} begin p:=link(p); {bypass the reference count} while info(p)>text_macro do begin r:=link(p); link(p):=null; show_token_list(p,null,l,0); link(p):=r; p:=r; if l>0 then l:=l-tally@+else return; end; {control printing of `\.{ETC.}'} @.ETC@> tally:=0; case info(p) of general_macro:print("->"); @.->@> primary_macro,secondary_macro,tertiary_macro:begin print_char("<"); print_cmd_mod(param_type,info(p)); print(">->"); end; expr_macro:print("->"); of_macro:print("of->"); suffix_macro:print("->"); text_macro:print("->"); end; {there are no other cases} show_token_list(link(p),q,l-tally,0); exit:end; @* \[15] Data structures for variables. The variables of \MP\ programs can be simple, like `\.x', or they can combine the structural properties of arrays and records, like `\.{x20a.b}'. A \MP\ user assigns a type to a variable like \.{x20a.b} by saying, for example, `\.{boolean} \.{x20a.b}'. It's time for us to study how such things are represented inside of the computer. Each variable value occupies two consecutive words, either in a two-word node called a value node, or as a two-word subfield of a larger node. One of those two words is called the |value| field; it is an integer, containing either a |scaled| numeric value or the representation of some other type of quantity. (It might also be subdivided into halfwords, in which case it is referred to by other names instead of |value|.) The other word is broken into subfields called |type|, |name_type|, and |link|. The |type| field is a quarterword that specifies the variable's type, and |name_type| is a quarterword from which \MP\ can reconstruct the variable's name (sometimes by using the |link| field as well). Thus, only 1.25 words are actually devoted to the value itself; the other three-quarters of a word are overhead, but they aren't wasted because they allow \MP\ to deal with sparse arrays and to provide meaningful diagnostics. In this section we shall be concerned only with the structural aspects of variables, not their values. Later parts of the program will change the |type| and |value| fields, but we shall treat those fields as black boxes whose contents should not be touched. However, if the |type| field is |structured|, there is no |value| field, and the second word is broken into two pointer fields called |attr_head| and |subscr_head|. Those fields point to additional nodes that contain structural information, as we shall see. @d subscr_head_loc(#) == #+1 {where |value|, |subscr_head| and |attr_head| are} @d attr_head(#) == info(subscr_head_loc(#)) {pointer to attribute info} @d subscr_head(#) == link(subscr_head_loc(#)) {pointer to subscript info} @d value_node_size=2 {the number of words in a value node} @ An attribute node is three words long. Two of these words contain |type| and |value| fields as described above, and the third word contains additional information: There is an |attr_loc| field, which contains the hash address of the token that names this attribute; and there's also a |parent| field, which points to the value node of |structured| type at the next higher level (i.e., at the level to which this attribute is subsidiary). The |name_type| in an attribute node is `|attr|'. The |link| field points to the next attribute with the same parent; these are arranged in increasing order, so that |attr_loc(link(p))>attr_loc(p)|. The final attribute node links to the constant |end_attr|, whose |attr_loc| field is greater than any legal hash address. The |attr_head| in the parent points to a node whose |name_type| is |structured_root|; this node represents the null attribute, i.e., the variable that is relevant when no attributes are attached to the parent. The |attr_head| node is either a value node, a subscript node, or an attribute node, depending on what the parent would be if it were not structured; but the subscript and attribute fields are ignored, so it effectively contains only the data of a value node. The |link| field in this special node points to an attribute node whose |attr_loc| field is zero; the latter node represents a collective subscript `\.{[]}' attached to the parent, and its |link| field points to the first non-special attribute node (or to |end_attr| if there are none). A subscript node likewise occupies three words, with |type| and |value| fields plus extra information; its |name_type| is |subscr|. In this case the third word is called the |subscript| field, which is a |scaled| integer. The |link| field points to the subscript node with the next larger subscript, if any; otherwise the |link| points to the attribute node for collective subscripts at this level. We have seen that the latter node contains an upward pointer, so that the parent can be deduced. The |name_type| in a parent-less value node is |root|, and the |link| is the hash address of the token that names this value. In other words, variables have a hierarchical structure that includes enough threads running around so that the program is able to move easily between siblings, parents, and children. An example should be helpful: (The reader is advised to draw a picture while reading the following description, since that will help to firm up the ideas.) Suppose that `\.x' and `\.{x.a}' and `\.{x[]b}' and `\.{x5}' and `\.{x20b}' have been mentioned in a user's program, where \.{x[]b} has been declared to be of \&{boolean} type. Let |h(x)|, |h(a)|, and |h(b)| be the hash addresses of \.x, \.a, and~\.b. Then |eq_type(h(x))=name| and |equiv(h(x))=p|, where |p|~is a two-word value node with |name_type(p)=root| and |link(p)=h(x)|. We have |type(p)=structured|, |attr_head(p)=q|, and |subscr_head(p)=r|, where |q| points to a value node and |r| to a subscript node. (Are you still following this? Use a pencil to draw a diagram.) The lone variable `\.x' is represented by |type(q)| and |value(q)|; furthermore |name_type(q)=structured_root| and |link(q)=q1|, where |q1| points to an attribute node representing `\.{x[]}'. Thus |name_type(q1)=attr|, |attr_loc(q1)=collective_subscript=0|, |parent(q1)=p|, |type(q1)=structured|, |attr_head(q1)=qq|, and |subscr_head(q1)=qq1|; |qq| is a value node with |type(qq)=numeric_type| (assuming that \.{x5} is numeric, because |qq| represents `\.{x[]}' with no further attributes), |name_type(qq)=structured_root|, and |link(qq)=qq1|. (Now pay attention to the next part.) Node |qq1| is an attribute node representing `\.{x[][]}', which has never yet occurred; its |type| field is |undefined|, and its |value| field is undefined. We have |name_type(qq1)=attr|, |attr_loc(qq1)=collective_subscript|, |parent(qq1)=q1|, and |link(qq1)=qq2|. Since |qq2| represents `\.{x[]b}', |type(qq2)=unknown_boolean|; also |attr_loc(qq2)=h(b)|, |parent(qq2)=q1|, |name_type(qq2)=attr|, |link(qq2)=end_attr|. (Maybe colored lines will help untangle your picture.) Node |r| is a subscript node with |type| and |value| representing `\.{x5}'; |name_type(r)=subscr|, |subscript(r)=5.0|, and |link(r)=r1| is another subscript node. To complete the picture, see if you can guess what |link(r1)| is; give up? It's~|q1|. Furthermore |subscript(r1)=20.0|, |name_type(r1)=subscr|, |type(r1)=structured|, |attr_head(r1)=qqq|, |subscr_head(r1)=qqq1|, and we finish things off with three more nodes |qqq|, |qqq1|, and |qqq2| hung onto~|r1|. (Perhaps you should start again with a larger sheet of paper.) The value of variable \.{x20b} appears in node~|qqq2|, as you can well imagine. If the example in the previous paragraph doesn't make things crystal clear, a glance at some of the simpler subroutines below will reveal how things work out in practice. The only really unusual thing about these conventions is the use of collective subscript attributes. The idea is to avoid repeating a lot of type information when many elements of an array are identical macros (for which distinct values need not be stored) or when they don't have all of the possible attributes. Branches of the structure below collective subscript attributes do not carry actual values except for macro identifiers; branches of the structure below subscript nodes do not carry significant information in their collective subscript attributes. @d attr_loc_loc(#)==#+2 {where the |attr_loc| and |parent| fields are} @d attr_loc(#)==info(attr_loc_loc(#)) {hash address of this attribute} @d parent(#)==link(attr_loc_loc(#)) {pointer to |structured| variable} @d subscript_loc(#)==#+2 {where the |subscript| field lives} @d subscript(#)==mem[subscript_loc(#)].sc {subscript of this variable} @d attr_node_size=3 {the number of words in an attribute node} @d subscr_node_size=3 {the number of words in a subscript node} @d collective_subscript=0 {code for the attribute `\.{[]}'} @= attr_loc(end_attr):=hash_end+1; parent(end_attr):=null; @ Variables of type \&{pair} will have values that point to four-word nodes containing two numeric values. The first of these values has |name_type=x_part_sector| and the second has |name_type=y_part_sector|; the |link| in the first points back to the node whose |value| points to this four-word node. Variables of type \&{transform} are similar, but in this case their |value| points to a 12-word node containing six values, identified by |x_part_sector|, |y_part_sector|, |xx_part_sector|, |xy_part_sector|, |yx_part_sector|, and |yy_part_sector|. Finally, variables of type \&{color} have three values in six words identified by |red_part_sector|, |green_part_sector|, and |blue_part_sector|. When an entire structured variable is saved, the |root| indication is temporarily replaced by |saved_root|. Some variables have no name; they just are used for temporary storage while expressions are being evaluated. We call them {\sl capsules}. @d x_part_loc(#)==# {where the \&{xpart} is found in a pair or transform node} @d y_part_loc(#)==#+2 {where the \&{ypart} is found in a pair or transform node} @d xx_part_loc(#)==#+4 {where the \&{xxpart} is found in a transform node} @d xy_part_loc(#)==#+6 {where the \&{xypart} is found in a transform node} @d yx_part_loc(#)==#+8 {where the \&{yxpart} is found in a transform node} @d yy_part_loc(#)==#+10 {where the \&{yypart} is found in a transform node} @d red_part_loc(#)==# {where the \&{redpart} is found in a color node} @d green_part_loc(#)==#+2 {where the \&{greenpart} is found in a color node} @d blue_part_loc(#)==#+4 {where the \&{bluepart} is found in a color node} @# @d pair_node_size=4 {the number of words in a pair node} @d transform_node_size=12 {the number of words in a transform node} @d color_node_size=6 {the number of words in a color node} @= @!big_node_size:array[transform_type..pair_type] of small_number; @!sector0:array[transform_type..pair_type] of small_number; @!sector_offset:array[x_part_sector..blue_part_sector] of small_number; @ The |sector0| array gives for each big node type, |name_type| values for its first subfield; the |sector_offset| array gives for each |name_type| value, the offset from the first subfield in words; and the |big_node_size| array gives the size in words for each type of big node. @= big_node_size[transform_type]:=transform_node_size; big_node_size[pair_type]:=pair_node_size; big_node_size[color_type]:=color_node_size; sector0[transform_type]:=x_part_sector; sector0[pair_type]:=x_part_sector; sector0[color_type]:=red_part_sector; for k:=x_part_sector to yy_part_sector do sector_offset[k]:=2*(k-x_part_sector); for k:=red_part_sector to blue_part_sector do sector_offset[k]:=2*(k-red_part_sector); @ If |type(p)=pair_type| or |transform_type| and if |value(p)=null|, the procedure call |init_big_node(p)| will allocate a pair or transform node for~|p|. The individual parts of such nodes are initially of type |independent|. @p procedure init_big_node(@!p:pointer); var @!q:pointer; {the new node} @!s:small_number; {its size} begin s:=big_node_size[type(p)]; q:=get_node(s); repeat s:=s-2; @; name_type(q+s):=halfp(s)+sector0[type(p)]; link(q+s):=null; until s=0; link(q):=p; value(p):=q; end; @ The |id_transform| function creates a capsule for the identity transformation. @p function id_transform:pointer; var @!p,@!q,@!r:pointer; {list manipulation registers} begin p:=get_node(value_node_size); type(p):=transform_type; name_type(p):=capsule; value(p):=null; init_big_node(p); q:=value(p); r:=q+transform_node_size; repeat r:=r-2; type(r):=known; value(r):=0; until r=q; value(xx_part_loc(q)):=unity; value(yy_part_loc(q)):=unity; id_transform:=p; end; @ Tokens are of type |tag_token| when they first appear, but they point to |null| until they are first used as the root of a variable. The following subroutine establishes the root node on such grand occasions. @p procedure new_root(@!x:pointer); var @!p:pointer; {the new node} begin p:=get_node(value_node_size); type(p):=undefined; name_type(p):=root; link(p):=x; equiv(x):=p; end; @ These conventions for variable representation are illustrated by the |print_variable_name| routine, which displays the full name of a variable given only a pointer to its two-word value packet. @p procedure print_variable_name(@!p:pointer); label found,exit; var @!q:pointer; {a token list that will name the variable's suffix} @!r:pointer; {temporary for token list creation} begin while name_type(p)>=x_part_sector do @; q:=null; while name_type(p)>saved_root do @; r:=get_avail; info(r):=link(p); link(r):=q; if name_type(p)=saved_root then print("(SAVED)"); @.SAVED@> show_token_list(r,null,el_gordo,tally); flush_token_list(r); exit:end; @ @= begin if name_type(p)=subscr then begin r:=new_num_tok(subscript(p)); repeat p:=link(p); until name_type(p)=attr; end else if name_type(p)=structured_root then begin p:=link(p); goto found; end else begin if name_type(p)<>attr then confusion("var"); @:this can't happen var}{\quad var@> r:=get_avail; info(r):=attr_loc(p); end; link(r):=q; q:=r; found: p:=parent(p); end @ @= begin case name_type(p) of x_part_sector: print_char("x"); y_part_sector: print_char("y"); xx_part_sector: print("xx"); xy_part_sector: print("xy"); yx_part_sector: print("yx"); yy_part_sector: print("yy"); red_part_sector: print("red"); green_part_sector: print("green"); blue_part_sector: print("blue"); capsule: begin print("%CAPSULE"); print_int(p-null); return; @.CAPSULE@> end; end; {there are no other cases} print("part "); p:=link(p-sector_offset[name_type(p)]); end @ The |interesting| function returns |true| if a given variable is not in a capsule, or if the user wants to trace capsules. @p function interesting(@!p:pointer):boolean; var @!t:small_number; {a |name_type|} begin if internal[tracing_capsules]>0 then interesting:=true else begin t:=name_type(p); if t>=x_part_sector then if t<>capsule then t:=name_type(link(p-sector_offset[t])); interesting:=(t<>capsule); end; end; @ Now here is a subroutine that converts an unstructured type into an equivalent structured type, by inserting a |structured| node that is capable of growing. This operation is done only when |name_type(p)=root|, |subscr|, or |attr|. The procedure returns a pointer to the new node that has taken node~|p|'s place in the structure. Node~|p| itself does not move, nor are its |value| or |type| fields changed in any way. @p function new_structure(@!p:pointer):pointer; var @!q,@!r:pointer; {list manipulation registers} begin case name_type(p) of root: begin q:=link(p); r:=get_node(value_node_size); equiv(q):=r; end; subscr: @; attr: @; othercases confusion("struct") @:this can't happen struct}{\quad struct@> endcases;@/ link(r):=link(p); type(r):=structured; name_type(r):=name_type(p); attr_head(r):=p; name_type(p):=structured_root;@/ q:=get_node(attr_node_size); link(p):=q; subscr_head(r):=q; parent(q):=r; type(q):=undefined; name_type(q):=attr; link(q):=end_attr; attr_loc(q):=collective_subscript; new_structure:=r; end; @ @= begin q:=p; repeat q:=link(q); until name_type(q)=attr; q:=parent(q); r:=subscr_head_loc(q); {|link(r)=subscr_head(q)|} repeat q:=r; r:=link(r); until r=p; r:=get_node(subscr_node_size); link(q):=r; subscript(r):=subscript(p); end @ If the attribute is |collective_subscript|, there are two pointers to node~|p|, so we must change both of them. @= begin q:=parent(p); r:=attr_head(q); repeat q:=r; r:=link(r); until r=p; r:=get_node(attr_node_size); link(q):=r;@/ mem[attr_loc_loc(r)]:=mem[attr_loc_loc(p)]; {copy |attr_loc| and |parent|} if attr_loc(p)=collective_subscript then begin q:=subscr_head_loc(parent(p)); while link(q)<>p do q:=link(q); link(q):=r; end; end @ The |find_variable| routine is given a pointer~|t| to a nonempty token list of suffixes; it returns a pointer to the corresponding two-word value. For example, if |t| points to token \.x followed by a numeric token containing the value~7, |find_variable| finds where the value of \.{x7} is stored in memory. This may seem a simple task, and it usually is, except when \.{x7} has never been referenced before. Indeed, \.x may never have even been subscripted before; complexities arise with respect to updating the collective subscript information. If a macro type is detected anywhere along path~|t|, or if the first item on |t| isn't a |tag_token|, the value |null| is returned. Otherwise |p| will be a non-null pointer to a node such that |undefined begin p:=info(t); t:=link(t); if eq_type(p) mod outer_tag<>tag_token then abort_find; if equiv(p)=null then new_root(p); p:=equiv(p); pp:=p; while t<>null do begin @; if t else @; t:=link(t); end; if type(pp)>=structured then if type(pp)=structured then pp:=attr_head(pp)@+else abort_find; if type(p)=structured then p:=attr_head(p); if type(p)=undefined then begin if type(pp)=undefined then begin type(pp):=numeric_type; value(pp):=null; end; type(p):=type(pp); value(p):=null; end; find_variable:=p; exit:end; @ Although |pp| and |p| begin together, they diverge when a subscript occurs; |pp|~stays in the collective line while |p|~goes through actual subscript values. @= if type(pp)<>structured then begin if type(pp)>structured then abort_find; ss:=new_structure(pp); if p=pp then p:=ss; pp:=ss; end; {now |type(pp)=structured|} if type(p)<>structured then {it cannot be |>structured|} p:=new_structure(p) {now |type(p)=structured|} @ We want this part of the program to be reasonably fast, in case there are @^inner loop@> lots of subscripts at the same level of the data structure. Therefore we store an ``infinite'' value in the word that appears at the end of the subscript list, even though that word isn't part of a subscript node. @= begin n:=value(t); pp:=link(attr_head(pp)); {now |attr_loc(pp)=collective_subscript|} q:=link(attr_head(p)); save_word:=mem[subscript_loc(q)]; subscript(q):=el_gordo; s:=subscr_head_loc(p); {|link(s)=subscr_head(p)|} repeat r:=s; s:=link(s); until n<=subscript(s); if n=subscript(s) then p:=s else begin p:=get_node(subscr_node_size); link(r):=p; link(p):=s; subscript(p):=n; name_type(p):=subscr; type(p):=undefined; end; mem[subscript_loc(q)]:=save_word; end @ @= begin n:=info(t); ss:=attr_head(pp); repeat rr:=ss; ss:=link(ss); until n<=attr_loc(ss); if n Parameter |p| points to the root information of the variable; parameter |t| points to a list of one-word nodes that represent suffixes, with |info=collective_subscript| for subscripts. @p @t\4@>@@;@/ @t\4@>@@; @t\4@>@@; @t\4@>@@; @t\4@>@@; procedure flush_variable(@!p,@!t:pointer;@!discard_suffixes:boolean); label exit; var @!q,@!r:pointer; {list manipulation} @!n:halfword; {attribute to match} begin while t<>null do begin if type(p)<>structured then return; n:=info(t); t:=link(t); if n=collective_subscript then begin r:=subscr_head_loc(p); q:=link(r); {|q=subscr_head(p)|} while name_type(q)=subscr do begin flush_variable(q,t,discard_suffixes); if t=null then if type(q)=structured then r:=q else begin link(r):=link(q); free_node(q,subscr_node_size); end else r:=q; q:=link(r); end; end; p:=attr_head(p); repeat r:=p; p:=link(p); until attr_loc(p)>=n; if attr_loc(p)<>n then return; end; if discard_suffixes then flush_below_variable(p) else begin if type(p)=structured then p:=attr_head(p); recycle_value(p); end; exit:end; @ The next procedure is simpler; it wipes out everything but |p| itself, which becomes undefined. @= procedure flush_below_variable(@!p:pointer); var @!q,@!r:pointer; {list manipulation registers} begin if type(p)<>structured then recycle_value(p) {this sets |type(p)=undefined|} else begin q:=subscr_head(p); while name_type(q)=subscr do begin flush_below_variable(q); r:=q; q:=link(q); free_node(r,subscr_node_size); end; r:=attr_head(p); q:=link(r); recycle_value(r); if name_type(p)<=saved_root then free_node(r,value_node_size) else free_node(r,subscr_node_size); {we assume that |subscr_node_size=attr_node_size|} repeat flush_below_variable(q); r:=q; q:=link(q); free_node(r,attr_node_size); until q=end_attr; type(p):=undefined; end; end; @ Just before assigning a new value to a variable, we will recycle the old value and make the old value undefined. The |und_type| routine determines what type of undefined value should be given, based on the current type before recycling. @p function und_type(@!p:pointer):small_number; begin case type(p) of undefined,vacuous:und_type:=undefined; boolean_type,unknown_boolean:und_type:=unknown_boolean; string_type,unknown_string:und_type:=unknown_string; pen_type,unknown_pen:und_type:=unknown_pen; path_type,unknown_path:und_type:=unknown_path; picture_type,unknown_picture:und_type:=unknown_picture; transform_type,color_type,pair_type,numeric_type:und_type:=type(p); known,dependent,proto_dependent,independent:und_type:=numeric_type; end; {there are no other cases} end; @ The |clear_symbol| routine is used when we want to redefine the equivalent of a symbolic token. It must remove any variable structure or macro definition that is currently attached to that symbol. If the |saving| parameter is true, a subsidiary structure is saved instead of destroyed. @p procedure clear_symbol(@!p:pointer;@!saving:boolean); var @!q:pointer; {|equiv(p)|} begin q:=equiv(p); case eq_type(p) mod outer_tag of defined_macro,secondary_primary_macro,tertiary_secondary_macro, expression_tertiary_macro: if not saving then delete_mac_ref(q); tag_token:if q<>null then if saving then name_type(q):=saved_root else begin flush_below_variable(q); free_node(q,value_node_size); end; othercases do_nothing endcases;@/ eqtb[p]:=eqtb[frozen_undefined]; end; @* \[16] Saving and restoring equivalents. The nested structure given by \&{begingroup} and \&{endgroup} allows |eqtb| entries to be saved and restored, so that temporary changes can be made without difficulty. When the user requests a current value to be saved, \MP\ puts that value into its ``save stack.'' An appearance of \&{endgroup} ultimately causes the old values to be removed from the save stack and put back in their former places. The save stack is a linked list containing three kinds of entries, distinguished by their |info| fields. If |p| points to a saved item, then \smallskip\hang |info(p)=0| stands for a group boundary; each \&{begingroup} contributes such an item to the save stack and each \&{endgroup} cuts back the stack until the most recent such entry has been removed. \smallskip\hang |info(p)=q|, where |1<=q<=hash_end|, means that |mem[p+1]| holds the former contents of |eqtb[q]|. Such save stack entries are generated by \&{save} commands or suitable \&{interim} commands. \smallskip\hang |info(p)=hash_end+q|, where |q>0|, means that |value(p)| is a |scaled| integer to be restored to internal parameter number~|q|. Such entries are generated by \&{interim} commands. \smallskip\noindent The global variable |save_ptr| points to the top item on the save stack. @d save_node_size=2 {number of words per non-boundary save-stack node} @d saved_equiv(#)==mem[#+1].hh {where an |eqtb| entry gets saved} @d save_boundary_item(#)==begin #:=get_avail; info(#):=0; link(#):=save_ptr; save_ptr:=#; end @=@!save_ptr:pointer; {the most recently saved item} @ @=save_ptr:=null; @ The |save_variable| routine is given a hash address |q|; it salts this address in the save stack, together with its current equivalent, then makes token~|q| behave as though it were brand new. Nothing is stacked when |save_ptr=null|, however; there's no way to remove things from the stack when the program is not inside a group, so there's no point in wasting the space. @p procedure save_variable(@!q:pointer); var @!p:pointer; {temporary register} begin if save_ptr<>null then begin p:=get_node(save_node_size); info(p):=q; link(p):=save_ptr; saved_equiv(p):=eqtb[q]; save_ptr:=p; end; clear_symbol(q,(save_ptr<>null)); end; @ Similarly, |save_internal| is given the location |q| of an internal quantity like |tracing_pens|. It creates a save stack entry of the third kind. @p procedure save_internal(@!q:halfword); var @!p:pointer; {new item for the save stack} begin if save_ptr<>null then begin p:=get_node(save_node_size); info(p):=hash_end+q; link(p):=save_ptr; value(p):=internal[q]; save_ptr:=p; end; end; @ At the end of a group, the |unsave| routine restores all of the saved equivalents in reverse order. This routine will be called only when there is at least one boundary item on the save stack. @p procedure unsave; var @!q:pointer; {index to saved item} @!p:pointer; {temporary register} begin while info(save_ptr)<>0 do begin q:=info(save_ptr); if q>hash_end then begin if internal[tracing_restores]>0 then begin begin_diagnostic; print_nl("{restoring "); print(int_name[q-(hash_end)]); print_char("="); print_scaled(value(save_ptr)); print_char("}"); end_diagnostic(false); end; internal[q-(hash_end)]:=value(save_ptr); end else begin if internal[tracing_restores]>0 then begin begin_diagnostic; print_nl("{restoring "); print(text(q)); print_char("}"); end_diagnostic(false); end; clear_symbol(q,false); eqtb[q]:=saved_equiv(save_ptr); if eq_type(q) mod outer_tag=tag_token then begin p:=equiv(q); if p<>null then name_type(p):=root; end; end; p:=link(save_ptr); free_node(save_ptr,save_node_size); save_ptr:=p; end; p:=link(save_ptr); free_avail(save_ptr); save_ptr:=p; end; @* \[17] Data structures for paths. When a \MP\ user specifies a path, \MP\ will create a list of knots and control points for the associated cubic spline curves. If the knots are $z_0$, $z_1$, \dots, $z_n$, there are control points $z_k^+$ and $z_{k+1}^-$ such that the cubic splines between knots $z_k$ and $z_{k+1}$ are defined by B\'ezier's formula @:Bezier}{B\'ezier, Pierre Etienne@> $$\eqalign{z(t)&=B(z_k,z_k^+,z_{k+1}^-,z_{k+1};t)\cr &=(1-t)^3z_k+3(1-t)^2tz_k^++3(1-t)t^2z_{k+1}^-+t^3z_{k+1}\cr}$$ for |0<=t<=1|. There is a 7-word node for each knot $z_k$, containing one word of control information and six words for the |x| and |y| coordinates of $z_k^-$ and $z_k$ and~$z_k^+$. The control information appears in the |left_type| and |right_type| fields, which each occupy a quarter of the first word in the node; they specify properties of the curve as it enters and leaves the knot. There's also a halfword |link| field, which points to the following knot. If the path is a closed contour, knots 0 and |n| are identical; i.e., the |link| in knot |n-1| points to knot~0. But if the path is not closed, the |left_type| of knot~0 and the |right_type| of knot~|n| are equal to |endpoint|. In the latter case the |link| in knot~|n| points to knot~0, and the control points $z_0^-$ and $z_n^+$ are not used. @d left_type(#) == mem[#].hh.b0 {characterizes the path entering this knot} @d right_type(#) == mem[#].hh.b1 {characterizes the path leaving this knot} @d endpoint=0 {|left_type| at path beginning and |right_type| at path end} @d x_coord(#) == mem[#+1].sc {the |x| coordinate of this knot} @d y_coord(#) == mem[#+2].sc {the |y| coordinate of this knot} @d left_x(#) == mem[#+3].sc {the |x| coordinate of previous control point} @d left_y(#) == mem[#+4].sc {the |y| coordinate of previous control point} @d right_x(#) == mem[#+5].sc {the |x| coordinate of next control point} @d right_y(#) == mem[#+6].sc {the |y| coordinate of next control point} @d x_loc(#) == #+1 {where the |x| coordinate is stored in a knot} @d y_loc(#) == #+2 {where the |y| coordinate is stored in a knot} @d knot_coord(#) == mem[#].sc {|x| or |y| coordinate given |x_loc| or |y_loc|} @d left_coord(#) == mem[#+2].sc {coordinate of previous control point given |x_loc| or |y_loc|} @d right_coord(#) == mem[#+4].sc {coordinate of next control point given |x_loc| or |y_loc|} @d knot_node_size=7 {number of words in a knot node} @ Before the B\'ezier control points have been calculated, the memory space they will ultimately occupy is taken up by information that can be used to compute them. There are four cases: \yskip \textindent{$\bullet$} If |right_type=open|, the curve should leave the knot in the same direction it entered; \MP\ will figure out a suitable direction. \yskip \textindent{$\bullet$} If |right_type=curl|, the curve should leave the knot in a direction depending on the angle at which it enters the next knot and on the curl parameter stored in |right_curl|. \yskip \textindent{$\bullet$} If |right_type=given|, the curve should leave the knot in a nonzero direction stored as an |angle| in |right_given|. \yskip \textindent{$\bullet$} If |right_type=explicit|, the B\'ezier control point for leaving this knot has already been computed; it is in the |right_x| and |right_y| fields. \yskip\noindent The rules for |left_type| are similar, but they refer to the curve entering the knot, and to \\{left} fields instead of \\{right} fields. Non-|explicit| control points will be chosen based on ``tension'' parameters in the |left_tension| and |right_tension| fields. The `\&{atleast}' option is represented by negative tension values. @!@:at_least_}{\&{atleast} primitive@> For example, the \MP\ path specification $$\.{z0..z1..tension atleast 1..\{curl 2\}z2..z3\{-1,-2\}..tension 3 and 4..p},$$ where \.p is the path `\.{z4..controls z45 and z54..z5}', will be represented by the six knots \def\lodash{\hbox to 1.1em{\thinspace\hrulefill\thinspace}} $$\vbox{\halign{#\hfil&&\qquad#\hfil\cr |left_type|&\\{left} info&|x_coord,y_coord|&|right_type|&\\{right} info\cr \noalign{\yskip} |endpoint|&\lodash$,\,$\lodash&$x_0,y_0$&|curl|&$1.0,1.0$\cr |open|&\lodash$,1.0$&$x_1,y_1$&|open|&\lodash$,-1.0$\cr |curl|&$2.0,-1.0$&$x_2,y_2$&|curl|&$2.0,1.0$\cr |given|&$d,1.0$&$x_3,y_3$&|given|&$d,3.0$\cr |open|&\lodash$,4.0$&$x_4,y_4$&|explicit|&$x_{45},y_{45}$\cr |explicit|&$x_{54},y_{54}$&$x_5,y_5$&|endpoint|&\lodash$,\,$\lodash\cr}}$$ Here |d| is the |angle| obtained by calling |n_arg(-unity,-two)|. Of course, this example is more complicated than anything a normal user would ever write. These types must satisfy certain restrictions because of the form of \MP's path syntax: (i)~|open| type never appears in the same node together with |endpoint|, |given|, or |curl|. (ii)~The |right_type| of a node is |explicit| if and only if the |left_type| of the following node is |explicit|. (iii)~|endpoint| types occur only at the ends, as mentioned above. @d left_curl==left_x {curl information when entering this knot} @d left_given==left_x {given direction when entering this knot} @d left_tension==left_y {tension information when entering this knot} @d right_curl==right_x {curl information when leaving this knot} @d right_given==right_x {given direction when leaving this knot} @d right_tension==right_y {tension information when leaving this knot} @d explicit=1 {|left_type| or |right_type| when control points are known} @d given=2 {|left_type| or |right_type| when a direction is given} @d curl=3 {|left_type| or |right_type| when a curl is desired} @d open=4 {|left_type| or |right_type| when \MP\ should choose the direction} @ Here is a routine that prints a given knot list in symbolic form. It illustrates the conventions discussed above, and checks for anomalies that might arise while \MP\ is being debugged. @= procedure pr_path(@!h:pointer); label done,done1; var @!p,@!q:pointer; {for list traversal} begin p:=h; repeat q:=link(p); if (p=null)or(q=null) then begin print_nl("???"); goto done; {this won't happen} @.???@> end; @; p:=q; if (p<>h)or(left_type(h)<>endpoint) then @; until p=h; if left_type(h)<>endpoint then print("cycle"); done:end; @ @= print_two(x_coord(p),y_coord(p)); case right_type(p) of endpoint: begin if left_type(p)=open then print("{open?}"); {can't happen} @.open?@> if (left_type(q)<>endpoint)or(q<>h) then q:=null; {force an error} goto done1; end; explicit: @; open: @; curl,given: @; othercases print("???") {can't happen} @.???@> endcases;@/ if left_type(q)<=explicit then print("..control?") {can't happen} @.control?@> else if (right_tension(p)<>unity)or(left_tension(q)<>unity) then @; done1: @ Since |n_sin_cos| produces |fraction| results, which we will print as if they were |scaled|, the magnitude of a |given| direction vector will be~4096. @= begin print_nl(" .."); if left_type(p)=given then begin n_sin_cos(left_given(p)); print_char("{"); print_scaled(n_cos); print_char(","); print_scaled(n_sin); print_char("}"); end else if left_type(p)=curl then begin print("{curl "); print_scaled(left_curl(p)); print_char("}"); end; end @ @= begin print("..tension "); if right_tension(p)<0 then print("atleast"); print_scaled(abs(right_tension(p))); if right_tension(p)<>left_tension(q) then begin print(" and "); if left_tension(q)<0 then print("atleast"); print_scaled(abs(left_tension(q))); end; end @ @= begin print("..controls "); print_two(right_x(p),right_y(p)); print(" and "); if left_type(q)<>explicit then print("??") {can't happen} @.??@> else print_two(left_x(q),left_y(q)); goto done1; end @ @= if (left_type(p)<>explicit)and(left_type(p)<>open) then print("{open?}") {can't happen} @.open?@> @ A curl of 1 is shown explicitly, so that the user sees clearly that \MP's default curl is present. The code here uses the fact that |left_curl==left_given| and |right_curl==right_given|. @= begin if left_type(p)=open then print("??"); {can't happen} @.??@> if right_type(p)=curl then begin print("{curl "); print_scaled(right_curl(p)); end else begin n_sin_cos(right_given(p)); print_char("{"); print_scaled(n_cos); print_char(","); print_scaled(n_sin); end; print_char("}"); end @ It is convenient to have another version of |pr_path| that prints the path as a diagnostic message. @= procedure print_path(@!h:pointer;@!s:str_number;@!nuline:boolean); begin print_diagnostic("Path",s,nuline); print_ln; @.Path at line...@> pr_path(h); end_diagnostic(true); end; @ If we want to duplicate a knot node, we can say |copy_knot|: @p function copy_knot(@!p:pointer):pointer; var @!q:pointer; {the copy} @!k:0..knot_node_size-1; {runs through the words of a knot node} begin q:=get_node(knot_node_size); for k:=0 to knot_node_size-1 do mem[q+k]:=mem[p+k]; copy_knot:=q; end; @ The |copy_path| routine makes a clone of a given path. @p function copy_path(@!p:pointer):pointer; var @!q,@!pp,@!qq:pointer; {for list manipulation} begin q:=copy_knot(p); qq:=q; pp:=link(p); while pp<>p do begin link(qq):=copy_knot(pp);@/ qq:=link(qq); pp:=link(pp); end; link(qq):=q; copy_path:=q; end; @ Similarly, there's a way to copy the {\sl reverse\/} of a path. This procedure returns a pointer to the first node of the copy, if the path is a cycle, but to the final node of a non-cyclic copy. The global variable |path_tail| will point to the final node of the original path; this trick makes it easier to implement `\&{doublepath}'. All node types are assumed to be |endpoint| or |explicit| only. @p function htap_ypoc(@!p:pointer):pointer; label exit; var @!q,@!pp,@!qq,@!rr:pointer; {for list manipulation} begin q:=get_node(knot_node_size); {this will correspond to |p|} qq:=q; pp:=p; loop@+ begin right_type(qq):=left_type(pp); left_type(qq):=right_type(pp);@/ x_coord(qq):=x_coord(pp); y_coord(qq):=y_coord(pp);@/ right_x(qq):=left_x(pp); right_y(qq):=left_y(pp);@/ left_x(qq):=right_x(pp); left_y(qq):=right_y(pp);@/ if link(pp)=p then begin link(q):=qq; path_tail:=pp; htap_ypoc:=q; return; end; rr:=get_node(knot_node_size); link(rr):=qq; qq:=rr; pp:=link(pp); end; exit:end; @ @= @!path_tail:pointer; {the node that links to the beginning of a path} @ When a cyclic list of knot nodes is no longer needed, it can be recycled by calling the following subroutine. @= procedure toss_knot_list(@!p:pointer); var @!q:pointer; {the node being freed} @!r:pointer; {the next node} begin q:=p; repeat r:=link(q); free_node(q,knot_node_size); q:=r; until q=p; end; @* \[18] Choosing control points. Now we must actually delve into one of \MP's more difficult routines, the |make_choices| procedure that chooses angles and control points for the splines of a curve when the user has not specified them explicitly. The parameter to |make_choices| points to a list of knots and path information, as described above. A path decomposes into independent segments at ``breakpoint'' knots, which are knots whose left and right angles are both prespecified in some way (i.e., their |left_type| and |right_type| aren't both open). @p @t\4@>@@; procedure make_choices(@!knots:pointer); label done; var @!h:pointer; {the first breakpoint} @!p,@!q:pointer; {consecutive breakpoints being processed} @@; begin check_arith; {make sure that |arith_error=false|} if internal[tracing_choices]>0 then print_path(knots,", before choices",true); @; @; p:=h; repeat @; until p=h; if internal[tracing_choices]>0 then print_path(knots,", after choices",true); if arith_error then @; end; @ @= begin print_err("Some number got too big"); @.Some number got too big@> help2("The path that I just computed is out of range.")@/ ("So it will probably look funny. Proceed, for a laugh."); put_get_error; arith_error:=false; end @ Two knots in a row with the same coordinates will always be joined by an explicit ``curve'' whose control points are identical with the knots. @= p:=knots; repeat q:=link(p); if x_coord(p)=x_coord(q) then if y_coord(p)=y_coord(q) then if right_type(p)>explicit then begin right_type(p):=explicit; if left_type(p)=open then begin left_type(p):=curl; left_curl(p):=unity; end; left_type(q):=explicit; if right_type(q)=open then begin right_type(q):=curl; right_curl(q):=unity; end; right_x(p):=x_coord(p); left_x(q):=x_coord(p);@/ right_y(p):=y_coord(p); left_y(q):=y_coord(p); end; p:=q; until p=knots @ If there are no breakpoints, it is necessary to compute the direction angles around an entire cycle. In this case the |left_type| of the first node is temporarily changed to |end_cycle|. @d end_cycle=open+1 @= h:=knots; loop@+ begin if left_type(h)<>open then goto done; if right_type(h)<>open then goto done; h:=link(h); if h=knots then begin left_type(h):=end_cycle; goto done; end; end; done: @ If |right_type(p)= q:=link(p); if right_type(p)>=given then begin while (left_type(q)=open)and(right_type(q)=open) do q:=link(q); @; end else if right_type(p)=endpoint then @; p:=q @ This step makes it possible to transform an explicitly computed path without checking the |left_type| and |right_type| fields. @= begin right_x(p):=x_coord(p); right_y(p):=y_coord(p);@/ left_x(q):=x_coord(q); left_y(q):=y_coord(q); end @ Before we can go further into the way choices are made, we need to consider the underlying theory. The basic ideas implemented in |make_choices| are due to John Hobby, who introduced the notion of ``mock curvature'' @^Hobby, John Douglas@> at a knot. Angles are chosen so that they preserve mock curvature when a knot is passed, and this has been found to produce excellent results. It is convenient to introduce some notations that simplify the necessary formulas. Let $d_{k,k+1}=\vert z\k-z_k\vert$ be the (nonzero) distance between knots |k| and |k+1|; and let $${z\k-z_k\over z_k-z_{k-1}}={d_{k,k+1}\over d_{k-1,k}}e^{i\psi_k}$$ so that a polygonal line from $z_{k-1}$ to $z_k$ to $z\k$ turns left through an angle of~$\psi_k$. We assume that $\vert\psi_k\vert\L180^\circ$. The control points for the spline from $z_k$ to $z\k$ will be denoted by $$\eqalign{z_k^+&=z_k+ \textstyle{1\over3}\rho_k e^{i\theta_k}(z\k-z_k),\cr z\k^-&=z\k- \textstyle{1\over3}\sigma\k e^{-i\phi\k}(z\k-z_k),\cr}$$ where $\rho_k$ and $\sigma\k$ are nonnegative ``velocity ratios'' at the beginning and end of the curve, while $\theta_k$ and $\phi\k$ are the corresponding ``offset angles.'' These angles satisfy the condition $$\theta_k+\phi_k+\psi_k=0,\eqno(*)$$ whenever the curve leaves an intermediate knot~|k| in the direction that it enters. @ Let $\alpha_k$ and $\beta\k$ be the reciprocals of the ``tension'' of the curve at its beginning and ending points. This means that $\rho_k=\alpha_k f(\theta_k,\phi\k)$ and $\sigma\k=\beta\k f(\phi\k,\theta_k)$, where $f(\theta,\phi)$ is \MP's standard velocity function defined in the |velocity| subroutine. The cubic spline $B(z_k^{\phantom+},z_k^+, z\k^-,z\k^{\phantom+};t)$ has curvature @^curvature@> $${2\sigma\k\sin(\theta_k+\phi\k)-6\sin\theta_k\over\rho_k^2d_{k,k+1}} \qquad{\rm and}\qquad {2\rho_k\sin(\theta_k+\phi\k)-6\sin\phi\k\over\sigma\k^2d_{k,k+1}}$$ at |t=0| and |t=1|, respectively. The mock curvature is the linear @^mock curvature@> approximation to this true curvature that arises in the limit for small $\theta_k$ and~$\phi\k$, if second-order terms are discarded. The standard velocity function satisfies $$f(\theta,\phi)=1+O(\theta^2+\theta\phi+\phi^2);$$ hence the mock curvatures are respectively $${2\beta\k(\theta_k+\phi\k)-6\theta_k\over\alpha_k^2d_{k,k+1}} \qquad{\rm and}\qquad {2\alpha_k(\theta_k+\phi\k)-6\phi\k\over\beta\k^2d_{k,k+1}}.\eqno(**)$$ @ The turning angles $\psi_k$ are given, and equation $(*)$ above determines $\phi_k$ when $\theta_k$ is known, so the task of angle selection is essentially to choose appropriate values for each $\theta_k$. When equation~$(*)$ is used to eliminate $\phi$~variables from $(**)$, we obtain a system of linear equations of the form $$A_k\theta_{k-1}+(B_k+C_k)\theta_k+D_k\theta\k=-B_k\psi_k-D_k\psi\k,$$ where $$A_k={\alpha_{k-1}\over\beta_k^2d_{k-1,k}}, \qquad B_k={3-\alpha_{k-1}\over\beta_k^2d_{k-1,k}}, \qquad C_k={3-\beta\k\over\alpha_k^2d_{k,k+1}}, \qquad D_k={\beta\k\over\alpha_k^2d_{k,k+1}}.$$ The tensions are always $3\over4$ or more, hence each $\alpha$ and~$\beta$ will be at most $4\over3$. It follows that $B_k\G{5\over4}A_k$ and $C_k\G{5\over4}D_k$; hence the equations are diagonally dominant; hence they have a unique solution. Moreover, in most cases the tensions are equal to~1, so that $B_k=2A_k$ and $C_k=2D_k$. This makes the solution numerically stable, and there is an exponential damping effect: The data at knot $k\pm j$ affects the angle at knot~$k$ by a factor of~$O(2^{-j})$. @ However, we still must consider the angles at the starting and ending knots of a non-cyclic path. These angles might be given explicitly, or they might be specified implicitly in terms of an amount of ``curl.'' Let's assume that angles need to be determined for a non-cyclic path starting at $z_0$ and ending at~$z_n$. Then equations of the form $$A_k\theta_{k-1}+(B_k+C_k)\theta_k+D_k\theta_{k+1}=R_k$$ have been given for $00$ and $C_0B_1-A_1D_0>0$ when $\gamma_0\G0$, hence the linear equations remain nonsingular. Similar considerations apply at the right end, when the final angle $\phi_n$ may or may not need to be determined. It is convenient to let $\psi_n=0$, hence $\theta_n=-\phi_n$. We either have an explicit equation $\theta_n=E_n$, or we have $$\bigl((3-\beta_n)\chi_n+\alpha_{n-1}\bigr)\theta_{n-1}+ (\beta_n\chi_n+3-\alpha_{n-1})\theta_n=0,\qquad \chi_n={\beta_n^2\gamma_n\over\alpha_{n-1}^2}.$$ When |make_choices| chooses angles, it must compute the coefficients of these linear equations, then solve the equations. To compute the coefficients, it is necessary to compute arctangents of the given turning angles~$\psi_k$. When the equations are solved, the chosen directions $\theta_k$ are put back into the form of control points by essentially computing sines and cosines. @ OK, we are ready to make the hard choices of |make_choices|. Most of the work is relegated to an auxiliary procedure called |solve_choices|, which has been introduced to keep |make_choices| from being extremely long. @= @; @; solve_choices(p,q,n) @ It's convenient to precompute quantities that will be needed several times later. The values of |delta_x[k]| and |delta_y[k]| will be the coordinates of $z\k-z_k$, and the magnitude of this vector will be |delta[k]=@t$d_{k,k+1}$@>|. The path angle $\psi_k$ between $z_k-z_{k-1}$ and $z\k-z_k$ will be stored in |psi[k]|. @= @!delta_x,@!delta_y,@!delta:array[0..path_size] of scaled; {knot differences} @!psi:array[1..path_size] of angle; {turning angles} @ @= @!k,@!n:0..path_size; {current and final knot numbers} @!s,@!t:pointer; {registers for list traversal} @!delx,@!dely:scaled; {directions where |open| meets |explicit|} @!sine,@!cosine:fraction; {trig functions of various angles} @ @= k:=0; s:=p; n:=path_size; repeat t:=link(s); delta_x[k]:=x_coord(t)-x_coord(s); delta_y[k]:=y_coord(t)-y_coord(s); delta[k]:=pyth_add(delta_x[k],delta_y[k]); if k>0 then begin sine:=make_fraction(delta_y[k-1],delta[k-1]); cosine:=make_fraction(delta_x[k-1],delta[k-1]); psi[k]:=n_arg(take_fraction(delta_x[k],cosine)+ take_fraction(delta_y[k],sine), take_fraction(delta_y[k],cosine)- take_fraction(delta_x[k],sine)); end; @:MetaPost capacity exceeded path size}{\quad path size@> incr(k); s:=t; if k=path_size then overflow("path size",path_size); if s=q then n:=k; until (k>=n)and(left_type(s)<>end_cycle); if k=n then psi[n]:=0@+else psi[k]:=psi[1] @ When we get to this point of the code, |right_type(p)| is either |given| or |curl| or |open|. If it is |open|, we must have |left_type(p)=end_cycle| or |left_type(p)=explicit|. In the latter case, the |open| type is converted to |given|; however, if the velocity coming into this knot is zero, the |open| type is converted to a |curl|, since we don't know the incoming direction. Similarly, |left_type(q)| is either |given| or |curl| or |open| or |end_cycle|. The |open| possibility is reduced either to |given| or to |curl|. @= if left_type(q)=open then begin delx:=right_x(q)-x_coord(q); dely:=right_y(q)-y_coord(q); if (delx=0)and(dely=0) then begin left_type(q):=curl; left_curl(q):=unity; end else begin left_type(q):=given; left_given(q):=n_arg(delx,dely); end; end; if (right_type(p)=open)and(left_type(p)=explicit) then begin delx:=x_coord(p)-left_x(p); dely:=y_coord(p)-left_y(p); if (delx=0)and(dely=0) then begin right_type(p):=curl; right_curl(p):=unity; end else begin right_type(p):=given; right_given(p):=n_arg(delx,dely); end; end @ Linear equations need to be solved whenever |n>1|; and also when |n=1| and exactly one of the breakpoints involves a curl. The simplest case occurs when |n=1| and there is a curl at both breakpoints; then we simply draw a straight line. But before coding up the simple cases, we might as well face the general case, since we must deal with it sooner or later, and since the general case is likely to give some insight into the way simple cases can be handled best. When there is no cycle, the linear equations to be solved form a tridiagonal system, and we can apply the standard technique of Gaussian elimination to convert that system to a sequence of equations of the form $$\theta_0+u_0\theta_1=v_0,\quad \theta_1+u_1\theta_2=v_1,\quad\ldots,\quad \theta_{n-1}+u_{n-1}\theta_n=v_{n-1},\quad \theta_n=v_n.$$ It is possible to do this diagonalization while generating the equations. Once $\theta_n$ is known, it is easy to determine $\theta_{n-1}$, \dots, $\theta_1$, $\theta_0$; thus, the equations will be solved. The procedure is slightly more complex when there is a cycle, but the basic idea will be nearly the same. In the cyclic case the right-hand sides will be $v_k+w_k\theta_0$ instead of simply $v_k$, and we will start the process off with $u_0=v_0=0$, $w_0=1$. The final equation will be not $\theta_n=v_n$ but $\theta_n+u_n\theta_1=v_n+w_n\theta_0$; an appropriate ending routine will take account of the fact that $\theta_n=\theta_0$ and eliminate the $w$'s from the system, after which the solution can be obtained as before. When $u_k$, $v_k$, and $w_k$ are being computed, the three pointer variables |r|, |s|,~|t| will point respectively to knots |k-1|, |k|, and~|k+1|. The $u$'s and $w$'s are scaled by $2^{28}$, i.e., they are of type |fraction|; the $\theta$'s and $v$'s are of type |angle|. @= @!theta:array[0..path_size] of angle; {values of $\theta_k$} @!uu:array[0..path_size] of fraction; {values of $u_k$} @!vv:array[0..path_size] of angle; {values of $v_k$} @!ww:array[0..path_size] of fraction; {values of $w_k$} @ Our immediate problem is to get the ball rolling by setting up the first equation or by realizing that no equations are needed, and to fit this initialization into a framework suitable for the overall computation. @= @t\4@>@@; procedure solve_choices(@!p,@!q:pointer;@!n:halfword); label found,exit; var @!k:0..path_size; {current knot number} @!r,@!s,@!t:pointer; {registers for list traversal} @@; begin k:=0; s:=p; loop@+ begin t:=link(s); if k=0 then @ else case left_type(s) of end_cycle,open:@; curl:@; given:@; end; {there are no other cases} r:=s; s:=t; incr(k); end; found:@; exit:end; @ On the first time through the loop, we have |k=0| and |r| is not yet defined. The first linear equation, if any, will have $A_0=B_0=0$. @= case right_type(s) of given: if left_type(t)=given then @ else @; curl: if left_type(t)=curl then @ else @; open: begin uu[0]:=0; vv[0]:=0; ww[0]:=fraction_one; end; {this begins a cycle} end {there are no other cases} @ The general equation that specifies equality of mock curvature at $z_k$ is $$A_k\theta_{k-1}+(B_k+C_k)\theta_k+D_k\theta\k=-B_k\psi_k-D_k\psi\k,$$ as derived above. We want to combine this with the already-derived equation $\theta_{k-1}+u_{k-1}\theta_k=v_{k-1}+w_{k-1}\theta_0$ in order to obtain a new equation $\theta_k+u_k\theta\k=v_k+w_k\theta_0$. This can be done by dividing the equation $$(B_k-u_{k-1}A_k+C_k)\theta_k+D_k\theta\k=-B_k\psi_k-D_k\psi\k-A_kv_{k-1} -A_kw_{k-1}\theta_0$$ by $B_k-u_{k-1}A_k+C_k$. The trick is to do this carefully with fixed-point arithmetic, avoiding the chance of overflow while retaining suitable precision. The calculations will be performed in several registers that provide temporary storage for intermediate quantities. @= @!aa,@!bb,@!cc,@!ff,@!acc:fraction; {temporary registers} @!dd,@!ee:scaled; {likewise, but |scaled|} @!lt,@!rt:scaled; {tension values} @ @= begin @; @; uu[k]:=take_fraction(ff,bb); @; if left_type(s)=end_cycle then @; end @ Since tension values are never less than 3/4, the values |aa| and |bb| computed here are never more than 4/5. @= if abs(right_tension(r))=unity then begin aa:=fraction_half; dd:=2*delta[k]; end else begin aa:=make_fraction(unity,3*abs(right_tension(r))-unity); dd:=take_fraction(delta[k], fraction_three-make_fraction(unity,abs(right_tension(r)))); end; if abs(left_tension(t))=unity then begin bb:=fraction_half; ee:=2*delta[k-1]; end else begin bb:=make_fraction(unity,3*abs(left_tension(t))-unity); ee:=take_fraction(delta[k-1], fraction_three-make_fraction(unity,abs(left_tension(t)))); end; cc:=fraction_one-take_fraction(uu[k-1],aa) @ The ratio to be calculated in this step can be written in the form $$\beta_k^2\cdot\\{ee}\over\beta_k^2\cdot\\{ee}+\alpha_k^2\cdot \\{cc}\cdot\\{dd},$$ because of the quantities just calculated. The values of |dd| and |ee| will not be needed after this step has been performed. @= dd:=take_fraction(dd,cc); lt:=abs(left_tension(s)); rt:=abs(right_tension(s)); if lt<>rt then {$\beta_k^{-1}\ne\alpha_k^{-1}$} if lt= acc:=-take_fraction(psi[k+1],uu[k]); if right_type(r)=curl then begin ww[k]:=0; vv[k]:=acc-take_fraction(psi[1],fraction_one-ff); end else begin ff:=make_fraction(fraction_one-ff,cc); {this is $B_k/(C_k+B_k-u_{k-1}A_k)<5$} acc:=acc-take_fraction(psi[k],ff); ff:=take_fraction(ff,aa); {this is $A_k/(C_k+B_k-u_{k-1}A_k)$} vv[k]:=acc-take_fraction(vv[k-1],ff); if ww[k-1]=0 then ww[k]:=0 else ww[k]:=-take_fraction(ww[k-1],ff); end @ When a complete cycle has been traversed, we have $\theta_k+u_k\theta\k= v_k+w_k\theta_0$, for |1<=k<=n|. We would like to determine the value of $\theta_n$ and reduce the system to the form $\theta_k+u_k\theta\k=v_k$ for |0<=k= begin aa:=0; bb:=fraction_one; {we have |k=n|} repeat decr(k); if k=0 then k:=n; aa:=vv[k]-take_fraction(aa,uu[k]); bb:=ww[k]-take_fraction(bb,uu[k]); until k=n; {now $\theta_n=\\{aa}+\\{bb}\cdot\theta_n$} aa:=make_fraction(aa,fraction_one-bb); theta[n]:=aa; vv[0]:=aa; for k:=1 to n-1 do vv[k]:=vv[k]+take_fraction(aa,ww[k]); goto found; end @ @d reduce_angle(#)==if abs(#)>one_eighty_deg then if #>0 then #:=#-three_sixty_deg@+else #:=#+three_sixty_deg @= begin theta[n]:=left_given(s)-n_arg(delta_x[n-1],delta_y[n-1]); reduce_angle(theta[n]); goto found; end @ @= begin vv[0]:=right_given(s)-n_arg(delta_x[0],delta_y[0]); reduce_angle(vv[0]); uu[0]:=0; ww[0]:=0; end @ @= begin cc:=right_curl(s); lt:=abs(left_tension(t)); rt:=abs(right_tension(s)); if (rt=unity)and(lt=unity) then uu[0]:=make_fraction(cc+cc+unity,cc+two) else uu[0]:=curl_ratio(cc,rt,lt); vv[0]:=-take_fraction(psi[1],uu[0]); ww[0]:=0; end @ @= begin cc:=left_curl(s); lt:=abs(left_tension(s)); rt:=abs(right_tension(r)); if (rt=unity)and(lt=unity) then ff:=make_fraction(cc+cc+unity,cc+two) else ff:=curl_ratio(cc,lt,rt); theta[n]:=-make_fraction(take_fraction(vv[n-1],ff), fraction_one-take_fraction(ff,uu[n-1])); goto found; end @ The |curl_ratio| subroutine has three arguments, which our previous notation encourages us to call $\gamma$, $\alpha^{-1}$, and $\beta^{-1}$. It is a somewhat tedious program to calculate $${(3-\alpha)\alpha^2\gamma+\beta^3\over \alpha^3\gamma+(3-\beta)\beta^2},$$ with the result reduced to 4 if it exceeds 4. (This reduction of curl is necessary only if the curl and tension are both large.) The values of $\alpha$ and $\beta$ will be at most~4/3. @= function curl_ratio(@!gamma,@!a_tension,@!b_tension:scaled):fraction; var @!alpha,@!beta,@!num,@!denom,@!ff:fraction; {registers} begin alpha:=make_fraction(unity,a_tension); beta:=make_fraction(unity,b_tension);@/ if alpha<=beta then begin ff:=make_fraction(alpha,beta); ff:=take_fraction(ff,ff); gamma:=take_fraction(gamma,ff);@/ beta:=beta div @'10000; {convert |fraction| to |scaled|} denom:=take_fraction(gamma,alpha)+three-beta; num:=take_fraction(gamma,fraction_three-alpha)+beta; end else begin ff:=make_fraction(beta,alpha); ff:=take_fraction(ff,ff); beta:=take_fraction(beta,ff) div @'10000; {convert |fraction| to |scaled|} denom:=take_fraction(gamma,alpha)+(ff div 1365)-beta; {$1365\approx 2^{12}/3$} num:=take_fraction(gamma,fraction_three-alpha)+beta; end; if num>=denom+denom+denom+denom then curl_ratio:=fraction_four else curl_ratio:=make_fraction(num,denom); end; @ We're in the home stretch now. @= for k:=n-1 downto 0 do theta[k]:=vv[k]-take_fraction(theta[k+1],uu[k]); s:=p; k:=0; repeat t:=link(s);@/ n_sin_cos(theta[k]); st:=n_sin; ct:=n_cos;@/ n_sin_cos(-psi[k+1]-theta[k+1]); sf:=n_sin; cf:=n_cos;@/ set_controls(s,t,k);@/ incr(k); s:=t; until k=n @ The |set_controls| routine actually puts the control points into a pair of consecutive nodes |p| and~|q|. Global variables are used to record the values of $\sin\theta$, $\cos\theta$, $\sin\phi$, and $\cos\phi$ needed in this calculation. @= @!st,@!ct,@!sf,@!cf:fraction; {sines and cosines} @ @= procedure set_controls(@!p,@!q:pointer;@!k:integer); var @!rr,@!ss:fraction; {velocities, divided by thrice the tension} @!lt,@!rt:scaled; {tensions} @!sine:fraction; {$\sin(\theta+\phi)$} begin lt:=abs(left_tension(q)); rt:=abs(right_tension(p)); rr:=velocity(st,ct,sf,cf,rt); ss:=velocity(sf,cf,st,ct,lt); if (right_tension(p)<0)or(left_tension(q)<0) then @; right_x(p):=x_coord(p)+take_fraction( take_fraction(delta_x[k],ct)-take_fraction(delta_y[k],st),rr); right_y(p):=y_coord(p)+take_fraction( take_fraction(delta_y[k],ct)+take_fraction(delta_x[k],st),rr); left_x(q):=x_coord(q)-take_fraction( take_fraction(delta_x[k],cf)+take_fraction(delta_y[k],sf),ss); left_y(q):=y_coord(q)-take_fraction( take_fraction(delta_y[k],cf)-take_fraction(delta_x[k],sf),ss); right_type(p):=explicit; left_type(q):=explicit; end; @ The boundedness conditions $\\{rr}\L\sin\phi\,/\sin(\theta+\phi)$ and $\\{ss}\L\sin\theta\,/\sin(\theta+\phi)$ are to be enforced if $\sin\theta$, $\sin\phi$, and $\sin(\theta+\phi)$ all have the same sign. Otherwise there is no ``bounding triangle.'' @!@:at_least_}{\&{atleast} primitive@> @= if((st>=0)and(sf>=0))or((st<=0)and(sf<=0)) then begin sine:=take_fraction(abs(st),cf)+take_fraction(abs(sf),ct); if sine>0 then begin sine:=take_fraction(sine,fraction_one+unity); {safety factor} if right_tension(p)<0 then if ab_vs_cd(abs(sf),fraction_one,rr,sine)<0 then rr:=make_fraction(abs(sf),sine); if left_tension(q)<0 then if ab_vs_cd(abs(st),fraction_one,ss,sine)<0 then ss:=make_fraction(abs(st),sine); end; end @ Only the simple cases remain to be handled. @= begin aa:=n_arg(delta_x[0],delta_y[0]);@/ n_sin_cos(right_given(p)-aa); ct:=n_cos; st:=n_sin;@/ n_sin_cos(left_given(q)-aa); cf:=n_cos; sf:=-n_sin;@/ set_controls(p,q,0); return; end @ @= begin right_type(p):=explicit; left_type(q):=explicit; lt:=abs(left_tension(q)); rt:=abs(right_tension(p)); if rt=unity then begin if delta_x[0]>=0 then right_x(p):=x_coord(p)+((delta_x[0]+1) div 3) else right_x(p):=x_coord(p)+((delta_x[0]-1) div 3); if delta_y[0]>=0 then right_y(p):=y_coord(p)+((delta_y[0]+1) div 3) else right_y(p):=y_coord(p)+((delta_y[0]-1) div 3); end else begin ff:=make_fraction(unity,3*rt); {$\alpha/3$} right_x(p):=x_coord(p)+take_fraction(delta_x[0],ff); right_y(p):=y_coord(p)+take_fraction(delta_y[0],ff); end; if lt=unity then begin if delta_x[0]>=0 then left_x(q):=x_coord(q)-((delta_x[0]+1) div 3) else left_x(q):=x_coord(q)-((delta_x[0]-1) div 3); if delta_y[0]>=0 then left_y(q):=y_coord(q)-((delta_y[0]+1) div 3) else left_y(q):=y_coord(q)-((delta_y[0]-1) div 3); end else begin ff:=make_fraction(unity,3*lt); {$\beta/3$} left_x(q):=x_coord(q)-take_fraction(delta_x[0],ff); left_y(q):=y_coord(q)-take_fraction(delta_y[0],ff); end; return; end @* \[19] Measuring paths. \MP's \&{llcorner}, \&{lrcorner}, \&{ulcorner}, and \&{urcorner} operators allow the user to measure the bounding box of anything that can go into a picture. It's easy to get rough bounds on the $x$ and $y$ extent of a path by just finding the bounding box of the knots and the control points. We need a more accurate version of the bounding box, but we can still use the easy estimate to save time by focusing on the interesting parts of the path. @ Computing an accurate bounding box involves a theme that will come up again and again. Given a Bernshte{\u\i}n polynomial @^Bernshte{\u\i}n, Serge{\u\i} Natanovich@> $$B(z_0,z_1,\ldots,z_n;t)=\sum_k{n\choose k}t^k(1-t)^{n-k}z_k,$$ we can conveniently bisect its range as follows: \smallskip \textindent{1)} Let $z_k^{(0)}=z_k$, for |0<=k<=n|. \smallskip \textindent{2)} Let $z_k^{(j+1)}={1\over2}(z_k^{(j)}+z\k^{(j)})$, for |0<=k=0 then begin if b>=0 then if c>0 then no_crossing else if (a=0)and(b=0) then no_crossing else one_crossing; if a=0 then zero_crossing; end else if a=0 then if b<=0 then zero_crossing; @; exit:end; @ The general bisection method is quite simple when $n=2$, hence |crossing_point| does not take much time. At each stage in the recursion we have a subinterval defined by |l| and~|j| such that $B(a,b,c;2^{-l}(j+t))=B(x_0,x_1,x_2;t)$, and we want to ``zero in'' on the subinterval where $x_0\G0$ and $\min(x_1,x_2)<0$. It is convenient for purposes of calculation to combine the values of |l| and~|j| in a single variable $d=2^l+j$, because the operation of bisection then corresponds simply to doubling $d$ and possibly adding~1. Furthermore it proves to be convenient to modify our previous conventions for bisection slightly, maintaining the variables $X_0=2^lx_0$, $X_1=2^l(x_0-x_1)$, and $X_2=2^l(x_1-x_2)$. With these variables the conditions $x_0\ge0$ and $\min(x_1,x_2)<0$ are equivalent to $\max(X_1,X_1+X_2)>X_0\ge0$. The following code maintains the invariant relations $0\L|x0|<\max(|x1|,|x1|+|x2|)$, $\vert|x1|\vert<2^{30}$, $\vert|x2|\vert<2^{30}$; it has been constructed in such a way that no arithmetic overflow will occur if the inputs satisfy $a<2^{30}$, $\vert a-b\vert<2^{30}$, and $\vert b-c\vert<2^{30}$. @= d:=1; x0:=a; x1:=a-b; x2:=b-c; repeat x:=half(x1+x2); if x1-x0>x0 then begin x2:=x; double(x0); double(d); end else begin xx:=x1+x-x0; if xx>x0 then begin x2:=x; double(x0); double(d); end else begin x0:=x0-xx; if x<=x0 then if x+x2<=x0 then no_crossing; x1:=x; d:=d+d+1; end; end; until d>=fraction_one; crossing_point:=d-fraction_one @ Here is a routine that computes the $x$ or $y$ coordinate of the point on a cubic corresponding to the |fraction| value~|t|. It is convenient to define a \.{WEB} macro |t_of_the_way| such that |t_of_the_way(a)(b)| expands to |a-(a-b)*t|, i.e., to |t[a,b]|. @d t_of_the_way_end(#)==#,t@=)@> @d t_of_the_way(#)==#-take_fraction@=(@>#-t_of_the_way_end @p function eval_cubic(@!p,@!q:pointer;t:fraction):scaled; var @!x1,@!x2,@!x3:scaled; {intermediate values} begin x1:=t_of_the_way(knot_coord(p))(right_coord(p)); x2:=t_of_the_way(right_coord(p))(left_coord(q)); x3:=t_of_the_way(left_coord(q))(knot_coord(q));@/ x1:=t_of_the_way(x1)(x2); x2:=t_of_the_way(x2)(x3); eval_cubic:=t_of_the_way(x1)(x2); end; @ The actual bounding box information is stored in global variables. Since it is convenient to address the $x$ and $y$ information separately, we define arrays indexed by |x_code..y_code| and use macros to give them more convenient names. @d x_code=0 {index for |minx| and |maxx|} @d y_code=1 {index for |miny| and |maxy|} @d minx==bbmin[x_code] @d maxx==bbmax[x_code] @d miny==bbmin[y_code] @d maxy==bbmax[y_code] @= @!bbmin,@!bbmax:array[x_code..y_code] of scaled; {the result of procedures that compute bounding box information} @ Now we're ready for the key part of the bounding box computation. The |bound_cubic| procedure updates |bbmin[c]| and |bbmax[c]| based on $$B(\hbox{|knot_coord(p)|}, \hbox{|right_coord(p)|}, \hbox{|left_coord(q)|}, \hbox{|knot_coord(q)|};t) $$ for $0; @; if wavy then begin del1:=right_coord(p)-knot_coord(p); del2:=left_coord(q)-right_coord(p); del3:=knot_coord(q)-left_coord(q); @; if del<0 then begin negate(del1); negate(del2); negate(del3); end; t:=crossing_point(del1,del2,del3); if t; end; end; @ @= if xbbmax[c] then bbmax[c]:=x @ @= wavy:=true; if bbmin[c]<=right_coord(p) then if right_coord(p)<=bbmax[c] then if bbmin[c]<=left_coord(q) then if left_coord(q)<=bbmax[c] then wavy:=false @ If |del1=del2=del3=0|, it's impossible to obey the title of this section. We just set |del=0| in that case. @= if del1<>0 then del:=del1 else if del2<>0 then del:=del2 else del:=del3; if del<>0 then begin dmax:=abs(del1); if abs(del2)>dmax then dmax:=abs(del2); if abs(del3)>dmax then dmax:=abs(del3); while dmax= begin x:=eval_cubic(p,q,t); @; del2:=t_of_the_way(del2)(del3); {now |0,del2,del3| represent the derivative on the remaining interval} if del2>0 then del2:=0; tt:=crossing_point(0,-del2,-del3); if tt; end @ @= begin x:=eval_cubic(p,q,t_of_the_way(tt)(fraction_one)); @; end @ Finding the bounding box of a path is basically a matter of applying |bound_cubic| twice for each pair of adjacent knots. @p procedure path_bbox(@!h:pointer); label exit; var @!p,@!q:pointer; {a pair of adjacent knots} begin minx:=x_coord(h); miny:=y_coord(h); maxx:=minx; maxy:=miny;@/ p:=h; repeat if right_type(p)=endpoint then return; q:=link(p);@/ bound_cubic(x_loc(p),x_loc(q),x_code); bound_cubic(y_loc(p),y_loc(q),y_code); p:=q; until p=h; exit:end; @ Another important way to measure a path is to find its arc length. This is best done by using the general bisection algorithm to subdivide the path until obtaining ``well behaved'' subpaths whose arc lengths can be approximated by simple means. Since the arc length is the integral with respect to time of the magnitude of the velocity, it is natural to use Simpson's rule for the approximation. @^Simpson's rule@> If $\dot B(t)$ is the spline velocity, Simpson's rule gives $$ \vb\dot B(0)\vb + 4\vb\dot B({1\over2})\vb + \vb\dot B(1)\vb \over 6 $$ for the arc length of a path of length~1. For a cubic spline $B(z_0,z_1,z_2,z_3;t)$, the time derivative $\dot B(t)$ is $3B(dz_0,dz_1,dz_2;t)$, where $dz_i=z_{i+1}-z_i$. Hence the arc length approximation is $$ {\vb dz_0\vb \over 2} + 2\vb dz_{02}\vb + {\vb dz_2\vb \over 2}, $$ where $$ dz_{02}={1\over2}\left({dz_0+dz_1\over 2}+{dz_1+dz_2\over 2}\right)$$ is the result of the bisection algorithm. @ The remaining problem is how to decide when a subpath is ``well behaved.'' This could be done via the theoretical error bound for Simpson's rule, @^Simpson's rule@> but this is impractical because it requires an estimate of the fourth derivative of the quantity being integrated. It is much easier to just perform a bisection step and see how much the arc length estimate changes. Since the error for Simpson's rule is proportional to the fourth power of the sample spacing, the remaining error is typically about $1\over16$ of the amount of the change. We say ``typically'' because the error has a pseudo-random behavior that could cause the two estimates to agree when each contain large errors. To protect against disasters such as undetected cusps, the bisection process should always continue until all the $dz_i$ vectors belong to a single $90^\circ$ sector. This ensures that no point on the spline can have velocity less than 70\% of the minimum of $\vb dz_0\vb$, $\vb dz_1\vb$ and $\vb dz_2\vb$. If such a spline happens to produce an erroneous arc length estimate that is little changed by bisection, the amount of the error is likely to be fairly small. We will try to arrange things so that freak accidents of this type do not destroy the inverse relationship between the \&{arclength} and \&{arctime} operations. @:arclength_}{\&{arclength} primitive@> @:arctime_}{\&{arctime} primitive@> @ The \&{arclength} and \&{arctime} operations are both based on a recursive @^recursion@> function that finds the arc length of a cubic spline given $dz_0$, $dz_1$, $dz_2$. This |arc_test| routine also takes an arc length goal |a_goal| and returns the time when the arc length reaches |a_goal| if there is such a time. Thus the return value is either an arc length less than |a_goal| or, if the arc length would be at least |a_goal|, it returns a time value decreased by |two|. This allows the caller to use the sign of the result to distinguish between arc lengths and time values. On certain types of overflow, it is possible for |a_goal| and the result of |arc_test| both to be |el_gordo|. Otherwise, the result is always less than |a_goal|. Rather than halving the control point coordinates on each recursive call to |arc_test|, it is better to keep them proportional to velocity on the original curve and halve the results instead. This means that recursive calls can potentially use larger error tolerances in their arc length estimates. How much larger depends on to what extent the errors behave as though they are independent of each other. To save computing time, we use optimistic assumptions and increase the tolerance by a factor of about $\sqrt2$ for each recursive call. In addition to the tolerance parameter, |arc_test| should also have parameters for ${1\over3}\vb\dot B(0)\vb$, ${2\over3}\vb\dot B({1\over2})\vb$, and ${1\over3}\vb\dot B(1)\vb$. These quantities are relatively expensive to compute and they are needed in different instances of |arc_test|. @p @t\4@>@@; function arc_test(@!dx0, @!dy0, @!dx1, @!dy1, @!dx2, @!dy2, @!v0, @!v02, @!v2, @!a_goal, @!tol:scaled): scaled; label exit; var simple: boolean; {are the control points confined to a $90^\circ$ sector?} @!dx01, @!dy01, @!dx12, @!dy12, @!dx02, @!dy02: scaled; {bisection results} @!v002, @!v022: scaled; {twice the velocity magnitudes at $t={1\over4}$ and $t={3\over4}$} @!arc: scaled; {best arc length estimate before recursion} @@; begin @; @; @; if simple and (abs(arc-v02-halfp(v0+v2)) <= tol) then if arc < a_goal then @+arc_test := arc else @ else @; exit:end; @ The |tol| value should by multiplied by $\sqrt 2$ before making recursive calls, but $1.5$ is an adequate approximation. It is best to avoid using |make_fraction| in this inner loop. @^inner loop@> @= begin @; tol := tol + halfp(tol); a := arc_test(dx0,dy0, dx01,dy01, dx02,dy02, v0, v002, halfp(v02), a_new, tol); if a<0 then @+arc_test := -halfp(two-a) else begin @; b := arc_test(dx02,dy02, dx12,dy12, dx2,dy2, halfp(v02), v022, v2, a_new, tol); if b<0 then @+arc_test := -halfp(-b) - half_unit else arc_test := a + half(b-a); end; end @ @= @!a, @!b: scaled; {results of recursive calls} @!a_new, @!a_aux: scaled; {the sum of these gives the |a_goal|} @ @= a_aux := el_gordo - a_goal; if a_goal > a_aux then begin a_aux := a_goal - a_aux; a_new := el_gordo; end else begin a_new := a_goal + a_goal; a_aux := 0; end @ There is no need to maintain |a_aux| at this point so we use it as a temporary to force the additions and subtractions to be done in an order that avoids overflow. @= if a > a_aux then begin a_aux := a_aux - a; a_new := a_new + a_aux; end @ This code assumes all {\it dx} and {\it dy} variables have magnitude less than |fraction_four|. To simplify the rest of the |arc_test| routine, we strengthen this assumption by requiring the norm of each $({\it dx},{\it dy})$ pair to obey this bound. Note that recursive calls will maintain this invariant. @= dx01 := half(dx0 + dx1); dx12 := half(dx1 + dx2); dx02 := half(dx01 + dx12);@/ dy01 := half(dy0 + dy1); dy12 := half(dy1 + dy2); dy02 := half(dy01 + dy12) @ We should be careful to keep |arc= v002 := pyth_add(dx01+half(dx0+dx02), dy01+half(dy0+dy02)); v022 := pyth_add(dx12+half(dx02+dx2), dy12+half(dy02+dy2)); tmp := halfp(v02+2); arc1 := v002 + half(halfp(v0+tmp) - v002); arc := v022 + half(halfp(v2+tmp) - v022); if (arc < el_gordo-arc1) then @+arc := arc+arc1 else begin arith_error := true; if a_goal=el_gordo then @+arc_test := el_gordo else arc_test := -two; return; end @ @= tmp, tmp2: scaled; {all purpose temporary registers} arc1: scaled; {arc length estimate for the first half} @ @= simple := (dx0>=0) and (dx1>=0) and (dx2>=0) or@| (dx0<=0) and (dx1<=0) and (dx2<=0); if simple then simple := (dy0>=0) and (dy1>=0) and (dy2>=0) or@| (dy0<=0) and (dy1<=0) and (dy2<=0); if not simple then begin simple := (dx0>=dy0) and (dx1>=dy1) and (dx2>=dy2) or@| (dx0<=dy0) and (dx1<=dy1) and (dx2<=dy2); if simple then simple := (-dx0>=dy0) and (-dx1>=dy1) and (-dx2>=dy2) or@| (-dx0<=dy0) and (-dx1<=dy1) and (-dx2<=dy2); end @ Since Simpson's rule is based on approximating the integrand by a parabola, @^Simpson's rule@> it is appropriate to use the same approximation to decide when the integral reaches the intermediate value |a_goal|. At this point $$\eqalign{ {\vb\dot B(0)\vb\over 3} &= \hbox{|v0|}, \qquad {\vb\dot B({1\over4})\vb\over 3} = {\hbox{|v002|}\over 2}, \qquad {\vb\dot B({1\over2})\vb\over 3} = {\hbox{|v02|}\over 2}, \cr {\vb\dot B({3\over4})\vb\over 3} &= {\hbox{|v022|}\over 2}, \qquad {\vb\dot B(1)\vb\over 3} = \hbox{|v2|} \cr } $$ and $$ {\vb\dot B(t)\vb\over 3} \approx \cases{B\left(\hbox{|v0|}, \hbox{|v002|}-{1\over 2}\hbox{|v0|}-{1\over 4}\hbox{|v02|}, {1\over 2}\hbox{|v02|}; 2t \right)& if $t\le{1\over 2}$\cr B\left({1\over 2}\hbox{|v02|}, \hbox{|v022|}-{1\over 4}\hbox{|v02|}-{1\over 2}\hbox{|v2|}, \hbox{|v2|}; 2t-1 \right)& if $t\ge{1\over 2}$.\cr} \eqno (*) $$ We can integrate $\vb\dot B(t)\vb$ by using $$\int 3B(a,b,c;\tau)\,dt = {B(0,a,a+b,a+b+c;\tau) + {\rm constant} \over {d\tau\over dt}}. $$ This construction allows us to find the time when the arc length reaches |a_goal| by solving a cubic equation of the form $$ B(0,a,a+b,a+b+c;\tau) = x, $$ where $\tau$ is $2t$ or $2t+1$, $x$ is |a_goal| or |a_goal-arc1|, and $a$, $b$, and $c$ are the Bernshte{\u\i}n coefficients from $(*)$ divided by @^Bernshte{\u\i}n, Serge{\u\i} Natanovich@> $d\tau\over dt$. We shall define a function |solve_rising_cubic| that finds $\tau$ given $a$, $b$, $c$, and $x$. @= begin tmp := (v02 + 2) div 4; if a_goal<=arc1 then begin tmp2 := halfp(v0); arc_test := halfp(solve_rising_cubic(tmp2, arc1-tmp2-tmp, tmp, a_goal)) - two; end else begin tmp2 := halfp(v2); arc_test := (half_unit - two) +@| halfp(solve_rising_cubic(tmp, arc-arc1-tmp-tmp2, tmp2, a_goal-arc1)); end; end @ Here is the |solve_rising_cubic| routine that finds the time~$t$ when $$ B(0, a, a+b, a+b+c; t) = x. $$ This routine is based on |crossing_point| but is simplified by the assumptions that $B(a,b,c;t)\ge0$ for $0\le t\le1$ and that |0<=x<=a+b+c|. If rounding error causes this condition to be violated slightly, we just ignore it and proceed with binary search. This finds a time when the function value reaches |x| and the slope is positive. @= function solve_rising_cubic(@!a, @!b, @!c, @!x: scaled): scaled; var @!ab, @!bc, @!ac: scaled; {bisection results} @!t: integer; {$2^k+q$ where unscaled answer is in $[q2^{-k},(q+1)2^{-k})$} @!xx: integer; {temporary for updating |x|} begin if (a<0) or (c<0) then confusion("rising?"); @:this can't happen rising?}{\quad rising?@> if x<=0 then solve_rising_cubic := 0 else if x >= a+b+c then solve_rising_cubic := unity else begin t := 1; @; repeat double(t); @; xx := x - a - ab - ac; if xx < -x then begin double(x); b:=ab; c:=ac; end else begin x := x + xx; a:=ac; b:=bc; t := t+1; end; until t >= unity; solve_rising_cubic := t - unity; end; end; @ @= ab := half(a+b); bc := half(b+c); ac := half(ab + bc) @ @d one_third_el_gordo==@'5252525252 {upper bound on |a|, |b|, and |c|} @= while (a>one_third_el_gordo) or@| (b>one_third_el_gordo) or@| (c>one_third_el_gordo) do begin a := halfp(a); b := half(b); c := halfp(c); x := halfp(x); end @ It is convenient to have a simpler interface to |arc_test| that requires no unnecessary arguments and ensures that each $({\it dx},{\it dy})$ pair has length less than |fraction_four|. @d arc_tol = 16 {quit when change in arc length estimate reaches this} @p function do_arc_test(@!dx0, @!dy0, @!dx1, @!dy1, @!dx2, @!dy2, @!a_goal: scaled): scaled; var @!v0, @!v1, @!v2: scaled; {length of each $({\it dx},{\it dy})$ pair} @!v02: scaled; {twice the norm of the quadratic at $t={1\over2}$} begin v0 := pyth_add(dx0,dy0); v1 := pyth_add(dx1,dy1); v2 := pyth_add(dx2,dy2); if (v0>=fraction_four) or (v1>=fraction_four) or (v2>=fraction_four) then begin arith_error := true; if a_goal=el_gordo then @+do_arc_test := el_gordo else do_arc_test := -two; end else begin v02 := pyth_add(dx1+half(dx0+dx2), dy1+half(dy0+dy2));@/ do_arc_test := arc_test(dx0,dy0, dx1,dy1, dx2,dy2,@| v0, v02, v2, a_goal, arc_tol); end; end; @ Now it is easy to find the arc length of an entire path. @p function get_arc_length(@!h: pointer): scaled; label done; var @!p, @!q: pointer; {for traversing the path} @!a, @!a_tot: scaled; {current and total arc lengths} begin a_tot := 0; p := h; while right_type(p)<>endpoint do begin q := link(p); a := do_arc_test(right_x(p)-x_coord(p), right_y(p)-y_coord(p),@| left_x(q)-right_x(p), left_y(q)-right_y(p),@| x_coord(q)-left_x(q), y_coord(q)-left_y(q), el_gordo); a_tot := slow_add(a, a_tot); if q=h then goto done @+else p:=q; end; done:check_arith; get_arc_length := a_tot; end; @ The inverse operation of finding the time on a path~|h| when the arc length reaches some value |arc0| can also be accomplished via |do_arc_test|. Some care is required to handle very large times or negative times on cyclic paths. For non-cyclic paths, |arc0| values that are negative or too large cause |get_arc_time| to return 0 or the length of path~|h|. If |arc0| is greater than the arc length of a cyclic path~|h|, the result is a time value greater than the length of the path. Since it could be much greater, we must be prepared to compute the arc length of path~|h| and divide this into |arc0| to find how many multiples of the length of path~|h| to add. @p function get_arc_time(@!h: pointer; @!arc0:scaled): scaled; label done; var @!p, @!q: pointer; {for traversing the path} @!t_tot: scaled; {accumulator for the result} @!t: scaled; {the result of |do_arc_test|} @!arc:scaled; {portion of |arc0| not used up so far} @!n: integer; {number of extra times to go around the cycle} begin if arc0<0 then @; if arc0=el_gordo then decr(arc0); t_tot := 0; arc := arc0; p := h; while (right_type(p)<>endpoint) and (arc>0) do begin q := link(p); t := do_arc_test(right_x(p)-x_coord(p), right_y(p)-y_coord(p),@| left_x(q)-right_x(p), left_y(q)-right_y(p),@| x_coord(q)-left_x(q), y_coord(q)-left_y(q), arc); @; if q=h then @; p := q; end; done: check_arith; get_arc_time := t_tot; end; @ @= if t<0 then begin t_tot := t_tot + t + two; arc := 0; end else begin t_tot := t_tot + unity; arc := arc - t; end @ @= begin if left_type(h)=endpoint then t_tot:=0 else begin p := htap_ypoc(h); t_tot := -get_arc_time(p, -arc0); toss_knot_list(p); end; goto done; end @ @= if arc>0 then begin n := arc div (arc0 - arc); arc := arc - n*(arc0 - arc); if t_tot > el_gordo div (n+1) then begin arith_error := true; t_tot := el_gordo; goto done; end; t_tot := (n + 1)*t_tot; end @* \[20] Data structures for pens. A Pen in \MP\ can be either elliptical or polygonal. Elliptical pens result in \ps\ \&{stroke} commands, while anything drawn with a polygonal pen is @:stroke}{\&{stroke} command@> converted into an area fill as described in the next part of this program. The mathematics behind this process is based on simple aspects of the theory of tracings developed by Leo Guibas, Lyle Ramshaw, and Jorge Stolfi [``A kinematic framework for computational geometry,'' Proc.\ IEEE Symp.\ Foundations of Computer Science {\bf 24} (1983), 100--111]. Polygonal pens are created from paths via \MP's \&{makepen} primitive. @:makepen_}{\&{makepen} primitive@> This path representation is almost sufficient for our purposes except that a pen path should always be a convex polygon with the vertices in counter-clockwise order. Since we will need to scan pen polygons both forward and backward, a pen should be represented as a doubly linked ring of knot nodes. There is room for the extra back pointer because we do not need the |left_type| or |right_type| fields. In fact, we don't need the |left_x|, |left_y|, |right_x|, or |right_y| fields either but we leave these alone so that certain procedures can operate on both pens and paths. In particular, pens can be copied using |copy_path| and recycled using |toss_knot_list|. @d knil==info {this replaces the |left_type| and |right_type| fields in a pen knot} @ The |make_pen| procedure turns a path into a pen by initializing the |knil| pointers and making sure the knots form a convex polygon. Thus each cubic in the given path becomes a straight line and the control points are ignored. If the path is not cyclic, the ends are connected by a straight line. @d copy_pen(#)==make_pen(copy_path(#),false) @p @@; function make_pen(h:pointer;@!need_hull:boolean):pointer; var @!p,@!q:pointer; {two consecutive knots} begin q:=h; repeat p:=q; q:=link(q); knil(q):=p; until q=h; if need_hull then begin h:=convex_hull(h); @; end; make_pen:=h; end; @ The only information required about an elliptical pen is the overall transformation that has been applied to the original \&{pencircle}. @:pencircle_}{\&{pencircle} primitive@> Since it suffices to keep track of how the three points $(0,0)$, $(1,0)$, and $(0,1)$ are transformed, an elliptical pen can be stored in a single knot node and transformed as if it were a path. @d pen_is_elliptical(#)==(#=link(#)) @p function get_pen_circle(@!diam:scaled):pointer; var @!h:pointer; {the knot node to return} begin h:=get_node(knot_node_size); link(h):=h; knil(h):=h;@/ x_coord(h):=0; y_coord(h):=0;@/ left_x(h):=diam; left_y(h):=0;@/ right_x(h):=0; right_y(h):=diam;@/ get_pen_circle:=h; end; @ If the polygon being returned by |make_pen| has only one vertex, it will be interpreted as an elliptical pen. This is no problem since a degenerate polygon can equally well be thought of as a degenerate ellipse. We need only initialize the |left_x|, |left_y|, |right_x|, and |right_y| fields. @= if pen_is_elliptical(h) then begin left_x(h):=x_coord(h); left_y(h):=y_coord(h);@/ right_x(h):=x_coord(h); right_y(h):=y_coord(h); end @ We have to cheat a little here but most operations on pens only use the first three words in each knot node. @^data structure assumptions@> @= x_coord(test_pen):=-half_unit; y_coord(test_pen):=0;@/ x_coord(test_pen+3):=half_unit; y_coord(test_pen+3):=0;@/ x_coord(test_pen+6):=0; y_coord(test_pen+6):=unity;@/ link(test_pen):=test_pen+3; link(test_pen+3):=test_pen+6; link(test_pen+6):=test_pen; knil(test_pen):=test_pen+6; knil(test_pen+3):=test_pen; knil(test_pen+6):=test_pen+3 @ Printing a polygonal pen is very much like printing a path @= procedure pr_pen(@!h:pointer); label done; var @!p,@!q:pointer; {for list traversal} begin if pen_is_elliptical(h) then @ else begin p:=h; repeat print_two(x_coord(p),y_coord(p)); print_nl(" .. "); @; until p=h; print("cycle"); end; done:end; @ @= q:=link(p); if (q=null) or (knil(q)<>p) then begin print_nl("???"); goto done; {this won't happen} @.???@> end; p:=q @ @= begin print("pencircle transformed ("); print_scaled(x_coord(h)); print_char(","); print_scaled(y_coord(h));@/ print_char(","); print_scaled(left_x(h)-x_coord(h)); print_char(","); print_scaled(right_x(h)-x_coord(h)); print_char(","); print_scaled(left_y(h)-y_coord(h));@/ print_char(","); print_scaled(right_y(h)-y_coord(h));@/ print_char(")"); end @ Here us another version of |pr_pen| that prints the pen as a diagnostic message. @= procedure print_pen(@!h:pointer;@!s:str_number;@!nuline:boolean); begin print_diagnostic("Pen",s,nuline); print_ln; @.Pen at line...@> pr_pen(h); end_diagnostic(true); end; @ Making a polygonal pen into a path involves restoring the |left_type| and |right_type| fields and setting the control points so as to make a polygonal path. @p procedure make_path(@!h:pointer); var @!p:pointer; {for traversing the knot list} @!k:small_number; {a loop counter} @@; begin if pen_is_elliptical(h) then @ else begin p:=h; repeat left_type(p):=explicit; right_type(p):=explicit;@/ @;@/ p:=link(p); until p=h; end; end; @ @= left_x(p):=x_coord(p); left_y(p):=y_coord(p);@/ right_x(p):=x_coord(p); right_y(p):=y_coord(p) @ We need an eight knot path to get a good approximation to an ellipse. @= begin @; p:=h; for k:=0 to 7 do begin @; if k=7 then link(p):=h @+else link(p):=get_node(knot_node_size); p:=link(p); end; end @ @= center_x:=x_coord(h); center_y:=y_coord(h);@/ width_x:=left_x(h)-center_x; width_y:=left_y(h)-center_y;@/ height_x:=right_x(h)-center_x; height_y:=right_y(h)-center_y @ @= @!center_x,@!center_y:scaled; {translation parameters for an elliptical pen} @!width_x,@!width_y:scaled; {the effect of a unit change in $x$} @!height_x,@!height_y:scaled; {the effect of a unit change in $y$} @!dx,@!dy:scaled; {the vector from knot |p| to its right control point} @!kk:integer; {|k| advanced $270^\circ$ around the ring (cf. $\sin\theta=\cos(\theta+270)$)} @ The only tricky thing here are the tables |half_cos| and |d_cos| used to find the point $k/8$ of the way around the circle and the direction vector to use there. @= kk:=(k+6)mod 8;@/ x_coord(p):=center_x+take_fraction(half_cos[k],width_x) +take_fraction(half_cos[kk],height_x); y_coord(p):=center_y+take_fraction(half_cos[k],width_y) +take_fraction(half_cos[kk],height_y); dx:=-take_fraction(d_cos[kk],width_x)+take_fraction(d_cos[k],height_x); dy:=-take_fraction(d_cos[kk],width_y)+take_fraction(d_cos[k],height_y); right_x(p):=x_coord(p)+dx; right_y(p):=y_coord(p)+dy;@/ left_x(p):=x_coord(p)-dx; left_y(p):=y_coord(p)-dy;@/ left_type(p):=explicit; right_type(p):=explicit @ @= half_cos:array[0..7] of fraction; {${1\over2}\cos(45k)$} d_cos:array[0..7] of fraction; {a magic constant times $\cos(45k)$} @ The magic constant for |d_cos| is the distance between $({1\over2},0)$ and $({1\over4}\sqrt2,{1\over4}\sqrt2)$ times the result of the |velocity| function for $\theta=\phi=22.5^\circ$. This comes out to be $$ d = {\sqrt{2-\sqrt2}\over 3+3\cos22.5^\circ} \approx 0.132608244919772. $$ @= half_cos[0]:=fraction_half; half_cos[1]:=94906266; {$2^{26}\sqrt2\approx94906265.62$} half_cos[2]:=0;@/ d_cos[0]:=35596755; {$2^{28}d\approx35596754.69$} d_cos[1]:=25170707; {$2^{27}\sqrt2\,d\approx25170706.63$} d_cos[2]:=0; for k:=3 to 4 do begin half_cos[k]:=-half_cos[4-k]; d_cos[k]:=-d_cos[4-k]; end; for k:=5 to 7 do begin half_cos[k]:=half_cos[8-k]; d_cos[k]:=d_cos[8-k]; end; @ The |convex_hull| function forces a pen polygon to be convex when it is returned by |make_pen| and after any subsequent transformation where rounding error might allow the convexity to be lost. The convex hull algorithm used here is described by F.~P. Preparata and M.~I. Shamos [{\sl Computational Geometry}, Springer-Verlag, 1985]. @= @@; function convex_hull(@!h:pointer):pointer; {Make a polygonal pen convex} label done1,done2,done3; var @!l,@!r:pointer; {the leftmost and rightmost knots} @!p,@!q:pointer; {knots being scanned} @!s:pointer; {the starting point for an upcoming scan} @!dx,@!dy:scaled; {a temporary pointer} begin if pen_is_elliptical(h) then convex_hull:=h else begin @; @; if l<>r then begin s:=link(r); @; @; @; @; end; if l<>link(l) then @; convex_hull:=l; end; end; @ All comparisons are done primarily on $x$ and secondarily on $y$. @= l:=h; p:=link(h); while p<>h do begin if x_coord(p)<=x_coord(l) then if (x_coord(p)= r:=h; p:=link(h); while p<>h do begin if x_coord(p)>=x_coord(r) then if (x_coord(p)>x_coord(r)) or (y_coord(p)>y_coord(r)) then r:=p; p:=link(p); end @ @= dx:=x_coord(r)-x_coord(l); dy:=y_coord(r)-y_coord(l); p:=link(l); while p<>r do begin q:=link(p); if ab_vs_cd(dx,y_coord(p)-y_coord(l),dy,x_coord(p)-x_coord(l))>0 then move_knot(p,r); p:=q; end @ The |move_knot| procedure removes |p| from a doubly linked list and inserts it after |q|. @ @= procedure move_knot(@!p,@!q:pointer); begin link(knil(p)):=link(p); knil(link(p)):=knil(p);@/ knil(p):=q; link(p):=link(q); link(q):=p; knil(link(p)):=p; end; @ @= p:=s; while p<>l do begin q:=link(p); if ab_vs_cd(dx,y_coord(p)-y_coord(l),dy,x_coord(p)-x_coord(l))<0 then move_knot(p,l); p:=q; end @ The list is likely to be in order already so we just do linear insertions. Secondary comparisons on $y$ ensure that the sort is consistent with the choice of |l| and |r|. @= p:=link(l); while p<>r do begin q:=knil(p); while x_coord(q)>x_coord(p) do q:=knil(q); while x_coord(q)=x_coord(p) do if y_coord(q)>y_coord(p) then q:=knil(q) else goto done1; done1: if q=knil(p) then p:=link(p) else begin p:=link(p); move_knot(knil(p),q); end; end @ @= p:=link(r); while p<>l do begin q:=knil(p); while x_coord(q)= begin p:=l; q:=link(l); loop @+begin dx:=x_coord(q)-x_coord(p); dy:=y_coord(q)-y_coord(p); p:=q; q:=link(q); if p=l then goto done3; if p<>r then if ab_vs_cd(dx,y_coord(q)-y_coord(p),dy,x_coord(q)-x_coord(p))<=0 then @; end; done3: do_nothing; end @ @= begin s:=knil(p); free_node(p,knot_node_size); link(s):=q; knil(q):=s; if s=l then p:=s else begin p:=knil(s); q:=s; end; end @ The |find_offset| procedure sets global variables |(cur_x,cur_y)| to the offset associated with the given direction |(x,y)|. If two different offsets apply, it chooses one of them. @p procedure find_offset(@!x,@!y:scaled;@!h:pointer); var @!p,@!q:pointer; {consecutive knots} @!wx,@!wy,@!hx,@!hy:scaled; {the transformation matrix for an elliptical pen} @!xx,@!yy:fraction; {untransformed offset for an elliptical pen} @!d:fraction; {a temporary register} begin if pen_is_elliptical(h) then @ else begin q:=h; repeat p:=q; q:=link(q); until ab_vs_cd(x_coord(q)-x_coord(p),y, y_coord(q)-y_coord(p),x)>=0; repeat p:=q; q:=link(q); until ab_vs_cd(x_coord(q)-x_coord(p),y, y_coord(q)-y_coord(p),x)<=0; cur_x:=x_coord(p); cur_y:=y_coord(p); end; end; @ @= @!cur_x,@!cur_y:scaled; {all-purpose return value registers} @ @= if (x=0) and (y=0) then begin cur_x:=x_coord(h); cur_y:=y_coord(h); @+end else begin @; while (abs(x); cur_x:=x_coord(h)+take_fraction(xx,wx)+take_fraction(yy,hx); cur_y:=y_coord(h)+take_fraction(xx,wy)+take_fraction(yy,hy); end @ @= wx:=left_x(h)-x_coord(h); wy:=left_y(h)-y_coord(h); hx:=right_x(h)-x_coord(h); hy:=right_y(h)-y_coord(h) @ @= yy:=-(take_fraction(x,hy)+take_fraction(y,-hx));@/ xx:=take_fraction(x,-wy)+take_fraction(y,wx);@/ d:=pyth_add(xx,yy);@/ if d>0 then begin xx:=half(make_fraction(xx,d)); yy:=half(make_fraction(yy,d)); end @ Finding the bounding box of a pen is easy except if the pen is elliptical. But we can handle that case by just calling |find_offset| twice. The answer is stored in the global variables |minx|, |maxx|, |miny|, and |maxy|. @p procedure pen_bbox(@!h:pointer); var @!p:pointer; {for scanning the knot list} begin if pen_is_elliptical(h) then @ else begin minx:=x_coord(h); maxx:=minx; miny:=y_coord(h); maxy:=miny;@/ p:=link(h); while p<>h do begin if x_coord(p)maxx then maxx:=x_coord(p); if y_coord(p)>maxy then maxy:=y_coord(p); p:=link(p); end; end; end; @ @= begin find_offset(0,fraction_one,h); maxx:=cur_x; minx:=2*x_coord(h)-cur_x;@/ find_offset(-fraction_one,0,h); maxy:=cur_y; miny:=2*y_coord(h)-cur_y; end @* \[21] Edge structures. Now we come to \MP's internal scheme for representing pictures. The representation is very different from \MF's edge structures because \MP\ pictures contain \ps\ graphics objects instead of pixel images. However, the basic idea is somewhat similar in that shapes are represented via their boundaries. The main purpose of edge structures is to keep track of graphical objects until it is time to translate them into \ps. Since \MP\ does not need to know anything about an edge structure other than how to translate it into \ps\ and how to find its bounding box, edge structures can be just linked lists of graphical objects. \MP\ has no easy way to determine whether two such objects overlap, but it suffices to draw the first one first and let the second one overwrite it if necessary. @ Let's consider the types of graphical objects one at a time. First of all, a filled contour is represented by a six-word node. The first word contains |type| and |link| fields, and the next four words contain a pointer to a cyclic path and the value to use for \ps' \&{currentrgbcolor} parameter. If a pen is used for filling |pen_p|, |ljoin_val| and |miterlim_val| give the relevant information. @d path_p(#)==link(#+1) {a pointer to the path that needs filling} @d pen_p(#)==info(#+1) {a pointer to the pen to fill or stroke with} @d obj_red_loc(#)==#+2 {the first of three locations for the color} @d red_val(#)==mem[#+2].sc {the red component of the color in the range $0\ldots1$} @d green_val(#)==mem[#+3].sc {the green component of the color in the range $0\ldots1$} @d blue_val(#)==mem[#+4].sc {the blue component of the color in the range $0\ldots1$} @d ljoin_val(#)==name_type(#) {the value of \&{linejoin}} @:linejoin_}{\&{linejoin} primitive@> @d miterlim_val(#)==mem[#+5].sc {the value of \&{miterlimit}} @:miterlimit_}{\&{miterlimit} primitive@> @d obj_color_part(#)==mem[#+2-red_part].sc {interpret an object pointer that has been offset by |red_part..blue_part|} @d fill_node_size=6 @d fill_code=1 @p function new_fill_node(@!p: pointer): pointer; {make a fill node for cyclic path |p| and color black} var @!t:pointer; {the new node} begin t:=get_node(fill_node_size); type(t):=fill_code; path_p(t):=p; pen_p(t):=null; {|null| means don't use a pen} red_val(t):=0; green_val(t):=0; blue_val(t):=0; @; new_fill_node:=t; end; @ @= if internal[linejoin]>unity then ljoin_val(t):=2 else if internal[linejoin]>0 then ljoin_val(t):=1 else ljoin_val(t):=0; if internal[miterlimit] @d dash_scale(#)==mem[#+7].sc {dash lengths are scaled by this factor} @d stroked_node_size=8 @d stroked_code=2 @p function new_stroked_node(@!p:pointer): pointer; {make a stroked node for path |p| with |pen_p(p)| temporarily |null|} var @!t:pointer; {the new node} begin t:=get_node(stroked_node_size); type(t):=stroked_code; path_p(t):=p; pen_p(t):=null; dash_p(t):=null; dash_scale(t):=unity; red_val(t):=0; green_val(t):=0; blue_val(t):=0; @; if internal[linecap]>unity then lcap_val(t):=2 else if internal[linecap]>0 then lcap_val(t):=1 else lcap_val(t):=0; new_stroked_node:=t; end; @ When a dashed line is computed in a transformed coordinate system, the dash lengths get scaled like the pen shape and we need to compensate for this. Since there is no unique scale factor for an arbitrary transformation, we use the the square root of the determinant. The properties of the determinant make it easier to maintain the |dash_scale|. The computation is fairly straight-forward except for the initialization of the scale factor |s|. The factor of 64 is needed because |square_rt| scales its result by $2^8$ while we need $2^{14}$ to counteract the effect of |take_fraction|. @= function sqrt_det(a,b,c,d:scaled):scaled; var @!maxabs:scaled; {$max(|a|,|b|,|c|,|d|)$} @!s:integer; {amount by which the result of |square_rt| needs to be scaled} begin @; s:=64; while (maxabs1) do begin double(a); double(b); double(c); double(d);@/ double(maxabs); s:=halfp(s); end; sqrt_det:=s*square_rt(abs(take_fraction(a,d)-take_fraction(b,c))); end; @# function get_pen_scale(p:pointer):scaled; begin get_pen_scale:=sqrt_det( left_x(p)-x_coord(p), right_x(p)-x_coord(p),@/ left_y(p)-y_coord(p), right_y(p)-y_coord(p)); end; @ @= maxabs:=abs(a); if abs(b)>maxabs then maxabs:=abs(b); if abs(c)>maxabs then maxabs:=abs(c); if abs(d)>maxabs then maxabs:=abs(d) @ When a picture contains text, this is represented by a fourteen-word node where the color information and |type| and |link| fields are augmented by additional fields that describe the text and how it is transformed. The |path_p| and |pen_p| pointers are replaced by a number that identifies the font and a string number that gives the text to be displayed. The |width|, |height|, and |depth| fields give the dimensions of the text at its design size, and the remaining six words give a transformation to be applied to the text. The |new_text_node| function initializes everything to default values so that the text comes out black with its reference point at the origin. @d text_p(#)==link(#+1) {a string pointer for the text to display} @d font_n(#)==info(#+1) {the font number} @d width_val(#)==mem[#+5].sc {unscaled width of the text} @d height_val(#)==mem[#+6].sc {unscaled height of the text} @d depth_val(#)==mem[#+7].sc {unscaled depth of the text} @d text_tx_loc(#)==#+8 {the first of six locations for transformation parameters} @d tx_val(#)==mem[#+8].sc {$x$ shift amount} @d ty_val(#)==mem[#+9].sc {$y$ shift amount} @d txx_val(#)==mem[#+10].sc {|txx| transformation parameter} @d txy_val(#)==mem[#+11].sc {|txy| transformation parameter} @d tyx_val(#)==mem[#+12].sc {|tyx| transformation parameter} @d tyy_val(#)==mem[#+13].sc {|tyy| transformation parameter} @d text_trans_part(#)==mem[#+8-x_part].sc {interpret a text node ponter that has been offset by |x_part..yy_part|} @d text_node_size=14 @d text_code=3 @p @@; function new_text_node(f,s:str_number):pointer; {make a text node for font |f| and text string |s|} var @!t:pointer; {the new node} begin t:=get_node(text_node_size); type(t):=text_code; text_p(t):=s; font_n(t):=find_font(f); {this identifies the font} red_val(t):=0; green_val(t):=0; blue_val(t):=0; tx_val(t):=0; ty_val(t):=0; txx_val(t):=unity; txy_val(t):=0; tyx_val(t):=0; tyy_val(t):=unity; set_text_box(t); {this finds the bounding box} new_text_node:=t; end; @ The last two types of graphical objects that can occur in an edge structure are clipping paths and \&{setbounds} paths. These are slightly more difficult @:set_bounds_}{\&{setbounds} primitive@> to implement because we must keep track of exactly what is being clipped or bounded when pictures get merged together. For this reason, each clipping or \&{setbounds} operation is represented by a pair of nodes: first comes a two-word node whose |path_p| gives the relevant path, then there is the list of objects to clip or bound followed by a two-word node whose second word is unused. Using at least two words for each graphical object node allows them all to be allocated and deallocated similarly with a global array |gr_object_size| to give the size in words for each object type. @d start_clip_size=2 @d start_clip_code=4 {|type| of a node that starts clipping} @d start_bounds_size=2 @d start_bounds_code=5 {|type| of a node that gives a \&{setbounds} path} @d stop_clip_size=2 {the second word is not used here} @d stop_clip_code=6 {|type| of a node that stops clipping} @d stop_bounds_size=2 {the second word is not used here} @d stop_bounds_code=7 {|type| of a node that stops \&{setbounds}} @# @d stop_type(#)==(#+2) {matching |type| for |start_clip_code| or |start_bounds_code|} @d has_color(#)==(type(#)=start_clip_code) @d is_stop(#)==(type(#)>=stop_clip_code) @p function new_bounds_node(@!p:pointer; c:small_number):pointer; {make a node of type |c| where |p| is the clipping or \&{setbounds} path} var @!t:pointer; {the new node} begin t:=get_node(gr_object_size[c]); type(t):=c; path_p(t):=p; new_bounds_node:=t; end; @ We need an array to keep track of the sizes of graphical objects. @= gr_object_size: array[fill_code..stop_bounds_code] of small_number; @ @= gr_object_size[fill_code]:=fill_node_size; gr_object_size[stroked_code]:=stroked_node_size; gr_object_size[text_code]:=text_node_size; gr_object_size[start_clip_code]:=start_clip_size; gr_object_size[stop_clip_code]:=stop_clip_size; gr_object_size[start_bounds_code]:=start_bounds_size; gr_object_size[stop_bounds_code]:=stop_bounds_size; @ All the essential information in an edge structure is encoded as a linked list of graphical objects as we have just seen, but it is helpful to add some redundant information. A single edge structure might be used as a dash pattern many times, and it would be nice to avoid scanning the same structure repeatedly. Thus, an edge structure known to be a suitable dash pattern has a header that gives a list of dashes in a sorted order designed for rapid translation into \ps. Each dash is represented by a three-word node containing the initial and final $x$~coordinates as well as the usual |link| field. The |link| fields points to the dash node with the next higher $x$-coordinates and the final link points to a special location called |null_dash|. (There should be no overlap between dashes). Since the $y$~coordinate of the dash pattern is needed to determine the period of repetition, this needs to be stored in the edge header along with a pointer to the list of dash nodes. @d start_x(#)==mem[#+1].sc {the starting $x$~coordinate in a dash node} @d stop_x(#)==mem[#+2].sc {the ending $x$~coordinate in a dash node} @d dash_node_size=3 @d dash_list==link {in an edge header this points to the first dash node} @d dash_y(#)==mem[#+1].sc {$y$ value for the dash list in an edge header} @ It is also convenient for an edge header to contain the bounding box information needed by the \&{llcorner} and \&{urcorner} operators so that this does not have to be recomputed unnecessarily. This is done by adding fields for the $x$~and $y$ extremes as well as a pointer that indicates how far the bounding box computation has gotten. Thus if the user asks for the bounding box and then adds some more text to the picture before asking for more bounding box information, the second computation need only look at the additional text. When the bounding box has not been computed, the |bblast| pointer points to a dummy link at the head of the graphical object list while the |minx_val| and |miny_val| fields contain |el_gordo| and the |maxx_val| and |maxy_val| fields contain |-el_gordo|. Since the bounding box of pictures containing objects of type |start_bounds_code| depends on the value of \&{truecorners}, the bounding box @:true_corners_}{\&{truecorners} primitive@> data might not be valid for all values of this parameter. Hence, the |bbtype| field is needed to keep track of this. @d minx_val(#)==mem[#+2].sc @d miny_val(#)==mem[#+3].sc @d maxx_val(#)==mem[#+4].sc @d maxy_val(#)==mem[#+5].sc @d bblast(#)==link(#+6) {last item considered in bounding box computation} @d bbtype(#)==info(#+6) {tells how bounding box data depends on \&{truecorners}} @d dummy_loc(#)==#+7 {where the object list begins in an edge header} @d no_bounds=0 {|bbtype| value when bounding box data is valid for all \&{truecorners} values} @d bounds_set=1 {|bbtype| value when bounding box data is for \&{truecorners}${}\le 0$} @d bounds_unset=2 {|bbtype| value when bounding box data is for \&{truecorners}${}>0$} @p procedure init_bbox(@!h:pointer); {Initialize the bounding box information in edge structure |h|} begin bblast(h):=dummy_loc(h); bbtype(h):=no_bounds; minx_val(h):=el_gordo; miny_val(h):=el_gordo; maxx_val(h):=-el_gordo; maxy_val(h):=-el_gordo; end; @ The only other entries in an edge header are a reference count in the first word and a pointer to the tail of the object list in the last word. @d obj_tail(#)==info(#+7) {points to the last entry in the object list} @d edge_header_size=8 @p procedure init_edges(@!h:pointer); {initialize an edge header to null values} begin dash_list(h):=null_dash; obj_tail(h):=dummy_loc(h); link(dummy_loc(h)):=null; ref_count(h):=null; init_bbox(h); end; @ Here is how edge structures are deleted. The process can be recursive because of the need to dereference edge structures that are used as dash patterns. @^recursion@> @d add_edge_ref(#)==incr(ref_count(#)) @d delete_edge_ref(#)==if ref_count(#)=null then toss_edges(#) else decr(ref_count(#)) @= @@; procedure toss_edges(@!h:pointer); var @!p,@!q:pointer; {pointers that scan the list being recycled} @!r:pointer; {an edge structure that object |p| refers to} begin flush_dash_list(h); q:=link(dummy_loc(h)); while (q<>null) do begin p:=q; q:=link(q); r:=toss_gr_object(p); if r<>null then delete_edge_ref(r); end; free_node(h,edge_header_size); end; @ @= procedure flush_dash_list(h:pointer); var @!p,@!q:pointer; {pointers that scan the list being recycled} begin q:=dash_list(h); while q<>null_dash do begin p:=q; q:=link(q); free_node(p,dash_node_size); end; dash_list(h):=null_dash; end; @ @= function toss_gr_object(@!p:pointer):pointer; {returns an edge structure that needs to be dereferenced} var @!e:pointer; {the edge structure to return} begin e:=null; @; free_node(p,gr_object_size[type(p)]);@/ toss_gr_object:=e; end; @ @= case type(p) of fill_code: begin toss_knot_list(path_p(p)); if pen_p(p)<>null then toss_knot_list(pen_p(p)); end; stroked_code: begin toss_knot_list(path_p(p)); if pen_p(p)<>null then toss_knot_list(pen_p(p)); e:=dash_p(p); end; text_code: delete_str_ref(text_p(p)); start_clip_code,start_bounds_code: toss_knot_list(path_p(p)); stop_clip_code,stop_bounds_code: do_nothing; end; {there are no other cases} @ If we use |add_edge_ref| to ``copy'' edge structures, the real copying needs to be done before making a significant change to an edge structure. Much of the work is done in a separate routine |copy_objects| that copies a list of graphical objects into a new edge header. @p @@; function private_edges(h:pointer):pointer; {make a private copy of the edge structure headed by |h|} var @!hh:pointer; {the edge header for the new copy} @!p,@!pp: pointer; {pointers for copying the dash list} begin if ref_count(h)=null then private_edges:=h else begin decr(ref_count(h)); hh:=copy_objects(link(dummy_loc(h)),null); @; @; private_edges:=hh; end; end; @ Here we use the fact that |dash_list(hh)=link(hh)|. @^data structure assumptions@> @= pp:=hh; p:=dash_list(h); while (p<>null_dash) do begin link(pp):=get_node(dash_node_size); pp:=link(pp);@/ start_x(pp):=start_x(p); stop_x(pp):=stop_x(p); p:=link(p); end; link(pp):=null_dash; dash_y(hh):=dash_y(h) @ @= minx_val(hh):=minx_val(h); miny_val(hh):=miny_val(h); maxx_val(hh):=maxx_val(h); maxy_val(hh):=maxy_val(h);@/ bbtype(hh):=bbtype(h); p:=dummy_loc(h); pp:=dummy_loc(hh); while(p<>bblast(h)) do begin if p=null then confusion("bblast"); @:this can't happen bblast}{\quad bblast@> p:=link(p); pp:=link(pp); end; bblast(hh):=pp @ Here is the promised routine for copying graphical objects into a new edge structure. It starts copying at object~|p| and stops just before object~|q|. If |q| is null, it copies the entire sublist headed at |p|. The resulting edge structure requires further initialization by |init_bbox|. @= function copy_objects(p, q:pointer):pointer; var @!hh: pointer; {the new edge header} @!pp:pointer; {the last newly copied object} @!k:small_number; {temporary register} begin hh:=get_node(edge_header_size); dash_list(hh):=null_dash; ref_count(hh):=null;@/ pp:=dummy_loc(hh); while (p<>q) do @; obj_tail(hh):=pp; link(pp):=null; copy_objects:=hh; end; @ @= begin k:=gr_object_size[type(p)];@/ link(pp):=get_node(k); pp:=link(pp); while (k>0) do begin decr(k); mem[pp+k]:=mem[p+k]; @+end; @; p:=link(p); end @ @= case type(p) of start_clip_code,start_bounds_code: path_p(pp):=copy_path(path_p(p)); fill_code: begin path_p(pp):=copy_path(path_p(p)); if pen_p(p)<>null then pen_p(pp):=copy_pen(pen_p(p)); end; stroked_code: begin path_p(pp):=copy_path(path_p(p)); pen_p(pp):=copy_pen(pen_p(p)); if dash_p(p)<>null then add_edge_ref(dash_p(pp)); end; text_code: add_str_ref(text_p(pp)); stop_clip_code,stop_bounds_code: do_nothing; end {there are no other cases} @ Here is one way to find an acceptable value for the second argument to |copy_objects|. Given a non-null graphical object list, |skip_1component| skips past one picture component, where a ``picture component'' is a single graphical object, or a start bounds or start clip object and everything up through the matching stop bounds or stop clip object. The macro version avoids procedure call overhead and error handling: |skip_component(p)(e)| advances |p| unless |p| points to a stop bounds or stop clip node, in which case it executes |e| instead. @d skip_component(#)==if not is_start_or_stop(#) then #:=link(#) else if not is_stop(#) then #:=skip_1component(#) else skipc_end @d skipc_end(#)==# @p function skip_1component(p:pointer):pointer; var @!lev:integer; {current nesting level} begin lev:=0; repeat if is_start_or_stop(p) then if is_stop(p) then decr(lev) @+else incr(lev); p:=link(p); until lev=0; skip_1component:=p; end; @ Here is a diagnostic routine for printing an edge structure in symbolic form. @= @@; procedure print_edges(@!h:pointer;@!s:str_number;@!nuline:boolean); var @!p:pointer; {a graphical object to be printed} @!hh,@!pp:pointer; {temporary pointers} @!scf:scaled; {a scale factor for the dash pattern} @!ok_to_dash:boolean; {|false| for polygonal pen strokes} begin print_diagnostic("Edge structure",s,nuline); p:=dummy_loc(h); while link(p)<>null do begin p:=link(p); print_ln; case type(p) of @@; othercases begin print("[unknown object type!]"); end endcases;@/ end; print_nl("End edges"); if p<>obj_tail(h) then print("?"); @.End edges?@> end_diagnostic(true); end; @ @= fill_code: begin print("Filled contour "); print_obj_color(p); print_char(":"); print_ln; pr_path(path_p(p)); print_ln; if (pen_p(p)<>null) then begin @; print(" with pen"); print_ln; pr_pen(pen_p(p)); end; end; @ @= case ljoin_val(p) of 0:begin print("mitered joins limited "); print_scaled(miterlim_val(p)); end; 1:print("round joins"); 2:print("beveled joins"); othercases print("?? joins"); @.??@> endcases @ For stroked nodes, we need to print |lcap_val(p)| as well. @= case lcap_val(p) of 0:print("butt"); 1:print("round"); 2:print("square"); othercases print("??") @.??@> endcases; print(" ends, "); @ @ Here is a routine that prints the color of a graphical object if it isn't black (the default color). @= @@; procedure print_obj_color(@!p:pointer); begin if (red_val(p)>0) or (green_val(p)>0) or (blue_val(p)>0) then begin print("colored "); print_compact_node(obj_red_loc(p),3); end; end; @ We also need a procedure for printing consecutive scaled values as if they were a known big node. @= procedure print_compact_node(@!p:pointer;k:small_number); var @!q:pointer; {last location to print} begin q:=p+k-1; print_char("("); while p<=q do begin print_scaled(mem[p].sc); if p= stroked_code: begin print("Filled pen stroke "); print_obj_color(p); print_char(":"); print_ln; pr_path(path_p(p)); if dash_p(p)<>null then begin print_nl("dashed ("); @; end; print_ln; @; print(" with pen"); print_ln; if pen_p(p)=null then print("???") {shouldn't happen} @.???@> else pr_pen(pen_p(p)); end; @ Normally, the |dash_list| field in an edge header is set to |null_dash| when it is not known to define a suitable dash pattern. This is disallowed here because the |dash_p| field should never point to such an edge header. Note that memory is allocated for |start_x(null_dash)| and we are free to give it any convenient value. @= ok_to_dash:=pen_is_elliptical(pen_p(p)); if not ok_to_dash then scf:=unity else scf:=dash_scale(p); hh:=dash_p(p); pp:=dash_list(hh); if (pp=null_dash) or (dash_y(hh)<0) then print(" ??") else begin start_x(null_dash):=start_x(pp)+dash_y(hh); while pp<>null_dash do begin print("on "); print_scaled(take_scaled(stop_x(pp)-start_x(pp),scf)); print(" off "); print_scaled(take_scaled(start_x(link(pp))-stop_x(pp),scf)); pp := link(pp); if pp<>null_dash then print_char(" "); end; print(") shifted "); print_scaled(-take_scaled(dash_offset(hh),scf)); if not ok_to_dash or (dash_y(hh)=0) then print(" (this will be ignored)"); end @ @= function dash_offset(h:pointer):scaled; var @!x:scaled; {the answer} begin if (dash_list(h)=null_dash) or (dash_y(h)<0) then confusion("dash0"); @:this can't happen dash0}{\quad dash0@> if dash_y(h)=0 then x:=0 else begin x:=-(start_x(dash_list(h)) mod dash_y(h)); if x<0 then x:=x+dash_y(h); end; dash_offset:=x; end; @ @= text_code: begin print_char(""""); print(text_p(p)); print(""" infont """); print(font_name[font_n(p)]); print_char(""""); print_ln; print_obj_color(p); print("transformed "); print_compact_node(text_tx_loc(p),6); end; @ @= start_clip_code: begin print("clipping path:"); print_ln; pr_path(path_p(p)); end; stop_clip_code: print("stop clipping"); @ @= start_bounds_code: begin print("setbounds path:"); print_ln; pr_path(path_p(p)); end; stop_bounds_code: print("end of setbounds"); @ To initialize the |dash_list| field in an edge header~|h|, we need a subroutine that scans an edge structure and tries to interpret it as a dash pattern. This can only be done when there are no filled regions or clipping paths and all the pen strokes have the same color. The first step is to let $y_0$ be the initial $y$~coordinate of the first pen stroke. Then we implicitly project all the pen stroke paths onto the line $y=y_0$ and require that there be no retracing. If the resulting paths cover a range of $x$~coordinates of length $\Delta x$, we set |dash_y(h)| to the length of the dash pattern by finding the maximum of $\Delta x$ and the absolute value of~$y_0$. @p @@; function make_dashes(h:pointer):pointer; {returns |h| or |null|} label exit, found, not_found; var @!p:pointer; {this scans the stroked nodes in the object list} @!y0:scaled; {the initial $y$ coordinate} @!p0:pointer; {if not |null| this points to the first stroked node} @!pp,@!qq,@!rr:pointer; {pointers into |path_p(p)|} @!d,@!dd:pointer; {pointers used to create the dash list} @@; begin if dash_list(h)<>null_dash then goto found; p0:=null; p:=link(dummy_loc(h)); while p<>null do begin if type(p)<>stroked_code then @; pp:=path_p(p); if p0=null then begin p0:=p; y0:=y_coord(pp); @+end; @; @; p:=link(p); end; if dash_list(h)=null_dash then goto not_found; {No error message} @; @; found:make_dashes:=h; return; not_found: @; exit:end; @ @= begin print_err("Picture is too complicated to use as a dash pattern"); help3("When you say `dashed p', picture p should not contain any")@/ ("text, filled regions, or clipping paths. This time it did")@/ ("so I'll just make it a solid line instead.");@/ put_get_error; goto not_found; end @ A similar error occurs when monotonicity fails. @= procedure x_retrace_error; begin print_err("Picture is too complicated to use as a dash pattern"); help3("When you say `dashed p', every path in p should be monotone")@/ ("in x and there must be no overlapping. This failed")@/ ("so I'll just make it a solid line instead."); put_get_error; end; @ We stash |p| in |info(d)| if |dash_p(p)<>0| so that subsequent processing can handle the case where the pen stroke |p| is itself dashed. @= @; rr:=pp; if link(pp)<>pp then repeat qq:=rr; rr:=link(rr); @; until right_type(rr)=endpoint; d:=get_node(dash_node_size); if dash_p(p)=0 then info(d):=0 @+else info(d):=p; if x_coord(pp)= x0:=x_coord(qq); x1:=right_x(qq); x2:=left_x(rr); x3:=x_coord(rr); if (x0>x1) or (x1>x2) or (x2>x3) then if (x00 then begin x_retrace_error; goto not_found; end; if (x_coord(pp)>x0) or (x0>x3) then if (x_coord(pp)= @!x0,@!x1,@!x2,@!x3:scaled; {$x$ coordinates of the segment from |qq| to |rr|} @ @= if (red_val(p)<>red_val(p0)) or@| (green_val(p)<>green_val(p0)) or (blue_val(p)<>blue_val(p0)) then begin print_err("Picture is too complicated to use as a dash pattern"); help3("When you say `dashed p', everything in picture p should")@/ ("be the same color. I can't handle your color changes")@/ ("so I'll just make it a solid line instead.");@/ put_get_error; goto not_found; end @ @= start_x(null_dash):=stop_x(d); dd:=h; {this makes |link(dd)=dash_list(h)|} while start_x(link(dd))h then if (stop_x(dd)>start_x(d)) then begin x_retrace_error; goto not_found; @+end; link(d):=link(dd); link(dd):=d @ @= d:=dash_list(h); while (link(d)<>null_dash) do d:=link(d); dd:=dash_list(h); dash_y(h):=stop_x(d)-start_x(dd); if abs(y0)>dash_y(h) then dash_y(h):=abs(y0) else if d<>dd then begin dash_list(h):=link(dd); stop_x(d):=stop_x(dd)+dash_y(h); free_node(dd,dash_node_size); end @ We get here when the argument is a null picture or when there is an error. Recovering from an error involves making |dash_list(h)| empty to indicate that |h| is not known to be a valid dash pattern. We also dereference |h| since it is not being used for the return value. @= flush_dash_list(h); delete_edge_ref(h); make_dashes:=null @ Having carefully saved the dashed stroked nodes in the corresponding dash nodes, we must be prepared to break up these dashes into smaller dashes. @= d:=h; {now |link(d)=dash_list(h)|} while link(d)<>null_dash do begin ds:=info(link(d)); if ds=null then d:=link(d) else begin hh:=dash_p(ds); hsf:=dash_scale(ds); if (hh=null) then confusion("dash1"); @:this can't happen dash0}{\quad dash1@> if dash_y(hh)=0 then d:=link(d) else begin if dash_list(hh)=null then confusion("dash1"); @:this can't happen dash0}{\quad dash1@> @; end; end; end @ @= @!dln:pointer; {|link(d)|} @!hh:pointer; {an edge header that tells how to break up |dln|} @!hsf:scaled; {the dash pattern from |hh| gets scaled by this} @!ds:pointer; {the stroked node from which |hh| and |hsf| are derived} @!xoff:scaled; {added to $x$ values in |dash_list(hh)| to match |dln|} @ @= dln:=link(d); dd:=dash_list(hh); xoff:=start_x(dln)-take_scaled(hsf,start_x(dd))- take_scaled(hsf,dash_offset(hh)); start_x(null_dash):=take_scaled(hsf,start_x(dd))+take_scaled(hsf,dash_y(hh)); stop_x(null_dash):=start_x(null_dash); @; while start_x(dln)<=stop_x(dln) do begin @; @; dd:=link(dd); start_x(dln):=xoff+take_scaled(hsf,start_x(dd)); end; link(d):=link(dln); free_node(dln,dash_node_size) @ The name of this module is a bit of a lie because we actually just find the first |dd| where |take_scaled(hsf,stop_x(dd))| is large enough to make an overlap possible. It could be that the unoffset version of dash |dln| falls in the gap between |dd| and its predecessor. @= while xoff+take_scaled(hsf,stop_x(dd))= if dd=null_dash then begin dd:=dash_list(hh); xoff:=xoff+take_scaled(hsf,dash_y(hh)); end @ At this point we already know that |start_x(dln)<=xoff+take_scaled(hsf,stop_x(dd))|. @= if xoff+take_scaled(hsf,start_x(dd))<=stop_x(dln) then begin link(d):=get_node(dash_node_size); d:=link(d); link(d):=dln; if start_x(dln)>xoff+take_scaled(hsf,start_x(dd)) then start_x(d):=start_x(dln) else start_x(d):=xoff+take_scaled(hsf,start_x(dd)); if stop_x(dln)maxx_val(h) then maxx_val(h):=maxx; if maxy>maxy_val(h) then maxy_val(h):=maxy; end; @ Here is a special routine for updating the bounding box information in edge header~|h| to account for the squared-off ends of a non-cyclic path~|p| that is to be stroked with the pen~|pp|. @p procedure box_ends(@!p, @!pp, @!h:pointer); label exit; var @!q:pointer; {a knot node adjacent to knot |p|} @!dx,@!dy:fraction; {a unit vector in the direction out of the path at~|p|} @!d:scaled; {a factor for adjusting the length of |(dx,dy)|} @!z:scaled; {a coordinate being tested against the bounding box} @!xx,@!yy:scaled; {the extreme pen vertex in the |(dx,dy)| direction} @!i:integer; {a loop counter} begin if right_type(p)<>endpoint then begin q:=link(p); loop @+begin @; d:=pyth_add(dx,dy); if d>0 then begin @; for i:=1 to 2 do begin @;@/ dx:=-dx; dy:=-dy; end; end; if right_type(p)=endpoint then return else @; end; end; exit: ; end; @ @= if q=link(p) then begin dx:=x_coord(p)-right_x(p); dy:=y_coord(p)-right_y(p); if (dx=0)and(dy=0) then begin dx:=x_coord(p)-left_x(q); dy:=y_coord(p)-left_y(q); end; end else begin dx:=x_coord(p)-left_x(p); dy:=y_coord(p)-left_y(p); if (dx=0)and(dy=0) then begin dx:=x_coord(p)-right_x(q); dy:=y_coord(p)-right_y(q); end; end; dx:=x_coord(p)-x_coord(q); dy:=y_coord(p)-y_coord(q) @ @= dx:=make_fraction(dx,d); dy:=make_fraction(dy,d);@/ find_offset(-dy,dx,pp); xx:=cur_x; yy:=cur_y @ @= find_offset(dx,dy,pp); d:=take_fraction(xx-cur_x,dx)+take_fraction(yy-cur_y,dy); if (d<0)and(i=1) or (d>0)and(i=2) then confusion("box_ends"); @:this can't happen box ends}{\quad\\{box_ends}@> z:=x_coord(p)+cur_x+take_fraction(d,dx); if zmaxx_val(h) then maxx_val(h):=z; z:=y_coord(p)+cur_y+take_fraction(d,dy); if zmaxy_val(h) then maxy_val(h):=z @ @= repeat q:=p; p:=link(p); until right_type(p)=endpoint @ The major difficulty in finding the bounding box of an edge structure is the effect of clipping paths. We treat them conservatively by only clipping to the clipping path's bounding box, but this still requires recursive calls to |set_bbox| in order to find the bounding box of @^recursion@> the objects to be clipped. Such calls are distinguished by the fact that the boolean parameter |top_level| is false. @p procedure set_bbox(@!h:pointer;top_level:boolean); label exit; var @!p:pointer; {a graphical object being considered} @!sminx,@!sminy,@!smaxx,@!smaxy:scaled; {for saving the bounding box during recursive calls} @!x0,@!x1,@!y0,@!y1:scaled; {temporary registers} @!lev:integer; {nesting level for |start_bounds_code| nodes} begin @; while link(bblast(h))<>null do begin p:=link(bblast(h)); bblast(h):=p; case type(p) of stop_clip_code: if top_level then confusion("bbox") @+else return; @:this can't happen bbox}{\quad bbox@> @@; end; {all cases are enumerated above} end; if not top_level then confusion("bbox"); exit:end; @ @= case bbtype(h) of no_bounds: do_nothing; bounds_set: if internal[true_corners]>0 then init_bbox(h); bounds_unset: if internal[true_corners]<=0 then init_bbox(h); end {there are no other cases} @ @= fill_code: begin path_bbox(path_p(p)); adjust_bbox(h); end; @ @= start_bounds_code: if internal[true_corners]>0 then bbtype(h):=bounds_unset else begin bbtype(h):=bounds_set; path_bbox(path_p(p)); adjust_bbox(h); @; end; stop_bounds_code: if internal[true_corners]<=0 then confusion("bbox2"); @:this can't happen bbox2}{\quad bbox2@> @ @= lev:=1; while lev<>0 do begin if link(p)=null then confusion("bbox2"); @:this can't happen bbox2}{\quad bbox2@> p:=link(p); if type(p)=start_bounds_code then incr(lev) else if type(p)=stop_bounds_code then decr(lev); end; bblast(h):=p @ It saves a lot of grief here to be slightly conservative and not account for omitted parts of dashed lines. We also don't worry about the material omitted when using butt end caps. The basic computation is for round end caps and |box_ends| augments it for square end caps. @= stroked_code: begin path_bbox(path_p(p)); x0:=minx; y0:=miny; x1:=maxx; y1:=maxy; pen_bbox(pen_p(p)); minx:=minx+x0; miny:=miny+y0; maxx:=maxx+x1; maxy:=maxy+y1; adjust_bbox(h); if (left_type(path_p(p))=endpoint)and(lcap_val(p)=2) then box_ends(path_p(p), pen_p(p), h); end; @ The height width and depth information stored in a text node determines a rectangle that needs to be transformed according to the transformation parameters stored in the text node. @= text_code: begin x1:=take_scaled(txx_val(p),width_val(p)); y0:=take_scaled(txy_val(p),-depth_val(p)); y1:=take_scaled(txy_val(p),height_val(p)); minx:=tx_val(p); maxx:=minx; if y0= start_clip_code: begin path_bbox(path_p(p));@/ x0:=minx; y0:=miny; x1:=maxx; y1:=maxy;@/ sminx:=minx_val(h); sminy:=miny_val(h); smaxx:=maxx_val(h); smaxy:=maxy_val(h);@/ @; @; minx:=sminx; miny:=sminy; maxx:=smaxx; maxy:=smaxy; adjust_bbox(h); end; @ @= minx_val(h):=el_gordo; miny_val(h):=el_gordo; maxx_val(h):=-el_gordo; maxy_val(h):=-el_gordo;@/ set_bbox(h,false) @ @= if minx_val(h)x1 then maxx_val(h):=x1; if maxy_val(h)>y1 then maxy_val(h):=y1 @* \[22] Finding an envelope. When \MP\ has a path and a polygonal pen, it needs to express the desired shape in terms of things \ps\ can understand. The present task is to compute a new path that describes the region to be filled. It is convenient to define this as a two step process where the first step is determining what offset to use for each segment of the path. @ Given a pointer |c| to a cyclic path, and a pointer~|h| to the first knot of a pen polygon, the |offset_prep| routine changes the path into cubics that are associated with particular pen offsets. Thus if the cubic between |p| and~|q| is associated with the |k|th offset and the cubic between |q| and~|r| has offset |l| then |info(q)=zero_off+l-k|. (The constant |zero_off| is added to because |l-k| could be negative.) After overwriting the type information with offset differences, we no longer have a true path so we refer to the knot list returned by |offset_prep| as an ``envelope spec.'' @!@^envelope spec@> Since an envelope spec only determines relative changes in pen offsets, |offset_prep| sets a global variable |spec_offset| to the relative change from |h| to the first offset. @d zero_off=16384 {added to offset changes to make them positive} @= spec_offset:integer; {number of pen edges between |h| and the initial offset} @ @p @t\4@>@@; function offset_prep(@!c,@!h:pointer):pointer; label not_found; var @!n:halfword; {the number of vertices in the pen polygon} @!p,@!q,@!r,@!w,@!ww:pointer; {for list manipulation} @!k_needed:integer; {amount to be added to |info(p)| when it is computed} @!w0:pointer; {a pointer to pen offset to use just before |p|} @!dxin,@!dyin:scaled; {the direction into knot |p|} @!turn_amt:integer; {change in pen offsets for the current cubic} @@; begin @; @; p:=c; k_needed:=0; repeat q:=link(p); @; @; until q=c; @; end; @ We shall want to keep track of where certain knots on the cyclic path wind up in the envelope spec. It doesn't suffice just to keep pointers to knot nodes because some nodes are deleted while removing dead cubics. Thus |offset_prep| updates the following pointers @= @!spec_p1,@!spec_p2:pointer; {pointers to distinguished knots} @ @= spec_p1:=null; spec_p2:=null; @ @= n:=0; p:=h; repeat incr(n); p:=link(p); until p=h @ Since the true incoming direction isn't known yet, we just pick a direction consistent with the pen offset~|h|. If this is wrong, it can be corrected later. @= dxin:=x_coord(link(h))-x_coord(knil(h)); dyin:=y_coord(link(h))-y_coord(knil(h)); if (dxin=0)and(dyin=0) then begin dxin:=y_coord(knil(h))-y_coord(h); dyin:=x_coord(h)-x_coord(knil(h)); end; w0:=h @ We must be careful not to remove the only cubic in a cycle. @= repeat r:=link(p); if x_coord(p)=right_x(p) then if y_coord(p)=right_y(p) then if x_coord(p)=left_x(r) then if y_coord(p)=left_y(r) then if x_coord(p)=x_coord(r) then if y_coord(p)=y_coord(r) then if r<>p then @; p:=r; until p=q @ @= begin k_needed:=info(p)-zero_off; if r=q then q:=p else begin info(p):=k_needed+info(r); k_needed:=0; end; if r=c then begin info(p):=info(c); c:=p; end; if r=spec_p1 then spec_p1:=p; if r=spec_p2 then spec_p2:=p; r:=p; remove_cubic(p); end @ Not setting the |info| field of the newly created knot allows the splitting routine to work for paths. @= procedure split_cubic(@!p:pointer;@!t:fraction); {splits the cubic after |p|} var @!v:scaled; {an intermediate value} @!q,@!r:pointer; {for list manipulation} begin q:=link(p); r:=get_node(knot_node_size); link(p):=r; link(r):=q;@/ left_type(r):=explicit; right_type(r):=explicit;@# v:=t_of_the_way(right_x(p))(left_x(q)); right_x(p):=t_of_the_way(x_coord(p))(right_x(p)); left_x(q):=t_of_the_way(left_x(q))(x_coord(q)); left_x(r):=t_of_the_way(right_x(p))(v); right_x(r):=t_of_the_way(v)(left_x(q)); x_coord(r):=t_of_the_way(left_x(r))(right_x(r));@# v:=t_of_the_way(right_y(p))(left_y(q)); right_y(p):=t_of_the_way(y_coord(p))(right_y(p)); left_y(q):=t_of_the_way(left_y(q))(y_coord(q)); left_y(r):=t_of_the_way(right_y(p))(v); right_y(r):=t_of_the_way(v)(left_y(q)); y_coord(r):=t_of_the_way(left_y(r))(right_y(r)); end; @ This does not set |info(p)| or |right_type(p)|. @= procedure remove_cubic(@!p:pointer); {removes the dead cubic following~|p|} var @!q:pointer; {the node that disappears} begin q:=link(p); link(p):=link(q);@/ right_x(p):=right_x(q); right_y(p):=right_y(q);@/ free_node(q,knot_node_size); end; @ Let $d\prec d'$ mean that the counter-clockwise angle from $d$ to~$d'$ is strictly between zero and $180^\circ$. Then we can define $d\preceq d'$ to mean that the angle could be zero or $180^\circ$. If $w_k=(u_k,v_k)$ is the $k$th pen offset, the $k$th pen edge direction is defined by the formula $$d_k=(u\k-u_k,\,v\k-v_k).$$ When listed by increasing $k$, these directions occur in counter-clockwise order so that $d_k\preceq d\k$ for all~$k$. The goal of |offset_prep| is to find an offset index~|k| to associate with each cubic, such that the direction $d(t)$ of the cubic satisfies $$d_{k-1}\preceq d(t)\preceq d_k\qquad\hbox{for $0\le t\le 1$.}\eqno(*)$$ We may have to split a cubic into many pieces before each piece corresponds to a unique offset. @= info(p):=zero_off+k_needed; k_needed:=0;@/ @; @; @; @; @; @;@/ w0:=pen_walk(w0,turn_amt); not_found: do_nothing @ @= function pen_walk(@!w:pointer;@!k:integer):pointer; {walk |k| steps around a pen from |w|} begin while k>0 do begin w:=link(w); decr(k); @+end; while k<0 do begin w:=knil(w); incr(k); @+end; pen_walk:=w; end; @ The direction of a cubic $B(z_0,z_1,z_2,z_3;t)=\bigl(x(t),y(t)\bigr)$ can be calculated from the quadratic polynomials ${1\over3}x'(t)=B(x_1-x_0,x_2-x_1,x_3-x_2;t)$ and ${1\over3}y'(t)=B(y_1-y_0,y_2-y_1,y_3-y_2;t)$. Since we may be calculating directions from several cubics split from the current one, it is desirable to do these calculations without losing too much precision. ``Scaled up'' values of the derivatives, which will be less tainted by accumulated errors than derivatives found from the cubics themselves, are maintained in local variables |x0|, |x1|, and |x2|, representing $X_0=2^l(x_1-x_0)$, $X_1=2^l(x_2-x_1)$, and $X_2=2^l(x_3-x_2)$; similarly |y0|, |y1|, and~|y2| represent $Y_0=2^l(y_1-y_0)$, $Y_1=2^l(y_2-y_1)$, and $Y_2=2^l(y_3-y_2)$. @= @!x0,@!x1,@!x2,@!y0,@!y1,@!y2:integer; {representatives of derivatives} @!t0,@!t1,@!t2:integer; {coefficients of polynomial for slope testing} @!du,@!dv,@!dx,@!dy:integer; {for directions of the pen and the curve} @!dx0,@!dy0:integer; {initial direction for the first cubic in the curve} @!max_coef:integer; {used while scaling} @!x0a,@!x1a,@!x2a,@!y0a,@!y1a,@!y2a:integer; {intermediate values} @!t:fraction; {where the derivative passes through zero} @!s:fraction; {a temporary value} @ @= x0:=right_x(p)-x_coord(p); x2:=x_coord(q)-left_x(q); x1:=left_x(q)-right_x(p); y0:=right_y(p)-y_coord(p); y2:=y_coord(q)-left_y(q); y1:=left_y(q)-right_y(p); max_coef:=abs(x0); if abs(x1)>max_coef then max_coef:=abs(x1); if abs(x2)>max_coef then max_coef:=abs(x2); if abs(y0)>max_coef then max_coef:=abs(y0); if abs(y1)>max_coef then max_coef:=abs(y1); if abs(y2)>max_coef then max_coef:=abs(y2); if max_coef=0 then goto not_found; while max_coef= procedure fin_offset_prep(@!p:pointer;@!w:pointer; @!x0,@!x1,@!x2,@!y0,@!y1,@!y2:integer;@!rise,@!turn_amt:integer); label exit; var @!ww:pointer; {for list manipulation} @!du,@!dv:scaled; {for slope calculation} @!t0,@!t1,@!t2:integer; {test coefficients} @!t:fraction; {place where the derivative passes a critical slope} @!s:fraction; {slope or reciprocal slope} @!v:integer; {intermediate value for updating |x0..y2|} @!q:pointer; {original |link(p)|} begin q:=link(p); loop @+begin if rise>0 then ww:=link(w) {a pointer to $w\k$} else ww:=knil(w); {a pointer to $w_{k-1}$} @; t:=crossing_point(t0,t1,t2); if t>=fraction_one then if turn_amt>0 then t:=fraction_one @+else return; @; w:=ww; end; exit:end; @ We want $B(\\{t0},\\{t1},\\{t2};t)$ to be the dot product of $d(t)$ with a $-90^\circ$ rotation of the vector from |w| to |ww|. This makes the resulting function cross from positive to negative when $d_{k-1}\preceq d(t)\preceq d_k$ begins to fail. @= du:=x_coord(ww)-x_coord(w); dv:=y_coord(ww)-y_coord(w); if abs(du)>=abs(dv) then begin s:=make_fraction(dv,du); t0:=take_fraction(x0,s)-y0; t1:=take_fraction(x1,s)-y1; t2:=take_fraction(x2,s)-y2; if du<0 then begin negate(t0); negate(t1); negate(t2); @+end end else begin s:=make_fraction(du,dv); t0:=x0-take_fraction(y0,s); t1:=x1-take_fraction(y1,s); t2:=x2-take_fraction(y2,s); if dv<0 then begin negate(t0); negate(t1); negate(t2); @+end end; if t0<0 then t0:=0 {should be positive without rounding error} @ The curve has crossed $d_k$ or $d_{k-1}$; its initial segment satisfies $(*)$, and it might cross again, yielding another solution of $(*)$. @= begin split_cubic(p,t); p:=link(p); info(p):=zero_off+rise; decr(turn_amt);@/ v:=t_of_the_way(x0)(x1); x1:=t_of_the_way(x1)(x2); x0:=t_of_the_way(v)(x1);@/ v:=t_of_the_way(y0)(y1); y1:=t_of_the_way(y1)(y2); y0:=t_of_the_way(v)(y1);@/ if turn_amt<0 then begin t1:=t_of_the_way(t1)(t2); if t1>0 then t1:=0; {without rounding error, |t1| would be |<=0|} t:=crossing_point(0,-t1,-t2); if t>fraction_one then t:=fraction_one; incr(turn_amt); if (t=fraction_one)and(link(p)<>q) then info(link(p)):=info(link(p))-rise else begin split_cubic(p,t); info(link(p)):=zero_off-rise;@/ v:=t_of_the_way(x1)(x2); x1:=t_of_the_way(x0)(x1); x2:=t_of_the_way(x1)(v);@/ v:=t_of_the_way(y1)(y2); y1:=t_of_the_way(y0)(y1); y2:=t_of_the_way(y1)(v);@/ end; end; end @ Now we must consider the general problem of |offset_prep|, when nothing is known about a given cubic. We start by finding its direction in the vicinity of |t=0|. If $z'(t)=0$, the given cubic is numerically unstable but |offset_prep| has not yet introduced any more numerical errors. Thus we can compute the true initial direction for the given cubic, even if it is almost degenerate. @= dx:=x0; dy:=y0; if dx=0 then if dy=0 then begin dx:=x1; dy:=y1; if dx=0 then if dy=0 then begin dx:=x2; dy:=y2; end; end; if p=c then begin dx0:=dx; dy0:=dy; @+end @ @= dxin:=x2; dyin:=y2; if dxin=0 then if dyin=0 then begin dxin:=x1; dyin:=y1; if dxin=0 then if dyin=0 then begin dxin:=x0; dyin:=y0; end; end @ The next step is to bracket the initial direction between consecutive edges of the pen polygon. We must be careful to turn clockwise only if this makes the turn less than $180^\circ$. (A $180^\circ$ turn must be counter-clockwise in order to make \&{doublepath} envelopes come out @:double_path_}{\&{doublepath} primitive@> right.) This code depends on |w0| being the offset for |(dxin,dyin)|. @= turn_amt:=get_turn_amt(w0, dx, dy, ab_vs_cd(dy,dxin,dx,dyin)>=0); w:=pen_walk(w0, turn_amt); w0:=w; info(p):=info(p)+turn_amt @ Decide how many pen offsets to go away from |w| in order to find the offset for |(dx,dy)|, going counterclockwise if |ccw| is |true|. This assumes that |w| is the offset for some direction $(x',y')$ from which the angle to |(dx,dy)| in the sense determined by |ccw| is less than or equal to $180^\circ$. If the pen polygon has only two edges, they could both be parallel to |(dx,dy)|. In this case, we must be careful to stop after crossing the first such edge in order to avoid an infinite loop. @= function get_turn_amt(@!w:pointer; @!dx,@!dy:scaled; ccw:boolean):integer; label done; var @!ww:pointer; {a neighbor of knot~|w|} @!s:integer; {turn amount so far} @!t:integer; {|ab_vs_cd| result} begin s:=0; if ccw then begin ww:=link(w); repeat t:=ab_vs_cd(dy,x_coord(ww)-x_coord(w),@| dx,y_coord(ww)-y_coord(w)); if t<0 then goto done; incr(s); w:=ww; ww:=link(ww); until t<=0; done: end else begin ww:=knil(w); while ab_vs_cd(dy,x_coord(w)-x_coord(ww),@| dx,y_coord(w)-y_coord(ww))<0 do begin decr(s); w:=ww; ww:=knil(ww); end; end; get_turn_amt:=s; end; @ When we're all done, the final offset is |w0| and the final curve direction is |(dxin,dyin)|. With this knowledge of the incoming direction at |c|, we can correct |info(c)| which was erroneously based on an incoming offset of~|h|. @d fix_by(#)==info(c):=info(c)+# @= spec_offset:=info(c)-zero_off; if link(c)=c then info(c):=zero_off+n else begin fix_by(k_needed); while w0<>h do begin fix_by(1); w0:=link(w0); @+end; while info(c)<=zero_off-n do fix_by(n); while info(c)>zero_off do fix_by(-n); if (info(c)<>zero_off)and(ab_vs_cd(dy0,dxin,dx0,dyin)>=0) then fix_by(n); end; offset_prep:=c @ Finally we want to reduce the general problem to situations that |fin_offset_prep| can handle. We split the cubic into at most three parts with respect to $d_{k-1}$, and apply |fin_offset_prep| to each part. @= ww:=knil(w); @; @; if t>fraction_one then fin_offset_prep(p,w,x0,x1,x2,y0,y1,y2,1,turn_amt) else begin split_cubic(p,t); r:=link(p);@/ x1a:=t_of_the_way(x0)(x1); x1:=t_of_the_way(x1)(x2); x2a:=t_of_the_way(x1a)(x1);@/ y1a:=t_of_the_way(y0)(y1); y1:=t_of_the_way(y1)(y2); y2a:=t_of_the_way(y1a)(y1);@/ fin_offset_prep(p,w,x0,x1a,x2a,y0,y1a,y2a,1,0); x0:=x2a; y0:=y2a; info(r):=zero_off-1; if turn_amt>=0 then begin t1:=t_of_the_way(t1)(t2); if t1>0 then t1:=0; t:=crossing_point(0,-t1,-t2); if t>fraction_one then t:=fraction_one; @; fin_offset_prep(r,ww,x0,x1,x2,y0,y1,y2,-1,0); end else fin_offset_prep(r,ww,x0,x1,x2,y0,y1,y2,-1,-1-turn_amt); end @ @= split_cubic(r,t); info(link(r)):=zero_off+1;@/ x1a:=t_of_the_way(x1)(x2); x1:=t_of_the_way(x0)(x1); x0a:=t_of_the_way(x1)(x1a);@/ y1a:=t_of_the_way(y1)(y2); y1:=t_of_the_way(y0)(y1); y0a:=t_of_the_way(y1)(y1a);@/ fin_offset_prep(link(r),w,x0a,x1a,x2,y0a,y1a,y2,1,turn_amt); x2:=x0a; y2:=y0a @ At this point, the direction of the incoming pen edge is |(-du,-dv)|. When the component of $d(t)$ perpendicular to |(-du,-dv)| crosses zero, we need to decide whether the directions are parallel or antiparallel. We can test this by finding the dot product of $d(t)$ and |(-du,-dv)|, but this should be avoided when the value of |turn_amt| already determines the answer. If |t2<0|, there is one crossing and it is antiparallel only if |turn_amt>=0|. If |turn_amt<0|, there should always be at least one crossing and the first crossing cannot be antiparallel. @= t:=crossing_point(t0,t1,t2); if turn_amt>=0 then if t2<0 then t:=fraction_one+1 else begin u0:=t_of_the_way(x0)(x1); u1:=t_of_the_way(x1)(x2); ss:=take_fraction(-du,t_of_the_way(u0)(u1));@/ v0:=t_of_the_way(y0)(y1); v1:=t_of_the_way(y1)(y2); ss:=ss+take_fraction(-dv,t_of_the_way(v0)(v1));@/ if ss<0 then t:=fraction_one+1; end else if t>fraction_one then t:=fraction_one; @ @= @!u0,@!u1,@!v0,@!v1:integer; {intermediate values for $d(t)$ calculation} @!ss:integer; {the part of the dot product computed so far} @!d_sign:-1..1; {sign of overall change in direction for this cubic} @ If the cubic almost has a cusp, it is a numerically ill-conditioned problem to decide which way it loops around but that's OK as long we're consistent. To make \&{doublepath} envelopes work properly, reversing the path should always change the sign of |turn_amt|. @= d_sign:=ab_vs_cd(dx,dyin, dxin,dy); if d_sign=0 then if dx=0 then if dy>0 then d_sign:=1 @+else d_sign:=-1 else if dx>0 then d_sign:=1 @+else d_sign:=-1; @; turn_amt:=get_turn_amt(w, dxin, dyin, d_sign>0); if ss<0 then turn_amt:=turn_amt-d_sign*n @ In order to be invariant under path reversal, the result of this computation should not change when |x0|, |y0|, $\ldots$ are all negated and |(x0,y0)| is then swapped with |(x2,y2)|. We make use of the identities |take_fraction(-a,-b)=take_fraction(a,b)| and |t_of_the_way(-a)(-b)=-(t_of_the_way(a)(b))|. @= t0:=half(take_fraction(x0,y2))-half(take_fraction(x2,y0));@/ t1:=half(take_fraction(x1,y0+y2))-half(take_fraction(y1,x0+x2));@/ if t0=0 then t0:=d_sign; {path reversal always negates |d_sign|} if t0>0 then begin t:=crossing_point(t0,t1,-t0); u0:=t_of_the_way(x0)(x1); u1:=t_of_the_way(x1)(x2);@/ v0:=t_of_the_way(y0)(y1); v1:=t_of_the_way(y1)(y2); end else begin t:=crossing_point(-t0,t1,t0); u0:=t_of_the_way(x2)(x1); u1:=t_of_the_way(x1)(x0);@/ v0:=t_of_the_way(y2)(y1); v1:=t_of_the_way(y1)(y0); end; ss:=take_fraction(x0+x2,t_of_the_way(u0)(u1))+@| take_fraction(y0+y2,t_of_the_way(v0)(v1)) @ Here's a routine that prints an envelope spec in symbolic form. It assumes that the |cur_pen| has not been walked around to the first offset. @p procedure print_spec(@!cur_spec,@!cur_pen:pointer;@!s:str_number); var @!p,@!q:pointer; {list traversal} @!w:pointer; {the current pen offset} begin print_diagnostic("Envelope spec",s,true); p:=cur_spec; w:=pen_walk(cur_pen,spec_offset); print_ln;@/ print_two(x_coord(cur_spec),y_coord(cur_spec)); print(" % beginning with offset "); print_two(x_coord(w),y_coord(w)); repeat repeat q:=link(p); @; p:=q; until (p=cur_spec) or (info(p)<>zero_off); if info(p)<>zero_off then @; until p=cur_spec; print_nl(" & cycle"); end_diagnostic(true); end; @ @= begin w:=pen_walk(w,info(p)-zero_off); print(" % "); if info(p)>zero_off then print("counter"); print("clockwise to offset "); print_two(x_coord(w),y_coord(w)); end @ @= begin print_nl(" ..controls "); print_two(right_x(p),right_y(p)); print(" and "); print_two(left_x(q),left_y(q)); print_nl(" .."); print_two(x_coord(q),y_coord(q)); end @ Once we have an envelope spec, the remaining task to construct the actual envelope by offsetting each cubic as determined by the |info| fields in the knots. First we use |offset_prep| to convert the |c| into an envelope spec. Then we add the offsets so that |c| becomes a cyclic path that represents the envelope. The |ljoin| and |miterlim| parameters control the treatment of points where the pen offset changes, and |lcap| controls the endpoints of a \&{doublepath}. The endpoints are easily located because |c| is given in undoubled form and then doubled in this procedure. We use |spec_p1| and |spec_p2| to keep track of the endpoints and treat them like very sharp corners. Butt end caps are treated like beveled joins; round end caps are treated like round joins; and square end caps are achieved by setting |join_type:=3|. None of these parameters apply to inside joins where the convolution tracing has retrograde lines. In such cases we use a simple connect-the-endpoints approach that is achieved by setting |join_type:=2|. @p @t\4@>@@; function make_envelope(@!c,@!h:pointer;@!ljoin,@!lcap:small_number; @!miterlim:scaled):pointer; label done; var @!p,@!q,@!r,@!q0:pointer; {for manipulating the path} @!join_type:0..3; {codes |0..3| for mitered, round, beveled, or square} @!w,@!w0:pointer; {the pen knot for the current offset} @!qx,@!qy:scaled; {unshifted coordinates of |q|} @!k,@!k0:halfword; {controls pen edge insertion} @@; begin spec_p1:=null; spec_p2:=null; if left_type(c)=endpoint then @; @; w:=h; p:=c; repeat q:=link(p); q0:=q; qx:=x_coord(q); qy:=y_coord(q); k:=info(q);@/ k0:=k; w0:=w; if k<>zero_off then @; @; while k<>zero_off do begin @; if (join_type=1)or(k=zero_off) then q:=insert_knot(q,qx+x_coord(w),qy+y_coord(w)); end; if q<>link(p) then @; p:=q; until q0=c; make_envelope:=c; end; @ @= c:=offset_prep(c,h); if internal[tracing_specs]>0 then print_spec(c,h,""); h:=pen_walk(h,spec_offset) @ Mitered and squared-off joins depend on path directions that are difficult to compute for degenerate cubics. The envelope spec computed by |offset_prep| can have degenerate cubics only if the entire cycle collapses to a single degenerate cubic. Setting |join_type:=2| in this case makes the computed envelope degenerate as well. @= if kspec_p1)and(q<>spec_p2) then join_type:=ljoin else if lcap=2 then join_type:=3 else join_type:=2-lcap; if (join_type=0)or(join_type=3) then begin @; if join_type=0 then @; end; end @ @= begin tmp:=take_fraction(miterlim,fraction_half+@| half(take_fraction(dxin,dxout)+take_fraction(dyin,dyout))); if tmp= @!dxin,@!dyin,@!dxout,@!dyout:fraction; {directions at |q| when square or mitered} @!tmp:scaled; {a temporary value} @ The coordinates of |p| have already been shifted unless |p| is the first knot in which case they get shifted at the very end. @= right_x(p):=right_x(p)+x_coord(w); right_y(p):=right_y(p)+y_coord(w);@/ left_x(q):=left_x(q)+x_coord(w); left_y(q):=left_y(q)+y_coord(w);@/ x_coord(q):=x_coord(q)+x_coord(w); y_coord(q):=y_coord(q)+y_coord(w);@/ left_type(q):=explicit; right_type(q):=explicit @ @= if k>zero_off then begin w:=link(w); decr(k); @+end else begin w:=knil(w); incr(k); @+end @ The cubic from |q| to the new knot at |(x,y)| becomes a line segment and the |right_x| and |right_y| fields of |r| are set from |q|. This is done in case the cubic containing these control points is ``yet to be examined.'' @= function insert_knot(@!q:pointer;@!x,@!y:scaled):pointer; {returns the inserted knot} var @!r:pointer; {the new knot} begin r:=get_node(knot_node_size); link(r):=link(q); link(q):=r;@/ right_x(r):=right_x(q); right_y(r):=right_y(q);@/ x_coord(r):=x; y_coord(r):=y;@/ right_x(q):=x_coord(q); right_y(q):=y_coord(q);@/ left_x(r):=x_coord(r); left_y(r):=y_coord(r);@/ left_type(r):=explicit; right_type(r):=explicit; insert_knot:=r; end; @ After setting |p:=link(p)|, either |join_type=1| or |q=link(p)|. @= begin p:=link(p); if (join_type=0)or(join_type=3) then begin if join_type=0 then @ else @; if r<>null then begin right_x(r):=x_coord(r); right_y(r):=y_coord(r); end; end; end @ For very small angles, adding a knot is unnecessary and would cause numerical problems, so we just set |r:=null| in that case. @= begin det:=take_fraction(dyout,dxin)-take_fraction(dxout,dyin); if abs(det)<26844 then r:=null {sine $<10^{-4}$} else begin tmp:=take_fraction(x_coord(q)-x_coord(p),dyout)-@| take_fraction(y_coord(q)-y_coord(p),dxout); tmp:=make_fraction(tmp,det); r:=insert_knot(p,x_coord(p)+take_fraction(tmp,dxin),@| y_coord(p)+take_fraction(tmp,dyin)); end; end @ @= @!det:fraction; {a determinant used for mitered join calculations} @ @= begin ht_x:=y_coord(w)-y_coord(w0); ht_y:=x_coord(w0)-x_coord(w); while (abs(ht_x); tmp:=make_fraction(max_ht,take_fraction(dxin,ht_x)+take_fraction(dyin,ht_y)); r:=insert_knot(p,x_coord(p)+take_fraction(tmp,dxin),@| y_coord(p)+take_fraction(tmp,dyin)); tmp:=make_fraction(max_ht,take_fraction(dxout,ht_x)+take_fraction(dyout,ht_y)); r:=insert_knot(r,x_coord(q)+take_fraction(tmp,dxout),@| y_coord(q)+take_fraction(tmp,dyout)); end @ @= @!ht_x,@!ht_y:fraction; {perpendicular to the segment from |p| to |q|} @!max_ht:scaled; {maximum height of the pen polygon above the |w0|-|w| line} @!kk:halfword; {keeps track of the pen vertices being scanned} @!ww:pointer; {the pen vertex being tested} @ The dot product of the vector from |w0| to |ww| with |(ht_x,ht_y)| ranges from zero to |max_ht|. @= max_ht:=0; kk:=zero_off; ww:=w; loop @+begin @; if kk=k0 then goto done; tmp:=take_fraction(x_coord(ww)-x_coord(w0),ht_x)+@| take_fraction(y_coord(ww)-y_coord(w0),ht_y); if tmp>max_ht then max_ht:=tmp; end; done:do_nothing @ @= if kk>k0 then begin ww:=link(ww); decr(kk); @+end else begin ww:=knil(ww); incr(kk); @+end @ @= begin spec_p1:=htap_ypoc(c); spec_p2:=path_tail; link(spec_p2):=link(spec_p1); link(spec_p1):=c;@/ remove_cubic(spec_p1); c:=spec_p1; if c<>link(c) then remove_cubic(spec_p2) else @; end @ @= begin left_type(c):=explicit; right_type(c):=explicit; left_x(c):=x_coord(c); left_y(c):=y_coord(c); right_x(c):=x_coord(c); right_y(c):=y_coord(c); end; @ In degenerate situations we might have to look at the knot preceding~|q|. That knot is |p| but if |p<>c|, its coordinates have already been offset by |w|. @= dxin:=x_coord(q)-left_x(q); dyin:=y_coord(q)-left_y(q); if (dxin=0)and(dyin=0) then begin dxin:=x_coord(q)-right_x(p); dyin:=y_coord(q)-right_y(p); if (dxin=0)and(dyin=0) then begin dxin:=x_coord(q)-x_coord(p); dyin:=y_coord(q)-y_coord(p); if p<>c then {the coordinates of |p| have been offset by |w|} begin dxin:=dxin+x_coord(w); dyin:=dyin+y_coord(w); end; end; end; tmp:=pyth_add(dxin,dyin); if tmp=0 then join_type:=2 else begin dxin:=make_fraction(dxin,tmp); dyin:=make_fraction(dyin,tmp); @; end @ If |q=c| then the coordinates of |r| and the control points between |q| and~|r| have already been offset by |h|. @= dxout:=right_x(q)-x_coord(q); dyout:=right_y(q)-y_coord(q); if (dxout=0)and(dyout=0) then begin r:=link(q); dxout:=left_x(r)-x_coord(q); dyout:=left_y(r)-y_coord(q); if (dxout=0)and(dyout=0) then begin dxout:=x_coord(r)-x_coord(q); dyout:=y_coord(r)-y_coord(q); end; end; if q=c then begin dxout:=dxout-x_coord(h); dyout:=dyout-y_coord(h); end; tmp:=pyth_add(dxout,dyout); if tmp=0 then confusion("degenerate spec"); @:this can't happen degerate spec}{\quad degenerate spec@> dxout:=make_fraction(dxout,tmp); dyout:=make_fraction(dyout,tmp) @* \[23] Direction and intersection times. A path of length $n$ is defined parametrically by functions $x(t)$ and $y(t)$, for |0<=t<=n|; we can regard $t$ as the ``time'' at which the path reaches the point $\bigl(x(t),y(t)\bigr)$. In this section of the program we shall consider operations that determine special times associated with given paths: the first time that a path travels in a given direction, and a pair of times at which two paths cross each other. @ Let's start with the easier task. The function |find_direction_time| is given a direction |(x,y)| and a path starting at~|h|. If the path never travels in direction |(x,y)|, the direction time will be~|-1|; otherwise it will be nonnegative. Certain anomalous cases can arise: If |(x,y)=(0,0)|, so that the given direction is undefined, the direction time will be~0. If $\bigl(x'(t), y'(t)\bigr)=(0,0)$, so that the path direction is undefined, it will be assumed to match any given direction at time~|t|. The routine solves this problem in nondegenerate cases by rotating the path and the given direction so that |(x,y)=(1,0)|; i.e., the main task will be to find when a given path first travels ``due east.'' @p function find_direction_time(@!x,@!y:scaled;@!h:pointer):scaled; label exit,found,not_found,done; var @!max:scaled; {$\max\bigl(\vert x\vert,\vert y\vert\bigr)$} @!p,@!q:pointer; {for list traversal} @!n:scaled; {the direction time at knot |p|} @!tt:scaled; {the direction time within a cubic} @@; begin @; n:=0; p:=h; loop@+ begin if right_type(p)=endpoint then goto not_found; q:=link(p); @; p:=q; n:=n+unity; end; not_found: find_direction_time:=-unity; return; found: find_direction_time:=n+tt; exit:end; @ @= if abs(x)0 then y:=fraction_one@+else y:=-fraction_one; end else if x=0 then begin find_direction_time:=0; return; end else begin y:=make_fraction(y,abs(x)); if x>0 then x:=fraction_one@+else x:=-fraction_one; end @ Since we're interested in the tangent directions, we work with the derivative $${\textstyle1\over3}B'(x_0,x_1,x_2,x_3;t)= B(x_1-x_0,x_2-x_1,x_3-x_2;t)$$ instead of $B(x_0,x_1,x_2,x_3;t)$ itself. The derived coefficients are also scaled up in order to achieve better accuracy. The given path may turn abruptly at a knot, and it might pass the critical tangent direction at such a time. Therefore we remember the direction |phi| in which the previous rotated cubic was traveling. (The value of |phi| will be undefined on the first cubic, i.e., when |n=0|.) @= tt:=0; @; if y1=0 then if x1>=0 then goto found; if n>0 then begin @; if p=h then goto not_found; end; if (x3<>0)or(y3<>0) then phi:=n_arg(x3,y3); @ @ @= @!x1,@!x2,@!x3,@!y1,@!y2,@!y3:scaled; {multiples of rotated derivatives} @!theta,@!phi:angle; {angles of exit and entry at a knot} @!t:fraction; {temp storage} @ @= x1:=right_x(p)-x_coord(p); x2:=left_x(q)-right_x(p); x3:=x_coord(q)-left_x(q);@/ y1:=right_y(p)-y_coord(p); y2:=left_y(q)-right_y(p); y3:=y_coord(q)-left_y(q);@/ max:=abs(x1); if abs(x2)>max then max:=abs(x2); if abs(x3)>max then max:=abs(x3); if abs(y1)>max then max:=abs(y1); if abs(y2)>max then max:=abs(y2); if abs(y3)>max then max:=abs(y3); if max=0 then goto found; while max= theta:=n_arg(x1,y1); if theta>=0 then if phi<=0 then if phi>=theta-one_eighty_deg then goto found; if theta<=0 then if phi>=0 then if phi<=theta+one_eighty_deg then goto found @ In this step we want to use the |crossing_point| routine to find the roots of the quadratic equation $B(y_1,y_2,y_3;t)=0$. Several complications arise: If the quadratic equation has a double root, the curve never crosses zero, and |crossing_point| will find nothing; this case occurs iff $y_1y_3=y_2^2$ and $y_1y_2<0$. If the quadratic equation has simple roots, or only one root, we may have to negate it so that $B(y_1,y_2,y_3;t)$ crosses from positive to negative at its first root. And finally, we need to do special things if $B(y_1,y_2,y_3;t)$ is identically zero. @ @= if x1<0 then if x2<0 then if x3<0 then goto done; if ab_vs_cd(y1,y3,y2,y2)=0 then @; if y1<=0 then if y1<0 then begin y1:=-y1; y2:=-y2; y3:=-y3; end else if y2>0 then begin y2:=-y2; y3:=-y3; end; @; done: @ The quadratic polynomial $B(y_1,y_2,y_3;t)$ begins |>=0| and has at most two roots, because we know that it isn't identically zero. It must be admitted that the |crossing_point| routine is not perfectly accurate; rounding errors might cause it to find a root when $y_1y_3>y_2^2$, or to miss the roots when $y_1y_3= t:=crossing_point(y1,y2,y3); if t>fraction_one then goto done; y2:=t_of_the_way(y2)(y3); x1:=t_of_the_way(x1)(x2); x2:=t_of_the_way(x2)(x3); x1:=t_of_the_way(x1)(x2); if x1>=0 then we_found_it; if y2>0 then y2:=0; tt:=t; t:=crossing_point(0,-y2,-y3); if t>fraction_one then goto done; x1:=t_of_the_way(x1)(x2); x2:=t_of_the_way(x2)(x3); if t_of_the_way(x1)(x2)>=0 then begin t:=t_of_the_way(tt)(fraction_one); we_found_it; end @ @= begin if ab_vs_cd(y1,y2,0,0)<0 then begin t:=make_fraction(y1,y1-y2); x1:=t_of_the_way(x1)(x2); x2:=t_of_the_way(x2)(x3); if t_of_the_way(x1)(x2)>=0 then we_found_it; end else if y3=0 then if y1=0 then @=0|@> else if x3>=0 then begin tt:=unity; goto found; end; goto done; end @ At this point we know that the derivative of |y(t)| is identically zero, and that |x1<0|; but either |x2>=0| or |x3>=0|, so there's some hope of traveling east. @=0|...@>= begin t:=crossing_point(-x1,-x2,-x3); if t<=fraction_one then we_found_it; if ab_vs_cd(x1,x3,x2,x2)<=0 then begin t:=make_fraction(x1,x1-x2); we_found_it; end; end @ The intersection of two cubics can be found by an interesting variant of the general bisection scheme described in the introduction to |crossing_point|.\ Given $w(t)=B(w_0,w_1,w_2,w_3;t)$ and $z(t)=B(z_0,z_1,z_2,z_3;t)$, we wish to find a pair of times $(t_1,t_2)$ such that $w(t_1)=z(t_2)$, if an intersection exists. First we find the smallest rectangle that encloses the points $\{w_0,w_1,w_2,w_3\}$ and check that it overlaps the smallest rectangle that encloses $\{z_0,z_1,z_2,z_3\}$; if not, the cubics certainly don't intersect. But if the rectangles do overlap, we bisect the intervals, getting new cubics $w'$ and~$w''$, $z'$~and~$z''$; the intersection routine first tries for an intersection between $w'$ and~$z'$, then (if unsuccessful) between $w'$ and~$z''$, then (if still unsuccessful) between $w''$ and~$z'$, finally (if thrice unsuccessful) between $w''$ and~$z''$. After $l$~successful levels of bisection we will have determined the intersection times $t_1$ and~$t_2$ to $l$~bits of accuracy. \def\submin{_{\rm min}} \def\submax{_{\rm max}} As before, it is better to work with the numbers $W_k=2^l(w_k-w_{k-1})$ and $Z_k=2^l(z_k-z_{k-1})$ rather than the coefficients $w_k$ and $z_k$ themselves. We also need one other quantity, $\Delta=2^l(w_0-z_0)$, to determine when the enclosing rectangles overlap. Here's why: The $x$~coordinates of~$w(t)$ are between $u\submin$ and $u\submax$, and the $x$~coordinates of~$z(t)$ are between $x\submin$ and $x\submax$, if we write $w_k=(u_k,v_k)$ and $z_k=(x_k,y_k)$ and $u\submin= \min(u_0,u_1,u_2,u_3)$, etc. These intervals of $x$~coordinates overlap if and only if $u\submin\L x\submax$ and $x\submin\L u\submax$. Letting $$U\submin=\min(0,U_1,U_1+U_2,U_1+U_2+U_3),\; U\submax=\max(0,U_1,U_1+U_2,U_1+U_2+U_3),$$ we have $u\submin=2^lu_0+U\submin$, etc.; the condition for overlap reduces to $$X\submin-U\submax\L 2^l(u_0-x_0)\L X\submax-U\submin.$$ Thus we want to maintain the quantity $2^l(u_0-x_0)$; similarly, the quantity $2^l(v_0-y_0)$ accounts for the $y$~coordinates. The coordinates of $\Delta=2^l(w_0-z_0)$ must stay bounded as $l$ increases, because of the overlap condition; i.e., we know that $X\submin$, $X\submax$, and their relatives are bounded, hence $X\submax- U\submin$ and $X\submin-U\submax$ are bounded. @ Incidentally, if the given cubics intersect more than once, the process just sketched will not necessarily find the lexicographically smallest pair $(t_1,t_2)$. The solution actually obtained will be smallest in ``shuffled order''; i.e., if $t_1=(.a_1a_2\ldots a_{16})_2$ and $t_2=(.b_1b_2\ldots b_{16})_2$, then we will minimize $a_1b_1a_2b_2\ldots a_{16}b_{16}$, not $a_1a_2\ldots a_{16}b_1b_2\ldots b_{16}$. Shuffled order agrees with lexicographic order if all pairs of solutions $(t_1,t_2)$ and $(t_1',t_2')$ have the property that $t_1= @!tol_step:0..6; {either 0 or 3, usually} @ We shall use an explicit stack to implement the recursive bisection method described above. The |bisect_stack| array will contain numerous 5-word packets like $(U_1,U_2,U_3,U\submin,U\submax)$, as well as 20-word packets comprising the 5-word packets for $U$, $V$, $X$, and~$Y$. The following macros define the allocation of stack positions to the quantities needed for bisection-intersection. @d stack_1(#)==bisect_stack[#] {$U_1$, $V_1$, $X_1$, or $Y_1$} @d stack_2(#)==bisect_stack[#+1] {$U_2$, $V_2$, $X_2$, or $Y_2$} @d stack_3(#)==bisect_stack[#+2] {$U_3$, $V_3$, $X_3$, or $Y_3$} @d stack_min(#)==bisect_stack[#+3] {$U\submin$, $V\submin$, $X\submin$, or $Y\submin$} @d stack_max(#)==bisect_stack[#+4] {$U\submax$, $V\submax$, $X\submax$, or $Y\submax$} @d int_packets=20 {number of words to represent $U_k$, $V_k$, $X_k$, and $Y_k$} @# @d u_packet(#)==#-5 @d v_packet(#)==#-10 @d x_packet(#)==#-15 @d y_packet(#)==#-20 @d l_packets==bisect_ptr-int_packets @d r_packets==bisect_ptr @d ul_packet==u_packet(l_packets) {base of $U'_k$ variables} @d vl_packet==v_packet(l_packets) {base of $V'_k$ variables} @d xl_packet==x_packet(l_packets) {base of $X'_k$ variables} @d yl_packet==y_packet(l_packets) {base of $Y'_k$ variables} @d ur_packet==u_packet(r_packets) {base of $U''_k$ variables} @d vr_packet==v_packet(r_packets) {base of $V''_k$ variables} @d xr_packet==x_packet(r_packets) {base of $X''_k$ variables} @d yr_packet==y_packet(r_packets) {base of $Y''_k$ variables} @# @d u1l==stack_1(ul_packet) {$U'_1$} @d u2l==stack_2(ul_packet) {$U'_2$} @d u3l==stack_3(ul_packet) {$U'_3$} @d v1l==stack_1(vl_packet) {$V'_1$} @d v2l==stack_2(vl_packet) {$V'_2$} @d v3l==stack_3(vl_packet) {$V'_3$} @d x1l==stack_1(xl_packet) {$X'_1$} @d x2l==stack_2(xl_packet) {$X'_2$} @d x3l==stack_3(xl_packet) {$X'_3$} @d y1l==stack_1(yl_packet) {$Y'_1$} @d y2l==stack_2(yl_packet) {$Y'_2$} @d y3l==stack_3(yl_packet) {$Y'_3$} @d u1r==stack_1(ur_packet) {$U''_1$} @d u2r==stack_2(ur_packet) {$U''_2$} @d u3r==stack_3(ur_packet) {$U''_3$} @d v1r==stack_1(vr_packet) {$V''_1$} @d v2r==stack_2(vr_packet) {$V''_2$} @d v3r==stack_3(vr_packet) {$V''_3$} @d x1r==stack_1(xr_packet) {$X''_1$} @d x2r==stack_2(xr_packet) {$X''_2$} @d x3r==stack_3(xr_packet) {$X''_3$} @d y1r==stack_1(yr_packet) {$Y''_1$} @d y2r==stack_2(yr_packet) {$Y''_2$} @d y3r==stack_3(yr_packet) {$Y''_3$} @# @d stack_dx==bisect_stack[bisect_ptr] {stacked value of |delx|} @d stack_dy==bisect_stack[bisect_ptr+1] {stacked value of |dely|} @d stack_tol==bisect_stack[bisect_ptr+2] {stacked value of |tol|} @d stack_uv==bisect_stack[bisect_ptr+3] {stacked value of |uv|} @d stack_xy==bisect_stack[bisect_ptr+4] {stacked value of |xy|} @d int_increment=int_packets+int_packets+5 {number of stack words per level} @= bisect_stack:array[0..bistack_size] of integer; bisect_ptr:0..bistack_size; @ @= if int_packets+17*int_increment>bistack_size then bad:=19; @ Computation of the min and max is a tedious but fairly fast sequence of instructions; exactly four comparisons are made in each branch. @d set_min_max(#)== if stack_1(#)<0 then if stack_3(#)>=0 then begin if stack_2(#)<0 then stack_min(#):=stack_1(#)+stack_2(#) else stack_min(#):=stack_1(#); stack_max(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_max(#)<0 then stack_max(#):=0; end else begin stack_min(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_min(#)>stack_1(#) then stack_min(#):=stack_1(#); stack_max(#):=stack_1(#)+stack_2(#); if stack_max(#)<0 then stack_max(#):=0; end else if stack_3(#)<=0 then begin if stack_2(#)>0 then stack_max(#):=stack_1(#)+stack_2(#) else stack_max(#):=stack_1(#); stack_min(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_min(#)>0 then stack_min(#):=0; end else begin stack_max(#):=stack_1(#)+stack_2(#)+stack_3(#); if stack_max(#)0 then stack_min(#):=0; end @ It's convenient to keep the current values of $l$, $t_1$, and $t_2$ in the integer form $2^l+2^lt_1$ and $2^l+2^lt_2$. The |cubic_intersection| routine uses global variables |cur_t| and |cur_tt| for this purpose; after successful completion, |cur_t| and |cur_tt| will contain |unity| plus the |scaled| values of $t_1$ and~$t_2$. The values of |cur_t| and |cur_tt| will be set to zero if |cubic_intersection| finds no intersection. The routine gives up and gives an approximate answer if it has backtracked more than 5000 times (otherwise there are cases where several minutes of fruitless computation would be possible). @d max_patience=5000 @= @!cur_t,@!cur_tt:integer; {controls and results of |cubic_intersection|} @!time_to_go:integer; {this many backtracks before giving up} @!max_t:integer; {maximum of $2^{l+1}$ so far achieved} @ The given cubics $B(w_0,w_1,w_2,w_3;t)$ and $B(z_0,z_1,z_2,z_3;t)$ are specified in adjacent knot nodes |(p,link(p))| and |(pp,link(pp))|, respectively. @p procedure cubic_intersection(@!p,@!pp:pointer); label continue, not_found, exit; var @!q,@!qq:pointer; {|link(p)|, |link(pp)|} begin time_to_go:=max_patience; max_t:=2; @; loop@+ begin continue: if delx-tol<=stack_max(x_packet(xy))-stack_min(u_packet(uv)) then if delx+tol>=stack_min(x_packet(xy))-stack_max(u_packet(uv)) then if dely-tol<=stack_max(y_packet(xy))-stack_min(v_packet(uv)) then if dely+tol>=stack_min(y_packet(xy))-stack_max(v_packet(uv)) then begin if cur_t>=max_t then begin if max_t=two then {we've done 17 bisections} begin cur_t:=halfp(cur_t+1); cur_tt:=halfp(cur_tt+1); return; end; double(max_t); appr_t:=cur_t; appr_tt:=cur_tt; end; @; goto continue; end; if time_to_go>0 then decr(time_to_go) else begin while appr_t; end; exit:end; @ The following variables are global, although they are used only by |cubic_intersection|, because it is necessary on some machines to split |cubic_intersection| up into two procedures. @= @!delx,@!dely:integer; {the components of $\Delta=2^l(w_0-z_0)$} @!tol:integer; {bound on the uncertainly in the overlap test} @!uv,@!xy:0..bistack_size; {pointers to the current packets of interest} @!three_l:integer; {|tol_step| times the bisection level} @!appr_t,@!appr_tt:integer; {best approximations known to the answers} @ We shall assume that the coordinates are sufficiently non-extreme that integer overflow will not occur. @= q:=link(p); qq:=link(pp); bisect_ptr:=int_packets;@/ u1r:=right_x(p)-x_coord(p); u2r:=left_x(q)-right_x(p); u3r:=x_coord(q)-left_x(q); set_min_max(ur_packet);@/ v1r:=right_y(p)-y_coord(p); v2r:=left_y(q)-right_y(p); v3r:=y_coord(q)-left_y(q); set_min_max(vr_packet);@/ x1r:=right_x(pp)-x_coord(pp); x2r:=left_x(qq)-right_x(pp); x3r:=x_coord(qq)-left_x(qq); set_min_max(xr_packet);@/ y1r:=right_y(pp)-y_coord(pp); y2r:=left_y(qq)-right_y(pp); y3r:=y_coord(qq)-left_y(qq); set_min_max(yr_packet);@/ delx:=x_coord(p)-x_coord(pp); dely:=y_coord(p)-y_coord(pp);@/ tol:=0; uv:=r_packets; xy:=r_packets; three_l:=0; cur_t:=1; cur_tt:=1 @ @= stack_dx:=delx; stack_dy:=dely; stack_tol:=tol; stack_uv:=uv; stack_xy:=xy; bisect_ptr:=bisect_ptr+int_increment;@/ double(cur_t); double(cur_tt);@/ u1l:=stack_1(u_packet(uv)); u3r:=stack_3(u_packet(uv)); u2l:=half(u1l+stack_2(u_packet(uv))); u2r:=half(u3r+stack_2(u_packet(uv))); u3l:=half(u2l+u2r); u1r:=u3l; set_min_max(ul_packet); set_min_max(ur_packet);@/ v1l:=stack_1(v_packet(uv)); v3r:=stack_3(v_packet(uv)); v2l:=half(v1l+stack_2(v_packet(uv))); v2r:=half(v3r+stack_2(v_packet(uv))); v3l:=half(v2l+v2r); v1r:=v3l; set_min_max(vl_packet); set_min_max(vr_packet);@/ x1l:=stack_1(x_packet(xy)); x3r:=stack_3(x_packet(xy)); x2l:=half(x1l+stack_2(x_packet(xy))); x2r:=half(x3r+stack_2(x_packet(xy))); x3l:=half(x2l+x2r); x1r:=x3l; set_min_max(xl_packet); set_min_max(xr_packet);@/ y1l:=stack_1(y_packet(xy)); y3r:=stack_3(y_packet(xy)); y2l:=half(y1l+stack_2(y_packet(xy))); y2r:=half(y3r+stack_2(y_packet(xy))); y3l:=half(y2l+y2r); y1r:=y3l; set_min_max(yl_packet); set_min_max(yr_packet);@/ uv:=l_packets; xy:=l_packets; double(delx); double(dely);@/ tol:=tol-three_l+tol_step; double(tol); three_l:=three_l+tol_step @ @= not_found: if odd(cur_tt) then if odd(cur_t) then @ else begin incr(cur_t); delx:=delx+stack_1(u_packet(uv))+stack_2(u_packet(uv)) +stack_3(u_packet(uv)); dely:=dely+stack_1(v_packet(uv))+stack_2(v_packet(uv)) +stack_3(v_packet(uv)); uv:=uv+int_packets; {switch from |l_packet| to |r_packet|} decr(cur_tt); xy:=xy-int_packets; {switch from |r_packet| to |l_packet|} delx:=delx+stack_1(x_packet(xy))+stack_2(x_packet(xy)) +stack_3(x_packet(xy)); dely:=dely+stack_1(y_packet(xy))+stack_2(y_packet(xy)) +stack_3(y_packet(xy)); end else begin incr(cur_tt); tol:=tol+three_l; delx:=delx-stack_1(x_packet(xy))-stack_2(x_packet(xy)) -stack_3(x_packet(xy)); dely:=dely-stack_1(y_packet(xy))-stack_2(y_packet(xy)) -stack_3(y_packet(xy)); xy:=xy+int_packets; {switch from |l_packet| to |r_packet|} end @ @= begin cur_t:=halfp(cur_t); cur_tt:=halfp(cur_tt); if cur_t=0 then return; bisect_ptr:=bisect_ptr-int_increment; three_l:=three_l-tol_step; delx:=stack_dx; dely:=stack_dy; tol:=stack_tol; uv:=stack_uv; xy:=stack_xy;@/ goto not_found; end @ The |path_intersection| procedure is much simpler. It invokes |cubic_intersection| in lexicographic order until finding a pair of cubics that intersect. The final intersection times are placed in |cur_t| and~|cur_tt|. @p procedure path_intersection(@!h,@!hh:pointer); label exit; var @!p,@!pp:pointer; {link registers that traverse the given paths} @!n,@!nn:integer; {integer parts of intersection times, minus |unity|} begin @; tol_step:=0; repeat n:=-unity; p:=h; repeat if right_type(p)<>endpoint then begin nn:=-unity; pp:=hh; repeat if right_type(pp)<>endpoint then begin cubic_intersection(p,pp); if cur_t>0 then begin cur_t:=cur_t+n; cur_tt:=cur_tt+nn; return; end; end; nn:=nn+unity; pp:=link(pp); until pp=hh; end; n:=n+unity; p:=link(p); until p=h; tol_step:=tol_step+3; until tol_step>3; cur_t:=-unity; cur_tt:=-unity; exit:end; @ @= if right_type(h)=endpoint then begin right_x(h):=x_coord(h); left_x(h):=x_coord(h); right_y(h):=y_coord(h); left_y(h):=y_coord(h); right_type(h):=explicit; end; if right_type(hh)=endpoint then begin right_x(hh):=x_coord(hh); left_x(hh):=x_coord(hh); right_y(hh):=y_coord(hh); left_y(hh):=y_coord(hh); right_type(hh):=explicit; end; @* \[24] Dynamic linear equations. \MP\ users define variables implicitly by stating equations that should be satisfied; the computer is supposed to be smart enough to solve those equations. And indeed, the computer tries valiantly to do so, by distinguishing five different types of numeric values: \smallskip\hang |type(p)=known| is the nice case, when |value(p)| is the |scaled| value of the variable whose address is~|p|. \smallskip\hang |type(p)=dependent| means that |value(p)| is not present, but |dep_list(p)| points to a {\sl dependency list\/} that expresses the value of variable~|p| as a |scaled| number plus a sum of independent variables with |fraction| coefficients. \smallskip\hang |type(p)=independent| means that |value(p)=64s+m|, where |s>0| is a ``serial number'' reflecting the time this variable was first used in an equation; also |0<=m<64|, and each dependent variable that refers to this one is actually referring to the future value of this variable times~$2^m$. (Usually |m=0|, but higher degrees of scaling are sometimes needed to keep the coefficients in dependency lists from getting too large. The value of~|m| will always be even.) \smallskip\hang |type(p)=numeric_type| means that variable |p| hasn't appeared in an equation before, but it has been explicitly declared to be numeric. \smallskip\hang |type(p)=undefined| means that variable |p| hasn't appeared before. \smallskip\noindent We have actually discussed these five types in the reverse order of their history during a computation: Once |known|, a variable never again becomes |dependent|; once |dependent|, it almost never again becomes |independent|; once |independent|, it never again becomes |numeric_type|; and once |numeric_type|, it never again becomes |undefined| (except of course when the user specifically decides to scrap the old value and start again). A backward step may, however, take place: Sometimes a |dependent| variable becomes |independent| again, when one of the independent variables it depends on is reverting to |undefined|. @d s_scale=64 {the serial numbers are multiplied by this factor} @d new_indep(#)== {create a new independent variable} begin type(#):=independent; serial_no:=serial_no+s_scale; value(#):=serial_no; end @= @!serial_no:integer; {the most recent serial number, times |s_scale|} @ @=new_indep(q+s) @ But how are dependency lists represented? It's simple: The linear combination $\alpha_1v_1+\cdots+\alpha_kv_k+\beta$ appears in |k+1| value nodes. If |q=dep_list(p)| points to this list, and if |k>0|, then |value(q)= @t$\alpha_1$@>| (which is a |fraction|); |info(q)| points to the location of $\alpha_1$; and |link(p)| points to the dependency list $\alpha_2v_2+\cdots+\alpha_kv_k+\beta$. On the other hand if |k=0|, then |value(q)=@t$\beta$@>| (which is |scaled|) and |info(q)=null|. The independent variables $v_1$, \dots,~$v_k$ have been sorted so that they appear in decreasing order of their |value| fields (i.e., of their serial numbers). \ (It is convenient to use decreasing order, since |value(null)=0|. If the independent variables were not sorted by serial number but by some other criterion, such as their location in |mem|, the equation-solving mechanism would be too system-dependent, because the ordering can affect the computed results.) The |link| field in the node that contains the constant term $\beta$ is called the {\sl final link\/} of the dependency list. \MP\ maintains a doubly-linked master list of all dependency lists, in terms of a permanently allocated node in |mem| called |dep_head|. If there are no dependencies, we have |link(dep_head)=dep_head| and |prev_dep(dep_head)=dep_head|; otherwise |link(dep_head)| points to the first dependent variable, say~|p|, and |prev_dep(p)=dep_head|. We have |type(p)=dependent|, and |dep_list(p)| points to its dependency list. If the final link of that dependency list occurs in location~|q|, then |link(q)| points to the next dependent variable (say~|r|); and we have |prev_dep(r)=q|, etc. @d dep_list(#)==link(value_loc(#)) {half of the |value| field in a |dependent| variable} @d prev_dep(#)==info(value_loc(#)) {the other half; makes a doubly linked list} @d dep_node_size=2 {the number of words per dependency node} @= serial_no:=0; link(dep_head):=dep_head; prev_dep(dep_head):=dep_head; info(dep_head):=null; dep_list(dep_head):=null; @ Actually the description above contains a little white lie. There's another kind of variable called |proto_dependent|, which is just like a |dependent| one except that the $\alpha$ coefficients in its dependency list are |scaled| instead of being fractions. Proto-dependency lists are mixed with dependency lists in the nodes reachable from |dep_head|. @ Here is a procedure that prints a dependency list in symbolic form. The second parameter should be either |dependent| or |proto_dependent|, to indicate the scaling of the coefficients. @= procedure print_dependency(@!p:pointer;@!t:small_number); label exit; var @!v:integer; {a coefficient} @!pp,@!q:pointer; {for list manipulation} begin pp:=p; loop@+ begin v:=abs(value(p)); q:=info(p); if q=null then {the constant term} begin if (v<>0)or(p=pp) then begin if value(p)>0 then if p<>pp then print_char("+"); print_scaled(value(p)); end; return; end; @; if type(q)<>independent then confusion("dep"); @:this can't happen dep}{\quad dep@> print_variable_name(q); v:=value(q) mod s_scale; while v>0 do begin print("*4"); v:=v-2; end; p:=link(p); end; exit:end; @ @= if value(p)<0 then print_char("-") else if p<>pp then print_char("+"); if t=dependent then v:=round_fraction(v); if v<>unity then print_scaled(v) @ The maximum absolute value of a coefficient in a given dependency list is returned by the following simple function. @p function max_coef(@!p:pointer):fraction; var @!x:fraction; {the maximum so far} begin x:=0; while info(p)<>null do begin if abs(value(p))>x then x:=abs(value(p)); p:=link(p); end; max_coef:=x; end; @ One of the main operations needed on dependency lists is to add a multiple of one list to the other; we call this |p_plus_fq|, where |p| and~|q| point to dependency lists and |f| is a fraction. If the coefficient of any independent variable becomes |coef_bound| or more, in absolute value, this procedure changes the type of that variable to `|independent_needing_fix|', and sets the global variable |fix_needed| to~|true|. The value of $|coef_bound|=\mu$ is chosen so that $\mu^2+\mu<8$; this means that the numbers we deal with won't get too large. (Instead of the ``optimum'' $\mu=(\sqrt{33}-1)/2\approx 2.3723$, the safer value 7/3 is taken as the threshold.) The changes mentioned in the preceding paragraph are actually done only if the global variable |watch_coefs| is |true|. But it usually is; in fact, it is |false| only when \MP\ is making a dependency list that will soon be equated to zero. Several procedures that act on dependency lists, including |p_plus_fq|, set the global variable |dep_final| to the final (constant term) node of the dependency list that they produce. @d coef_bound==@'4525252525 {|fraction| approximation to 7/3} @d independent_needing_fix=0 @= @!fix_needed:boolean; {does at least one |independent| variable need scaling?} @!watch_coefs:boolean; {should we scale coefficients that exceed |coef_bound|?} @!dep_final:pointer; {location of the constant term and final link} @ @= fix_needed:=false; watch_coefs:=true; @ The |p_plus_fq| procedure has a fourth parameter, |t|, that should be set to |proto_dependent| if |p| is a proto-dependency list. In this case |f| will be |scaled|, not a |fraction|. Similarly, the fifth parameter~|tt| should be |proto_dependent| if |q| is a proto-dependency list. List |q| is unchanged by the operation; but list |p| is totally destroyed. The final link of the dependency list or proto-dependency list returned by |p_plus_fq| is the same as the original final link of~|p|. Indeed, the constant term of the result will be located in the same |mem| location as the original constant term of~|p|. Coefficients of the result are assumed to be zero if they are less than a certain threshold. This compensates for inevitable rounding errors, and tends to make more variables `|known|'. The threshold is approximately $10^{-5}$ in the case of normal dependency lists, $10^{-4}$ for proto-dependencies. @d fraction_threshold=2685 {a |fraction| coefficient less than this is zeroed} @d half_fraction_threshold=1342 {half of |fraction_threshold|} @d scaled_threshold=8 {a |scaled| coefficient less than this is zeroed} @d half_scaled_threshold=4 {half of |scaled_threshold|} @= function p_plus_fq(@!p:pointer;@!f:integer;@!q:pointer; @!t,@!tt:small_number):pointer; label done; var @!pp,@!qq:pointer; {|info(p)| and |info(q)|, respectively} @!r,@!s:pointer; {for list manipulation} @!threshold:integer; {defines a neighborhood of zero} @!v:integer; {temporary register} begin if t=dependent then threshold:=fraction_threshold else threshold:=scaled_threshold; r:=temp_head; pp:=info(p); qq:=info(q); loop@+ if pp=qq then if pp=null then goto done else @ else if value(pp) else begin link(r):=p; r:=p; p:=link(p); pp:=info(p); end; done: if t=dependent then value(p):=slow_add(value(p),take_fraction(value(q),f)) else value(p):=slow_add(value(p),take_scaled(value(q),f)); link(r):=p; dep_final:=p; p_plus_fq:=link(temp_head); end; @ @= begin if tt=dependent then v:=value(p)+take_fraction(f,value(q)) else v:=value(p)+take_scaled(f,value(q)); value(p):=v; s:=p; p:=link(p); if abs(v)=coef_bound then if watch_coefs then begin type(qq):=independent_needing_fix; fix_needed:=true; end; link(r):=s; r:=s; end; pp:=info(p); q:=link(q); qq:=info(q); end @ @= begin if tt=dependent then v:=take_fraction(f,value(q)) else v:=take_scaled(f,value(q)); if abs(v)>halfp(threshold) then begin s:=get_node(dep_node_size); info(s):=qq; value(s):=v; if abs(v)>=coef_bound then if watch_coefs then begin type(qq):=independent_needing_fix; fix_needed:=true; end; link(r):=s; r:=s; end; q:=link(q); qq:=info(q); end @ It is convenient to have another subroutine for the special case of |p_plus_fq| when |f=1.0|. In this routine lists |p| and |q| are both of the same type~|t| (either |dependent| or |proto_dependent|). @p function p_plus_q(@!p:pointer;@!q:pointer;@!t:small_number):pointer; label done; var @!pp,@!qq:pointer; {|info(p)| and |info(q)|, respectively} @!r,@!s:pointer; {for list manipulation} @!threshold:integer; {defines a neighborhood of zero} @!v:integer; {temporary register} begin if t=dependent then threshold:=fraction_threshold else threshold:=scaled_threshold; r:=temp_head; pp:=info(p); qq:=info(q); loop@+ if pp=qq then if pp=null then goto done else @ else if value(pp)= begin v:=value(p)+value(q); value(p):=v; s:=p; p:=link(p); pp:=info(p); if abs(v)=coef_bound then if watch_coefs then begin type(qq):=independent_needing_fix; fix_needed:=true; end; link(r):=s; r:=s; end; q:=link(q); qq:=info(q); end @ A somewhat simpler routine will multiply a dependency list by a given constant~|v|. The constant is either a |fraction| less than |fraction_one|, or it is |scaled|. In the latter case we might be forced to convert a dependency list to a proto-dependency list. Parameters |t0| and |t1| are the list types before and after; they should agree unless |t0=dependent| and |t1=proto_dependent| and |v_is_scaled=true|. @p function p_times_v(@!p:pointer;@!v:integer; @!t0,@!t1:small_number;@!v_is_scaled:boolean):pointer; var @!r,@!s:pointer; {for list manipulation} @!w:integer; {tentative coefficient} @!threshold:integer; @!scaling_down:boolean; begin if t0<>t1 then scaling_down:=true@+else scaling_down:=not v_is_scaled; if t1=dependent then threshold:=half_fraction_threshold else threshold:=half_scaled_threshold; r:=temp_head; while info(p)<>null do begin if scaling_down then w:=take_fraction(v,value(p)) else w:=take_scaled(v,value(p)); if abs(w)<=threshold then begin s:=link(p); free_node(p,dep_node_size); p:=s; end else begin if abs(w)>=coef_bound then begin fix_needed:=true; type(info(p)):=independent_needing_fix; end; link(r):=p; r:=p; value(p):=w; p:=link(p); end; end; link(r):=p; if v_is_scaled then value(p):=take_scaled(value(p),v) else value(p):=take_fraction(value(p),v); p_times_v:=link(temp_head); end; @ Similarly, we sometimes need to divide a dependency list by a given |scaled| constant. @= function p_over_v(@!p:pointer;@!v:scaled; @!t0,@!t1:small_number):pointer; var @!r,@!s:pointer; {for list manipulation} @!w:integer; {tentative coefficient} @!threshold:integer; @!scaling_down:boolean; begin if t0<>t1 then scaling_down:=true@+else scaling_down:=false; if t1=dependent then threshold:=half_fraction_threshold else threshold:=half_scaled_threshold; r:=temp_head; while info(p)<>null do begin if scaling_down then if abs(v)<@'2000000 then w:=make_scaled(value(p),v*@'10000) else w:=make_scaled(round_fraction(value(p)),v) else w:=make_scaled(value(p),v); if abs(w)<=threshold then begin s:=link(p); free_node(p,dep_node_size); p:=s; end else begin if abs(w)>=coef_bound then begin fix_needed:=true; type(info(p)):=independent_needing_fix; end; link(r):=p; r:=p; value(p):=w; p:=link(p); end; end; link(r):=p; value(p):=make_scaled(value(p),v); p_over_v:=link(temp_head); end; @ Here's another utility routine for dependency lists. When an independent variable becomes dependent, we want to remove it from all existing dependencies. The |p_with_x_becoming_q| function computes the dependency list of~|p| after variable~|x| has been replaced by~|q|. This procedure has basically the same calling conventions as |p_plus_fq|: List~|q| is unchanged; list~|p| is destroyed; the constant node and the final link are inherited from~|p|; and the fourth parameter tells whether or not |p| is |proto_dependent|. However, the global variable |dep_final| is not altered if |x| does not occur in list~|p|. @p function p_with_x_becoming_q(@!p,@!x,@!q:pointer;@!t:small_number):pointer; var @!r,@!s:pointer; {for list manipulation} @!v:integer; {coefficient of |x|} @!sx:integer; {serial number of |x|} begin s:=p; r:=temp_head; sx:=value(x); while value(info(s))>sx do begin r:=s; s:=link(s); end; if info(s)<>x then p_with_x_becoming_q:=p else begin link(temp_head):=p; link(r):=link(s); v:=value(s); free_node(s,dep_node_size); p_with_x_becoming_q:=p_plus_fq(link(temp_head),v,q,t,dependent); end; end; @ Here's a simple procedure that reports an error when a variable has just received a known value that's out of the required range. @= procedure val_too_big(@!x:scaled); begin if internal[warning_check]>0 then begin print_err("Value is too large ("); print_scaled(x); print_char(")"); @.Value is too large@> help4("The equation I just processed has given some variable")@/ ("a value of 4096 or more. Continue and I'll try to cope")@/ ("with that big value; but it might be dangerous.")@/ ("(Set warningcheck:=0 to suppress this message.)"); error; end; end; @ When a dependent variable becomes known, the following routine removes its dependency list. Here |p| points to the variable, and |q| points to the dependency list (which is one node long). @= procedure make_known(@!p,@!q:pointer); var @!t:dependent..proto_dependent; {the previous type} begin prev_dep(link(q)):=prev_dep(p); link(prev_dep(p)):=link(q); t:=type(p); type(p):=known; value(p):=value(q); free_node(q,dep_node_size); if abs(value(p))>=fraction_one then val_too_big(value(p)); if internal[tracing_equations]>0 then if interesting(p) then begin begin_diagnostic; print_nl("#### "); @:]]]\#\#\#\#_}{\.{\#\#\#\#}@> print_variable_name(p); print_char("="); print_scaled(value(p)); end_diagnostic(false); end; if cur_exp=p then if cur_type=t then begin cur_type:=known; cur_exp:=value(p); free_node(p,value_node_size); end; end; @ The |fix_dependencies| routine is called into action when |fix_needed| has been triggered. The program keeps a list~|s| of independent variables whose coefficients must be divided by~4. In unusual cases, this fixup process might reduce one or more coefficients to zero, so that a variable will become known more or less by default. @= procedure fix_dependencies; label done; var @!p,@!q,@!r,@!s,@!t:pointer; {list manipulation registers} @!x:pointer; {an independent variable} begin r:=link(dep_head); s:=null; while r<>dep_head do begin t:=r; @; r:=link(q); if q=dep_list(t) then make_known(t,q); end; while s<>null do begin p:=link(s); x:=info(s); free_avail(s); s:=p; type(x):=independent; value(x):=value(x)+2; end; fix_needed:=false; end; @ @d independent_being_fixed=1 {this variable already appears in |s|} @= r:=value_loc(t); {|link(r)=dep_list(t)|} loop@+ begin q:=link(r); x:=info(q); if x=null then goto done; if type(x)<=independent_being_fixed then begin if type(x)28 then single_dependency:=const_dependency(0) else begin q:=get_node(dep_node_size); value(q):=two_to_the[28-m]; info(q):=p;@/ link(q):=const_dependency(0); single_dependency:=q; end; end; @ We sometimes need to make an exact copy of a dependency list. @p function copy_dep_list(@!p:pointer):pointer; label done; var @!q:pointer; {the new dependency list} begin q:=get_node(dep_node_size); dep_final:=q; loop@+ begin info(dep_final):=info(p); value(dep_final):=value(p); if info(dep_final)=null then goto done; link(dep_final):=get_node(dep_node_size); dep_final:=link(dep_final); p:=link(p); end; done:copy_dep_list:=q; end; @ But how do variables normally become known? Ah, now we get to the heart of the equation-solving mechanism. The |linear_eq| procedure is given a |dependent| or |proto_dependent| list,~|p|, in which at least one independent variable appears. It equates this list to zero, by choosing an independent variable with the largest coefficient and making it dependent on the others. The newly dependent variable is eliminated from all current dependencies, thereby possibly making other dependent variables known. The given list |p| is, of course, totally destroyed by all this processing. @p procedure linear_eq(@!p:pointer;@!t:small_number); var @!q,@!r,@!s:pointer; {for link manipulation} @!x:pointer; {the variable that loses its independence} @!n:integer; {the number of times |x| had been halved} @!v:integer; {the coefficient of |x| in list |p|} @!prev_r:pointer; {lags one step behind |r|} @!final_node:pointer; {the constant term of the new dependency list} @!w:integer; {a tentative coefficient} begin @; x:=info(q); n:=value(x) mod s_scale;@/ @; if internal[tracing_equations]>0 then @; @; @; if fix_needed then fix_dependencies; end; @ @= q:=p; r:=link(p); v:=value(q); while info(r)<>null do begin if abs(value(r))>abs(v) then begin q:=r; v:=value(r); end; r:=link(r); end @ Here we want to change the coefficients from |scaled| to |fraction|, except in the constant term. In the common case of a trivial equation like `\.{x=3.14}', we will have |v=-fraction_one|, |q=p|, and |t=dependent|. @= s:=temp_head; link(s):=p; r:=p; repeat if r=q then begin link(s):=link(r); free_node(r,dep_node_size); end else begin w:=make_fraction(value(r),v); if abs(w)<=half_fraction_threshold then begin link(s):=link(r); free_node(r,dep_node_size); end else begin value(r):=-w; s:=r; end; end; r:=link(s); until info(r)=null; if t=proto_dependent then value(r):=-make_scaled(value(r),v) else if v<>-fraction_one then value(r):=-make_fraction(value(r),v); final_node:=r; p:=link(temp_head) @ @= if interesting(x) then begin begin_diagnostic; print_nl("## "); print_variable_name(x); @:]]]\#\#_}{\.{\#\#}@> w:=n; while w>0 do begin print("*4"); w:=w-2; end; print_char("="); print_dependency(p,dependent); end_diagnostic(false); end @ @= prev_r:=dep_head; r:=link(dep_head); while r<>dep_head do begin s:=dep_list(r); q:=p_with_x_becoming_q(s,x,p,type(r)); if info(q)=null then make_known(r,q) else begin dep_list(r):=q; repeat q:=link(q); until info(q)=null; prev_r:=q; end; r:=link(prev_r); end @ @= if n>0 then @; if info(p)=null then begin type(x):=known; value(x):=value(p); if abs(value(x))>=fraction_one then val_too_big(value(x)); free_node(p,dep_node_size); if cur_exp=x then if cur_type=independent then begin cur_exp:=value(x); cur_type:=known; free_node(x,value_node_size); end; end else begin type(x):=dependent; dep_final:=final_node; new_dep(x,p); if cur_exp=x then if cur_type=independent then cur_type:=dependent; end @ @= begin s:=temp_head; link(temp_head):=p; r:=p; repeat if n>30 then w:=0 else w:=value(r) div two_to_the[n]; if (abs(w)<=half_fraction_threshold)and(info(r)<>null) then begin link(s):=link(r); free_node(r,dep_node_size); end else begin value(r):=w; s:=r; end; r:=link(s); until info(s)=null; p:=link(temp_head); end @ The |check_mem| procedure, which is used only when \MP\ is being debugged, makes sure that the current dependency lists are well formed. @= q:=dep_head; p:=link(q); while p<>dep_head do begin if prev_dep(p)<>q then begin print_nl("Bad PREVDEP at "); print_int(p); @.Bad PREVDEP...@> end; p:=dep_list(p); loop @+begin r:=info(p); q:=p; p:=link(q); if r=null then goto done3; if value(info(p))>=value(r) then begin print_nl("Out of order at "); print_int(p); @.Out of order...@> end; end; done3: do_nothing; end @* \[25] Dynamic nonlinear equations. Variables of numeric type are maintained by the general scheme of independent, dependent, and known values that we have just studied; and the components of pair and transform variables are handled in the same way. But \MP\ also has five other types of values: \&{boolean}, \&{string}, \&{pen}, \&{path}, and \&{picture}; what about them? Equations are allowed between nonlinear quantities, but only in a simple form. Two variables that haven't yet been assigned values are either equal to each other, or they're not. Before a boolean variable has received a value, its type is |unknown_boolean|; similarly, there are variables whose type is |unknown_string|, |unknown_pen|, |unknown_path|, and |unknown_picture|. In such cases the value is either |null| (which means that no other variables are equivalent to this one), or it points to another variable of the same undefined type. The pointers in the latter case form a cycle of nodes, which we shall call a ``ring.'' Rings of undefined variables may include capsules, which arise as intermediate results within expressions or as \&{expr} parameters to macros. When one member of a ring receives a value, the same value is given to all the other members. In the case of paths and pictures, this implies making separate copies of a potentially large data structure; users should restrain their enthusiasm for such generality, unless they have lots and lots of memory space. @ The following procedure is called when a capsule node is being added to a ring (e.g., when an unknown variable is mentioned in an expression). @p function new_ring_entry(@!p:pointer):pointer; var q:pointer; {the new capsule node} begin q:=get_node(value_node_size); name_type(q):=capsule; type(q):=type(p); if value(p)=null then value(q):=p@+else value(q):=value(p); value(p):=q; new_ring_entry:=q; end; @ Conversely, we might delete a capsule or a variable before it becomes known. The following procedure simply detaches a quantity from its ring, without recycling the storage. @= procedure ring_delete(@!p:pointer); var @!q:pointer; begin q:=value(p); if q<>null then if q<>p then begin while value(q)<>p do q:=value(q); value(q):=value(p); end; end; @ Eventually there might be an equation that assigns values to all of the variables in a ring. The |nonlinear_eq| subroutine does the necessary propagation of values. If the parameter |flush_p| is |true|, node |p| itself needn't receive a value, it will soon be recycled. @p procedure nonlinear_eq(@!v:integer;@!p:pointer;@!flush_p:boolean); var @!t:small_number; {the type of ring |p|} @!q,@!r:pointer; {link manipulation registers} begin t:=type(p)-unknown_tag; q:=value(p); if flush_p then type(p):=vacuous@+else p:=q; repeat r:=value(q); type(q):=t; case t of boolean_type: value(q):=v; string_type: begin value(q):=v; add_str_ref(v); end; pen_type: value(q):=copy_pen(v); path_type: value(q):=copy_path(v); picture_type: begin value(q):=v; add_edge_ref(v); end; end; {there ain't no more cases} q:=r; until q=p; end; @ If two members of rings are equated, and if they have the same type, the |ring_merge| procedure is called on to make them equivalent. @p procedure ring_merge(@!p,@!q:pointer); label exit; var @!r:pointer; {traverses one list} begin r:=value(p); while r<>p do begin if r=q then begin @; return; end; r:=value(r); end; r:=value(p); value(p):=value(q); value(q):=r; exit:end; @ @= begin print_err("Redundant equation");@/ @.Redundant equation@> help2("I already knew that this equation was true.")@/ ("But perhaps no harm has been done; let's continue.");@/ put_get_error; end @* \[26] Introduction to the syntactic routines. Let's pause a moment now and try to look at the Big Picture. The \MP\ program consists of three main parts: syntactic routines, semantic routines, and output routines. The chief purpose of the syntactic routines is to deliver the user's input to the semantic routines, while parsing expressions and locating operators and operands. The semantic routines act as an interpreter responding to these operators, which may be regarded as commands. And the output routines are periodically called on to produce compact font descriptions that can be used for typesetting or for making interim proof drawings. We have discussed the basic data structures and many of the details of semantic operations, so we are good and ready to plunge into the part of \MP\ that actually controls the activities. Our current goal is to come to grips with the |get_next| procedure, which is the keystone of \MP's input mechanism. Each call of |get_next| sets the value of three variables |cur_cmd|, |cur_mod|, and |cur_sym|, representing the next input token. $$\vbox{\halign{#\hfil\cr \hbox{|cur_cmd| denotes a command code from the long list of codes given earlier;}\cr \hbox{|cur_mod| denotes a modifier of the command code;}\cr \hbox{|cur_sym| is the hash address of the symbolic token that was just scanned,}\cr \hbox{\qquad or zero in the case of a numeric or string or capsule token.}\cr}}$$ Underlying this external behavior of |get_next| is all the machinery necessary to convert from character files to tokens. At a given time we may be only partially finished with the reading of several files (for which \&{input} was specified), and partially finished with the expansion of some user-defined macros and/or some macro parameters, and partially finished reading some text that the user has inserted online, and so on. When reading a character file, the characters must be converted to tokens; comments and blank spaces must be removed, numeric and string tokens must be evaluated. To handle these situations, which might all be present simultaneously, \MP\ uses various stacks that hold information about the incomplete activities, and there is a finite state control for each level of the input mechanism. These stacks record the current state of an implicitly recursive process, but the |get_next| procedure is not recursive. @= @!cur_cmd: eight_bits; {current command set by |get_next|} @!cur_mod: integer; {operand of current command} @!cur_sym: halfword; {hash address of current symbol} @ The |print_cmd_mod| routine prints a symbolic interpretation of a command code and its modifier. It consists of a rather tedious sequence of print commands, and most of it is essentially an inverse to the |primitive| routine that enters a \MP\ primitive into |hash| and |eqtb|. Therefore almost all of this procedure appears elsewhere in the program, together with the corresponding |primitive| calls. @= procedure print_cmd_mod(@!c,@!m:integer); begin case c of @t\4@>@@/ othercases print("[unknown command code!]") endcases; end; @ Here is a procedure that displays a given command in braces, in the user's transcript file. @d show_cur_cmd_mod==show_cmd_mod(cur_cmd,cur_mod) @p procedure show_cmd_mod(@!c,@!m:integer); begin begin_diagnostic; print_nl("{"); print_cmd_mod(c,m); print_char("}"); end_diagnostic(false); end; @* \[27] Input stacks and states. The state of \MP's input mechanism appears in the input stack, whose entries are records with five fields, called |index|, |start|, |loc|, |limit|, and |name|. The top element of this stack is maintained in a global variable for which no subscripting needs to be done; the other elements of the stack appear in an array. Hence the stack is declared thus: @= @!in_state_record = record @!index_field: quarterword; @!start_field,@!loc_field, @!limit_field, @!name_field: halfword; end; @ @= @!input_stack : array[0..stack_size] of in_state_record; @!input_ptr : 0..stack_size; {first unused location of |input_stack|} @!max_in_stack: 0..stack_size; {largest value of |input_ptr| when pushing} @!cur_input : in_state_record; {the ``top'' input state} @ We've already defined the special variable |@!loc==cur_input.loc_field| in our discussion of basic input-output routines. The other components of |cur_input| are defined in the same way: @d index==cur_input.index_field {reference for buffer information} @d start==cur_input.start_field {starting position in |buffer|} @d limit==cur_input.limit_field {end of current line in |buffer|} @d name==cur_input.name_field {name of the current file} @ Let's look more closely now at the five control variables (|index|,~|start|,~|loc|,~|limit|,~|name|), assuming that \MP\ is reading a line of characters that have been input from some file or from the user's terminal. There is an array called |buffer| that acts as a stack of all lines of characters that are currently being read from files, including all lines on subsidiary levels of the input stack that are not yet completed. \MP\ will return to the other lines when it is finished with the present input file. (Incidentally, on a machine with byte-oriented addressing, it would be appropriate to combine |buffer| with the |str_pool| array, letting the buffer entries grow downward from the top of the string pool and checking that these two tables don't bump into each other.) The line we are currently working on begins in position |start| of the buffer; the next character we are about to read is |buffer[loc]|; and |limit| is the location of the last character present. We always have |loc<=limit|. For convenience, |buffer[limit]| has been set to |"%"|, so that the end of a line is easily sensed. The |name| variable is a string number that designates the name of the current file, if we are reading an ordinary text file. Special codes |is_term..max_spec_src| indicate other sources of input text. @d is_term=0 {|name| value when reading from the terminal for normal input} @d is_read=1 {|name| value when executing a \&{readstring} or \&{readfrom}} @d is_scantok=2 {|name| value when reading text generated by \&{scantokens}} @d max_spec_src=is_scantok @ Additional information about the current line is available via the |index| variable, which counts how many lines of characters are present in the buffer below the current level. We have |index=0| when reading from the terminal and prompting the user for each line; then if the user types, e.g., `\.{input figs}', we will have |index=1| while reading the file \.{figs.mp}. However, it does not follow that |index| is the same as the input stack pointer, since many of the levels on the input stack may come from token lists and some |index| values may correspond to \.{MPX} files that are not currently on the stack. The global variable |in_open| is equal to the highest |index| value counting \.{MPX} files but excluding token-list input levels. Thus, the number of partially read lines in the buffer is |in_open+1| and we have |in_open>=index| when we are not reading a token list. If we are not currently reading from the terminal, we are reading from the file variable |input_file[index]|. We use the notation |terminal_input| as a convenient abbreviation for |name=is_term|, and |cur_file| as an abbreviation for |input_file[index]|. When \MP\ is not reading from the terminal, the global variable |line| contains the line number in the current file, for use in error messages. More precisely, |line| is a macro for |line_stack[index]| and the |line_stack| array gives the line number for each file in the |input_file| array. When an \.{MPX} file is opened the file name is stored in the |mpx_name| array so that the name doesn't get lost when the file is temporarily removed from the input stack. Thus when |input_file[k]| is an \.{MPX} file, its name is |mpx_name[k]| and it contains translated \TeX\ pictures for |input_file[k-1]|. Since this is not an \.{MPX} file, we have $$ \hbox{|mpx_name[k-1]<=absent|}. $$ This |name| field is set to |finished| when |input_file[k]| is completely read. If more information about the input state is needed, it can be included in small arrays like those shown here. For example, the current page or segment number in the input file might be put into a variable |@!page|, that is really a macro for the current entry in `\ignorespaces|@!page_stack:array[0..max_in_open] of integer|\unskip' by analogy with |line_stack|. @^system dependencies@> @d terminal_input==(name=is_term) {are we reading from the terminal?} @d cur_file==input_file[index] {the current |alpha_file| variable} @d line==line_stack[index] {current line number in the current source file} @d in_name==iname_stack[index] {a string used to construct \.{MPX} file names} @d in_area==iarea_stack[index] {another string for naming \.{MPX} files} @d absent=1 {|name_field| value for unused |mpx_in_stack| entries} @d mpx_reading==(mpx_name[index]>absent) {when reading a file, is it an \.{MPX} file?} @d finished=0 {|name_field| value when the corresponding \.{MPX} file is finished} @= @!in_open : 0..max_in_open; {the number of lines in the buffer, less one} @!open_parens : 0..max_in_open; {the number of open text files} @!input_file : array[1..max_in_open] of alpha_file; @!line_stack : array[0..max_in_open] of integer; {the line number for each file} @!iname_stack : array[0..max_in_open] of str_number; {used for naming \.{MPX} files} @!iarea_stack : array[0..max_in_open] of str_number; {used for naming \.{MPX} files} @!mpx_name : array[0..max_in_open] of halfword; @ However, all this discussion about input state really applies only to the case that we are inputting from a file. There is another important case, namely when we are currently getting input from a token list. In this case |index>max_in_open|, and the conventions about the other state variables are different: \yskip\hang|loc| is a pointer to the current node in the token list, i.e., the node that will be read next. If |loc=null|, the token list has been fully read. \yskip\hang|start| points to the first node of the token list; this node may or may not contain a reference count, depending on the type of token list involved. \yskip\hang|token_type|, which takes the place of |index| in the discussion above, is a code number that explains what kind of token list is being scanned. \yskip\hang|name| points to the |eqtb| address of the control sequence being expanded, if the current token list is a macro not defined by \&{vardef}. Macros defined by \&{vardef} have |name=null|; their name can be deduced by looking at their first two parameters. \yskip\hang|param_start|, which takes the place of |limit|, tells where the parameters of the current macro or loop text begin in the |param_stack|. \yskip\noindent The |token_type| can take several values, depending on where the current token list came from: \yskip \indent|forever_text|, if the token list being scanned is the body of a \&{forever} loop; \indent|loop_text|, if the token list being scanned is the body of a \&{for} or \&{forsuffixes} loop; \indent|parameter|, if a \&{text} or \&{suffix} parameter is being scanned; \indent|backed_up|, if the token list being scanned has been inserted as `to be read again'. \indent|inserted|, if the token list being scanned has been inserted as part of error recovery; \indent|macro|, if the expansion of a user-defined symbolic token is being scanned. \yskip\noindent The token list begins with a reference count if and only if |token_type= macro|. @^reference counts@> @d token_type==index {type of current token list} @d token_state==(index>max_in_open) {are we scanning a token list?} @d file_state==(index<=max_in_open) {are we scanning a file line?} @d param_start==limit {base of macro parameters in |param_stack|} @d forever_text=max_in_open+1 {|token_type| code for loop texts} @d loop_text=max_in_open+2 {|token_type| code for loop texts} @d parameter=max_in_open+3 {|token_type| code for parameter texts} @d backed_up=max_in_open+4 {|token_type| code for texts to be reread} @d inserted=max_in_open+5 {|token_type| code for inserted texts} @d macro=max_in_open+6 {|token_type| code for macro replacement texts} @ The |param_stack| is an auxiliary array used to hold pointers to the token lists for parameters at the current level and subsidiary levels of input. This stack grows at a different rate from the others. @= @!param_stack:array [0..param_size] of pointer; {token list pointers for parameters} @!param_ptr:0..param_size; {first unused entry in |param_stack|} @!max_param_stack:integer; {largest value of |param_ptr|} @ Notice that the |line| isn't valid when |token_state| is true because it depends on |index|. If we really need to know the line number for the topmost file in the index stack we use the following function. If a page number or other information is needed, this routine should be modified to compute it as well. @^system dependencies@> @= function true_line: integer; var @!k:0..stack_size; {an index into the input stack} begin if file_state and (name>max_spec_src) then true_line:=line else begin k:=input_ptr; while (k>0)and(input_stack[k].index_field>max_in_open)or@| (input_stack[k].name_field<=max_spec_src) do decr(k); true_line:=line_stack[k]; end; end; @ Thus, the ``current input state'' can be very complicated indeed; there can be many levels and each level can arise in a variety of ways. The |show_context| procedure, which is used by \MP's error-reporting routine to print out the current input state on all levels down to the most recent line of characters from an input file, illustrates most of these conventions. The global variable |file_ptr| contains the lowest level that was displayed by this procedure. @= @!file_ptr:0..stack_size; {shallowest level shown by |show_context|} @ The status at each level is indicated by printing two lines, where the first line indicates what was read so far and the second line shows what remains to be read. The context is cropped, if necessary, so that the first line contains at most |half_error_line| characters, and the second contains at most |error_line|. Non-current input levels whose |token_type| is `|backed_up|' are shown only if they have not been fully read. @p procedure show_context; {prints where the scanner is} label done; var @!old_setting:0..max_selector; {saved |selector| setting} @@/ begin file_ptr:=input_ptr; input_stack[file_ptr]:=cur_input; {store current state} loop@+begin cur_input:=input_stack[file_ptr]; {enter into the context} @; if file_state then if (name>max_spec_src) or (file_ptr=0) then goto done; decr(file_ptr); end; done: cur_input:=input_stack[input_ptr]; {restore original state} end; @ @= if (file_ptr=input_ptr) or file_state or (token_type<>backed_up) or (loc<>null) then {we omit backed-up token lists that have already been read} begin tally:=0; {get ready to count characters} old_setting:=selector; if file_state then begin @; @; end else begin @; @; end; selector:=old_setting; {stop pseudoprinting} @; end @ This routine should be changed, if necessary, to give the best possible indication of where the current line resides in the input file. For example, on some systems it is best to print both a page and line number. @^system dependencies@> @= if name>max_spec_src then begin print_nl("l."); print_int(true_line); end else if terminal_input then if file_ptr=0 then print_nl("<*>") @+else print_nl("") else if name=is_scantok then print_nl("") else print_nl(""); print_char(" ") @ @= case token_type of forever_text: print_nl(" "); loop_text: @; parameter: print_nl(" "); backed_up: if loc=null then print_nl(" ") else print_nl(" "); inserted: print_nl(" "); macro: begin print_ln; if name<>null then print(text(name)) else @; print("->"); end; othercases print_nl("?") {this should never happen} @.?\relax@> endcases @ The parameter that corresponds to a loop text is either a token list (in the case of \&{forsuffixes}) or a ``capsule'' (in the case of \&{for}). We'll discuss capsules later; for now, all we need to know is that the |link| field in a capsule parameter is |void| and that |print_exp(p,0)| displays the value of capsule~|p| in abbreviated form. @d void==null+1 {a null pointer different from |null|} @= begin print_nl("null then if link(p)=void then print_exp(p,0) {we're in a \&{for} loop} else show_token_list(p,null,20,tally); print(")> "); end @ The first two parameters of a macro defined by \&{vardef} will be token lists representing the macro's prefix and ``at point.'' By putting these together, we get the macro's full name. @= begin p:=param_stack[param_start]; if p=null then show_token_list(param_stack[param_start+1],null,20,tally) else begin q:=p; while link(q)<>null do q:=link(q); link(q):=param_stack[param_start+1]; show_token_list(p,null,20,tally); link(q):=null; end; end @ Now it is necessary to explain a little trick. We don't want to store a long string that corresponds to a token list, because that string might take up lots of memory; and we are printing during a time when an error message is being given, so we dare not do anything that might overflow one of \MP's tables. So `pseudoprinting' is the answer: We enter a mode of printing that stores characters into a buffer of length |error_line|, where character $k+1$ is placed into \hbox{|trick_buf[k mod error_line]|} if |k(error_line, tally+1+error_line-half_error_line)|. At the end of the pseudoprinting, the values of |first_count|, |tally|, and |trick_count| give us all the information we need to print the two lines, and all of the necessary text is in |trick_buf|. Namely, let |l| be the length of the descriptive information that appears on the first line. The length of the context information gathered for that line is |k=first_count|, and the length of the context information gathered for line~2 is $m=\min(|tally|, |trick_count|)-k$. If |l+k<=h|, where |h=half_error_line|, we print |trick_buf[0..k-1]| after the descriptive information on line~1, and set |n:=l+k|; here |n| is the length of line~1. If $l+k>h$, some cropping is necessary, so we set |n:=h| and print `\.{...}' followed by $$\hbox{|trick_buf[(l+k-h+3)..k-1]|,}$$ where subscripts of |trick_buf| are circular modulo |error_line|. The second line consists of |n|~spaces followed by |trick_buf[k..(k+m-1)]|, unless |n+m>error_line|; in the latter case, further cropping is done. This is easier to program than to explain. @= @!i:0..buf_size; {index into |buffer|} @!l:integer; {length of descriptive information on line 1} @!m:integer; {context information gathered for line 2} @!n:0..error_line; {length of line 1} @!p: integer; {starting or ending place in |trick_buf|} @!q: integer; {temporary index} @ The following code tells the print routines to gather the desired information. @d begin_pseudoprint== begin l:=tally; tally:=0; selector:=pseudo; trick_count:=1000000; end @d set_trick_count== begin first_count:=tally; trick_count:=tally+1+error_line-half_error_line; if trick_count= if trick_count=1000000 then set_trick_count; {|set_trick_count| must be performed} if tallyerror_line then print("...") @ But the trick is distracting us from our current goal, which is to understand the input state. So let's concentrate on the data structures that are being pseudoprinted as we finish up the |show_context| procedure. @= begin_pseudoprint; if limit>0 then for i:=start to limit-1 do begin if i=loc then set_trick_count; print(buffer[i]); end @ @= begin_pseudoprint; if token_type<>macro then show_token_list(start,loc,100000,0) else show_macro(start,loc,100000) @ Here is the missing piece of |show_token_list| that is activated when the token beginning line~2 is about to be shown: @=set_trick_count @* \[28] Maintaining the input stacks. The following subroutines change the input status in commonly needed ways. First comes |push_input|, which stores the current state and creates a new level (having, initially, the same properties as the old). @d push_input==@t@> {enter a new input level, save the old} begin if input_ptr>max_in_stack then begin max_in_stack:=input_ptr; if input_ptr=stack_size then overflow("input stack size",stack_size); @:MetaPost capacity exceeded input stack size}{\quad input stack size@> end; input_stack[input_ptr]:=cur_input; {stack the record} incr(input_ptr); end @ And of course what goes up must come down. @d pop_input==@t@> {leave an input level, re-enter the old} begin decr(input_ptr); cur_input:=input_stack[input_ptr]; end @ Here is a procedure that starts a new level of token-list input, given a token list |p| and its type |t|. If |t=macro|, the calling routine should set |name|, reset~|loc|, and increase the macro's reference count. @d back_list(#)==begin_token_list(#,backed_up) {backs up a simple token list} @p procedure begin_token_list(@!p:pointer;@!t:quarterword); begin push_input; start:=p; token_type:=t; param_start:=param_ptr; loc:=p; end; @ When a token list has been fully scanned, the following computations should be done as we leave that level of input. @^inner loop@> @p procedure end_token_list; {leave a token-list input level} label done; var @!p:pointer; {temporary register} begin if token_type>=backed_up then {token list to be deleted} if token_type<=inserted then begin flush_token_list(start); goto done; end else delete_mac_ref(start); {update reference count} while param_ptr>param_start do {parameters must be flushed} begin decr(param_ptr); p:=param_stack[param_ptr]; if p<>null then if link(p)=void then {it's an \&{expr} parameter} begin recycle_value(p); free_node(p,value_node_size); end else flush_token_list(p); {it's a \&{suffix} or \&{text} parameter} end; done: pop_input; check_interrupt; end; @ The contents of |cur_cmd,cur_mod,cur_sym| are placed into an equivalent token by the |cur_tok| routine. @^inner loop@> @p @t\4@>@@;@/ function cur_tok:pointer; var @!p:pointer; {a new token node} @!save_type:small_number; {|cur_type| to be restored} @!save_exp:integer; {|cur_exp| to be restored} begin if cur_sym=0 then if cur_cmd=capsule_token then begin save_type:=cur_type; save_exp:=cur_exp; make_exp_copy(cur_mod); p:=stash_cur_exp; link(p):=null; cur_type:=save_type; cur_exp:=save_exp; end else begin p:=get_node(token_node_size); value(p):=cur_mod; name_type(p):=token; if cur_cmd=numeric_token then type(p):=known else type(p):=string_type; end else begin fast_get_avail(p); info(p):=cur_sym; end; cur_tok:=p; end; @ Sometimes \MP\ has read too far and wants to ``unscan'' what it has seen. The |back_input| procedure takes care of this by putting the token just scanned back into the input stream, ready to be read again. If |cur_sym<>0|, the values of |cur_cmd| and |cur_mod| are irrelevant. @p procedure back_input; {undoes one token of input} var @!p:pointer; {a token list of length one} begin p:=cur_tok; while token_state and(loc=null) do end_token_list; {conserve stack space} back_list(p); end; @ The |back_error| routine is used when we want to restore or replace an offending token just before issuing an error message. We disable interrupts during the call of |back_input| so that the help message won't be lost. @p procedure back_error; {back up one token and call |error|} begin OK_to_interrupt:=false; back_input; OK_to_interrupt:=true; error; end; @# procedure ins_error; {back up one inserted token and call |error|} begin OK_to_interrupt:=false; back_input; token_type:=inserted; OK_to_interrupt:=true; error; end; @ The |begin_file_reading| procedure starts a new level of input for lines of characters to be read from a file, or as an insertion from the terminal. It does not take care of opening the file, nor does it set |loc| or |limit| or |line|. @^system dependencies@> @p procedure begin_file_reading; begin if in_open=max_in_open then overflow("text input levels",max_in_open); @:MetaPost capacity exceeded text input levels}{\quad text input levels@> if first=buf_size then overflow("buffer size",buf_size); @:MetaPost capacity exceeded buffer size}{\quad buffer size@> incr(in_open); push_input; index:=in_open; mpx_name[index]:=absent; start:=first; name:=is_term; {|terminal_input| is now |true|} end; @ Conversely, the variables must be downdated when such a level of input is finished. Any associated \.{MPX} file must also be closed and popped off the file stack. @p procedure end_file_reading; begin if in_open>index then if (mpx_name[in_open]=absent)or(name<=max_spec_src) then confusion("endinput") @:this can't happen endinput}{\quad endinput@> else begin a_close(input_file[in_open]); {close an \.{MPX} file} delete_str_ref(mpx_name[in_open]); decr(in_open); end; first:=start; if index<>in_open then confusion("endinput"); if name>max_spec_src then begin a_close(cur_file); delete_str_ref(name); delete_str_ref(in_name); delete_str_ref(in_area); end; pop_input; decr(in_open); end; @ Here is a function that tries to resume input from an \.{MPX} file already associated with the current input file. It returns |false| if this doesn't work. @p function begin_mpx_reading:boolean; begin if in_open<>index+1 then begin_mpx_reading:=false else begin if mpx_name[in_open]<=absent then confusion("mpx"); @:this can't happen mpx}{\quad mpx@> if first=buf_size then overflow("buffer size",buf_size); @:MetaPost capacity exceeded buffer size}{\quad buffer size@> push_input; index:=in_open; start:=first; name:=mpx_name[in_open]; add_str_ref(name); @; begin_mpx_reading:=true; end; end; @ This procedure temporarily stops reading an \.{MPX} file. @p procedure end_mpx_reading; begin if in_open<>index then confusion("mpx"); @:this can't happen mpx}{\quad mpx@> if loc; first:=start; pop_input; end; @ Here we enforce a restriction that simplifies the input stacks considerably. This should not inconvenience the user because \.{MPX} files are generated by an auxiliary program called \.{DVItoMP}. @ @= begin print_err("`mpxbreak' must be at the end of a line"); help4("This file contains picture expressions for btex...etex")@/ ("blocks. Such files are normally generated automatically")@/ ("but this one seems to be messed up. I'm going to ignore")@/ ("the rest of this line.");@/ error; end @ In order to keep the stack from overflowing during a long sequence of inserted `\.{show}' commands, the following routine removes completed error-inserted lines from memory. @p procedure clear_for_error_prompt; begin while file_state and terminal_input and@| (input_ptr>0)and(loc=limit) do end_file_reading; print_ln; clear_terminal; end; @ To get \MP's whole input mechanism going, we perform the following actions. @= begin input_ptr:=0; max_in_stack:=0; in_open:=0; open_parens:=0; max_buf_stack:=0; param_ptr:=0; max_param_stack:=0; first:=1; start:=1; index:=0; line:=0; name:=is_term; mpx_name[0]:=absent; force_eof:=false; if not init_terminal then goto final_end; limit:=last; first:=last+1; {|init_terminal| has set |loc| and |last|} end; @* \[29] Getting the next token. The heart of \MP's input mechanism is the |get_next| procedure, which we shall develop in the next few sections of the program. Perhaps we shouldn't actually call it the ``heart,'' however; it really acts as \MP's eyes and mouth, reading the source files and gobbling them up. And it also helps \MP\ to regurgitate stored token lists that are to be processed again. The main duty of |get_next| is to input one token and to set |cur_cmd| and |cur_mod| to that token's command code and modifier. Furthermore, if the input token is a symbolic token, that token's |hash| address is stored in |cur_sym|; otherwise |cur_sym| is set to zero. Underlying this simple description is a certain amount of complexity because of all the cases that need to be handled. However, the inner loop of |get_next| is reasonably short and fast. @ Before getting into |get_next|, we need to consider a mechanism by which \MP\ helps keep errors from propagating too far. Whenever the program goes into a mode where it keeps calling |get_next| repeatedly until a certain condition is met, it sets |scanner_status| to some value other than |normal|. Then if an input file ends, or if an `\&{outer}' symbol appears, an appropriate error recovery will be possible. The global variable |warning_info| helps in this error recovery by providing additional information. For example, |warning_info| might indicate the name of a macro whose replacement text is being scanned. @d normal=0 {|scanner_status| at ``quiet times''} @d skipping=1 {|scanner_status| when false conditional text is being skipped} @d flushing=2 {|scanner_status| when junk after a statement is being ignored} @d absorbing=3 {|scanner_status| when a \&{text} parameter is being scanned} @d var_defining=4 {|scanner_status| when a \&{vardef} is being scanned} @d op_defining=5 {|scanner_status| when a macro \&{def} is being scanned} @d loop_defining=6 {|scanner_status| when a \&{for} loop is being scanned} @d tex_flushing=7 {|scanner_status| when skipping \TeX\ material} @= @!scanner_status:normal..tex_flushing; {are we scanning at high speed?} @!warning_info:integer; {if so, what else do we need to know, in case an error occurs?} @ @= scanner_status:=normal; @ The following subroutine is called when an `\&{outer}' symbolic token has been scanned or when the end of a file has been reached. These two cases are distinguished by |cur_sym|, which is zero at the end of a file. @p function check_outer_validity:boolean; var @!p:pointer; {points to inserted token list} begin if scanner_status=normal then check_outer_validity:=true else if scanner_status=tex_flushing then @ else begin deletions_allowed:=false; @; if scanner_status>skipping then @ else begin print_err("Incomplete if; all text was ignored after line "); @.Incomplete if...@> print_int(warning_info);@/ help3("A forbidden `outer' token occurred in skipped text.")@/ ("This kind of error happens when you say `if...' and forget")@/ ("the matching `fi'. I've inserted a `fi'; this might work."); if cur_sym=0 then help_line[2]:=@| "The file ended while I was skipping conditional text."; cur_sym:=frozen_fi; ins_error; end; deletions_allowed:=true; check_outer_validity:=false; end; end; @ @= if cur_sym<>0 then check_outer_validity:=true else begin deletions_allowed:=false; print_err("TeX mode didn't end; all text was ignored after line "); print_int(warning_info); help2("The file ended while I was looking for the `etex' to")@/ ("finish this TeX material. I've inserted `etex' now.");@/ cur_sym := frozen_etex; ins_error;@/ deletions_allowed:=true; check_outer_validity:=false; end @ @= if cur_sym<>0 then begin p:=get_avail; info(p):=cur_sym; back_list(p); {prepare to read the symbolic token again} end @ @= begin runaway; {print the definition-so-far} if cur_sym=0 then print_err("File ended") @.File ended while scanning...@> else begin print_err("Forbidden token found"); @.Forbidden token found...@> end; print(" while scanning "); help4("I suspect you have forgotten an `enddef',")@/ ("causing me to read past where you wanted me to stop.")@/ ("I'll try to recover; but if the error is serious,")@/ ("you'd better type `E' or `X' now and fix your file.");@/ case scanner_status of @t\4@>@@; end; {there are no other cases} ins_error; end @ As we consider various kinds of errors, it is also appropriate to change the first line of the help message just given; |help_line[3]| points to the string that might be changed. @= flushing: begin print("to the end of the statement"); help_line[3]:="A previous error seems to have propagated,"; cur_sym:=frozen_semicolon; end; absorbing: begin print("a text argument"); help_line[3]:="It seems that a right delimiter was left out,"; if warning_info=0 then cur_sym:=frozen_end_group else begin cur_sym:=frozen_right_delimiter; equiv(frozen_right_delimiter):=warning_info; end; end; var_defining, op_defining: begin print("the definition of "); if scanner_status=op_defining then print(text(warning_info)) else print_variable_name(warning_info); cur_sym:=frozen_end_def; end; loop_defining: begin print("the text of a "); print(text(warning_info)); print(" loop"); help_line[3]:="I suspect you have forgotten an `endfor',"; cur_sym:=frozen_end_for; end; @ The |runaway| procedure displays the first part of the text that occurred when \MP\ began its special |scanner_status|, if that text has been saved. @= procedure runaway; begin if scanner_status>flushing then begin print_nl("Runaway "); case scanner_status of absorbing: print("text?"); var_defining,op_defining: print("definition?"); loop_defining: print("loop?"); end; {there are no other cases} print_ln; show_token_list(link(hold_head),null,error_line-10,0); end; end; @ We need to mention a procedure that may be called by |get_next|. @p procedure@?firm_up_the_line; forward; @ And now we're ready to take the plunge into |get_next| itself. Note that the behavior depends on the |scanner_status| because percent signs and double quotes need to be passed over when skipping TeX material. @d switch=25 {a label in |get_next|} @d start_numeric_token=85 {another} @d start_decimal_token=86 {and another} @d fin_numeric_token=87 {and still another, although |goto| is considered harmful} @p procedure get_next; {sets |cur_cmd|, |cur_mod|, |cur_sym| to next token} @^inner loop@> label restart, {go here to get the next input token} exit, {go here when the next input token has been got} common_ending, {go here to finish getting a symbolic token} found, {go here when the end of a symbolic token has been found} switch, {go here to branch on the class of an input character} start_numeric_token,start_decimal_token,fin_numeric_token,done; {go here at crucial stages when scanning a number} var @!k:0..buf_size; {an index into |buffer|} @!c:ASCII_code; {the current character in the buffer} @!class:ASCII_code; {its class number} @!n,@!f:integer; {registers for decimal-to-binary conversion} begin restart: cur_sym:=0; if file_state then @ else @; common_ending: @; exit:end; @ When a symbolic token is declared to be `\&{outer}', its command code is increased by |outer_tag|. @^inner loop@> @= cur_cmd:=eq_type(cur_sym); cur_mod:=equiv(cur_sym); if cur_cmd>=outer_tag then if check_outer_validity then cur_cmd:=cur_cmd-outer_tag else goto restart @ A percent sign appears in |buffer[limit]|; this makes it unnecessary to have a special test for end-of-line. @^inner loop@> @= begin switch: c:=buffer[loc]; incr(loc); class:=char_class[c]; case class of digit_class: goto start_numeric_token; period_class: begin class:=char_class[buffer[loc]]; if class>period_class then goto switch else if class end; space_class: goto switch; percent_class: begin if scanner_status=tex_flushing then if loc; check_interrupt; goto switch; end; string_class: if scanner_status=tex_flushing then goto switch else @; isolated_classes: begin k:=loc-1; goto found; end; invalid_class: if scanner_status=tex_flushing then goto switch else @; othercases do_nothing {letters, etc.} endcases;@/ k:=loc-1; while char_class[buffer[loc]]=class do incr(loc); goto found; start_numeric_token:@; start_decimal_token:@; fin_numeric_token:@; found: cur_sym:=id_lookup(k,loc-k); end @ We go to |restart| instead of to |switch|, because |state| might equal |token_list| after the error has been dealt with (cf.\ |clear_for_error_prompt|). @= begin print_err("Text line contains an invalid character"); @.Text line contains...@> help2("A funny symbol that I can't read has just been input.")@/ ("Continue, and I'll forget that it ever happened.");@/ deletions_allowed:=false; error; deletions_allowed:=true; goto restart; end @ @= begin if buffer[loc]="""" then cur_mod:="" else begin k:=loc; buffer[limit+1]:=""""; repeat incr(loc); until buffer[loc]=""""; if loc>limit then @; if loc=k+1 then cur_mod:=buffer[k] else begin str_room(loc-k); repeat append_char(buffer[k]); incr(k); until k=loc; cur_mod:=make_string; end; end; incr(loc); cur_cmd:=string_token; return; end @ We go to |restart| after this error message, not to |switch|, because the |clear_for_error_prompt| routine might have reinstated |token_state| after |error| has finished. @= begin loc:=limit; {the next character to be read on this line will be |"%"|} print_err("Incomplete string token has been flushed"); @.Incomplete string token...@> help3("Strings should finish on the same line as they began.")@/ ("I've deleted the partial string; you might want to")@/ ("insert another by typing, e.g., `I""new string""'.");@/ deletions_allowed:=false; error; deletions_allowed:=true; goto restart; end @ @= n:=c-"0"; while char_class[buffer[loc]]=digit_class do begin if n<32768 then n:=10*n+buffer[loc]-"0"; incr(loc); end; if buffer[loc]="." then if char_class[buffer[loc+1]]=digit_class then goto done; f:=0; goto fin_numeric_token; done: incr(loc) @ @= k:=0; repeat if k<17 then {digits for |k>=17| cannot affect the result} begin dig[k]:=buffer[loc]-"0"; incr(k); end; incr(loc); until char_class[buffer[loc]]<>digit_class; f:=round_decimals(k); if f=unity then begin incr(n); f:=0; end @ @= if n<32768 then @ else if scanner_status<>tex_flushing then begin print_err("Enormous number has been reduced"); @.Enormous number...@> help2("I can't handle numbers bigger than 32767.99998;")@/ ("so I've changed your constant to that maximum amount.");@/ deletions_allowed:=false; error; deletions_allowed:=true; cur_mod:=el_gordo; end; cur_cmd:=numeric_token; return @ @= begin cur_mod:=n*unity+f; if cur_mod>=fraction_one then if (internal[warning_check]>0) and (scanner_status<>tex_flushing) then begin print_err("Number is too large ("); print_scaled(cur_mod); print_char(")"); help3("It is at least 4096. Continue and I'll try to cope")@/ ("with that big value; but it might be dangerous.")@/ ("(Set warningcheck:=0 to suppress this message.)"); error; end; end @ Let's consider now what happens when |get_next| is looking at a token list. @^inner loop@> @= if loc>=hi_mem_min then {one-word token} begin cur_sym:=info(loc); loc:=link(loc); {move to next} if cur_sym>=expr_base then if cur_sym>=suffix_base then @ else begin cur_cmd:=capsule_token; cur_mod:=param_stack[param_start+cur_sym-(expr_base)]; cur_sym:=0; return; end; end else if loc>null then @ else begin {we are done with this token list} end_token_list; goto restart; {resume previous level} end @ @= begin if cur_sym>=text_base then cur_sym:=cur_sym-param_size; {|param_size=text_base-suffix_base|} begin_token_list(param_stack[param_start+cur_sym-(suffix_base)],parameter); goto restart; end @ @= begin if name_type(loc)=token then begin cur_mod:=value(loc); if type(loc)=known then cur_cmd:=numeric_token else begin cur_cmd:=string_token; add_str_ref(cur_mod); end; end else begin cur_mod:=loc; cur_cmd:=capsule_token; end; loc:=link(loc); return; end @ All of the easy branches of |get_next| have now been taken care of. There is one more branch. @= if name>max_spec_src then @ else begin if input_ptr>0 then {text was inserted during error recovery or by \&{scantokens}} begin end_file_reading; goto restart; {resume previous level} end; if selectornonstop_mode then begin if limit=start then {previous line was empty} print_nl("(Please type a command or say `end')"); @.Please type...@> print_ln; first:=start; prompt_input("*"); {input on-line into |buffer|} @.*\relax@> limit:=last; buffer[limit]:="%"; first:=limit+1; loc:=start; end else fatal_error("*** (job aborted, no legal end found)"); @.job aborted@> {nonstop mode, which is intended for overnight batch processing, never waits for on-line input} end @ The global variable |force_eof| is normally |false|; it is set |true| by an \&{endinput} command. @= @!force_eof:boolean; {should the next \&{input} be aborted early?} @ We must decrement |loc| in order to leave the buffer in a valid state when an error condition causes us to |goto restart| without calling |end_file_reading|. @= begin incr(line); first:=start; if not force_eof then begin if input_ln(cur_file,true) then {not end of file} firm_up_the_line {this sets |limit|} else force_eof:=true; end; if force_eof then begin force_eof:=false; decr(loc); if mpx_reading then @ else begin print_char(")"); decr(open_parens); update_terminal; {show user that file has been read} end_file_reading; {resume previous level} if check_outer_validity then goto restart @+else goto restart; end end; buffer[limit]:="%"; first:=limit+1; loc:=start; {ready to read} end @ We should never actually come to the end of an \.{MPX} file because such files should have an \&{mpxbreak} after the translation of the last \&{btex}$\,\ldots\,$\&{etex} block. @= begin mpx_name[index]:=finished; print_err("mpx file ended unexpectedly"); help4("The file had too few picture expressions for btex...etex")@/ ("blocks. Such files are normally generated automatically")@/ ("but this one got messed up. You might want to insert a")@/ ("picture expression now.");@/ deletions_allowed:=false; error; deletions_allowed:=true; cur_sym:=frozen_mpx_break; goto common_ending; end @ Sometimes we want to make it look as though we have just read a blank line without really doing so. @= last:=first; limit:=last; {simulate |input_ln| and |firm_up_the_line|} buffer[limit]:="%"; first:=limit+1; loc:=start @ If the user has set the |pausing| parameter to some positive value, and if nonstop mode has not been selected, each line of input is displayed on the terminal and the transcript file, followed by `\.{=>}'. \MP\ waits for a response. If the response is null (i.e., if nothing is typed except perhaps a few blank spaces), the original line is accepted as it stands; otherwise the line typed is used instead of the line in the file. @p procedure firm_up_the_line; var @!k:0..buf_size; {an index into |buffer|} begin limit:=last; if internal[pausing]>0 then if interaction>nonstop_mode then begin wake_up_terminal; print_ln; if start"); {wait for user response} @.=>@> if last>first then begin for k:=first to last-1 do {move line down in buffer} buffer[k+start-first]:=buffer[k]; limit:=start+last-first; end; end; end; @* \[30] Dealing with \TeX\ material. The \&{btex}$\,\ldots\,$\&{etex} and \&{verbatimtex}$\,\ldots\,$\&{etex} features need to be implemented at a low level in the scanning process so that \MP\ can stay in synch with the a preprocessor that treats blocks of \TeX\ material as they occur in the input file without trying to expand \MP\ macros. Thus we need a special version of |get_next| that does not expand macros and such but does handle \&{btex}, \&{verbatimtex}, etc. The special version of |get_next| is called |get_t_next|. It works by flushing \&{btex}$\,\ldots\,$\&{etex} and \&{verbatimtex}\allowbreak $\,\ldots\,$\&{etex} blocks, switching to the \.{MPX} file when it sees \&{btex}, and switching back when it sees \&{mpxbreak}. @d btex_code=0 @d verbatim_code=1 @ @= primitive("btex",start_tex,btex_code);@/ @!@:btex_}{\&{btex} primitive@> primitive("verbatimtex",start_tex,verbatim_code); @!@:verbatimtex_}{\&{verbatimtex} primitive@> primitive("etex",etex_marker,0); eqtb[frozen_etex]:=eqtb[cur_sym];@/ @!@:etex_}{\&{etex} primitive@> primitive("mpxbreak",mpx_break,0); eqtb[frozen_mpx_break]:=eqtb[cur_sym];@/ @!@:mpx_break_}{\&{mpxbreak} primitive@> @ @= start_tex: if m=btex_code then print("btex") else print("verbatimtex"); etex_marker: print("etex"); mpx_break: print("mpxbreak"); @ Actually, |get_t_next| is a macro that avoids procedure overhead except in the unusual case where \&{btex}, \&{verbatimtex}, \&{etex}, or \&{mpxbreak} is encountered. @d get_t_next==begin get_next; if cur_cmd<=max_pre_command then t_next; end @d TeX_flush=65 {go here to flush to the next ``\&{etex}''} @p procedure@?start_mpx_input; forward;@t\2@> procedure t_next; label TeX_flush, common_ending; var @!old_status:normal..loop_defining; {saves the |scanner_status|} @!old_info:integer; {saves the |warning_info|} begin while cur_cmd<=max_pre_command do begin if cur_cmd=mpx_break then if not file_state or (mpx_name[index]=absent) then @ else begin end_mpx_reading; goto TeX_flush; end else if cur_cmd=start_tex then if token_state or (name<=max_spec_src) then @ else if mpx_reading then @ else if (cur_mod<>verbatim_code)and(mpx_name[index]<>finished) then begin if not begin_mpx_reading then start_mpx_input; end else goto TeX_flush else @; goto common_ending; TeX_flush: @; common_ending: get_next; end; end; @ We could be in the middle of an operation such as skipping false conditional text when \TeX\ material is encountered, so we must be careful to save the |scanner_status|. @= old_status:=scanner_status; old_info:=warning_info; scanner_status:=tex_flushing; warning_info:=line; repeat get_next; until cur_cmd=etex_marker; scanner_status:=old_status; warning_info:=old_info @ @= begin print_err("An mpx file cannot contain btex or verbatimtex blocks"); help4("This file contains picture expressions for btex...etex")@/ ("blocks. Such files are normally generated automatically")@/ ("but this one seems to be messed up. I'll just keep going")@/ ("and hope for the best.");@/ error; end @ @= begin print_err("You can only use `btex' or `verbatimtex' in a file"); help3("I'll have to ignore this preprocessor command because it")@/ ("only works when there is a file to preprocess. You might")@/ ("want to delete everything up to the next `etex`.");@/ error; end @ @= begin print_err("Misplaced mpxbreak"); help2("I'll ignore this preprocessor command because it")@/ ("doesn't belong here");@/ error; end @ @= begin print_err("Extra etex will be ignored"); help1("There is no btex or verbatimtex for this to match");@/ error; end @* \[31] Scanning macro definitions. \MP\ has a variety of ways to tuck tokens away into token lists for later use: Macros can be defined with \&{def}, \&{vardef}, \&{primarydef}, etc.; repeatable code can be defined with \&{for}, \&{forever}, \&{forsuffixes}. All such operations are handled by the routines in this part of the program. The modifier part of each command code is zero for the ``ending delimiters'' like \&{enddef} and \&{endfor}. @d start_def=1 {command modifier for \&{def}} @d var_def=2 {command modifier for \&{vardef}} @d end_def=0 {command modifier for \&{enddef}} @d start_forever=1 {command modifier for \&{forever}} @d end_for=0 {command modifier for \&{endfor}} @= primitive("def",macro_def,start_def);@/ @!@:def_}{\&{def} primitive@> primitive("vardef",macro_def,var_def);@/ @!@:var_def_}{\&{vardef} primitive@> primitive("primarydef",macro_def,secondary_primary_macro);@/ @!@:primary_def_}{\&{primarydef} primitive@> primitive("secondarydef",macro_def,tertiary_secondary_macro);@/ @!@:secondary_def_}{\&{secondarydef} primitive@> primitive("tertiarydef",macro_def,expression_tertiary_macro);@/ @!@:tertiary_def_}{\&{tertiarydef} primitive@> primitive("enddef",macro_def,end_def); eqtb[frozen_end_def]:=eqtb[cur_sym];@/ @!@:end_def_}{\&{enddef} primitive@> @# primitive("for",iteration,expr_base);@/ @!@:for_}{\&{for} primitive@> primitive("forsuffixes",iteration,suffix_base);@/ @!@:for_suffixes_}{\&{forsuffixes} primitive@> primitive("forever",iteration,start_forever);@/ @!@:forever_}{\&{forever} primitive@> primitive("endfor",iteration,end_for); eqtb[frozen_end_for]:=eqtb[cur_sym];@/ @!@:end_for_}{\&{endfor} primitive@> @ @= macro_def:if m<=var_def then if m=start_def then print("def") else if m0 then begin @; if cur_cmd=terminator then @ else if cur_cmd=macro_special then @; end; link(p):=cur_tok; p:=link(p); end; done: link(p):=tail_end; flush_node_list(subst_list); scan_toks:=link(hold_head); end; @ @= begin q:=subst_list; while q<>null do begin if info(q)=cur_sym then begin cur_sym:=value(q); cur_cmd:=relax; goto found; end; q:=link(q); end; found:end @ @= if cur_mod>0 then incr(balance) else begin decr(balance); if balance=0 then goto done; end @ Four commands are intended to be used only within macro texts: \&{quote}, \.{\#\AT!}, \.{\AT!}, and \.{\AT!\#}. They are variants of a single command code called |macro_special|. @d quote=0 {|macro_special| modifier for \&{quote}} @d macro_prefix=1 {|macro_special| modifier for \.{\#\AT!}} @d macro_at=2 {|macro_special| modifier for \.{\AT!}} @d macro_suffix=3 {|macro_special| modifier for \.{\AT!\#}} @= primitive("quote",macro_special,quote);@/ @!@:quote_}{\&{quote} primitive@> primitive("#@@",macro_special,macro_prefix);@/ @!@:]]]\#\AT!_}{\.{\#\AT!} primitive@> primitive("@@",macro_special,macro_at);@/ @!@:]]]\AT!_}{\.{\AT!} primitive@> primitive("@@#",macro_special,macro_suffix);@/ @!@:]]]\AT!\#_}{\.{\AT!\#} primitive@> @ @= macro_special: case m of macro_prefix: print("#@@"); macro_at: print_char("@@"); macro_suffix: print("@@#"); othercases print("quote") endcases; @ @= begin if cur_mod=quote then get_t_next else if cur_mod<=suffix_count then cur_sym:=suffix_base-1+cur_mod; end @ Here is a routine that's used whenever a token will be redefined. If the user's token is unredefinable, the `|frozen_inaccessible|' token is substituted; the latter is redefinable but essentially impossible to use, hence \MP's tables won't get fouled up. @p procedure get_symbol; {sets |cur_sym| to a safe symbol} label restart; begin restart: get_t_next; if (cur_sym=0)or(cur_sym>frozen_inaccessible) then begin print_err("Missing symbolic token inserted"); @.Missing symbolic token...@> help3("Sorry: You can't redefine a number, string, or expr.")@/ ("I've inserted an inaccessible symbol so that your")@/ ("definition will be completed without mixing me up too badly."); if cur_sym>0 then help_line[2]:="Sorry: You can't redefine my error-recovery tokens." else if cur_cmd=string_token then delete_str_ref(cur_mod); cur_sym:=frozen_inaccessible; ins_error; goto restart; end; end; @ Before we actually redefine a symbolic token, we need to clear away its former value, if it was a variable. The following stronger version of |get_symbol| does that. @p procedure get_clear_symbol; begin get_symbol; clear_symbol(cur_sym,false); end; @ Here's another little subroutine; it checks that an equals sign or assignment sign comes along at the proper place in a macro definition. @p procedure check_equals; begin if cur_cmd<>equals then if cur_cmd<>assignment then begin missing_err("=");@/ @.Missing `='@> help5("The next thing in this `def' should have been `=',")@/ ("because I've already looked at the definition heading.")@/ ("But don't worry; I'll pretend that an equals sign")@/ ("was present. Everything from here to `enddef'")@/ ("will be the replacement text of this macro."); back_error; end; end; @ A \&{primarydef}, \&{secondarydef}, or \&{tertiarydef} is rather easily handled now that we have |scan_toks|. In this case there are two parameters, which will be \.{EXPR0} and \.{EXPR1} (i.e., |expr_base| and |expr_base+1|). @p procedure make_op_def; var @!m:command_code; {the type of definition} @!p,@!q,@!r:pointer; {for list manipulation} begin m:=cur_mod;@/ get_symbol; q:=get_node(token_node_size); info(q):=cur_sym; value(q):=expr_base;@/ get_clear_symbol; warning_info:=cur_sym;@/ get_symbol; p:=get_node(token_node_size); info(p):=cur_sym; value(p):=expr_base+1; link(p):=q;@/ get_t_next; check_equals;@/ scanner_status:=op_defining; q:=get_avail; ref_count(q):=null; r:=get_avail; link(q):=r; info(r):=general_macro; link(r):=scan_toks(macro_def,p,null,0); scanner_status:=normal; eq_type(warning_info):=m; equiv(warning_info):=q; get_x_next; end; @ Parameters to macros are introduced by the keywords \&{expr}, \&{suffix}, \&{text}, \&{primary}, \&{secondary}, and \&{tertiary}. @= primitive("expr",param_type,expr_base);@/ @!@:expr_}{\&{expr} primitive@> primitive("suffix",param_type,suffix_base);@/ @!@:suffix_}{\&{suffix} primitive@> primitive("text",param_type,text_base);@/ @!@:text_}{\&{text} primitive@> primitive("primary",param_type,primary_macro);@/ @!@:primary_}{\&{primary} primitive@> primitive("secondary",param_type,secondary_macro);@/ @!@:secondary_}{\&{secondary} primitive@> primitive("tertiary",param_type,tertiary_macro);@/ @!@:tertiary_}{\&{tertiary} primitive@> @ @= param_type:if m>=expr_base then if m=expr_base then print("expr") else if m=suffix_base then print("suffix") else print("text") else if m@@; @t\4@>@@; procedure scan_def; var @!m:start_def..var_def; {the type of definition} @!n:0..3; {the number of special suffix parameters} @!k:0..param_size; {the total number of parameters} @!c:general_macro..text_macro; {the kind of macro we're defining} @!r:pointer; {parameter-substitution list} @!q:pointer; {tail of the macro token list} @!p:pointer; {temporary storage} @!base:halfword; {|expr_base|, |suffix_base|, or |text_base|} @!l_delim,@!r_delim:pointer; {matching delimiters} begin m:=cur_mod; c:=general_macro; link(hold_head):=null;@/ q:=get_avail; ref_count(q):=null; r:=null;@/ @; k:=n; if cur_cmd=left_delimiter then @; if cur_cmd=param_type then @; check_equals; p:=get_avail; info(p):=c; link(q):=p; @; scanner_status:=normal; get_x_next; end; @ We don't put `|frozen_end_group|' into the replacement text of a \&{vardef}, because the user may want to redefine `\.{endgroup}'. @= if m=start_def then link(p):=scan_toks(macro_def,r,null,n) else begin q:=get_avail; info(q):=bg_loc; link(p):=q; p:=get_avail; info(p):=eg_loc; link(q):=scan_toks(macro_def,r,p,n); end; if warning_info=bad_vardef then flush_token_list(value(bad_vardef)) @ @= @!bg_loc,@!eg_loc:1..hash_end; {hash addresses of `\.{begingroup}' and `\.{endgroup}'} @ @= if m=start_def then begin get_clear_symbol; warning_info:=cur_sym; get_t_next; scanner_status:=op_defining; n:=0; eq_type(warning_info):=defined_macro; equiv(warning_info):=q; end else begin p:=scan_declared_variable; flush_variable(equiv(info(p)),link(p),true); warning_info:=find_variable(p); flush_list(p); if warning_info=null then @; scanner_status:=var_defining; n:=2; if cur_cmd=macro_special then if cur_mod=macro_suffix then {\.{\AT!\#}} begin n:=3; get_t_next; end; type(warning_info):=unsuffixed_macro-2+n; value(warning_info):=q; end {|suffixed_macro=unsuffixed_macro+1|} @ @= begin print_err("This variable already starts with a macro"); @.This variable already...@> help2("After `vardef a' you can't say `vardef a.b'.")@/ ("So I'll have to discard this definition.");@/ error; warning_info:=bad_vardef; end @ @= name_type(bad_vardef):=root; link(bad_vardef):=frozen_bad_vardef; equiv(frozen_bad_vardef):=bad_vardef; eq_type(frozen_bad_vardef):=tag_token; @ @= repeat l_delim:=cur_sym; r_delim:=cur_mod; get_t_next; if (cur_cmd=param_type)and(cur_mod>=expr_base) then base:=cur_mod else begin print_err("Missing parameter type; `expr' will be assumed"); @.Missing parameter type@> help1("You should've had `expr' or `suffix' or `text' here."); back_error; base:=expr_base; end; @; check_delimiter(l_delim,r_delim); get_t_next; until cur_cmd<>left_delimiter @ @= repeat link(q):=get_avail; q:=link(q); info(q):=base+k;@/ get_symbol; p:=get_node(token_node_size); value(p):=base+k; info(p):=cur_sym; if k=param_size then overflow("parameter stack size",param_size); @:MetaPost capacity exceeded parameter stack size}{\quad parameter stack size@> incr(k); link(p):=r; r:=p; get_t_next; until cur_cmd<>comma @ @= begin p:=get_node(token_node_size); if cur_mod=min_command|. In other words, |get_x_next| expands macros and removes conditionals or iterations or input instructions that might be present. It follows that |get_x_next| might invoke itself recursively. In fact, there is massive recursion, since macro expansion can involve the scanning of arbitrarily complex expressions, which in turn involve macro expansion and conditionals, etc. @^recursion@> Therefore it's necessary to declare a whole bunch of |forward| procedures at this point, and to insert some other procedures that will be invoked by |get_x_next|. @p procedure@?scan_primary; forward;@t\2@> procedure@?scan_secondary; forward;@t\2@> procedure@?scan_tertiary; forward;@t\2@> procedure@?scan_expression; forward;@t\2@> procedure@?scan_suffix; forward;@t\2@>@/ @t\4@>@@;@/ procedure@?get_boolean; forward;@t\2@> procedure@?pass_text; forward;@t\2@> procedure@?conditional; forward;@t\2@> procedure@?start_input; forward;@t\2@> procedure@?begin_iteration; forward;@t\2@> procedure@?resume_iteration; forward;@t\2@> procedure@?stop_iteration; forward;@t\2@> @ An auxiliary subroutine called |expand| is used by |get_x_next| when it has to do exotic expansion commands. @p procedure expand; var @!p:pointer; {for list manipulation} @!k:integer; {something that we hope is |<=buf_size|} @!j:pool_pointer; {index into |str_pool|} begin if internal[tracing_commands]>unity then if cur_cmd<>defined_macro then show_cur_cmd_mod; case cur_cmd of if_test:conditional; {this procedure is discussed in Part 36 below} fi_or_else:@; input:@; iteration:if cur_mod=end_for then @ else begin_iteration; {this procedure is discussed in Part 37 below} repeat_loop: @; exit_test: @; relax: do_nothing; expand_after: @; scan_tokens: @; defined_macro:macro_call(cur_mod,null,cur_sym); end; {there are no other cases} end; @ @= begin print_err("Extra `endfor'"); @.Extra `endfor'@> help2("I'm not currently working on a for loop,")@/ ("so I had better not try to end anything.");@/ error; end @ The processing of \&{input} involves the |start_input| subroutine, which will be declared later; the processing of \&{endinput} is trivial. @= primitive("input",input,0);@/ @!@:input_}{\&{input} primitive@> primitive("endinput",input,1);@/ @!@:end_input_}{\&{endinput} primitive@> @ @= input: if m=0 then print("input")@+else print("endinput"); @ @= if cur_mod>0 then force_eof:=true else start_input @ We'll discuss the complicated parts of loop operations later. For now it suffices to know that there's a global variable called |loop_ptr| that will be |null| if no loop is in progress. @= begin while token_state and(loc=null) do end_token_list; {conserve stack space} if loop_ptr=null then begin print_err("Lost loop"); @.Lost loop@> help2("I'm confused; after exiting from a loop, I still seem")@/ ("to want to repeat it. I'll try to forget the problem.");@/ error; end else resume_iteration; {this procedure is in Part 37 below} end @ @= begin get_boolean; if internal[tracing_commands]>unity then show_cmd_mod(nullary,cur_exp); if cur_exp=true_code then if loop_ptr=null then begin print_err("No loop is in progress"); @.No loop is in progress@> help1("Why say `exitif' when there's nothing to exit from?"); if cur_cmd=semicolon then error@+else back_error; end else @ else if cur_cmd<>semicolon then begin missing_err(";");@/ @.Missing `;'@> help2("After `exitif ' I expect to see a semicolon.")@/ ("I shall pretend that one was there."); back_error; end; end @ Here we use the fact that |forever_text| is the only |token_type| that is less than |loop_text|. @= begin p:=null; repeat if file_state then end_file_reading else begin if token_type<=loop_text then p:=start; end_token_list; end; until p<>null; if p<>info(loop_ptr) then fatal_error("*** (loop confusion)"); @.loop confusion@> stop_iteration; {this procedure is in Part 34 below} end @ @= begin get_t_next; p:=cur_tok; get_t_next; if cur_cmd= begin get_x_next; scan_primary; if cur_type<>string_type then begin disp_err(null,"Not a string"); @.Not a string@> help2("I'm going to flush this expression, since")@/ ("scantokens should be followed by a known string."); put_get_flush_error(0); end else begin back_input; if length(cur_exp)>0 then @; end; end @ @= begin begin_file_reading; name:=is_scantok; k:=first+length(cur_exp); if k>=max_buf_stack then begin if k>=buf_size then begin max_buf_stack:=buf_size; overflow("buffer size",buf_size); @:MetaPost capacity exceeded buffer size}{\quad buffer size@> end; max_buf_stack:=k+1; end; j:=str_start[cur_exp]; limit:=k; while first @p procedure get_x_next; var @!save_exp:pointer; {a capsule to save |cur_type| and |cur_exp|} begin get_t_next; if cur_cmd=min_command; unstash_cur_exp(save_exp); {that restores |cur_type| and |cur_exp|} end; end; @ Now let's consider the |macro_call| procedure, which is used to start up all user-defined macros. Since the arguments to a macro might be expressions, |macro_call| is recursive. @^recursion@> The first parameter to |macro_call| points to the reference count of the token list that defines the macro. The second parameter contains any arguments that have already been parsed (see below). The third parameter points to the symbolic token that names the macro. If the third parameter is |null|, the macro was defined by \&{vardef}, so its name can be reconstructed from the prefix and ``at'' arguments found within the second parameter. What is this second parameter? It's simply a linked list of one-word items, whose |info| fields point to the arguments. In other words, if |arg_list=null|, no arguments have been scanned yet; otherwise |info(arg_list)| points to the first scanned argument, and |link(arg_list)| points to the list of further arguments (if any). Arguments of type \&{expr} are so-called capsules, which we will discuss later when we concentrate on expressions; they can be recognized easily because their |link| field is |void|. Arguments of type \&{suffix} and \&{text} are token lists without reference counts. @ After argument scanning is complete, the arguments are moved to the |param_stack|. (They can't be put on that stack any sooner, because the stack is growing and shrinking in unpredictable ways as more arguments are being acquired.) Then the macro body is fed to the scanner; i.e., the replacement text of the macro is placed at the top of the \MP's input stack, so that |get_t_next| will proceed to read it next. @= @t\4@>@@; @t\4@>@@; @t\4@>@@; procedure macro_call(@!def_ref,@!arg_list,@!macro_name:pointer); {invokes a user-defined control sequence} label found; var @!r:pointer; {current node in the macro's token list} @!p,@!q:pointer; {for list manipulation} @!n:integer; {the number of arguments} @!l_delim,@!r_delim:pointer; {a delimiter pair} @!tail:pointer; {tail of the argument list} begin r:=link(def_ref); add_mac_ref(def_ref); if arg_list=null then n:=0 else @; if internal[tracing_macros]>0 then @; @; @; end; @ @= begin begin_diagnostic; print_ln; print_macro_name(arg_list,macro_name); if n=3 then print("@@#"); {indicate a suffixed macro} show_macro(def_ref,null,100000); if arg_list<>null then begin n:=0; p:=arg_list; repeat q:=info(p); print_arg(q,n,0); incr(n); p:=link(p); until p=null; end; end_diagnostic(false); end @ @= procedure print_macro_name(@!a,@!n:pointer); var @!p,@!q:pointer; {they traverse the first part of |a|} begin if n<>null then print(text(n)) else begin p:=info(a); if p=null then print(text(info(info(link(a))))) else begin q:=p; while link(q)<>null do q:=link(q); link(q):=info(link(a)); show_token_list(p,null,1000,0); link(q):=null; end; end; end; @ @= procedure print_arg(@!q:pointer;@!n:integer;@!b:pointer); begin if link(q)=void then print_nl("(EXPR") else if (btext_macro) then print_nl("(SUFFIX") else print_nl("(TEXT"); print_int(n); print(")<-"); if link(q)=void then print_exp(q,1) else show_token_list(q,null,1000,0); end; @ @= begin n:=1; tail:=arg_list; while link(tail)<>null do begin incr(n); tail:=link(tail); end; end @ @= cur_cmd:=comma+1; {anything |<>comma| will do} while info(r)>=expr_base do begin @; r:=link(r); end; if cur_cmd=comma then begin print_err("Too many arguments to "); @.Too many arguments...@> print_macro_name(arg_list,macro_name); print_char(";"); print_nl(" Missing `"); print(text(r_delim)); @.Missing `)'...@> print("' has been inserted"); help3("I'm going to assume that the comma I just read was a")@/ ("right delimiter, and then I'll begin expanding the macro.")@/ ("You might want to delete some tokens before continuing."); error; end; if info(r)<>general_macro then @; r:=link(r) @ At this point, the reader will find it advisable to review the explanation of token list format that was presented earlier, paying special attention to the conventions that apply only at the beginning of a macro's token list. On the other hand, the reader will have to take the expression-parsing aspects of the following program on faith; we will explain |cur_type| and |cur_exp| later. (Several things in this program depend on each other, and it's necessary to jump into the circle somewhere.) @= if cur_cmd<>comma then begin get_x_next; if cur_cmd<>left_delimiter then begin print_err("Missing argument to "); @.Missing argument...@> print_macro_name(arg_list,macro_name); help3("That macro has more parameters than you thought.")@/ ("I'll continue by pretending that each missing argument")@/ ("is either zero or null."); if info(r)>=suffix_base then begin cur_exp:=null; cur_type:=token_list; end else begin cur_exp:=0; cur_type:=known; end; back_error; cur_cmd:=right_delimiter; goto found; end; l_delim:=cur_sym; r_delim:=cur_mod; end; @; if cur_cmd<>comma then @; found: @ @ @= if (cur_cmd<>right_delimiter)or(cur_mod<>l_delim) then if info(link(r))>=expr_base then begin missing_err(","); @.Missing `,'@> help3("I've finished reading a macro argument and am about to")@/ ("read another; the arguments weren't delimited correctly.")@/ ("You might want to delete some tokens before continuing."); back_error; cur_cmd:=comma; end else begin missing_err(text(r_delim)); @.Missing `)'@> help2("I've gotten to the end of the macro parameter list.")@/ ("You might want to delete some tokens before continuing."); back_error; end @ A \&{suffix} or \&{text} parameter will be have been scanned as a token list pointed to by |cur_exp|, in which case we will have |cur_type=token_list|. @= begin p:=get_avail; if cur_type=token_list then info(p):=cur_exp else info(p):=stash_cur_exp; if internal[tracing_macros]>0 then begin begin_diagnostic; print_arg(info(p),n,info(r)); end_diagnostic(false); end; if arg_list=null then arg_list:=p else link(tail):=p; tail:=p; incr(n); end @ @= if info(r)>=text_base then scan_text_arg(l_delim,r_delim) else begin get_x_next; if info(r)>=suffix_base then scan_suffix else scan_expression; end @ The parameters to |scan_text_arg| are either a pair of delimiters or zero; the latter case is for undelimited text arguments, which end with the first semicolon or \&{endgroup} or \&{end} that is not contained in a group. @= procedure scan_text_arg(@!l_delim,@!r_delim:pointer); label done; var @!balance:integer; {excess of |l_delim| over |r_delim|} @!p:pointer; {list tail} begin warning_info:=l_delim; scanner_status:=absorbing; p:=hold_head; balance:=1; link(hold_head):=null; loop@+ begin get_t_next; if l_delim=0 then @ else @; link(p):=cur_tok; p:=link(p); end; done: cur_exp:=link(hold_head); cur_type:=token_list; scanner_status:=normal; end; @ @= begin if cur_cmd=right_delimiter then begin if cur_mod=l_delim then begin decr(balance); if balance=0 then goto done; end; end else if cur_cmd=left_delimiter then if cur_mod=r_delim then incr(balance); end @ @= begin if end_of_statement then {|cur_cmd=semicolon|, |end_group|, or |stop|} begin if balance=1 then goto done else if cur_cmd=end_group then decr(balance); end else if cur_cmd=begin_group then incr(balance); end @ @= begin if info(r)suffix_macro then if (cur_cmd=equals)or(cur_cmd=assignment) then get_x_next; end; case info(r) of primary_macro:scan_primary; secondary_macro:scan_secondary; tertiary_macro:scan_tertiary; expr_macro:scan_expression; of_macro:@; suffix_macro:@; text_macro:scan_text_arg(0,0); end; {there are no other cases} back_input; @; end @ @= begin scan_expression; p:=get_avail; info(p):=stash_cur_exp; if internal[tracing_macros]>0 then begin begin_diagnostic; print_arg(info(p),n,0); end_diagnostic(false); end; if arg_list=null then arg_list:=p@+else link(tail):=p; tail:=p;incr(n); if cur_cmd<>of_token then begin missing_err("of"); print(" for "); @.Missing `of'@> print_macro_name(arg_list,macro_name); help1("I've got the first argument; will look now for the other."); back_error; end; get_x_next; scan_primary; end @ @= begin if cur_cmd<>left_delimiter then l_delim:=null else begin l_delim:=cur_sym; r_delim:=cur_mod; get_x_next; end; scan_suffix; if l_delim<>null then begin if(cur_cmd<>right_delimiter)or(cur_mod<>l_delim) then begin missing_err(text(r_delim)); @.Missing `)'@> help2("I've gotten to the end of the macro parameter list.")@/ ("You might want to delete some tokens before continuing."); back_error; end; get_x_next; end; end @ Before we put a new token list on the input stack, it is wise to clean off all token lists that have recently been depleted. Then a user macro that ends with a call to itself will not require unbounded stack space. @= while token_state and(loc=null) do end_token_list; {conserve stack space} if param_ptr+n>max_param_stack then begin max_param_stack:=param_ptr+n; if max_param_stack>param_size then overflow("parameter stack size",param_size); @:MetaPost capacity exceeded parameter stack size}{\quad parameter stack size@> end; begin_token_list(def_ref,macro); name:=macro_name; loc:=r; if n>0 then begin p:=arg_list; repeat param_stack[param_ptr]:=info(p); incr(param_ptr); p:=link(p); until p=null; flush_list(arg_list); end @ It's sometimes necessary to put a single argument onto |param_stack|. The |stack_argument| subroutine does this. @p procedure stack_argument(@!p:pointer); begin if param_ptr=max_param_stack then begin incr(max_param_stack); if max_param_stack>param_size then overflow("parameter stack size",param_size); @:MetaPost capacity exceeded parameter stack size}{\quad parameter stack size@> end; param_stack[param_ptr]:=p; incr(param_ptr); end; @* \[33] Conditional processing. Let's consider now the way \&{if} commands are handled. Conditions can be inside conditions, and this nesting has a stack that is independent of other stacks. Four global variables represent the top of the condition stack: |cond_ptr| points to pushed-down entries, if~any; |cur_if| tells whether we are processing \&{if} or \&{elseif}; |if_limit| specifies the largest code of a |fi_or_else| command that is syntactically legal; and |if_line| is the line number at which the current conditional began. If no conditions are currently in progress, the condition stack has the special state |cond_ptr=null|, |if_limit=normal|, |cur_if=0|, |if_line=0|. Otherwise |cond_ptr| points to a two-word node; the |type|, |name_type|, and |link| fields of the first word contain |if_limit|, |cur_if|, and |cond_ptr| at the next level, and the second word contains the corresponding |if_line|. @d if_node_size=2 {number of words in stack entry for conditionals} @d if_line_field(#)==mem[#+1].int @d if_code=1 {code for \&{if} being evaluated} @d fi_code=2 {code for \&{fi}} @d else_code=3 {code for \&{else}} @d else_if_code=4 {code for \&{elseif}} @= @!cond_ptr:pointer; {top of the condition stack} @!if_limit:normal..else_if_code; {upper bound on |fi_or_else| codes} @!cur_if:small_number; {type of conditional being worked on} @!if_line:integer; {line where that conditional began} @ @= cond_ptr:=null; if_limit:=normal; cur_if:=0; if_line:=0; @ @= primitive("if",if_test,if_code);@/ @!@:if_}{\&{if} primitive@> primitive("fi",fi_or_else,fi_code); eqtb[frozen_fi]:=eqtb[cur_sym];@/ @!@:fi_}{\&{fi} primitive@> primitive("else",fi_or_else,else_code);@/ @!@:else_}{\&{else} primitive@> primitive("elseif",fi_or_else,else_if_code);@/ @!@:else_if_}{\&{elseif} primitive@> @ @= if_test,fi_or_else: case m of if_code:print("if"); fi_code:print("fi"); else_code:print("else"); othercases print("elseif") endcases; @ Here is a procedure that ignores text until coming to an \&{elseif}, \&{else}, or \&{fi} at level zero of $\&{if}\ldots\&{fi}$ nesting. After it has acted, |cur_mod| will indicate the token that was found. \MP's smallest two command codes are |if_test| and |fi_or_else|; this makes the skipping process a bit simpler. @p procedure pass_text; label done; var l:integer; begin scanner_status:=skipping; l:=0; warning_info:=true_line; loop@+ begin get_t_next; if cur_cmd<=fi_or_else then if cur_cmd; end; done: scanner_status:=normal; end; @ @= if cur_cmd=string_token then delete_str_ref(cur_mod) @ When we begin to process a new \&{if}, we set |if_limit:=if_code|; then if \&{elseif} or \&{else} or \&{fi} occurs before the current \&{if} condition has been evaluated, a colon will be inserted. A construction like `\.{if fi}' would otherwise get \MP\ confused. @= begin p:=get_node(if_node_size); link(p):=cond_ptr; type(p):=if_limit; name_type(p):=cur_if; if_line_field(p):=if_line; cond_ptr:=p; if_limit:=if_code; if_line:=true_line; cur_if:=if_code; end @ @= begin p:=cond_ptr; if_line:=if_line_field(p); cur_if:=name_type(p); if_limit:=type(p); cond_ptr:=link(p); free_node(p,if_node_size); end @ Here's a procedure that changes the |if_limit| code corresponding to a given value of |cond_ptr|. @p procedure change_if_limit(@!l:small_number;@!p:pointer); label exit; var q:pointer; begin if p=cond_ptr then if_limit:=l {that's the easy case} else begin q:=cond_ptr; loop@+ begin if q=null then confusion("if"); @:this can't happen if}{\quad if@> if link(q)=p then begin type(q):=l; return; end; q:=link(q); end; end; exit:end; @ The user is supposed to put colons into the proper parts of conditional statements. Therefore, \MP\ has to check for their presence. @p procedure check_colon; begin if cur_cmd<>colon then begin missing_err(":");@/ @.Missing `:'@> help2("There should've been a colon after the condition.")@/ ("I shall pretend that one was there.");@; back_error; end; end; @ A condition is started when the |get_x_next| procedure encounters an |if_test| command; in that case |get_x_next| calls |conditional|, which is a recursive procedure. @^recursion@> @p procedure conditional; label exit,done,reswitch,found; var @!save_cond_ptr:pointer; {|cond_ptr| corresponding to this conditional} @!new_if_limit:fi_code..else_if_code; {future value of |if_limit|} @!p:pointer; {temporary register} begin @;@+save_cond_ptr:=cond_ptr; reswitch: get_boolean; new_if_limit:=else_if_code; if internal[tracing_commands]>unity then @; found: check_colon; if cur_exp=true_code then begin change_if_limit(new_if_limit,save_cond_ptr); return; {wait for \&{elseif}, \&{else}, or \&{fi}} end; @; done: cur_if:=cur_mod; if_line:=true_line; if cur_mod=fi_code then @ else if cur_mod=else_if_code then goto reswitch else begin cur_exp:=true_code; new_if_limit:=fi_code; get_x_next; goto found; end; exit:end; @ In a construction like `\&{if} \&{if} \&{true}: $0=1$: \\{foo} \&{else}: \\{bar} \&{fi}', the first \&{else} that we come to after learning that the \&{if} is false is not the \&{else} we're looking for. Hence the following curious logic is needed. @= loop@+ begin pass_text; if cond_ptr=save_cond_ptr then goto done else if cur_mod=fi_code then @; end @ @= begin begin_diagnostic; if cur_exp=true_code then print("{true}")@+else print("{false}"); end_diagnostic(false); end @ The processing of conditionals is complete except for the following code, which is actually part of |get_x_next|. It comes into play when \&{elseif}, \&{else}, or \&{fi} is scanned. @= if cur_mod>if_limit then if if_limit=if_code then {condition not yet evaluated} begin missing_err(":"); @.Missing `:'@> back_input; cur_sym:=frozen_colon; ins_error; end else begin print_err("Extra "); print_cmd_mod(fi_or_else,cur_mod); @.Extra else@> @.Extra elseif@> @.Extra fi@> help1("I'm ignoring this; it doesn't match any if."); error; end else begin while cur_mod<>fi_code do pass_text; {skip to \&{fi}} @; end @* \[34] Iterations. To bring our treatment of |get_x_next| to a close, we need to consider what \MP\ does when it sees \&{for}, \&{forsuffixes}, and \&{forever}. There's a global variable |loop_ptr| that keeps track of the \&{for} loops that are currently active. If |loop_ptr=null|, no loops are in progress; otherwise |info(loop_ptr)| points to the iterative text of the current (innermost) loop, and |link(loop_ptr)| points to the data for any other loops that enclose the current one. A loop-control node also has two other fields, called |loop_type| and |loop_list|, whose contents depend on the type of loop: \yskip\indent|loop_type(loop_ptr)=null| means that |loop_list(loop_ptr)| points to a list of one-word nodes whose |info| fields point to the remaining argument values of a suffix list and expression list. \yskip\indent|loop_type(loop_ptr)=void| means that the current loop is `\&{forever}'. \yskip\indent|loop_type(loop_ptr)=progression_flag| means that |p=loop_list(loop_ptr)| points to a ``progression node'' and |value(p)|, |step_size(p)|, and |final_value(p)| contain the data for an arithmetic progression. \yskip\indent|loop_type(loop_ptr)=p>void| means that |p| points to an edge header and |loop_list(loop_ptr)| points into the graphical object list for that edge header. \yskip\noindent In the case of a progression node, the first word is not used because the link field of words in the dynamic memory area cannot be arbitrary. @d loop_list_loc(#)==#+1 {where the |loop_list| field resides} @d loop_type(#)==info(loop_list_loc(#)) {the type of \&{for} loop} @d loop_list(#)==link(loop_list_loc(#)) {the remaining list elements} @d loop_node_size=2 {the number of words in a loop control node} @d progression_node_size=4 {the number of words in a progression node} @d step_size(#)==mem[#+2].sc {the step size in an arithmetic progression} @d final_value(#)==mem[#+3].sc {the final value in an arithmetic progression} @d progression_flag==null+2 {|loop_type| value when |loop_list| points to a progression node} @= @!loop_ptr:pointer; {top of the loop-control-node stack} @ @= loop_ptr:=null; @ If the expressions that define an arithmetic progression in a \&{for} loop don't have known numeric values, the |bad_for| subroutine screams at the user. @p procedure bad_for(@!s:str_number); begin disp_err(null,"Improper "); {show the bad expression above the message} @.Improper...replaced by 0@> print(s); print(" has been replaced by 0"); help4("When you say `for x=a step b until c',")@/ ("the initial value `a' and the step size `b'")@/ ("and the final value `c' must have known numeric values.")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); put_get_flush_error(0); end; @ Here's what \MP\ does when \&{for}, \&{forsuffixes}, or \&{forever} has just been scanned. (This code requires slight familiarity with expression-parsing routines that we have not yet discussed; but it seems to belong in the present part of the program, even though the original author didn't write it until later. The reader may wish to come back to it.) @p procedure begin_iteration; label continue,done; var @!m:halfword; {|expr_base| (\&{for}) or |suffix_base| (\&{forsuffixes})} @!n:halfword; {hash address of the current symbol} @!s:pointer; {the new loop-control node} @!p:pointer; {substitution list for |scan_toks|} @!q:pointer; {link manipulation register} @!pp:pointer; {a new progression node} begin m:=cur_mod; n:=cur_sym; s:=get_node(loop_node_size); if m=start_forever then begin loop_type(s):=void; p:=null; get_x_next; end else begin get_symbol; p:=get_node(token_node_size); info(p):=cur_sym; value(p):=m;@/ get_x_next; if cur_cmd=within_token then @ else begin @= if cur_cmd<>colon then begin missing_err(":");@/ @.Missing `:'@> help3("The next thing in this loop should have been a `:'.")@/ ("So I'll pretend that a colon was present;")@/ ("everything from here to `endfor' will be iterated."); back_error; end @ We append a special |frozen_repeat_loop| token in place of the `\&{endfor}' at the end of the loop. This will come through \MP's scanner at the proper time to cause the loop to be repeated. (If the user tries some shenanigan like `\&{for} $\ldots$ \&{let} \&{endfor}', he will be foiled by the |get_symbol| routine, which keeps frozen tokens unchanged. Furthermore the |frozen_repeat_loop| is an \&{outer} token, so it won't be lost accidentally.) @ @= q:=get_avail; info(q):=frozen_repeat_loop; scanner_status:=loop_defining; warning_info:=n; info(s):=scan_toks(iteration,p,q,0); scanner_status:=normal;@/ link(s):=loop_ptr; loop_ptr:=s @ @= eq_type(frozen_repeat_loop):=repeat_loop+outer_tag; text(frozen_repeat_loop):=" ENDFOR"; @ The loop text is inserted into \MP's scanning apparatus by the |resume_iteration| routine. @p procedure resume_iteration; label not_found,exit; var @!p,@!q:pointer; {link registers} begin p:=loop_type(loop_ptr); if p=progression_flag then begin p:=loop_list(loop_ptr); {now |p| points to a progression node} cur_exp:=value(p); if @ then goto not_found; cur_type:=known; q:=stash_cur_exp; {make |q| an \&{expr} argument} value(p):=cur_exp+step_size(p); {set |value(p)| for the next iteration} end else if p=null then begin p:=loop_list(loop_ptr); if p=null then goto not_found; loop_list(loop_ptr):=link(p); q:=info(p); free_avail(p); end else if p=void then begin begin_token_list(info(loop_ptr),forever_text); return; end else @; begin_token_list(info(loop_ptr),loop_text); stack_argument(q); if internal[tracing_commands]>unity then @; return; not_found:stop_iteration; exit:end; @ @= ((step_size(p)>0)and(cur_exp>final_value(p)))or@| ((step_size(p)<0)and(cur_exp= begin begin_diagnostic; print_nl("{loop value="); @.loop value=n@> if (q<>null)and(link(q)=void) then print_exp(q,1) else show_token_list(q,null,50,0); print_char("}"); end_diagnostic(false); end @ @= begin q:=loop_list(loop_ptr); if q=null then goto not_found; skip_component(q)(goto not_found); cur_exp:=copy_objects(loop_list(loop_ptr),q); init_bbox(cur_exp); cur_type:=picture_type;@/ loop_list(loop_ptr):=q; q:=stash_cur_exp; end @ A level of loop control disappears when |resume_iteration| has decided not to resume, or when an \&{exitif} construction has removed the loop text from the input stack. @p procedure stop_iteration; var @!p,@!q:pointer; {the usual} begin p:=loop_type(loop_ptr); if p=progression_flag then free_node(loop_list(loop_ptr),progression_node_size) else if p=null then begin q:=loop_list(loop_ptr); while q<>null do begin p:=info(q); if p<>null then if link(p)=void then {it's an \&{expr} parameter} begin recycle_value(p); free_node(p,value_node_size); end else flush_token_list(p); {it's a \&{suffix} or \&{text} parameter} p:=q; q:=link(q); free_avail(p); end; end else if p>progression_flag then delete_edge_ref(p); p:=loop_ptr; loop_ptr:=link(p); flush_token_list(info(p)); free_node(p,loop_node_size); end; @ Now that we know all about loop control, we can finish up the missing portion of |begin_iteration| and we'll be done. The following code is performed after the `\.=' has been scanned in a \&{for} construction (if |m=expr_base|) or a \&{forsuffixes} construction (if |m=suffix_base|). @= loop_type(s):=null; q:=loop_list_loc(s); link(q):=null; {|link(q)=loop_list(s)|} repeat get_x_next; if m<>expr_base then scan_suffix else begin if cur_cmd>=colon then if cur_cmd<=comma then goto continue; scan_expression; if cur_cmd=step_token then if q=loop_list_loc(s) then @; cur_exp:=stash_cur_exp; end; link(q):=get_avail; q:=link(q); info(q):=cur_exp; cur_type:=vacuous; continue: until cur_cmd<>comma; done: @ @= begin if cur_type<>known then bad_for("initial value"); pp:=get_node(progression_node_size); value(pp):=cur_exp;@/ get_x_next; scan_expression; if cur_type<>known then bad_for("step size"); step_size(pp):=cur_exp; if cur_cmd<>until_token then begin missing_err("until");@/ @.Missing `until'@> help2("I assume you meant to say `until' after `step'.")@/ ("So I'll look for the final value and colon next."); back_error; end; get_x_next; scan_expression; if cur_type<>known then bad_for("final value"); final_value(pp):=cur_exp; loop_list(s):=pp; loop_type(s):=progression_flag; goto done; end @ The last case is when we have just seen ``\&{within}'', and we need to parse a picture expression and prepare to iterate over it. @= begin get_x_next; scan_expression; @; loop_type(s):=cur_exp; cur_type:=vacuous;@/ q:=link(dummy_loc(cur_exp)); if q<> null then if is_start_or_stop(q) then if skip_1component(q)=null then q:=link(q); loop_list(s):=q; end @ @= if cur_type<>picture_type then begin disp_err(null,"Improper iteration spec has been replaced by nullpicture"); help1("When you say `for x in p', p must be a known picture."); put_get_flush_error(get_node(edge_header_size)); init_edges(cur_exp); cur_type:=picture_type; end @* \[35] File names. It's time now to fret about file names. Besides the fact that different operating systems treat files in different ways, we must cope with the fact that completely different naming conventions are used by different groups of people. The following programs show what is required for one particular operating system; similar routines for other systems are not difficult to devise. @^system dependencies@> \MP\ assumes that a file name has three parts: the name proper; its ``extension''; and a ``file area'' where it is found in an external file system. The extension of an input file is assumed to be `\.{.mp}' unless otherwise specified; it is `\.{.log}' on the transcript file that records each run of \MP; it is `\.{.tfm}' on the font metric files that describe characters in any fonts created by \MP; it is `\.{.ps}' or `.{\it nnn}' for some number {\it nnn} on the \ps\ output files; and it is `\.{.mem}' on the mem files written by \.{INIMP} to initialize \MP. The file area can be arbitrary on input files, but files are usually output to the user's current area. If an input file cannot be found on the specified area, \MP\ will look for it on a special system area; this special area is intended for commonly used input files. Simple uses of \MP\ refer only to file names that have no explicit extension or area. For example, a person usually says `\.{input} \.{cmr10}' instead of `\.{input} \.{cmr10.new}'. Simple file names are best, because they make the \MP\ source files portable; whenever a file name consists entirely of letters and digits, it should be treated in the same way by all implementations of \MP. However, users need the ability to refer to other files in their environment, especially when responding to error messages concerning unopenable files; therefore we want to let them use the syntax that appears in their favorite operating system. @ \MP\ uses the same conventions that have proved to be satisfactory for \TeX\ and \MF. In order to isolate the system-dependent aspects of file names, @^system dependencies@> the system-independent parts of \MP\ are expressed in terms of three system-dependent procedures called |begin_name|, |more_name|, and |end_name|. In essence, if the user-specified characters of the file name are $c_1\ldots c_n$, the system-independent driver program does the operations $$|begin_name|;\,|more_name|(c_1);\,\ldots\,;|more_name|(c_n); \,|end_name|.$$ These three procedures communicate with each other via global variables. Afterwards the file name will appear in the string pool as three strings called |cur_name|\penalty10000\hskip-.05em, |cur_area|, and |cur_ext|; the latter two are null (i.e., |""|), unless they were explicitly specified by the user. Actually the situation is slightly more complicated, because \MP\ needs to know when the file name ends. The |more_name| routine is a function (with side effects) that returns |true| on the calls |more_name|$(c_1)$, \dots, |more_name|$(c_{n-1})$. The final call |more_name|$(c_n)$ returns |false|; or, it returns |true| and $c_n$ is the last character on the current input line. In other words, |more_name| is supposed to return |true| unless it is sure that the file name has been completely scanned; and |end_name| is supposed to be able to finish the assembly of |cur_name|, |cur_area|, and |cur_ext| regardless of whether $|more_name|(c_n)$ returned |true| or |false|. @= @!cur_name:str_number; {name of file just scanned} @!cur_area:str_number; {file area just scanned, or \.{""}} @!cur_ext:str_number; {file extension just scanned, or \.{""}} @ It is easier to maintain reference counts if we assign initial values. @= cur_name:=""; cur_area:=""; cur_ext:=""; @ The file names we shall deal with for illustrative purposes have the following structure: If the name contains `\.>' or `\.:', the file area consists of all characters up to and including the final such character; otherwise the file area is null. If the remaining file name contains `\..', the file extension consists of all such characters from the first remaining `\..' to the end, otherwise the file extension is null. @^system dependencies@> We can scan such file names easily by using two global variables that keep track of the occurrences of area and extension delimiters. Note that these variables cannot be of type |pool_pointer| because a string pool compaction could occur while scanning a file name. @= @!area_delimiter:integer; {most recent `\.>' or `\.:' relative to |str_start[str_ptr]|} @!ext_delimiter:integer; {the relevant `\..', if any} @ Input files that can't be found in the user's area may appear in standard system areas called |MP_area| and |MF_area|. (The latter is used when the file extension is |".mf"|.) The standard system area for font metric files to be read is |MP_font_area|. This system area name will, of course, vary from place to place. @^system dependencies@> @d MP_area=="MPinputs:" @.MPinputs@> @d MF_area=="MFinputs:" @.MFinputs@> @d MP_font_area=="TeXfonts:" @.TeXfonts@> @ Here now is the first of the system-dependent routines for file name scanning. @^system dependencies@> @= procedure begin_name; begin delete_str_ref(cur_name); delete_str_ref(cur_area); delete_str_ref(cur_ext);@/ area_delimiter:=-1; ext_delimiter:=-1; end; @ And here's the second. @^system dependencies@> @= function more_name(@!c:ASCII_code):boolean; begin if c=" " then more_name:=false else begin if (c=">")or(c=":") then begin area_delimiter:=pool_ptr-str_start[str_ptr]; ext_delimiter:=-1; end else if (c=".")and(ext_delimiter<0) then ext_delimiter:=pool_ptr-str_start[str_ptr]; str_room(1); append_char(c); {contribute |c| to the current string} more_name:=true; end; end; @ The third. @^system dependencies@> @= procedure end_name; var a,@!n,@!e:pool_pointer; {length of area, name, and extension} begin e:=pool_ptr-str_start[str_ptr]; {total length} if ext_delimiter<0 then ext_delimiter:=e; a:=area_delimiter+1; n:=ext_delimiter-a; e:=e-ext_delimiter; if a=0 then cur_area:="" else begin cur_area:=make_string; chop_last_string(str_start[cur_area]+a); end; if n=0 then cur_name:="" else begin cur_name:=make_string; chop_last_string(str_start[cur_name]+n); end; if e=0 then cur_ext:="" @+ else cur_ext:=make_string; end; @ Conversely, here is a routine that takes three strings and prints a file name that might have produced them. (The routine is system dependent, because some operating systems put the file area last instead of first.) @^system dependencies@> @= procedure print_file_name(@!n,@!a,@!e:integer); begin print(a); print(n); print(e); end; @ Another system-dependent routine is needed to convert three internal \MP\ strings to the |name_of_file| value that is used to open files. The present code allows both lowercase and uppercase letters in the file name. @^system dependencies@> @d append_to_name(#)==begin c:=#; incr(k); if k<=file_name_size then name_of_file[k]:=xchr[c]; end @= procedure pack_file_name(@!n,@!a,@!e:str_number); var @!k:integer; {number of positions filled in |name_of_file|} @!c: ASCII_code; {character being packed} @!j:pool_pointer; {index into |str_pool|} begin k:=0; for j:=str_start[a] to str_stop(a)-1 do append_to_name(so(str_pool[j])); for j:=str_start[n] to str_stop(n)-1 do append_to_name(so(str_pool[j])); for j:=str_start[e] to str_stop(e)-1 do append_to_name(so(str_pool[j])); if k<=file_name_size then name_length:=k@+else name_length:=file_name_size; for k:=name_length+1 to file_name_size do name_of_file[k]:=' '; end; @ A messier routine is also needed, since mem file names must be scanned before \MP's string mechanism has been initialized. We shall use the global variable |MP_mem_default| to supply the text for default system areas and extensions related to mem files. @^system dependencies@> @d mem_default_length=15 {length of the |MP_mem_default| string} @d mem_area_length=6 {length of its area part} @d mem_ext_length=4 {length of its `\.{.mem}' part} @d mem_extension=".mem" {the extension, as a \.{WEB} constant} @= @!MP_mem_default:packed array[1..mem_default_length] of char; @ @= MP_mem_default:='MPlib:plain.mem'; @.MPlib@> @.plain@> @^system dependencies@> @ @= if mem_default_length>file_name_size then bad:=20; @ Here is the messy routine that was just mentioned. It sets |name_of_file| from the first |n| characters of |MP_mem_default|, followed by |buffer[a..b]|, followed by the last |mem_ext_length| characters of |MP_mem_default|. We dare not give error messages here, since \MP\ calls this routine before the |error| routine is ready to roll. Instead, we simply drop excess characters, since the error will be detected in another way when a strange file name isn't found. @^system dependencies@> @p procedure pack_buffered_name(@!n:small_number;@!a,@!b:integer); var @!k:integer; {number of positions filled in |name_of_file|} @!c: ASCII_code; {character being packed} @!j:integer; {index into |buffer| or |MP_mem_default|} begin if n+b-a+1+mem_ext_length>file_name_size then b:=a+file_name_size-n-1-mem_ext_length; k:=0; for j:=1 to n do append_to_name(xord[MP_mem_default[j]]); for j:=a to b do append_to_name(buffer[j]); for j:=mem_default_length-mem_ext_length+1 to mem_default_length do append_to_name(xord[MP_mem_default[j]]); if k<=file_name_size then name_length:=k@+else name_length:=file_name_size; for k:=name_length+1 to file_name_size do name_of_file[k]:=' '; end; @ Here is the only place we use |pack_buffered_name|. This part of the program becomes active when a ``virgin'' \MP\ is trying to get going, just after the preliminary initialization, or when the user is substituting another mem file by typing `\.\&' after the initial `\.{**}' prompt. The buffer contains the first line of input in |buffer[loc..(last-1)]|, where |loc" "|. @= function open_mem_file:boolean; label found,exit; var @!j:0..buf_size; {the first space after the file name} begin j:=loc; if buffer[loc]="&" then begin incr(loc); j:=loc; buffer[last]:=" "; while buffer[j]<>" " do incr(j); pack_buffered_name(0,loc,j-1); {try first without the system file area} if w_open_in(mem_file) then goto found; pack_buffered_name(mem_area_length,loc,j-1); {now try the system mem file area} if w_open_in(mem_file) then goto found; wake_up_terminal; wterm_ln('Sorry, I can''t find that mem file;',' will try PLAIN.'); @.Sorry, I can't find...@> update_terminal; end; {now pull out all the stops: try for the system \.{plain} file} pack_buffered_name(mem_default_length-mem_ext_length,1,0); if not w_open_in(mem_file) then begin wake_up_terminal; wterm_ln('I can''t find the PLAIN mem file!'); @.I can't find PLAIN...@> @.plain@> open_mem_file:=false; return; end; found:loc:=j; open_mem_file:=true; exit:end; @ Operating systems often make it possible to determine the exact name (and possible version number) of a file that has been opened. The following routine, which simply makes a \MP\ string from the value of |name_of_file|, should ideally be changed to deduce the full name of file~|f|, which is the file most recently opened, if it is possible to do this in a \PASCAL\ program. @^system dependencies@> This routine might be called after string memory has overflowed, hence we check for this before calling `|str_room|'. @p function make_name_string:str_number; var @!k:1..file_name_size; {index into |name_of_file|} begin if str_overflowed then make_name_string:="?" else begin str_room(name_length); for k:=1 to name_length do append_char(xord[name_of_file[k]]); make_name_string:=make_string; end; end; function a_make_name_string(var @!f:alpha_file):str_number; begin a_make_name_string:=make_name_string; end; function b_make_name_string(var @!f:byte_file):str_number; begin b_make_name_string:=make_name_string; end; function w_make_name_string(var @!f:word_file):str_number; begin w_make_name_string:=make_name_string; end; @ Now let's consider the ``driver'' routines by which \MP\ deals with file names in a system-independent manner. First comes a procedure that looks for a file name in the input by taking the information from the input buffer. (We can't use |get_next|, because the conversion to tokens would destroy necessary information.) This procedure doesn't allow semicolons or percent signs to be part of file names, because of other conventions of \MP. {\sl The {\logos METAFONT\/}book} doesn't use semicolons or percents immediately after file names, but some users no doubt will find it natural to do so; therefore system-dependent changes to allow such characters in file names should probably be made with reluctance, and only when an entire file name that includes special characters is ``quoted'' somehow. @^system dependencies@> @p procedure scan_file_name; label done; begin begin_name; while buffer[loc]=" " do incr(loc); loop@+begin if (buffer[loc]=";")or(buffer[loc]="%") then goto done; if not more_name(buffer[loc]) then goto done; incr(loc); end; done: end_name; end; @ Here is another version that takes its input from a string. @= procedure str_scan_file(@!s:str_number); label done; var @!p,@!q:pool_pointer; {current position and stopping point} begin begin_name; p:=str_start[s]; q:=str_stop(s); while p= @!job_name:str_number; {principal file name} @!log_opened:boolean; {has the transcript file been opened?} @!log_name:str_number; {full name of the log file} @ Initially |job_name=0|; it becomes nonzero as soon as the true name is known. We have |job_name=0| if and only if the `\.{log}' file has not been opened, except of course for a short time just after |job_name| has become nonzero. @=job_name:=0; log_opened:=false; @ Here is a routine that manufactures the output file names, assuming that |job_name<>0|. It ignores and changes the current settings of |cur_area| and |cur_ext|. @d pack_cur_name==pack_file_name(cur_name,cur_area,cur_ext) @p procedure pack_job_name(@!s:str_number); {|s = ".log"|, |".mem"|, |".ps"|, or .\\{nnn}} begin add_str_ref(s); delete_str_ref(cur_name); delete_str_ref(cur_area); delete_str_ref(cur_ext);@/ cur_area:=""; cur_ext:=s; cur_name:=job_name; pack_cur_name; end; @ If some trouble arises when \MP\ tries to open a file, the following routine calls upon the user to supply another file name. Parameter~|s| is used in the error message to identify the type of file; parameter~|e| is the default extension if none is given. Upon exit from the routine, variables |cur_name|, |cur_area|, |cur_ext|, and |name_of_file| are ready for another attempt at file opening. @p procedure prompt_file_name(@!s,@!e:str_number); label done; var @!k:0..buf_size; {index into |buffer|} begin if interaction=scroll_mode then wake_up_terminal; if s="input file name" then print_err("I can't find file `") @.I can't find file x@> else print_err("I can't write on file `"); @.I can't write on file x@> print_file_name(cur_name,cur_area,cur_ext); print("'."); if e="" then show_context; print_nl("Please type another "); print(s); @.Please type...@> if interaction clear_terminal; prompt_input(": "); @; if cur_ext="" then cur_ext:=e; pack_cur_name; end; @ @= begin begin_name; k:=first; while (buffer[k]=" ")and(k; log_name:=a_make_name_string(log_file); selector:=log_only; log_opened:=true; @; input_stack[input_ptr]:=cur_input; {make sure bottom level is in memory} print_nl("**"); @.**@> l:=input_stack[0].limit_field-1; {last position of first line} for k:=1 to l do print(buffer[k]); print_ln; {now the transcript file contains the first line of input} selector:=old_setting+2; {|log_only| or |term_and_log|} end; @ Sometimes |open_log_file| is called at awkward moments when \MP\ is unable to print error messages or even to |show_context|. The |prompt_file_name| routine can result in a |fatal_error|, but the |error| routine will not be invoked because |log_opened| will be false. The normal idea of |batch_mode| is that nothing at all should be written on the terminal. However, in the unusual case that no log file could be opened, we make an exception and allow an explanatory message to be seen. Incidentally, the program always refers to the log file as a `\.{transcript file}', because some systems cannot use the extension `\.{.log}' for this file. @= begin selector:=term_only; prompt_file_name("transcript file name",".log"); end @ @= begin wlog(banner); print(mem_ident); print(" "); print_int(round_unscaled(internal[day])); print_char(" "); months:='JANFEBMARAPRMAYJUNJULAUGSEPOCTNOVDEC'; m:=round_unscaled(internal[month]); for k:=3*m-2 to 3*m do wlog(months[k]); print_char(" "); print_int(round_unscaled(internal[year])); print_char(" "); m:=round_unscaled(internal[time]); print_dd(m div 60); print_char(":"); print_dd(m mod 60); end @ The |try_extension| function tries to open an input file determined by |cur_name|, |cur_area|, and the argument |ext|. It returns |false| if it can't find the file in |cur_area| or the appropriate system area. @p function try_extension(@!ext:str_number):boolean; begin pack_file_name(cur_name,cur_area,ext); in_name:=cur_name; in_area:=cur_area; if a_open_in(cur_file) then try_extension:=true else begin if str_vs_str(ext,".mf")=0 then in_area:=MF_area else in_area:=MP_area; pack_file_name(cur_name,in_area,ext); try_extension:=a_open_in(cur_file); end; end; @ After all calls to |try_extension|, we must make sure that we count references for |in_name| and |in_area| if they match |cur_name| and/or |cur_area|. @= if in_name=cur_name then add_str_ref(cur_name); if in_area=cur_area then add_str_ref(cur_area) @ Let's turn now to the procedure that is used to initiate file reading when an `\.{input}' command is being processed. @p procedure start_input; {\MP\ will \.{input} something} label done; begin @; loop@+ begin begin_file_reading; {set up |cur_file| and new level of input} if cur_ext="" then if try_extension(".mp") then goto done else if try_extension("") then goto done else if try_extension(".mf") then goto done else do_nothing else if try_extension(cur_ext) then goto done; end_file_reading; {remove the level that didn't work} prompt_file_name("input file name",""); end; done: name:=a_make_name_string(cur_file); @; if job_name=0 then begin job_name:=cur_name; str_ref[job_name]:=max_str_ref; open_log_file; end; {|open_log_file| doesn't |show_context|, so |limit| and |loc| needn't be set to meaningful values yet} if term_offset+length(name)>max_print_line-2 then print_ln else if (term_offset>0)or(file_offset>0) then print_char(" "); print_char("("); incr(open_parens); print(name); update_terminal; @; @; end; @ This code should be omitted if |a_make_name_string| returns something other than just a copy of its argument and the full file name is needed for opening \.{MPX} files or implementing the switch-to-editor option. @^system dependencies@> @= flush_string(name); name:=cur_name; cur_name:=0 @ Here we have to remember to tell the |input_ln| routine not to start with a |get|. If the file is empty, it is considered to contain a single blank line. @^system dependencies@> @= begin line:=1; if input_ln(cur_file,false) then do_nothing; firm_up_the_line; buffer[limit]:="%"; first:=limit+1; loc:=start; end @ @= while token_state and(loc=null) do end_token_list; if token_state then begin print_err("File names can't appear within macros"); @.File names can't...@> help3("Sorry...I've converted what follows to tokens,")@/ ("possibly garbaging the name you gave.")@/ ("Please delete the tokens and insert the name again.");@/ error; end; if file_state then scan_file_name else begin cur_name:=""; cur_ext:=""; cur_area:=""; end @ Sometimes we need to deal with two file names at once. This procedure copies the given string into a special array for an old file name. @p procedure copy_old_name(s:str_number); var @!k:integer; {number of positions filled in |old_file_name|} @!j:pool_pointer; {index into |str_pool|} begin k:=0; for j:=str_start[s] to str_stop(s)-1 do begin incr(k); if k<=file_name_size then old_file_name[k]:=xchr[so(str_pool[j])]; end; if k<=file_name_size then old_name_length:=k else old_name_length:=file_name_size; for k:=old_name_length+1 to file_name_size do @+old_file_name[k]:=' '; end; @ @= @!old_file_name : packed array[1..file_name_size] of char; {analogous to |name_of_file|} @!old_name_length : 0..file_name_size; {this many relevant characters followed by blanks} @ The following simple routine starts reading the \.{MPX} file associated with the current input file. @p procedure start_mpx_input; label exit,not_found; var k:1..file_name_size; begin pack_file_name(in_name,in_area,".mpx"); @; begin_file_reading; if not a_open_in(cur_file) then begin end_file_reading; goto not_found; end; name:=a_make_name_string(cur_file); mpx_name[index]:=name; add_str_ref(name); @; return; not_found: @; exit:end; @ This should ideally be changed to do whatever is necessary to create the \.{MPX} file given by |name_of_file| if it does not exist or if it is out of date. This requires invoking \.{MPtoTeX} on the |old_file_name| and passing the results through \TeX\ and \.{DVItoMP}. (It is possible to use a completely different typesetting program if suitable postprocessor is available to perform the function of \.{DVItoMP}.) @^system dependencies@> @= copy_old_name(name) {System-dependent code should be added here} @ @= if interaction=error_stop_mode then wake_up_terminal; print_nl(">> "); for k:=1 to old_name_length do print(xord[old_file_name[k]]); print_nl(">> "); for k:=1 to name_length do print(xord[name_of_file[k]]); print_nl("! Unable to make mpx file"); help4("The two files given above are one of your source files")@/ ("and an auxiliary file I need to read to find out what your")@/ ("btex..etex blocks mean. If you don't know why I had trouble,")@/ ("try running it manually through MPtoTeX, TeX, and DVItoMP"); succumb; @ The last file-opening commands are for files accessed via the \&{readfrom} @:read_from_}{\&{readfrom} primitive@> operator and the \&{write} command. Such files are stored in separate arrays. @:write_}{\&{write} primitive@> @= readf_index = 0..max_read_files; write_index = 0..max_write_files; @ @= rd_file:array [readf_index] of alpha_file; {\&{readfrom} files} rd_fname:array [readf_index] of str_number; {corresponding file name or 0 if file not open} read_files:readf_index; {number of valid entries in the above arrays} wr_file:array [write_index] of alpha_file; {\&{write} files} wr_fname:array [write_index] of str_number; {corresponding file name or 0 if file not open} write_files:write_index; {number of valid entries in the above arrays} @ @= read_files:=0; write_files:=0; @ This routine starts reading the file named by string~|s| without setting |loc|, |limit|, or |name|. It returns |false| if the file is empty or cannot be opened. Otherwise it updates |rd_file[n]| and |rd_fname[n]|. @p function start_read_input(s:str_number; n:readf_index):boolean; label exit,not_found; begin str_scan_file(s); pack_cur_name; begin_file_reading; if not a_open_in(rd_file[n]) then goto not_found; if not input_ln(rd_file[n],false) then begin a_close(rd_file[n]); goto not_found; end; rd_fname[n]:=s; add_str_ref(s); start_read_input:=true; return; not_found: end_file_reading; start_read_input:=false; exit:end; @ Open |wr_file[n]| using file name~|s| and update |wr_fname[n]|. @p procedure open_write_file(s:str_number; n:readf_index); begin str_scan_file(s); pack_cur_name; while not a_open_out(wr_file[n]) do prompt_file_name("file name for write output",""); wr_fname[n]:=s; add_str_ref(s); end; @* \[36] Introduction to the parsing routines. We come now to the central nervous system that sparks many of \MP's activities. By evaluating expressions, from their primary constituents to ever larger subexpressions, \MP\ builds the structures that ultimately define complete pictures or fonts of type. Four mutually recursive subroutines are involved in this process: We call them $$\hbox{|scan_primary|, |scan_secondary|, |scan_tertiary|, and |scan_expression|.}$$ @^recursion@> Each of them is parameterless and begins with the first token to be scanned already represented in |cur_cmd|, |cur_mod|, and |cur_sym|. After execution, the value of the primary or secondary or tertiary or expression that was found will appear in the global variables |cur_type| and |cur_exp|. The token following the expression will be represented in |cur_cmd|, |cur_mod|, and |cur_sym|. Technically speaking, the parsing algorithms are ``LL(1),'' more or less; backup mechanisms have been added in order to provide reasonable error recovery. @= @!cur_type:small_number; {the type of the expression just found} @!cur_exp:integer; {the value of the expression just found} @ @= cur_exp:=0; @ Many different kinds of expressions are possible, so it is wise to have precise descriptions of what |cur_type| and |cur_exp| mean in all cases: \smallskip\hang |cur_type=vacuous| means that this expression didn't turn out to have a value at all, because it arose from a \&{begingroup}$\,\ldots\,$\&{endgroup} construction in which there was no expression before the \&{endgroup}. In this case |cur_exp| has some irrelevant value. \smallskip\hang |cur_type=boolean_type| means that |cur_exp| is either |true_code| or |false_code|. \smallskip\hang |cur_type=unknown_boolean| means that |cur_exp| points to a capsule node that is in the ring of variables equivalent to at least one undefined boolean variable. \smallskip\hang |cur_type=string_type| means that |cur_exp| is a string number (i.e., an integer in the range |0<=cur_exp= function stash_cur_exp:pointer; var @!p:pointer; {the capsule that will be returned} begin case cur_type of unknown_types,transform_type,color_type,pair_type,dependent,proto_dependent, independent:p:=cur_exp; othercases begin p:=get_node(value_node_size); name_type(p):=capsule; type(p):=cur_type; value(p):=cur_exp; end endcases;@/ cur_type:=vacuous; link(p):=void; stash_cur_exp:=p; end; @ The inverse of |stash_cur_exp| is the following procedure, which deletes an unnecessary capsule and puts its contents into |cur_type| and |cur_exp|. The program steps of \MP\ can be divided into two categories: those in which |cur_type| and |cur_exp| are ``alive'' and those in which they are ``dead,'' in the sense that |cur_type| and |cur_exp| contain relevant information or not. It's important not to ignore them when they're alive, and it's important not to pay attention to them when they're dead. There's also an intermediate category: If |cur_type=vacuous|, then |cur_exp| is irrelevant, hence we can proceed without caring if |cur_type| and |cur_exp| are alive or dead. In such cases we say that |cur_type| and |cur_exp| are {\sl dormant}. It is permissible to call |get_x_next| only when they are alive or dormant. The \\{stash} procedure above assumes that |cur_type| and |cur_exp| are alive or dormant. The \\{unstash} procedure assumes that they are dead or dormant; it resuscitates them. @= procedure unstash_cur_exp(@!p:pointer); begin cur_type:=type(p); case cur_type of unknown_types,transform_type,color_type,pair_type,dependent,proto_dependent, independent: cur_exp:=p; othercases begin cur_exp:=value(p); free_node(p,value_node_size); end endcases;@/ end; @ The following procedure prints the values of expressions in an abbreviated format. If its first parameter |p| is null, the value of |(cur_type,cur_exp)| is displayed; otherwise |p| should be a capsule containing the desired value. The second parameter controls the amount of output. If it is~0, dependency lists will be abbreviated to `\.{linearform}' unless they consist of a single term. If it is greater than~1, complicated structures (pens, pictures, and paths) will be displayed in full. @= @t\4@>@@; @t\4@>@@; procedure print_exp(@!p:pointer;@!verbosity:small_number); var @!restore_cur_exp:boolean; {should |cur_exp| be restored?} @!t:small_number; {the type of the expression} @!v:integer; {the value of the expression} @!q:pointer; {a big node being displayed} begin if p<>null then restore_cur_exp:=false else begin p:=stash_cur_exp; restore_cur_exp:=true; end; t:=type(p); if t; if restore_cur_exp then unstash_cur_exp(p); end; @ @= case t of vacuous:print("vacuous"); boolean_type:if v=true_code then print("true")@+else print("false"); unknown_types,numeric_type:@; string_type:begin print_char(""""); print(v); print_char(""""); end; pen_type,path_type,picture_type:@; transform_type,color_type,pair_type:if v=null then print_type(t) else @; known:print_scaled(v); dependent,proto_dependent:print_dp(t,v,verbosity); independent:print_variable_name(p); othercases confusion("exp") @:this can't happen exp}{\quad exp@> endcases @ @= begin print_char("("); q:=v+big_node_size[t]; repeat if type(v)=known then print_scaled(value(v)) else if type(v)=independent then print_variable_name(v) else print_dp(type(v),dep_list(v),verbosity); v:=v+2; if v<>q then print_char(","); until v=q; print_char(")"); end @ Values of type \&{picture}, \&{path}, and \&{pen} are displayed verbosely in the log file only, unless the user has given a positive value to \\{tracingonline}. @= if verbosity<=1 then print_type(t) else begin if selector=term_and_log then if internal[tracing_online]<=0 then begin selector:=term_only; print_type(t); print(" (see the transcript file)"); selector:=term_and_log; end; case t of pen_type:print_pen(v,"",false); path_type:print_path(v,"",false); picture_type:print_edges(v,"",false); end; {there are no other cases} end @ @= procedure print_dp(@!t:small_number;@!p:pointer;@!verbosity:small_number); var @!q:pointer; {the node following |p|} begin q:=link(p); if (info(q)=null) or (verbosity>0) then print_dependency(p,t) else print("linearform"); end; @ The displayed name of a variable in a ring will not be a capsule unless the ring consists entirely of capsules. @= begin print_type(t); if v<>null then begin print_char(" "); while (name_type(v)=capsule) and (v<>p) do v:=value(v); print_variable_name(v); end; end @ When errors are detected during parsing, it is often helpful to display an expression just above the error message, using |exp_err| or |disp_err| instead of |print_err|. @d exp_err(#)==disp_err(null,#) {displays the current expression} @= procedure disp_err(@!p:pointer;@!s:str_number); begin if interaction=error_stop_mode then wake_up_terminal; print_nl(">> "); @.>>@> print_exp(p,1); {``medium verbose'' printing of the expression} if s<>"" then begin print_nl("! "); print(s); @.!\relax@> end; end; @ If |cur_type| and |cur_exp| contain relevant information that should be recycled, we will use the following procedure, which changes |cur_type| to |known| and stores a given value in |cur_exp|. We can think of |cur_type| and |cur_exp| as either alive or dormant after this has been done, because |cur_exp| will not contain a pointer value. @= procedure flush_cur_exp(@!v:scaled); begin case cur_type of unknown_types,transform_type,color_type,pair_type,@| dependent,proto_dependent,independent: begin recycle_value(cur_exp); free_node(cur_exp,value_node_size); end; string_type:delete_str_ref(cur_exp); pen_type,path_type: toss_knot_list(cur_exp); picture_type:delete_edge_ref(cur_exp); othercases do_nothing endcases;@/ cur_type:=known; cur_exp:=v; end; @ There's a much more general procedure that is capable of releasing the storage associated with any two-word value packet. @= procedure recycle_value(@!p:pointer); label done; var @!t:small_number; {a type code} @!v:integer; {a value} @!vv:integer; {another value} @!q,@!r,@!s,@!pp:pointer; {link manipulation registers} begin t:=type(p); if t; dependent,proto_dependent:@; independent:@; token_list,structured:confusion("recycle"); @:this can't happen recycle}{\quad recycle@> unsuffixed_macro,suffixed_macro:delete_mac_ref(value(p)); end; {there are no other cases} type(p):=undefined; end; @ @= if v<>null then begin q:=v+big_node_size[t]; repeat q:=q-2; recycle_value(q); until q=v; free_node(v,big_node_size[t]); end @ @= begin q:=dep_list(p); while info(q)<>null do q:=link(q); link(prev_dep(p)):=link(q); prev_dep(link(q)):=prev_dep(p); link(q):=null; flush_node_list(dep_list(p)); end @ When an independent variable disappears, it simply fades away, unless something depends on it. In the latter case, a dependent variable whose coefficient of dependence is maximal will take its place. The relevant algorithm is due to Ignacio~A. Zabala, who implemented it as part of his Ph.D. thesis (Stanford University, December 1982). @^Zabala Salelles, Ignacio Andres@> For example, suppose that variable $x$ is being recycled, and that the only variables depending on~$x$ are $y=2x+a$ and $z=x+b$. In this case we want to make $y$ independent and $z=.5y-.5a+b$; no other variables will depend on~$y$. If $\\{tracingequations}>0$ in this situation, we will print `\.{\#\#\# -2x=-y+a}'. There's a slight complication, however: An independent variable $x$ can occur both in dependency lists and in proto-dependency lists. This makes it necessary to be careful when deciding which coefficient is maximal. Furthermore, this complication is not so slight when a proto-dependent variable is chosen to become independent. For example, suppose that $y=2x+100a$ is proto-dependent while $z=x+b$ is dependent; then we must change $z=.5y-50a+b$ to a proto-dependency, because of the large coefficient `50'. In order to deal with these complications without wasting too much time, we shall link together the occurrences of~$x$ among all the linear dependencies, maintaining separate lists for the dependent and proto-dependent cases. @= begin max_c[dependent]:=0; max_c[proto_dependent]:=0;@/ max_link[dependent]:=null; max_link[proto_dependent]:=null;@/ q:=link(dep_head); while q<>dep_head do begin s:=value_loc(q); {now |link(s)=dep_list(q)|} loop@+ begin r:=link(s); if info(r)=null then goto done; if info(r)<>p then s:=r else begin t:=type(q); link(s):=link(r); info(r):=q; if abs(value(r))>max_c[t] then @ else begin link(r):=max_link[t]; max_link[t]:=r; end; end; end; done: q:=link(r); end; if (max_c[dependent]>0)or(max_c[proto_dependent]>0) then @; end @ The code for independency removal makes use of three two-word arrays. @= @!max_c:array[dependent..proto_dependent] of integer; {max coefficient magnitude} @!max_ptr:array[dependent..proto_dependent] of pointer; {where |p| occurs with |max_c|} @!max_link:array[dependent..proto_dependent] of pointer; {other occurrences of |p|} @ @= begin if max_c[t]>0 then begin link(max_ptr[t]):=max_link[t]; max_link[t]:=max_ptr[t]; end; max_c[t]:=abs(value(r)); max_ptr[t]:=r; end @ @= begin if (max_c[dependent] div @'10000 >= max_c[proto_dependent]) then t:=dependent else t:=proto_dependent; @; t:=dependent+proto_dependent-t; {complement |t|} if max_c[t]>0 then {we need to pick up an unchosen dependency} begin link(max_ptr[t]):=max_link[t]; max_link[t]:=max_ptr[t]; end; if t<>dependent then @ else @; flush_node_list(s); if fix_needed then fix_dependencies; check_arith; end @ Let |s=max_ptr[t]|. At this point we have $|value|(s)=\pm|max_c|[t]$, and |info(s)| points to the dependent variable~|pp| of type~|t| from whose dependency list we have removed node~|s|. We must reinsert node~|s| into the dependency list, with coefficient $-1.0$, and with |pp| as the new independent variable. Since |pp| will have a larger serial number than any other variable, we can put node |s| at the head of the list. @= s:=max_ptr[t]; pp:=info(s); v:=value(s); if t=dependent then value(s):=-fraction_one@+else value(s):=-unity; r:=dep_list(pp); link(s):=r; while info(r)<>null do r:=link(r); q:=link(r); link(r):=null; prev_dep(q):=prev_dep(pp); link(prev_dep(pp)):=q; new_indep(pp); if cur_exp=pp then if cur_type=t then cur_type:=independent; if internal[tracing_equations]>0 then @ @ Now $(-v)$ times the formerly independent variable~|p| is being replaced by the dependency list~|s|. @= if interesting(p) then begin begin_diagnostic; print_nl("### "); @:]]]\#\#\#_}{\.{\#\#\#}@> if v>0 then print_char("-"); if t=dependent then vv:=round_fraction(max_c[dependent]) else vv:=max_c[proto_dependent]; if vv<>unity then print_scaled(vv); print_variable_name(p); while value(p) mod s_scale>0 do begin print("*4"); value(p):=value(p)-2; end; if t=dependent then print_char("=")@+else print(" = "); print_dependency(s,t); end_diagnostic(false); end @ Finally, there are dependent and proto-dependent variables whose dependency lists must be brought up to date. @= for t:=dependent to proto_dependent do begin r:=max_link[t]; while r<>null do begin q:=info(r); dep_list(q):=p_plus_fq(dep_list(q),@| make_fraction(value(r),-v),s,t,dependent); if dep_list(q)=dep_final then make_known(q,dep_final); q:=r; r:=link(r); free_node(q,dep_node_size); end; end @ @= for t:=dependent to proto_dependent do begin r:=max_link[t]; while r<>null do begin q:=info(r); if t=dependent then {for safety's sake, we change |q| to |proto_dependent|} begin if cur_exp=q then if cur_type=dependent then cur_type:=proto_dependent; dep_list(q):=p_over_v(dep_list(q),unity,dependent,proto_dependent); type(q):=proto_dependent; value(r):=round_fraction(value(r)); end; dep_list(q):=p_plus_fq(dep_list(q),@| make_scaled(value(r),-v),s,proto_dependent,proto_dependent); if dep_list(q)=dep_final then make_known(q,dep_final); q:=r; r:=link(r); free_node(q,dep_node_size); end; end @ Here are some routines that provide handy combinations of actions that are often needed during error recovery. For example, `|flush_error|' flushes the current expression, replaces it by a given value, and calls |error|. Errors often are detected after an extra token has already been scanned. The `\\{put\_get}' routines put that token back before calling |error|; then they get it back again. (Or perhaps they get another token, if the user has changed things.) @= procedure flush_error(@!v:scaled);@+begin error; flush_cur_exp(v);@+end; @# procedure@?back_error; forward;@t\2@>@/ procedure@?get_x_next; forward;@t\2@>@/ @# procedure put_get_error;@+begin back_error; get_x_next;@+end; @# procedure put_get_flush_error(@!v:scaled);@+begin put_get_error; flush_cur_exp(v);@+end; @ A global variable |var_flag| is set to a special command code just before \MP\ calls |scan_expression|, if the expression should be treated as a variable when this command code immediately follows. For example, |var_flag| is set to |assignment| at the beginning of a statement, because we want to know the {\sl location\/} of a variable at the left of `\.{:=}', not the {\sl value\/} of that variable. The |scan_expression| subroutine calls |scan_tertiary|, which calls |scan_secondary|, which calls |scan_primary|, which sets |var_flag:=0|. In this way each of the scanning routines ``knows'' when it has been called with a special |var_flag|, but |var_flag| is usually zero. A variable preceding a command that equals |var_flag| is converted to a token list rather than a value. Furthermore, an `\.{=}' sign following an expression with |var_flag=assignment| is not considered to be a relation that produces boolean expressions. @= @!var_flag:0..max_command_code; {command that wants a variable} @ @= var_flag:=0; @* \[37] Parsing primary expressions. The first parsing routine, |scan_primary|, is also the most complicated one, since it involves so many different cases. But each case---with one exception---is fairly simple by itself. When |scan_primary| begins, the first token of the primary to be scanned should already appear in |cur_cmd|, |cur_mod|, and |cur_sym|. The values of |cur_type| and |cur_exp| should be either dead or dormant, as explained earlier. If |cur_cmd| is not between |min_primary_command| and |max_primary_command|, inclusive, a syntax error will be signaled. @= procedure scan_primary; label restart, done, done1, done2; var @!p,@!q,@!r:pointer; {for list manipulation} @!c:quarterword; {a primitive operation code} @!my_var_flag:0..max_command_code; {initial value of |my_var_flag|} @!l_delim,@!r_delim:pointer; {hash addresses of a delimiter pair} @@; begin my_var_flag:=var_flag; var_flag:=0; restart:check_arith; @; case cur_cmd of left_delimiter:@; begin_group:@; string_token:@; numeric_token:@; nullary:@; unary,type_name,cycle,plus_or_minus:@; primary_binary:@; str_op:@; internal_quantity:@; capsule_token:make_exp_copy(cur_mod); tag_token:@; othercases begin bad_exp("A primary"); goto restart; @.A primary expression...@> end endcases;@/ get_x_next; {the routines |goto done| if they don't want this} done: if cur_cmd=left_bracket then if cur_type>=known then @; end; @ Errors at the beginning of expressions are flagged by |bad_exp|. @p procedure bad_exp(@!s:str_number); var save_flag:0..max_command_code; begin print_err(s); print(" expression can't begin with `"); print_cmd_mod(cur_cmd,cur_mod); print_char("'"); help4("I'm afraid I need some sort of value in order to continue,")@/ ("so I've tentatively inserted `0'. You may want to")@/ ("delete this zero and insert something else;")@/ ("see Chapter 27 of The METAFONTbook for an example."); @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> back_input; cur_sym:=0; cur_cmd:=numeric_token; cur_mod:=0; ins_error;@/ save_flag:=var_flag; var_flag:=0; get_x_next; var_flag:=save_flag; end; @ @= debug if panicking then check_mem(false);@+gubed@;@/ if interrupt<>0 then if OK_to_interrupt then begin back_input; check_interrupt; get_x_next; end @ @= begin l_delim:=cur_sym; r_delim:=cur_mod; get_x_next; scan_expression; if (cur_cmd=comma) and (cur_type>=known) then @ else check_delimiter(l_delim,r_delim); end @ The |stash_in| subroutine puts the current (numeric) expression into a field within a ``big node.'' @p procedure stash_in(@!p:pointer); var @!q:pointer; {temporary register} begin type(p):=cur_type; if cur_type=known then value(p):=cur_exp else begin if cur_type=independent then @ else begin mem[value_loc(p)]:=mem[value_loc(cur_exp)]; {|dep_list(p):=dep_list(cur_exp)| and |prev_dep(p):=prev_dep(cur_exp)|} link(prev_dep(p)):=p; end; free_node(cur_exp,value_node_size); end; cur_type:=vacuous; end; @ In rare cases the current expression can become |independent|. There may be many dependency lists pointing to such an independent capsule, so we can't simply move it into place within a big node. Instead, we copy it, then recycle it. @ @= begin q:=single_dependency(cur_exp); if q=dep_final then begin type(p):=known; value(p):=0; free_node(q,dep_node_size); end else begin type(p):=dependent; new_dep(p,q); end; recycle_value(cur_exp); end @ This code uses the fact that |red_part_loc| and |green_part_loc| are synonymous with |x_part_loc| and |y_part_loc|. @= begin p:=stash_cur_exp; get_x_next; scan_expression; @; q:=get_node(value_node_size); name_type(q):=capsule; if cur_cmd=comma then type(q):=color_type else type(q):=pair_type; init_big_node(q); r:=value(q); stash_in(y_part_loc(r)); unstash_cur_exp(p); stash_in(x_part_loc(r)); if cur_cmd=comma then @; check_delimiter(l_delim,r_delim); cur_type:=type(q); cur_exp:=q; end @ @= if cur_type help4("I've started to scan a pair `(a,b)' or a color `(a,b,c)';")@/ ("but after finding a nice `a' I found a `b' that isn't")@/ ("of numeric type. So I've changed that part to zero.")@/ ("(The b that I didn't like appears above the error message.)"); put_get_flush_error(0); end @ @= begin get_x_next; scan_expression; if cur_type help3("I've just scanned a color `(r,g,b)'; but the `b' isn't")@/ ("of numeric type. So I've changed that part to zero.")@/ ("(The b that I didn't like appears above the error message.)");@/ put_get_flush_error(0); end; stash_in(blue_part_loc(r)); end @ The local variable |group_line| keeps track of the line where a \&{begingroup} command occurred; this will be useful in an error message if the group doesn't actually end. @= @!group_line:integer; {where a group began} @ @= begin group_line:=true_line; if internal[tracing_commands]>0 then show_cur_cmd_mod; save_boundary_item(p); repeat do_statement; {ends with |cur_cmd>=semicolon|} until cur_cmd<>semicolon; if cur_cmd<>end_group then begin print_err("A group begun on line "); @.A group...never ended@> print_int(group_line); print(" never ended"); help2("I saw a `begingroup' back there that hasn't been matched")@/ ("by `endgroup'. So I've inserted `endgroup' now."); back_error; cur_cmd:=end_group; end; unsave; {this might change |cur_type|, if independent variables are recycled} if internal[tracing_commands]>0 then show_cur_cmd_mod; end @ @= begin cur_type:=string_type; cur_exp:=cur_mod; end @ Later we'll come to procedures that perform actual operations like addition, square root, and so on; our purpose now is to do the parsing. But we might as well mention those future procedures now, so that the suspense won't be too bad: \smallskip |do_nullary(c)| does primitive operations that have no operands (e.g., `\&{true}' or `\&{pencircle}'); \smallskip |do_unary(c)| applies a primitive operation to the current expression; \smallskip |do_binary(p,c)| applies a primitive operation to the capsule~|p| and the current expression. @=do_nullary(cur_mod) @ @= begin c:=cur_mod; get_x_next; scan_primary; do_unary(c); goto done; end @ A numeric token might be a primary by itself, or it might be the numerator of a fraction composed solely of numeric tokens, or it might multiply the primary that follows (provided that the primary doesn't begin with a plus sign or a minus sign). The code here uses the facts that |max_primary_command=plus_or_minus| and |max_primary_command-1=numeric_token|. If a fraction is found that is less than unity, we try to retain higher precision when we use it in scalar multiplication. @= @!num,@!denom:scaled; {for primaries that are fractions, like `1/2'} @ @= begin cur_exp:=cur_mod; cur_type:=known; get_x_next; if cur_cmd<>slash then begin num:=0; denom:=0; end else begin get_x_next; if cur_cmd<>numeric_token then begin back_input; cur_cmd:=slash; cur_mod:=over; cur_sym:=frozen_slash; goto done; end; num:=cur_exp; denom:=cur_mod; if denom=0 then @ else cur_exp:=make_scaled(num,denom); check_arith; get_x_next; end; if cur_cmd>=min_primary_command then if cur_cmdplus_or_minus|} begin p:=stash_cur_exp; scan_primary; if (abs(num)>=abs(denom))or(cur_type= begin print_err("Division by zero"); @.Division by zero@> help1("I'll pretend that you meant to divide by 1."); error; end @ @= begin c:=cur_mod; get_x_next; scan_expression; if cur_cmd<>of_token then begin missing_err("of"); print(" for "); print_cmd_mod(primary_binary,c); @.Missing `of'@> help1("I've got the first argument; will look now for the other."); back_error; end; p:=stash_cur_exp; get_x_next; scan_primary; do_binary(p,c); goto done; end @ @= begin get_x_next; scan_suffix; old_setting:=selector; selector:=new_string; show_token_list(cur_exp,null,100000,0); flush_token_list(cur_exp); cur_exp:=make_string; selector:=old_setting; cur_type:=string_type; goto done; end @ If an internal quantity appears all by itself on the left of an assignment, we return a token list of length one, containing the address of the internal quantity plus |hash_end|. (This accords with the conventions of the save stack, as described earlier.) @= begin q:=cur_mod; if my_var_flag=assignment then begin get_x_next; if cur_cmd=assignment then begin cur_exp:=get_avail; info(cur_exp):=q+hash_end; cur_type:=token_list; goto done; end; back_input; end; cur_type:=known; cur_exp:=internal[q]; end @ The most difficult part of |scan_primary| has been saved for last, since it was necessary to build up some confidence first. We can now face the task of scanning a variable. As we scan a variable, we build a token list containing the relevant names and subscript values, simultaneously following along in the ``collective'' structure to see if we are actually dealing with a macro instead of a value. The local variables |pre_head| and |post_head| will point to the beginning of the prefix and suffix lists; |tail| will point to the end of the list that is currently growing. Another local variable, |tt|, contains partial information about the declared type of the variable-so-far. If |tt>=unsuffixed_macro|, the relation |tt=type(q)| will always hold. If |tt=undefined|, the routine doesn't bother to update its information about type. And if |undefined= @!pre_head,@!post_head,@!tail:pointer; {prefix and suffix list variables} @!tt:small_number; {approximation to the type of the variable-so-far} @!t:pointer; {a token} @!macro_ref:pointer; {reference count for a suffixed macro} @ @= begin fast_get_avail(pre_head); tail:=pre_head; post_head:=null; tt:=vacuous; loop@+ begin t:=cur_tok; link(tail):=t; if tt<>undefined then begin @; if tt>=unsuffixed_macro then @; end; get_x_next; tail:=t; if cur_cmd=left_bracket then @; if cur_cmd>max_suffix_token then goto done1; if cur_cmd; end @ @= begin link(tail):=null; if tt>unsuffixed_macro then {|tt=suffixed_macro|} begin post_head:=get_avail; tail:=post_head; link(tail):=t;@/ tt:=undefined; macro_ref:=value(q); add_mac_ref(macro_ref); end else @; end @ @= begin get_x_next; scan_expression; if cur_cmd<>right_bracket then @ else begin if cur_type<>known then bad_subscript; cur_cmd:=numeric_token; cur_mod:=cur_exp; cur_sym:=0; end; end @ The left bracket that we thought was introducing a subscript might have actually been the left bracket in a mediation construction like `\.{x[a,b]}'. So we don't issue an error message at this point; but we do want to back up so as to avoid any embarrassment about our incorrect assumption. @= begin back_input; {that was the token following the current expression} back_expr; cur_cmd:=left_bracket; cur_mod:=0; cur_sym:=frozen_left_bracket; end @ Here's a routine that puts the current expression back to be read again. @p procedure back_expr; var @!p:pointer; {capsule token} begin p:=stash_cur_exp; link(p):=null; back_list(p); end; @ Unknown subscripts lead to the following error message. @p procedure bad_subscript; begin exp_err("Improper subscript has been replaced by zero"); @.Improper subscript...@> help3("A bracketed subscript must have a known numeric value;")@/ ("unfortunately, what I found was the value that appears just")@/ ("above this error message. So I'll try a zero subscript."); flush_error(0); end; @ Every time we call |get_x_next|, there's a chance that the variable we've been looking at will disappear. Thus, we cannot safely keep |q| pointing into the variable structure; we need to start searching from the root each time. @= @^inner loop@> begin p:=link(pre_head); q:=info(p); tt:=undefined; if eq_type(q) mod outer_tag=tag_token then begin q:=equiv(q); if q=null then goto done2; loop@+ begin p:=link(p); if p=null then begin tt:=type(q); goto done2; end; if type(q)<>structured then goto done2; q:=link(attr_head(q)); {the |collective_subscript| attribute} if p>=hi_mem_min then {it's not a subscript} begin repeat q:=link(q); until attr_loc(q)>=info(p); if attr_loc(q)>info(p) then goto done2; end; end; end; done2:end @ How do things stand now? Well, we have scanned an entire variable name, including possible subscripts and/or attributes; |cur_cmd|, |cur_mod|, and |cur_sym| represent the token that follows. If |post_head=null|, a token list for this variable name starts at |link(pre_head)|, with all subscripts evaluated. But if |post_head<>null|, the variable turned out to be a suffixed macro; |pre_head| is the head of the prefix list, while |post_head| is the head of a token list containing both `\.{\AT!}' and the suffix. Our immediate problem is to see if this variable still exists. (Variable structures can change drastically whenever we call |get_x_next|; users aren't supposed to do this, but the fact that it is possible means that we must be cautious.) The following procedure prints an error message when a variable unexpectedly disappears. Its help message isn't quite right for our present purposes, but we'll be able to fix that up. @p procedure obliterated(@!q:pointer); begin print_err("Variable "); show_token_list(q,null,1000,0); print(" has been obliterated"); @.Variable...obliterated@> help5("It seems you did a nasty thing---probably by accident,")@/ ("but nevertheless you nearly hornswoggled me...")@/ ("While I was evaluating the right-hand side of this")@/ ("command, something happened, and the left-hand side")@/ ("is no longer a variable! So I won't change anything."); end; @ If the variable does exist, we also need to check for a few other special cases before deciding that a plain old ordinary variable has, indeed, been scanned. @= if post_head<>null then @; q:=link(pre_head); free_avail(pre_head); if cur_cmd=my_var_flag then begin cur_type:=token_list; cur_exp:=q; goto done; end; p:=find_variable(q); if p<>null then make_exp_copy(p) else begin obliterated(q);@/ help_line[2]:="While I was evaluating the suffix of this variable,"; help_line[1]:="something was redefined, and it's no longer a variable!"; help_line[0]:="In order to get back on my feet, I've inserted `0' instead."; put_get_flush_error(0); end; flush_node_list(q); goto done @ The only complication associated with macro calling is that the prefix and ``at'' parameters must be packaged in an appropriate list of lists. @= begin p:=get_avail; info(pre_head):=link(pre_head); link(pre_head):=p; info(p):=t; macro_call(value(q),pre_head,null); get_x_next; goto restart; end @ If the ``variable'' that turned out to be a suffixed macro no longer exists, we don't care, because we have reserved a pointer (|macro_ref|) to its token list. @= begin back_input; p:=get_avail; q:=link(post_head); info(pre_head):=link(pre_head); link(pre_head):=post_head; info(post_head):=q; link(post_head):=p; info(p):=link(q); link(q):=null; macro_call(macro_ref,pre_head,null); decr(ref_count(macro_ref)); get_x_next; goto restart; end @ Our remaining job is simply to make a copy of the value that has been found. Some cases are harder than others, but complexity arises solely because of the multiplicity of possible cases. @= @t\4@>@@; procedure make_exp_copy(@!p:pointer); label restart; var @!q,@!r,@!t:pointer; {registers for list manipulation} begin restart: cur_type:=type(p); case cur_type of vacuous,boolean_type,known:cur_exp:=value(p); unknown_types:cur_exp:=new_ring_entry(p); string_type:begin cur_exp:=value(p); add_str_ref(cur_exp); end; picture_type:begin cur_exp:=value(p);add_edge_ref(cur_exp); end; pen_type:cur_exp:=copy_pen(value(p)); path_type:cur_exp:=copy_path(value(p)); transform_type,color_type,pair_type:@; dependent,proto_dependent:encapsulate(copy_dep_list(dep_list(p))); numeric_type:begin new_indep(p); goto restart; end; independent: begin q:=single_dependency(p); if q=dep_final then begin cur_type:=known; cur_exp:=0; free_node(q,value_node_size); end else begin cur_type:=dependent; encapsulate(q); end; end; othercases confusion("copy") @:this can't happen copy}{\quad copy@> endcases; end; @ The |encapsulate| subroutine assumes that |dep_final| is the tail of dependency list~|p|. @= procedure encapsulate(@!p:pointer); begin cur_exp:=get_node(value_node_size); type(cur_exp):=cur_type; name_type(cur_exp):=capsule; new_dep(cur_exp,p); end; @ The most tedious case arises when the user refers to a \&{pair}, \&{color}, or \&{transform} variable; we must copy several fields, each of which can be |independent|, |dependent|, |proto_dependent|, or |known|. @= begin if value(p)=null then init_big_node(p); t:=get_node(value_node_size); name_type(t):=capsule; type(t):=cur_type; init_big_node(t);@/ q:=value(p)+big_node_size[cur_type]; r:=value(t)+big_node_size[cur_type]; repeat q:=q-2; r:=r-2; install(r,q); until q=value(p); cur_exp:=t; end @ The |install| procedure copies a numeric field~|q| into field~|r| of a big node that will be part of a capsule. @= procedure install(@!r,@!q:pointer); var p:pointer; {temporary register} begin if type(q)=known then begin value(r):=value(q); type(r):=known; end else if type(q)=independent then begin p:=single_dependency(q); if p=dep_final then begin type(r):=known; value(r):=0; free_node(p,value_node_size); end else begin type(r):=dependent; new_dep(r,p); end; end else begin type(r):=type(q); new_dep(r,copy_dep_list(dep_list(q))); end; end; @ Expressions of the form `\.{a[b,c]}' are converted into `\.{b+a*(c-b)}', without checking the types of \.b~or~\.c, provided that \.a is numeric. @= begin p:=stash_cur_exp; get_x_next; scan_expression; if cur_cmd<>comma then begin @; unstash_cur_exp(p); end else begin q:=stash_cur_exp; get_x_next; scan_expression; if cur_cmd<>right_bracket then begin missing_err("]");@/ @.Missing `]'@> help3("I've scanned an expression of the form `a[b,c',")@/ ("so a right bracket should have come next.")@/ ("I shall pretend that one was there.");@/ back_error; end; r:=stash_cur_exp; make_exp_copy(q);@/ do_binary(r,minus); do_binary(p,times); do_binary(q,plus); get_x_next; end; end @ Here is a comparatively simple routine that is used to scan the \&{suffix} parameters of a macro. @= procedure scan_suffix; label done; var @!h,@!t:pointer; {head and tail of the list being built} @!p:pointer; {temporary register} begin h:=get_avail; t:=h; loop@+ begin if cur_cmd=left_bracket then @; if cur_cmd=numeric_token then p:=new_num_tok(cur_mod) else if (cur_cmd=tag_token)or(cur_cmd=internal_quantity) then begin p:=get_avail; info(p):=cur_sym; end else goto done; link(t):=p; t:=p; get_x_next; end; done: cur_exp:=link(h); free_avail(h); cur_type:=token_list; end; @ @= begin get_x_next; scan_expression; if cur_type<>known then bad_subscript; if cur_cmd<>right_bracket then begin missing_err("]");@/ @.Missing `]'@> help3("I've seen a `[' and a subscript value, in a suffix,")@/ ("so a right bracket should have come next.")@/ ("I shall pretend that one was there.");@/ back_error; end; cur_cmd:=numeric_token; cur_mod:=cur_exp; end @* \[38] Parsing secondary and higher expressions. After the intricacies of |scan_primary|\kern-1pt, the |scan_secondary| routine is refreshingly simple. It's not trivial, but the operations are relatively straightforward; the main difficulty is, again, that expressions and data structures might change drastically every time we call |get_x_next|, so a cautious approach is mandatory. For example, a macro defined by \&{primarydef} might have disappeared by the time its second argument has been scanned; we solve this by increasing the reference count of its token list, so that the macro can be called even after it has been clobbered. @= procedure scan_secondary; label restart,continue; var @!p:pointer; {for list manipulation} @!c,@!d:halfword; {operation codes or modifiers} @!mac_name:pointer; {token defined with \&{primarydef}} begin restart:if(cur_cmdmax_primary_command) then bad_exp("A secondary"); @.A secondary expression...@> scan_primary; continue: if cur_cmd<=max_secondary_command then if cur_cmd>=min_secondary_command then begin p:=stash_cur_exp; c:=cur_mod; d:=cur_cmd; if d=secondary_primary_macro then begin mac_name:=cur_sym; add_mac_ref(c); end; get_x_next; scan_primary; if d<>secondary_primary_macro then do_binary(p,c) else begin back_input; binary_mac(p,c,mac_name); decr(ref_count(c)); get_x_next; goto restart; end; goto continue; end; end; @ The following procedure calls a macro that has two parameters, |p| and |cur_exp|. @p procedure binary_mac(@!p,@!c,@!n:pointer); var @!q,@!r:pointer; {nodes in the parameter list} begin q:=get_avail; r:=get_avail; link(q):=r;@/ info(q):=p; info(r):=stash_cur_exp;@/ macro_call(c,q,n); end; @ The next procedure, |scan_tertiary|, is pretty much the same deal. @= procedure scan_tertiary; label restart,continue; var @!p:pointer; {for list manipulation} @!c,@!d:halfword; {operation codes or modifiers} @!mac_name:pointer; {token defined with \&{secondarydef}} begin restart:if(cur_cmdmax_primary_command) then bad_exp("A tertiary"); @.A tertiary expression...@> scan_secondary; continue: if cur_cmd<=max_tertiary_command then if cur_cmd>=min_tertiary_command then begin p:=stash_cur_exp; c:=cur_mod; d:=cur_cmd; if d=tertiary_secondary_macro then begin mac_name:=cur_sym; add_mac_ref(c); end; get_x_next; scan_secondary; if d<>tertiary_secondary_macro then do_binary(p,c) else begin back_input; binary_mac(p,c,mac_name); decr(ref_count(c)); get_x_next; goto restart; end; goto continue; end; end; @ Finally we reach the deepest level in our quartet of parsing routines. This one is much like the others; but it has an extra complication from paths, which materialize here. @d continue_path=25 {a label inside of |scan_expression|} @d finish_path=26 {another} @= procedure scan_expression; label restart,done,continue,continue_path,finish_path,exit; var @!p,@!q,@!r,@!pp,@!qq:pointer; {for list manipulation} @!c,@!d:halfword; {operation codes or modifiers} @!my_var_flag:0..max_command_code; {initial value of |var_flag|} @!mac_name:pointer; {token defined with \&{tertiarydef}} @!cycle_hit:boolean; {did a path expression just end with `\&{cycle}'?} @!x,@!y:scaled; {explicit coordinates or tension at a path join} @!t:endpoint..open; {knot type following a path join} begin my_var_flag:=var_flag; restart:if(cur_cmdmax_primary_command) then bad_exp("An"); @.An expression...@> scan_tertiary; continue: if cur_cmd<=max_expression_command then if cur_cmd>=min_expression_command then if (cur_cmd<>equals)or(my_var_flag<>assignment) then begin p:=stash_cur_exp; c:=cur_mod; d:=cur_cmd; if d=expression_tertiary_macro then begin mac_name:=cur_sym; add_mac_ref(c); end; if (d else begin get_x_next; scan_tertiary; if d<>expression_tertiary_macro then do_binary(p,c) else begin back_input; binary_mac(p,c,mac_name); decr(ref_count(c)); get_x_next; goto restart; end; end; goto continue; end; exit:end; @ The reader should review the data structure conventions for paths before hoping to understand the next part of this code. @= begin cycle_hit:=false; @; continue_path: @; if cur_cmd=cycle then @ else begin scan_tertiary; @; end; @; if cur_cmd>=min_expression_command then if cur_cmd<=ampersand then if not cycle_hit then goto continue_path; finish_path: @; end @ @= begin unstash_cur_exp(p); if cur_type=pair_type then p:=new_knot else if cur_type=path_type then p:=cur_exp else return; q:=p; while link(q)<>p do q:=link(q); if left_type(p)<>endpoint then {open up a cycle} begin r:=copy_knot(p); link(q):=r; q:=r; end; left_type(p):=open; right_type(q):=open; end @ A pair of numeric values is changed into a knot node for a one-point path when \MP\ discovers that the pair is part of a path. @p@t\4@>@@; function new_knot:pointer; {convert a pair to a knot with two endpoints} var @!q:pointer; {the new node} begin q:=get_node(knot_node_size); left_type(q):=endpoint; right_type(q):=endpoint; link(q):=q;@/ known_pair; x_coord(q):=cur_x; y_coord(q):=cur_y; new_knot:=q; end; @ The |known_pair| subroutine sets |cur_x| and |cur_y| to the components of the current expression, assuming that the current expression is a pair of known numerics. Unknown components are zeroed, and the current expression is flushed. @= procedure known_pair; var @!p:pointer; {the pair node} begin if cur_type<>pair_type then begin exp_err("Undefined coordinates have been replaced by (0,0)"); @.Undefined coordinates...@> help5("I need x and y numbers for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> ("you might want to type `I ???' now.)"); put_get_flush_error(0); cur_x:=0; cur_y:=0; end else begin p:=value(cur_exp); @; flush_cur_exp(0); end; end; @ @= if type(x_part_loc(p))=known then cur_x:=value(x_part_loc(p)) else begin disp_err(x_part_loc(p), "Undefined x coordinate has been replaced by 0"); @.Undefined coordinates...@> help5("I need a `known' x value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> ("you might want to type `I ???' now.)"); put_get_error; recycle_value(x_part_loc(p)); cur_x:=0; end; if type(y_part_loc(p))=known then cur_y:=value(y_part_loc(p)) else begin disp_err(y_part_loc(p), "Undefined y coordinate has been replaced by 0"); help5("I need a `known' y value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ ("you might want to type `I ???' now.)"); put_get_error; recycle_value(y_part_loc(p)); cur_y:=0; end @ At this point |cur_cmd| is either |ampersand|, |left_brace|, or |path_join|. @= if cur_cmd=left_brace then @; d:=cur_cmd; if d=path_join then @ else if d<>ampersand then goto finish_path; get_x_next; if cur_cmd=left_brace then @ else if right_type(q)<>explicit then begin t:=open; x:=0; end @ The |scan_direction| subroutine looks at the directional information that is enclosed in braces, and also scans ahead to the following character. A type code is returned, either |open| (if the direction was $(0,0)$), or |curl| (if the direction was a curl of known value |cur_exp|), or |given| (if the direction is given by the |angle| value that now appears in |cur_exp|). There's nothing difficult about this subroutine, but the program is rather lengthy because a variety of potential errors need to be nipped in the bud. @p function scan_direction:small_number; var @!t:given..open; {the type of information found} @!x:scaled; {an |x| coordinate} begin get_x_next; if cur_cmd=curl_command then @ else @; if cur_cmd<>right_brace then begin missing_err("}");@/ @.Missing `\char`\}'@> help3("I've scanned a direction spec for part of a path,")@/ ("so a right brace should have come next.")@/ ("I shall pretend that one was there.");@/ back_error; end; get_x_next; scan_direction:=t; end; @ @= begin get_x_next; scan_expression; if (cur_type<>known)or(cur_exp<0) then begin exp_err("Improper curl has been replaced by 1"); @.Improper curl@> help1("A curl must be a known, nonnegative number."); put_get_flush_error(unity); end; t:=curl; end @ @= begin scan_expression; if cur_type>pair_type then @ else known_pair; if (cur_x=0)and(cur_y=0) then t:=open else begin t:=given; cur_exp:=n_arg(cur_x,cur_y); end; end @ @= begin if cur_type<>known then begin exp_err("Undefined x coordinate has been replaced by 0"); @.Undefined coordinates...@> help5("I need a `known' x value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> ("you might want to type `I ???' now.)"); put_get_flush_error(0); end; x:=cur_exp; if cur_cmd<>comma then begin missing_err(",");@/ @.Missing `,'@> help2("I've got the x coordinate of a path direction;")@/ ("will look for the y coordinate next."); back_error; end; get_x_next; scan_expression; if cur_type<>known then begin exp_err("Undefined y coordinate has been replaced by 0"); help5("I need a `known' y value for this part of the path.")@/ ("The value I found (see above) was no good;")@/ ("so I'll try to keep going by using zero instead.")@/ ("(Chapter 27 of The METAFONTbook explains that")@/ ("you might want to type `I ???' now.)"); put_get_flush_error(0); end; cur_y:=cur_exp; cur_x:=x; end @ At this point |right_type(q)| is usually |open|, but it may have been set to some other value by a previous splicing operation. We must maintain the value of |right_type(q)| in unusual cases such as `\.{..z1\{z2\}\&\{z3\}z1\{0,0\}..}'. @= begin t:=scan_direction; if t<>open then begin right_type(q):=t; right_given(q):=cur_exp; if left_type(q)=open then begin left_type(q):=t; left_given(q):=cur_exp; end; {note that |left_given(q)=left_curl(q)|} end; end @ Since |left_tension| and |left_y| share the same position in knot nodes, and since |left_given| is similarly equivalent to |left_x|, we use |x| and |y| to hold the given direction and tension information when there are no explicit control points. @= begin t:=scan_direction; if right_type(q)<>explicit then x:=cur_exp else t:=explicit; {the direction information is superfluous} end @ @= begin get_x_next; if cur_cmd=tension then @ else if cur_cmd=controls then @ else begin right_tension(q):=unity; y:=unity; back_input; {default tension} goto done; end; if cur_cmd<>path_join then begin missing_err("..");@/ @.Missing `..'@> help1("A path join command should end with two dots."); back_error; end; done:end @ @= begin get_x_next; y:=cur_cmd; if cur_cmd=at_least then get_x_next; scan_primary; @; if y=at_least then negate(cur_exp); right_tension(q):=cur_exp; if cur_cmd=and_command then begin get_x_next; y:=cur_cmd; if cur_cmd=at_least then get_x_next; scan_primary; @; if y=at_least then negate(cur_exp); end; y:=cur_exp; end @ @d min_tension==three_quarter_unit @= if (cur_type<>known)or(cur_exp help1("The expression above should have been a number >=3/4."); put_get_flush_error(unity); end @ @= begin right_type(q):=explicit; t:=explicit; get_x_next; scan_primary;@/ known_pair; right_x(q):=cur_x; right_y(q):=cur_y; if cur_cmd<>and_command then begin x:=right_x(q); y:=right_y(q); end else begin get_x_next; scan_primary;@/ known_pair; x:=cur_x; y:=cur_y; end; end @ @= begin if cur_type<>path_type then pp:=new_knot else pp:=cur_exp; qq:=pp; while link(qq)<>pp do qq:=link(qq); if left_type(pp)<>endpoint then {open up a cycle} begin r:=copy_knot(pp); link(qq):=r; qq:=r; end; left_type(pp):=open; right_type(qq):=open; end @ If a person tries to define an entire path by saying `\.{(x,y)\&cycle}', we silently change the specification to `\.{(x,y)..cycle}', since a cycle shouldn't have length zero. @= begin cycle_hit:=true; get_x_next; pp:=p; qq:=p; if d=ampersand then if p=q then begin d:=path_join; right_tension(q):=unity; y:=unity; end; end @ @= begin if d=ampersand then if (x_coord(q)<>x_coord(pp))or(y_coord(q)<>y_coord(pp)) then begin print_err("Paths don't touch; `&' will be changed to `..'"); @.Paths don't touch@> help3("When you join paths `p&q', the ending point of p")@/ ("must be exactly equal to the starting point of q.")@/ ("So I'm going to pretend that you said `p..q' instead."); put_get_error; d:=path_join; right_tension(q):=unity; y:=unity; end; @; if d=ampersand then @ else begin @; link(q):=pp; left_y(pp):=y; if t<>open then begin left_x(pp):=x; left_type(pp):=t; end; end; q:=qq; end @ @= if right_type(q)=open then if (left_type(q)=curl)or(left_type(q)=given) then begin right_type(q):=left_type(q); right_given(q):=left_given(q); end @ @= if right_type(pp)=open then if (t=curl)or(t=given) then begin right_type(pp):=t; right_given(pp):=x; end @ @= begin if left_type(q)=open then if right_type(q)=open then begin left_type(q):=curl; left_curl(q):=unity; end; if right_type(pp)=open then if t=open then begin right_type(pp):=curl; right_curl(pp):=unity; end; right_type(q):=right_type(pp); link(q):=link(pp);@/ right_x(q):=right_x(pp); right_y(q):=right_y(pp); free_node(pp,knot_node_size); if qq=pp then qq:=q; end @ @= if cycle_hit then begin if d=ampersand then p:=q; end else begin left_type(p):=endpoint; if right_type(p)=open then begin right_type(p):=curl; right_curl(p):=unity; end; right_type(q):=endpoint; if left_type(q)=open then begin left_type(q):=curl; left_curl(q):=unity; end; link(q):=p; end; make_choices(p); cur_type:=path_type; cur_exp:=p @ Finally, we sometimes need to scan an expression whose value is supposed to be either |true_code| or |false_code|. @= procedure get_boolean; begin get_x_next; scan_expression; if cur_type<>boolean_type then begin exp_err("Undefined condition will be treated as `false'"); @.Undefined condition...@> help2("The expression shown above should have had a definite")@/ ("true-or-false value. I'm changing it to `false'.");@/ put_get_flush_error(false_code); cur_type:=boolean_type; end; end; @* \[39] Doing the operations. The purpose of parsing is primarily to permit people to avoid piles of parentheses. But the real work is done after the structure of an expression has been recognized; that's when new expressions are generated. We turn now to the guts of \MP, which handles individual operators that have come through the parsing mechanism. We'll start with the easy ones that take no operands, then work our way up to operators with one and ultimately two arguments. In other words, we will write the three procedures |do_nullary|, |do_unary|, and |do_binary| that are invoked periodically by the expression scanners. First let's make sure that all of the primitive operators are in the hash table. Although |scan_primary| and its relatives made use of the \\{cmd} code for these operators, the \\{do} routines base everything on the \\{mod} code. For example, |do_binary| doesn't care whether the operation it performs is a |primary_binary| or |secondary_binary|, etc. @= primitive("true",nullary,true_code);@/ @!@:true_}{\&{true} primitive@> primitive("false",nullary,false_code);@/ @!@:false_}{\&{false} primitive@> primitive("nullpicture",nullary,null_picture_code);@/ @!@:null_picture_}{\&{nullpicture} primitive@> primitive("nullpen",nullary,null_pen_code);@/ @!@:null_pen_}{\&{nullpen} primitive@> primitive("jobname",nullary,job_name_op);@/ @!@:job_name_}{\&{jobname} primitive@> primitive("readstring",nullary,read_string_op);@/ @!@:read_string_}{\&{readstring} primitive@> primitive("pencircle",nullary,pen_circle);@/ @!@:pen_circle_}{\&{pencircle} primitive@> primitive("normaldeviate",nullary,normal_deviate);@/ @!@:normal_deviate_}{\&{normaldeviate} primitive@> primitive("readfrom",unary,read_from_op);@/ @!@:read_from_}{\&{readfrom} primitive@> primitive("closefrom",unary,close_from_op);@/ @!@:close_from_}{\&{closefrom} primitive@> primitive("odd",unary,odd_op);@/ @!@:odd_}{\&{odd} primitive@> primitive("known",unary,known_op);@/ @!@:known_}{\&{known} primitive@> primitive("unknown",unary,unknown_op);@/ @!@:unknown_}{\&{unknown} primitive@> primitive("not",unary,not_op);@/ @!@:not_}{\&{not} primitive@> primitive("decimal",unary,decimal);@/ @!@:decimal_}{\&{decimal} primitive@> primitive("reverse",unary,reverse);@/ @!@:reverse_}{\&{reverse} primitive@> primitive("makepath",unary,make_path_op);@/ @!@:make_path_}{\&{makepath} primitive@> primitive("makepen",unary,make_pen_op);@/ @!@:make_pen_}{\&{makepen} primitive@> primitive("oct",unary,oct_op);@/ @!@:oct_}{\&{oct} primitive@> primitive("hex",unary,hex_op);@/ @!@:hex_}{\&{hex} primitive@> primitive("ASCII",unary,ASCII_op);@/ @!@:ASCII_}{\&{ASCII} primitive@> primitive("char",unary,char_op);@/ @!@:char_}{\&{char} primitive@> primitive("length",unary,length_op);@/ @!@:length_}{\&{length} primitive@> primitive("turningnumber",unary,turning_op);@/ @!@:turning_number_}{\&{turningnumber} primitive@> primitive("xpart",unary,x_part);@/ @!@:x_part_}{\&{xpart} primitive@> primitive("ypart",unary,y_part);@/ @!@:y_part_}{\&{ypart} primitive@> primitive("xxpart",unary,xx_part);@/ @!@:xx_part_}{\&{xxpart} primitive@> primitive("xypart",unary,xy_part);@/ @!@:xy_part_}{\&{xypart} primitive@> primitive("yxpart",unary,yx_part);@/ @!@:yx_part_}{\&{yxpart} primitive@> primitive("yypart",unary,yy_part);@/ @!@:yy_part_}{\&{yypart} primitive@> primitive("redpart",unary,red_part);@/ @!@:red_part_}{\&{redpart} primitive@> primitive("greenpart",unary,green_part);@/ @!@:green_part_}{\&{greenpart} primitive@> primitive("bluepart",unary,blue_part);@/ @!@:blue_part_}{\&{bluepart} primitive@> primitive("fontpart",unary,font_part);@/ @!@:font_part_}{\&{fontpart} primitive@> primitive("textpart",unary,text_part);@/ @!@:text_part_}{\&{textpart} primitive@> primitive("pathpart",unary,path_part);@/ @!@:path_part_}{\&{pathpart} primitive@> primitive("penpart",unary,pen_part);@/ @!@:pen_part_}{\&{penpart} primitive@> primitive("dashpart",unary,dash_part);@/ @!@:dash_part_}{\&{dashpart} primitive@> primitive("sqrt",unary,sqrt_op);@/ @!@:sqrt_}{\&{sqrt} primitive@> primitive("mexp",unary,m_exp_op);@/ @!@:m_exp_}{\&{mexp} primitive@> primitive("mlog",unary,m_log_op);@/ @!@:m_log_}{\&{mlog} primitive@> primitive("sind",unary,sin_d_op);@/ @!@:sin_d_}{\&{sind} primitive@> primitive("cosd",unary,cos_d_op);@/ @!@:cos_d_}{\&{cosd} primitive@> primitive("floor",unary,floor_op);@/ @!@:floor_}{\&{floor} primitive@> primitive("uniformdeviate",unary,uniform_deviate);@/ @!@:uniform_deviate_}{\&{uniformdeviate} primitive@> primitive("charexists",unary,char_exists_op);@/ @!@:char_exists_}{\&{charexists} primitive@> primitive("fontsize",unary,font_size);@/ @!@:font_size_}{\&{fontsize} primitive@> primitive("llcorner",unary,ll_corner_op);@/ @!@:ll_corner_}{\&{llcorner} primitive@> primitive("lrcorner",unary,lr_corner_op);@/ @!@:lr_corner_}{\&{lrcorner} primitive@> primitive("ulcorner",unary,ul_corner_op);@/ @!@:ul_corner_}{\&{ulcorner} primitive@> primitive("urcorner",unary,ur_corner_op);@/ @!@:ur_corner_}{\&{urcorner} primitive@> primitive("arclength",unary,arc_length);@/ @!@:arc_length_}{\&{arclength} primitive@> primitive("angle",unary,angle_op);@/ @!@:angle_}{\&{angle} primitive@> primitive("cycle",cycle,cycle_op);@/ @!@:cycle_}{\&{cycle} primitive@> primitive("stroked",unary,stroked_op);@/ @!@:stroked_}{\&{stroked} primitive@> primitive("filled",unary,filled_op);@/ @!@:filled_}{\&{filled} primitive@> primitive("textual",unary,textual_op);@/ @!@:textual_}{\&{textual} primitive@> primitive("clipped",unary,clipped_op);@/ @!@:clipped_}{\&{clipped} primitive@> primitive("bounded",unary,bounded_op);@/ @!@:bounded_}{\&{bounded} primitive@> primitive("+",plus_or_minus,plus);@/ @!@:+ }{\.{+} primitive@> primitive("-",plus_or_minus,minus);@/ @!@:- }{\.{-} primitive@> primitive("*",secondary_binary,times);@/ @!@:* }{\.{*} primitive@> primitive("/",slash,over); eqtb[frozen_slash]:=eqtb[cur_sym];@/ @!@:/ }{\.{/} primitive@> primitive("++",tertiary_binary,pythag_add);@/ @!@:++_}{\.{++} primitive@> primitive("+-+",tertiary_binary,pythag_sub);@/ @!@:+-+_}{\.{+-+} primitive@> primitive("or",tertiary_binary,or_op);@/ @!@:or_}{\&{or} primitive@> primitive("and",and_command,and_op);@/ @!@:and_}{\&{and} primitive@> primitive("<",expression_binary,less_than);@/ @!@:< }{\.{<} primitive@> primitive("<=",expression_binary,less_or_equal);@/ @!@:<=_}{\.{<=} primitive@> primitive(">",expression_binary,greater_than);@/ @!@:> }{\.{>} primitive@> primitive(">=",expression_binary,greater_or_equal);@/ @!@:>=_}{\.{>=} primitive@> primitive("=",equals,equal_to);@/ @!@:= }{\.{=} primitive@> primitive("<>",expression_binary,unequal_to);@/ @!@:<>_}{\.{<>} primitive@> primitive("substring",primary_binary,substring_of);@/ @!@:substring_}{\&{substring} primitive@> primitive("subpath",primary_binary,subpath_of);@/ @!@:subpath_}{\&{subpath} primitive@> primitive("directiontime",primary_binary,direction_time_of);@/ @!@:direction_time_}{\&{directiontime} primitive@> primitive("point",primary_binary,point_of);@/ @!@:point_}{\&{point} primitive@> primitive("precontrol",primary_binary,precontrol_of);@/ @!@:precontrol_}{\&{precontrol} primitive@> primitive("postcontrol",primary_binary,postcontrol_of);@/ @!@:postcontrol_}{\&{postcontrol} primitive@> primitive("penoffset",primary_binary,pen_offset_of);@/ @!@:pen_offset_}{\&{penoffset} primitive@> primitive("arctime",primary_binary,arc_time_of);@/ @!@:arc_time_of_}{\&{arctime} primitive@> primitive("&",ampersand,concatenate);@/ @!@:!!!}{\.{\&} primitive@> primitive("rotated",secondary_binary,rotated_by);@/ @!@:rotated_}{\&{rotated} primitive@> primitive("slanted",secondary_binary,slanted_by);@/ @!@:slanted_}{\&{slanted} primitive@> primitive("scaled",secondary_binary,scaled_by);@/ @!@:scaled_}{\&{scaled} primitive@> primitive("shifted",secondary_binary,shifted_by);@/ @!@:shifted_}{\&{shifted} primitive@> primitive("transformed",secondary_binary,transformed_by);@/ @!@:transformed_}{\&{transformed} primitive@> primitive("xscaled",secondary_binary,x_scaled);@/ @!@:x_scaled_}{\&{xscaled} primitive@> primitive("yscaled",secondary_binary,y_scaled);@/ @!@:y_scaled_}{\&{yscaled} primitive@> primitive("zscaled",secondary_binary,z_scaled);@/ @!@:z_scaled_}{\&{zscaled} primitive@> primitive("infont",secondary_binary,in_font);@/ @!@:in_font_}{\&{infont} primitive@> primitive("intersectiontimes",tertiary_binary,intersect);@/ @!@:intersection_times_}{\&{intersectiontimes} primitive@> @ @= nullary,unary,primary_binary,secondary_binary,tertiary_binary, expression_binary,cycle,plus_or_minus,slash,ampersand,equals,and_command: print_op(m); @ OK, let's look at the simplest \\{do} procedure first. @p @t\4@>@@; procedure do_nullary(@!c:quarterword); begin check_arith; if internal[tracing_commands]>two then show_cmd_mod(nullary,c); case c of true_code,false_code:begin cur_type:=boolean_type; cur_exp:=c; end; null_picture_code:begin cur_type:=picture_type; cur_exp:=get_node(edge_header_size); init_edges(cur_exp); end; null_pen_code:begin cur_type:=pen_type; cur_exp:=get_pen_circle(0); end; normal_deviate:begin cur_type:=known; cur_exp:=norm_rand; end; pen_circle:begin cur_type:=pen_type; cur_exp:=get_pen_circle(unity); end; job_name_op: begin if job_name=0 then open_log_file; cur_type:=string_type; cur_exp:=job_name; end; read_string_op:@; end; {there are no other cases} check_arith; end; @ @= begin if interaction<=nonstop_mode then fatal_error("*** (cannot readstring in nonstop modes)"); begin_file_reading; name:=is_read; limit:=start; prompt_input(""); finish_read; end @ @= procedure finish_read; {copy |buffer| line to |cur_exp|} var @!k:pool_pointer; begin str_room(last-start); for k:=start to last-1 do append_char(buffer[k]); end_file_reading; cur_type:=string_type; cur_exp:=make_string; end; @ Things get a bit more interesting when there's an operand. The operand to |do_unary| appears in |cur_type| and |cur_exp|. @p @t\4@>@@; procedure do_unary(@!c:quarterword); var @!p,@!q,@!r:pointer; {for list manipulation} @!x:integer; {a temporary register} begin check_arith; if internal[tracing_commands]>two then @; case c of plus:if cur_type; @t\4@>@@; end; {there are no other cases} check_arith; end; @ The |nice_pair| function returns |true| if both components of a pair are known. @= function nice_pair(@!p:integer;@!t:quarterword):boolean; label exit; begin if t=pair_type then begin p:=value(p); if type(x_part_loc(p))=known then if type(y_part_loc(p))=known then begin nice_pair:=true; return; end; end; nice_pair:=false; exit:end; @ The |nice_color_or_pair| function is analogous except that it also accepts fully known colors. @= function nice_color_or_pair(@!p:integer;@!t:quarterword):boolean; label exit; var @!q,@!r:pointer; {for scanning the big node} begin if (t<>pair_type)and(t<>color_type) then nice_color_or_pair:=false else begin q:=value(p); r:=q+big_node_size[type(p)]; repeat r:=r-2; if type(r)<>known then begin nice_color_or_pair:=false; return; end; until r=q; nice_color_or_pair:=true; end; exit:end; @ @= procedure print_known_or_unknown_type(@!t:small_number;@!v:integer); begin print_char("("); if t>known then print("unknown numeric") else begin if (t=pair_type)or(t=color_type) then if not nice_color_or_pair(v,t) then print("unknown "); print_type(t); end; print_char(")"); end; @ @= procedure bad_unary(@!c:quarterword); begin exp_err("Not implemented: "); print_op(c); @.Not implemented...@> print_known_or_unknown_type(cur_type,cur_exp); help3("I'm afraid I don't know how to apply that operation to that")@/ ("particular type. Continue, and I'll simply return the")@/ ("argument (shown above) as the result of the operation."); put_get_error; end; @ @= begin begin_diagnostic; print_nl("{"); print_op(c); print_char("(");@/ print_exp(null,0); {show the operand, but not verbosely} print(")}"); end_diagnostic(false); end @ Negation is easy except when the current expression is of type |independent|, or when it is a pair with one or more |independent| components. It is tempting to argue that the negative of an independent variable is an independent variable, hence we don't have to do anything when negating it. The fallacy is that other dependent variables pointing to the current expression must change the sign of their coefficients if we make no change to the current expression. Instead, we work around the problem by copying the current expression and recycling it afterwards (cf.~the |stash_in| routine). @= case cur_type of color_type,pair_type,independent: begin q:=cur_exp; make_exp_copy(q); if cur_type=dependent then negate_dep_list(dep_list(cur_exp)) else if cur_type<=pair_type then {|color_type| or |pair_type|} begin p:=value(cur_exp); r:=p+big_node_size[cur_type]; repeat r:=r-2; if type(r)=known then negate(value(r)) else negate_dep_list(dep_list(r)); until r=p; end; {if |cur_type=known| then |cur_exp=0|} recycle_value(q); free_node(q,value_node_size); end; dependent,proto_dependent:negate_dep_list(dep_list(cur_exp)); known:negate(cur_exp); othercases bad_unary(minus) endcases @ @= procedure negate_dep_list(@!p:pointer); label exit; begin loop@+begin negate(value(p)); if info(p)=null then return; p:=link(p); end; exit:end; @ @= not_op: if cur_type<>boolean_type then bad_unary(not_op) else cur_exp:=true_code+false_code-cur_exp; @ @d three_sixty_units==23592960 {that's |360*unity|} @d boolean_reset(#)==if # then cur_exp:=true_code@+else cur_exp:=false_code @= sqrt_op,m_exp_op,m_log_op,sin_d_op,cos_d_op,floor_op, uniform_deviate,odd_op,char_exists_op:@t@>@;@/ if cur_type<>known then bad_unary(c) else case c of sqrt_op:cur_exp:=square_rt(cur_exp); m_exp_op:cur_exp:=m_exp(cur_exp); m_log_op:cur_exp:=m_log(cur_exp); sin_d_op,cos_d_op:begin n_sin_cos((cur_exp mod three_sixty_units)*16); if c=sin_d_op then cur_exp:=round_fraction(n_sin) else cur_exp:=round_fraction(n_cos); end; floor_op:cur_exp:=floor_scaled(cur_exp); uniform_deviate:cur_exp:=unif_rand(cur_exp); odd_op: begin boolean_reset(odd(round_unscaled(cur_exp))); cur_type:=boolean_type; end; char_exists_op:@; end; {there are no other cases} @ @= angle_op:if nice_pair(cur_exp,cur_type) then begin p:=value(cur_exp); x:=n_arg(value(x_part_loc(p)),value(y_part_loc(p))); if x>=0 then flush_cur_exp((x+8)div 16) else flush_cur_exp(-((-x+8)div 16)); end else bad_unary(angle_op); @ If the current expression is a pair, but the context wants it to be a path, we call |pair_to_path|. @= procedure pair_to_path; begin cur_exp:=new_knot; cur_type:=path_type; end; @ @= x_part,y_part:if (cur_type=pair_type)or(cur_type=transform_type) then take_part(c) else if cur_type=picture_type then take_pict_part(c) else bad_unary(c); xx_part,xy_part,yx_part,yy_part: if cur_type=transform_type then take_part(c) else if cur_type=picture_type then take_pict_part(c) else bad_unary(c); red_part,green_part,blue_part: if cur_type=color_type then take_part(c) else if cur_type=picture_type then take_pict_part(c) else bad_unary(c); @ In the following procedure, |cur_exp| points to a capsule, which points to a big node. We want to delete all but one part of the big node. @= procedure take_part(@!c:quarterword); var @!p:pointer; {the big node} begin p:=value(cur_exp); value(temp_val):=p; type(temp_val):=cur_type; link(p):=temp_val; free_node(cur_exp,value_node_size); make_exp_copy(p+sector_offset[c+x_part_sector-x_part]); recycle_value(temp_val); end; @ @= name_type(temp_val):=capsule; @ @= font_part,text_part,path_part,pen_part,dash_part: if cur_type=picture_type then take_pict_part(c) else bad_unary(c); @ @= procedure@?scale_edges; forward;@t\2@>@;@/ procedure take_pict_part(@!c:quarterword); label exit, not_found; var @!p:pointer; {first graphical object in |cur_exp|} begin p:=link(dummy_loc(cur_exp)); if p<>null then begin case c of x_part,y_part,xx_part,xy_part,yx_part,yy_part: if type(p)=text_code then flush_cur_exp(text_trans_part(p+c)) else goto not_found; red_part,green_part,blue_part: if has_color(p) then flush_cur_exp(obj_color_part(p+c)) else goto not_found; @@; end; {all cases have been enumerated} return; end; not_found:@; exit:end; @ @= text_part: if type(p)<>text_code then goto not_found else begin flush_cur_exp(text_p(p)); add_str_ref(cur_exp); cur_type:=string_type; end; font_part: if type(p)<>text_code then goto not_found else begin flush_cur_exp(font_name[font_n(p)]); add_str_ref(cur_exp); cur_type:=string_type; end; path_part:if type(p)=text_code then goto not_found else if is_stop(p) then confusion("pict") @:this can't happen pict}{\quad pict@> else begin flush_cur_exp(copy_path(path_p(p))); cur_type:=path_type; end; pen_part: if not has_pen(p) then goto not_found else if pen_p(p)=null then goto not_found else begin flush_cur_exp(copy_pen(pen_p(p))); cur_type:=pen_type; end; dash_part: if type(p)<>stroked_code then goto not_found else if dash_p(p)=null then goto not_found else begin add_edge_ref(dash_p(p));@/ se_sf:=dash_scale(p); se_pic:=dash_p(p); scale_edges; flush_cur_exp(se_pic); cur_type:=picture_type; end; @ Since |scale_edges| had to be declared |forward|, it had to be declared as a parameterless procedure even though it really takes two arguments and updates one of them. Hence the following globals are needed. @= @!se_pic:pointer; {edge header used and updated by |scale_edges|} @!se_sf:scaled; {the scale factor argument to |scale_edges|} @ @= case c of text_part,font_part: begin flush_cur_exp(""); cur_type:=string_type; end; path_part: begin flush_cur_exp(get_node(knot_node_size)); left_type(cur_exp):=endpoint; right_type(cur_exp):=endpoint; link(cur_exp):=cur_exp; x_coord(cur_exp):=0; y_coord(cur_exp):=0; cur_type:=path_type; end; pen_part: begin flush_cur_exp(get_pen_circle(0)); cur_type:=pen_type; end; dash_part: begin flush_cur_exp(get_node(edge_header_size)); init_edges(cur_exp); cur_type:=picture_type; end; othercases flush_cur_exp(0) endcases @ @= char_op: if cur_type<>known then bad_unary(char_op) else begin cur_exp:=round_unscaled(cur_exp) mod 256; cur_type:=string_type; if cur_exp<0 then cur_exp:=cur_exp+256; end; decimal: if cur_type<>known then bad_unary(decimal) else begin old_setting:=selector; selector:=new_string; print_scaled(cur_exp); cur_exp:=make_string; selector:=old_setting; cur_type:=string_type; end; oct_op,hex_op,ASCII_op: if cur_type<>string_type then bad_unary(c) else str_to_num(c); font_size: if cur_type<>string_type then bad_unary(font_size) else @; @ @= procedure str_to_num(@!c:quarterword); {converts a string to a number} var @!n:integer; {accumulator} @!m:ASCII_code; {current character} @!k:pool_pointer; {index into |str_pool|} @!b:8..16; {radix of conversion} @!bad_char:boolean; {did the string contain an invalid digit?} begin if c=ASCII_op then if length(cur_exp)=0 then n:=-1 else n:=so(str_pool[str_start[cur_exp]]) else begin if c=oct_op then b:=8@+else b:=16; n:=0; bad_char:=false; for k:=str_start[cur_exp] to str_stop(cur_exp)-1 do begin m:=so(str_pool[k]); if (m>="0")and(m<="9") then m:=m-"0" else if (m>="A")and(m<="F") then m:=m-"A"+10 else if (m>="a")and(m<="f") then m:=m-"a"+10 else begin bad_char:=true; m:=0; end; if m>=b then begin bad_char:=true; m:=0; end; if n<32768 div b then n:=n*b+m@+else n:=32767; end; @=4096|@>; end; flush_cur_exp(n*unity); end; @ @= if bad_char then begin exp_err("String contains illegal digits"); @.String contains illegal digits@> if c=oct_op then help1("I zeroed out characters that weren't in the range 0..7.") else help1("I zeroed out characters that weren't hex digits."); put_get_error; end; if (n>4095) then if internal[warning_check]>0 then begin print_err("Number too large ("); print_int(n); print_char(")"); @.Number too large@> help2("I have trouble with numbers greater than 4095; watch out.")@/ ("(Set warningcheck:=0 to suppress this message.)"); put_get_error; end @ The length operation is somewhat unusual in that it applies to a variety of different types of operands. @= length_op: case cur_type of string_type: flush_cur_exp(length(cur_exp)*unity); path_type: flush_cur_exp(path_length); known: cur_exp:=abs(cur_exp); picture_type: flush_cur_exp(pict_length); othercases if nice_pair(cur_exp,cur_type) then flush_cur_exp(pyth_add(value(x_part_loc(value(cur_exp))),@| value(y_part_loc(value(cur_exp))))) else bad_unary(c) endcases; @ @= function path_length:scaled; {computes the length of the current path} var @!n:scaled; {the path length so far} @!p:pointer; {traverser} begin p:=cur_exp; if left_type(p)=endpoint then n:=-unity@+else n:=0; repeat p:=link(p); n:=n+unity; until p=cur_exp; path_length:=n; end; @ @= function pict_length:scaled; {counts interior components in picture |cur_exp|} label found; var @!n:scaled; {the count so far} @!p:pointer; {traverser} begin n:=0; p:=link(dummy_loc(cur_exp)); if p<>null then begin if is_start_or_stop(p) then if skip_1component(p)=null then p:=link(p); while p<>null do begin skip_component(p)(goto found); n:=n+unity; end; end; found:pict_length:=n; end; @ The turning number is computed only with respect to a triangular pen whose @:turning_number_}{\&{turningnumber} primitive@> vertices are $(0,1)$ and $(\pm{1\over2},0)$. The choice of pen isn't supposed to matter but rounding error could make a difference if the path has a cusp. @= turning_op:if cur_type=pair_type then flush_cur_exp(0) else if cur_type<>path_type then bad_unary(turning_op) else if left_type(cur_exp)=endpoint then flush_cur_exp(0) {not a cyclic path} else begin cur_exp:=offset_prep(cur_exp,test_pen); if internal[tracing_specs]>unity then print_spec(cur_exp,test_pen," (for turningnumber)"); flush_cur_exp(count_turns(cur_exp)); end; @ @= function count_turns(@!c:pointer):scaled; var @!p:pointer; {a knot in envelope spec |c|} @!t:integer; {total pen offset changes counted} begin t:=0; p:=c; repeat t:=t+info(p)-zero_off; p:=link(p); until p=c; count_turns:=(t div 3)*unity; end; @ @d type_test_end== flush_cur_exp(true_code) else flush_cur_exp(false_code); cur_type:=boolean_type; end @d type_range_end(#)==(cur_type<=#) then type_test_end @d type_range(#)==begin if (cur_type>=#) and type_range_end @d type_test(#)==begin if cur_type=# then type_test_end @= boolean_type: type_range(boolean_type)(unknown_boolean); string_type: type_range(string_type)(unknown_string); pen_type: type_range(pen_type)(unknown_pen); path_type: type_range(path_type)(unknown_path); picture_type: type_range(picture_type)(unknown_picture); transform_type,color_type,pair_type: type_test(c); numeric_type: type_range(known)(independent); known_op,unknown_op: test_known(c); @ @= procedure test_known(@!c:quarterword); label done; var @!b:true_code..false_code; {is the current expression known?} @!p,@!q:pointer; {locations in a big node} begin b:=false_code; case cur_type of vacuous,boolean_type,string_type,pen_type,path_type,picture_type, known: b:=true_code; transform_type,color_type,pair_type:begin p:=value(cur_exp); q:=p+big_node_size[cur_type]; repeat q:=q-2; if type(q)<>known then goto done; until q=p; b:=true_code; done: end; othercases do_nothing endcases; if c=known_op then flush_cur_exp(b) else flush_cur_exp(true_code+false_code-b); cur_type:=boolean_type; end; @ @= cycle_op: begin if cur_type<>path_type then flush_cur_exp(false_code) else if left_type(cur_exp)<>endpoint then flush_cur_exp(true_code) else flush_cur_exp(false_code); cur_type:=boolean_type; end; @ @= arc_length: begin if cur_type=pair_type then pair_to_path; if cur_type<>path_type then bad_unary(arc_length) else flush_cur_exp(get_arc_length(cur_exp)); end; @ Here we use the fact that |c-filled_op+fill_code| is the desired graphical object |type|. @^data structure assumptions@> @= filled_op,stroked_op,textual_op,clipped_op,bounded_op: begin if cur_type<>picture_type then flush_cur_exp(false_code) else if link(dummy_loc(cur_exp))=null then flush_cur_exp(false_code) else if type(link(dummy_loc(cur_exp)))=c+fill_code-filled_op then flush_cur_exp(true_code) else flush_cur_exp(false_code); cur_type:=boolean_type; end; @ @= make_pen_op: begin if cur_type=pair_type then pair_to_path; if cur_type<>path_type then bad_unary(make_pen_op) else begin cur_type:=pen_type; cur_exp:=make_pen(cur_exp,true); end; end; make_path_op: if cur_type<>pen_type then bad_unary(make_path_op) else begin cur_type:=path_type; make_path(cur_exp); end; reverse: if cur_type=path_type then begin p:=htap_ypoc(cur_exp); if right_type(p)=endpoint then p:=link(p); toss_knot_list(cur_exp); cur_exp:=p; end else if cur_type=pair_type then pair_to_path else bad_unary(reverse); @ The |pair_value| routine changes the current expression to a given ordered pair of values. @= procedure pair_value(@!x,@!y:scaled); var @!p:pointer; {a pair node} begin p:=get_node(value_node_size); flush_cur_exp(p); cur_type:=pair_type; type(p):=pair_type; name_type(p):=capsule; init_big_node(p); p:=value(p);@/ type(x_part_loc(p)):=known; value(x_part_loc(p)):=x;@/ type(y_part_loc(p)):=known; value(y_part_loc(p)):=y;@/ end; @ @= ll_corner_op: if not get_cur_bbox then bad_unary(ll_corner_op) else pair_value(minx,miny); lr_corner_op: if not get_cur_bbox then bad_unary(lr_corner_op) else pair_value(maxx,miny); ul_corner_op: if not get_cur_bbox then bad_unary(ul_corner_op) else pair_value(minx,maxy); ur_corner_op: if not get_cur_bbox then bad_unary(ur_corner_op) else pair_value(maxx,maxy); @ Here is a function that sets |minx|, |maxx|, |miny|, |maxy| to the bounding box of the current expression. The boolean result is |false| if the expression has the wrong type. @= function get_cur_bbox: boolean; label exit; begin case cur_type of picture_type: begin set_bbox(cur_exp,true); if minx_val(cur_exp)>maxx_val(cur_exp) then begin minx:=0; maxx:=0; miny:=0; maxy:=0; end else begin minx:=minx_val(cur_exp); maxx:=maxx_val(cur_exp); miny:=miny_val(cur_exp); maxy:=maxy_val(cur_exp); end; end; path_type: path_bbox(cur_exp); pen_type: pen_bbox(cur_exp); othercases begin get_cur_bbox:=false; return; end endcases;@/ get_cur_bbox:=true; exit:end; @ @= read_from_op,close_from_op: if cur_type<>string_type then bad_unary(c) else do_read_or_close(c); @ Here is a routine that interprets |cur_exp| as a file name and tries to read a line from the file or to close the file. @d close_file=46 {go here when closing the file} @= procedure do_read_or_close(@!c:quarterword); label exit, continue, found, not_found, close_file; var @!n,@!n0:readf_index; {indices for searching |rd_fname|} begin @; begin_file_reading; name:=is_read; if input_ln(rd_file[n],true) then goto found; end_file_reading; not_found:@; return; close_file:flush_cur_exp(0); cur_type:=vacuous; return; found:flush_cur_exp(0); finish_read; exit:end; @ Free slots in the |rd_file| and |rd_fname| arrays are marked with 0's in |rd_fname|. @= n:=read_files; n0:=read_files; repeat continue:if n>0 then decr(n) else if c=close_from_op then goto close_file else @; if rd_fname[n]=0 then begin n0:=n; goto continue; end; until str_vs_str(cur_exp,rd_fname[n])=0; if c=close_from_op then begin a_close(rd_file[n]); goto not_found; end @ @= begin if n0=read_files then if read_files= delete_str_ref(rd_fname[n]); rd_fname[n]:=0; if n=read_files-1 then read_files:=n; if c=close_from_op then goto close_file; @; flush_cur_exp(eof_line); cur_type:=string_type @ Since the |eof_line| string contains a non-printable character, it must be initialized at run time and stored in a global variable. @= eof_line:str_number; {string denoting end-of-file or 0 if uninitialized} @ @= eof_line:=0; @ @= if eof_line=0 then begin append_char(0); eof_line:=make_string; str_ref[eof_line]:=max_str_ref; end @ Finally, we have the operations that combine a capsule~|p| with the current expression. @p @t\4@>@@; procedure do_binary(@!p:pointer;@!c:quarterword); label done,done1,exit; var @!q,@!r,@!rr:pointer; {for list manipulation} @!old_p,@!old_exp:pointer; {capsules to recycle} @!v:integer; {for numeric manipulation} begin check_arith; if internal[tracing_commands]>two then @; @; @; case c of plus,minus:@; @t\4@>@@; end; {there are no other cases} recycle_value(p); free_node(p,value_node_size); {|return| to avoid this} exit:check_arith; @; end; @ @= procedure bad_binary(@!p:pointer;@!c:quarterword); begin disp_err(p,""); exp_err("Not implemented: "); @.Not implemented...@> if c>=min_of then print_op(c); print_known_or_unknown_type(type(p),p); if c>=min_of then print("of")@+else print_op(c); print_known_or_unknown_type(cur_type,cur_exp);@/ help3("I'm afraid I don't know how to apply that operation to that")@/ ("combination of types. Continue, and I'll return the second")@/ ("argument (see above) as the result of the operation."); put_get_error; end; @ @= begin begin_diagnostic; print_nl("{("); print_exp(p,0); {show the operand, but not verbosely} print_char(")"); print_op(c); print_char("(");@/ print_exp(null,0); print(")}"); end_diagnostic(false); end @ Several of the binary operations are potentially complicated by the fact that |independent| values can sneak into capsules. For example, we've seen an instance of this difficulty in the unary operation of negation. In order to reduce the number of cases that need to be handled, we first change the two operands (if necessary) to rid them of |independent| components. The original operands are put into capsules called |old_p| and |old_exp|, which will be recycled after the binary operation has been safely carried out. @= if old_p<>null then begin recycle_value(old_p); free_node(old_p,value_node_size); end; if old_exp<>null then begin recycle_value(old_exp); free_node(old_exp,value_node_size); end @ A big node is considered to be ``tarnished'' if it contains at least one independent component. We will define a simple function called `|tarnished|' that returns |null| if and only if its argument is not tarnished. @= case type(p) of transform_type,color_type,pair_type: old_p:=tarnished(p); independent: old_p:=void; othercases old_p:=null endcases; if old_p<>null then begin q:=stash_cur_exp; old_p:=p; make_exp_copy(old_p); p:=stash_cur_exp; unstash_cur_exp(q); end; @ @= case cur_type of transform_type,color_type,pair_type:old_exp:=tarnished(cur_exp); independent:old_exp:=void; othercases old_exp:=null endcases; if old_exp<>null then begin old_exp:=cur_exp; make_exp_copy(old_exp); end @ @= function tarnished(@!p:pointer):pointer; label exit; var @!q:pointer; {beginning of the big node} @!r:pointer; {current position in the big node} begin q:=value(p); r:=q+big_node_size[type(p)]; repeat r:=r-2; if type(r)=independent then begin tarnished:=void; return; end; until r=q; tarnished:=null; exit:end; @ @= if (cur_typepair_type)and(type(p)>pair_type) then add_or_subtract(p,null,c) else if cur_type<>type(p) then bad_binary(p,c) else begin q:=value(p); r:=value(cur_exp); rr:=r+big_node_size[cur_type]; while r= @t\4@>@@; procedure add_or_subtract(@!p,@!q:pointer;@!c:quarterword); label done,exit; var @!s,@!t:small_number; {operand types} @!r:pointer; {list traverser} @!v:integer; {second operand value} begin if q=null then begin t:=cur_type; if t; end else begin if c=minus then negate_dep_list(v); @; end; exit:end; @ @= r:=dep_list(p); while info(r)<>null do r:=link(r); value(r):=slow_add(value(r),v); if q=null then begin q:=get_node(value_node_size); cur_exp:=q; cur_type:=type(p); name_type(q):=capsule; end; dep_list(q):=dep_list(p); type(q):=type(p); prev_dep(q):=prev_dep(p); link(prev_dep(p)):=q; type(p):=known; {this will keep the recycler from collecting non-garbage} @ We prefer |dependent| lists to |proto_dependent| ones, because it is nice to retain the extra accuracy of |fraction| coefficients. But we have to handle both kinds, and mixtures too. @= if type(p)=known then @ else begin s:=type(p); r:=dep_list(p); if t=dependent then begin if s=dependent then if max_coef(r)+max_coef(v); end @ @= begin while info(v)<>null do v:=link(v); value(v):=slow_add(value(p),value(v)); end @ @= if q<>null then dep_finish(v,q,t) else begin cur_type:=t; dep_finish(v,null,t); end @ Here's the current situation: The dependency list |v| of type |t| should either be put into the current expression (if |q=null|) or into location |q| within a pair node (otherwise). The destination (|cur_exp| or |q|) formerly held a dependency list with the same final pointer as the list |v|. @= procedure dep_finish(@!v,@!q:pointer;@!t:small_number); var @!p:pointer; {the destination} @!vv:scaled; {the value, if it is |known|} begin if q=null then p:=cur_exp@+else p:=q; dep_list(p):=v; type(p):=t; if info(v)=null then begin vv:=value(v); if q=null then flush_cur_exp(vv) else begin recycle_value(p); type(q):=known; value(q):=vv; end; end else if q=null then cur_type:=t; if fix_needed then fix_dependencies; end; @ Let's turn now to the six basic relations of comparison. @= less_than,less_or_equal,greater_than,greater_or_equal,equal_to,unequal_to: begin check_arith; {at this point |arith_error| should be |false|?} if (cur_type>pair_type)and(type(p)>pair_type) then add_or_subtract(p,null,minus) {|cur_exp:=(p)-cur_exp|} else if cur_type<>type(p) then begin bad_binary(p,c); goto done; end else if cur_type=string_type then flush_cur_exp(str_vs_str(value(p),cur_exp)) else if (cur_type=unknown_string)or(cur_type=unknown_boolean) then @ else if (cur_type<=pair_type)and(cur_type>=transform_type) then @ else if cur_type=boolean_type then flush_cur_exp(cur_exp-value(p)) else begin bad_binary(p,c); goto done; end; @; done: arith_error:=false; {ignore overflow in comparisons} end; @ @= if cur_type<>known then begin if cur_type put_get_flush_error(false_code); end else case c of less_than: boolean_reset(cur_exp<0); less_or_equal: boolean_reset(cur_exp<=0); greater_than: boolean_reset(cur_exp>0); greater_or_equal: boolean_reset(cur_exp>=0); equal_to: boolean_reset(cur_exp=0); unequal_to: boolean_reset(cur_exp<>0); end; {there are no other cases} cur_type:=boolean_type @ When two unknown strings are in the same ring, we know that they are equal. Otherwise, we don't know whether they are equal or not, so we make no change. @= begin q:=value(cur_exp); while (q<>cur_exp)and(q<>p) do q:=value(q); if q=p then flush_cur_exp(0); end @ @= begin q:=value(p); r:=value(cur_exp); rr:=r+big_node_size[cur_type]-2; loop@+ begin add_or_subtract(q,r,minus); if type(r)<>known then goto done1; if value(r)<>0 then goto done1; if r=rr then goto done1; q:=q+2; r:=r+2; end; done1:take_part(name_type(r)+x_part-x_part_sector); end @ Here we use the sneaky fact that |and_op-false_code=or_op-true_code|. @= and_op,or_op: if (type(p)<>boolean_type)or(cur_type<>boolean_type) then bad_binary(p,c) else if value(p)=c+false_code-and_op then cur_exp:=value(p); @ @= times: if (cur_type else if (nice_color_or_pair(p,type(p))and(cur_type>pair_type)) or(nice_color_or_pair(cur_exp,cur_type)and(type(p)>pair_type)) then begin hard_times(p); return; end else bad_binary(p,times); @ @= begin if type(p)=known then begin v:=value(p); free_node(p,value_node_size); end else begin v:=cur_exp; unstash_cur_exp(p); end; if cur_type=known then cur_exp:=take_scaled(cur_exp,v) else if (cur_type=pair_type)or(cur_type=color_type) then begin p:=value(cur_exp)+big_node_size[cur_type]; repeat p:=p-2; dep_mult(p,v,true); until p=value(cur_exp); end else dep_mult(null,v,true); return; end @ @= procedure dep_mult(@!p:pointer;@!v:integer;@!v_is_scaled:boolean); label exit; var @!q:pointer; {the dependency list being multiplied by |v|} @!s,@!t:small_number; {its type, before and after} begin if p=null then q:=cur_exp else if type(p)<>known then q:=p else begin if v_is_scaled then value(p):=take_scaled(value(p),v) else value(p):=take_fraction(value(p),v); return; end; t:=type(q); q:=dep_list(q); s:=t; if t=dependent then if v_is_scaled then if ab_vs_cd(max_coef(q),abs(v),coef_bound-1,unity)>=0 then t:=proto_dependent; q:=p_times_v(q,v,s,t,v_is_scaled); dep_finish(q,p,t); exit:end; @ Here is a routine that is similar to |times|; but it is invoked only internally, when |v| is a |fraction| whose magnitude is at most~1, and when |cur_type>=color_type|. @p procedure frac_mult(@!n,@!d:scaled); {multiplies |cur_exp| by |n/d|} var @!p:pointer; {a pair node} @!old_exp:pointer; {a capsule to recycle} @!v:fraction; {|n/d|} begin if internal[tracing_commands]>two then @; case cur_type of transform_type,color_type,pair_type:old_exp:=tarnished(cur_exp); independent:old_exp:=void; othercases old_exp:=null endcases; if old_exp<>null then begin old_exp:=cur_exp; make_exp_copy(old_exp); end; v:=make_fraction(n,d); if cur_type=known then cur_exp:=take_fraction(cur_exp,v) else if cur_type<=pair_type then begin p:=value(cur_exp)+big_node_size[cur_type]; repeat p:=p-2; dep_mult(p,v,false); until p=value(cur_exp); end else dep_mult(null,v,false); if old_exp<>null then begin recycle_value(old_exp); free_node(old_exp,value_node_size); end end; @ @= begin begin_diagnostic; print_nl("{("); print_scaled(n); print_char("/"); print_scaled(d); print(")*("); print_exp(null,0); print(")}"); end_diagnostic(false); end @ The |hard_times| routine multiplies a nice color or pair by a dependency list. @= procedure hard_times(@!p:pointer); label done; var @!q:pointer; {a copy of the dependent variable |p|} @!r:pointer; {a component of the big node for the nice color or pair} @!v:scaled; {the known value for |r|} begin if type(p)<=pair_type then begin q:=stash_cur_exp; unstash_cur_exp(p); p:=q; end; {now |cur_type=pair_type| or |cur_type=color_type|} r:=value(cur_exp)+big_node_size[cur_type]; loop @+begin r:=r-2; v:=value(r); type(r):=type(p); if r=value(cur_exp) then goto done; new_dep(r,copy_dep_list(dep_list(p))); dep_mult(r,v,true); end; done:mem[value_loc(r)]:=mem[value_loc(p)]; link(prev_dep(p)):=r; free_node(p,value_node_size); dep_mult(r,v,true); end; @ @= over: if (cur_type<>known)or(type(p) else begin if cur_type=known then cur_exp:=make_scaled(cur_exp,v) else if cur_type<=pair_type then begin p:=value(cur_exp)+big_node_size[cur_type]; repeat p:=p-2; dep_div(p,v); until p=value(cur_exp); end else dep_div(null,v); end; return; end; @ @= procedure dep_div(@!p:pointer;@!v:scaled); label exit; var @!q:pointer; {the dependency list being divided by |v|} @!s,@!t:small_number; {its type, before and after} begin if p=null then q:=cur_exp else if type(p)<>known then q:=p else begin value(p):=make_scaled(value(p),v); return; end; t:=type(q); q:=dep_list(q); s:=t; if t=dependent then if ab_vs_cd(max_coef(q),unity,coef_bound-1,abs(v))>=0 then t:=proto_dependent; q:=p_over_v(q,v,s,t); dep_finish(q,p,t); exit:end; @ @= begin exp_err("Division by zero"); @.Division by zero@> help2("You're trying to divide the quantity shown above the error")@/ ("message by zero. I'm going to divide it by one instead."); put_get_error; end @ @= pythag_add,pythag_sub: if (cur_type=known)and(type(p)=known) then if c=pythag_add then cur_exp:=pyth_add(value(p),cur_exp) else cur_exp:=pyth_sub(value(p),cur_exp) else bad_binary(p,c); @ The next few sections of the program deal with affine transformations of coordinate data. @= rotated_by,slanted_by,scaled_by,shifted_by,transformed_by, x_scaled,y_scaled,z_scaled: @t@>@;@/ if type(p)=path_type then begin path_trans(c)(p); return; end else if type(p)=pen_type then begin pen_trans(c)(p); cur_exp:=convex_hull(cur_exp); {rounding error could destroy convexity} return; end else if (type(p)=pair_type)or(type(p)=transform_type) then big_trans(p,c) else if type(p)=picture_type then begin do_edges_trans(p,c); return; end else bad_binary(p,c); @ Let |c| be one of the eight transform operators. The procedure call |set_up_trans(c)| first changes |cur_exp| to a transform that corresponds to |c| and the original value of |cur_exp|. (In particular, |cur_exp| doesn't change at all if |c=transformed_by|.) Then, if all components of the resulting transform are |known|, they are moved to the global variables |txx|, |txy|, |tyx|, |tyy|, |tx|, |ty|; and |cur_exp| is changed to the known value zero. @= procedure set_up_trans(@!c:quarterword); label done,exit; var @!p,@!q,@!r:pointer; {list manipulation registers} begin if (c<>transformed_by)or(cur_type<>transform_type) then @; @; exit:end; @ @= @!txx,@!txy,@!tyx,@!tyy,@!tx,@!ty:scaled; {current transform coefficients} @ @= begin p:=stash_cur_exp; cur_exp:=id_transform; cur_type:=transform_type; q:=value(cur_exp); case c of @@; end; {there are no other cases} disp_err(p,"Improper transformation argument"); @.Improper transformation argument@> help3("The expression shown above has the wrong type,")@/ ("so I can't transform anything using it.")@/ ("Proceed, and I'll omit the transformation."); put_get_error; done: recycle_value(p); free_node(p,value_node_size); end @ @= q:=value(cur_exp); r:=q+transform_node_size; repeat r:=r-2; if type(r)<>known then return; until r=q; txx:=value(xx_part_loc(q)); txy:=value(xy_part_loc(q)); tyx:=value(yx_part_loc(q)); tyy:=value(yy_part_loc(q)); tx:=value(x_part_loc(q)); ty:=value(y_part_loc(q)); flush_cur_exp(0) @ @= rotated_by:if type(p)=known then @; slanted_by:if type(p)>pair_type then begin install(xy_part_loc(q),p); goto done; end; scaled_by:if type(p)>pair_type then begin install(xx_part_loc(q),p); install(yy_part_loc(q),p); goto done; end; shifted_by:if type(p)=pair_type then begin r:=value(p); install(x_part_loc(q),x_part_loc(r)); install(y_part_loc(q),y_part_loc(r)); goto done; end; x_scaled:if type(p)>pair_type then begin install(xx_part_loc(q),p); goto done; end; y_scaled:if type(p)>pair_type then begin install(yy_part_loc(q),p); goto done; end; z_scaled:if type(p)=pair_type then @; transformed_by:do_nothing; @ @= begin n_sin_cos((value(p) mod three_sixty_units)*16); value(xx_part_loc(q)):=round_fraction(n_cos); value(yx_part_loc(q)):=round_fraction(n_sin); value(xy_part_loc(q)):=-value(yx_part_loc(q)); value(yy_part_loc(q)):=value(xx_part_loc(q)); goto done; end @ @= begin r:=value(p); install(xx_part_loc(q),x_part_loc(r)); install(yy_part_loc(q),x_part_loc(r)); install(yx_part_loc(q),y_part_loc(r)); if type(y_part_loc(r))=known then negate(value(y_part_loc(r))) else negate_dep_list(dep_list(y_part_loc(r))); install(xy_part_loc(q),y_part_loc(r)); goto done; end @ Procedure |set_up_known_trans| is like |set_up_trans|, but it insists that the transformation be entirely known. @= procedure set_up_known_trans(@!c:quarterword); begin set_up_trans(c); if cur_type<>known then begin exp_err("Transform components aren't all known"); @.Transform components...@> help3("I'm unable to apply a partially specified transformation")@/ ("except to a fully known pair or transform.")@/ ("Proceed, and I'll omit the transformation."); put_get_flush_error(0); txx:=unity; txy:=0; tyx:=0; tyy:=unity; tx:=0; ty:=0; end; end; @ Here's a procedure that applies the transform |txx..ty| to a pair of coordinates in locations |p| and~|q|. @= procedure trans(@!p,@!q:pointer); var @!v:scaled; {the new |x| value} begin v:=take_scaled(mem[p].sc,txx)+take_scaled(mem[q].sc,txy)+tx; mem[q].sc:=take_scaled(mem[p].sc,tyx)+take_scaled(mem[q].sc,tyy)+ty; mem[p].sc:=v; end; @ The simplest transformation procedure applies a transform to all coordinates of a path. The |path_trans(c)(p)| macro applies a transformation defined by |cur_exp| and the transform operator |c| to the path~|p|. @d path_trans(#)==begin set_up_known_trans(#); path_trans_end @d path_trans_end(#)==unstash_cur_exp(#); do_path_trans(cur_exp); end @= procedure do_path_trans(@!p:pointer); label exit; var @!q:pointer; {list traverser} begin q:=p; repeat if left_type(q)<>endpoint then trans(q+3,q+4); {that's |left_x| and |left_y|} trans(q+1,q+2); {that's |x_coord| and |y_coord|} if right_type(q)<>endpoint then trans(q+5,q+6); {that's |right_x| and |right_y|} @^data structure assumptions@> q:=link(q); until q=p; exit:end; @ Transforming a pen is very similar, except that there are no |left_type| and |right_type| fields. @d pen_trans(#)==begin set_up_known_trans(#); pen_trans_end @d pen_trans_end(#)==unstash_cur_exp(#); do_pen_trans(cur_exp); end @= procedure do_pen_trans(@!p:pointer); label exit; var @!q:pointer; {list traverser} begin if pen_is_elliptical(p) then begin trans(p+3,p+4); {that's |left_x| and |left_y|} trans(p+5,p+6); {that's |right_x| and |right_y|} end; q:=p; repeat trans(q+1,q+2); {that's |x_coord| and |y_coord|} @^data structure assumptions@> q:=link(q); until q=p; exit:end; @ The next transformation procedure applies to edge structures. It will do any transformation, but the results may be substandard if the picture contains text that uses downloaded bitmap fonts. The binary action procedure is |do_edges_trans|, but we also need a function that just scales a picture. That routine is |scale_edges|. Both it and the underlying routine |edges_trans| should be thought of as procedures that update an edge structure |h|, except that they have to return a (possibly new) structure because of the need to call |private_edges|. @= function edges_trans(@!h:pointer):pointer; label done1; var @!q:pointer; {the object being transformed} @!r,@!s:pointer; {for list manipulation} @!sx,@!sy:scaled; {saved transformation parameters} @!sqdet:scaled; {square root of determinant for |dash_scale|} @!sgndet:integer; {sign of the determinant} @!v:scaled; {a temporary value} begin h:=private_edges(h);@/ sqdet:=sqrt_det(txx,txy,tyx,tyy); sgndet:=ab_vs_cd(txx,tyy,txy,tyx); if dash_list(h)<>null_dash then @; @; q:=link(dummy_loc(h)); while q<>null do begin @;@/ q:=link(q); end; edges_trans:=h; end; @# procedure do_edges_trans(@!p:pointer;@!c:quarterword); begin set_up_known_trans(c); value(p):=edges_trans(value(p)); unstash_cur_exp(p); end; @# procedure scale_edges; begin txx:=se_sf; tyy:=se_sf; txy:=0; tyx:=0; tx:=0; ty:=0; se_pic:=edges_trans(se_pic); end; @ @= if (txy<>0)or(tyx<>0)or(ty<>0)or(abs(txx)<>abs(tyy)) then flush_dash_list(h) else begin if txx<0 then @; @; dash_y(h):=take_scaled(dash_y(h),abs(tyy)); end @ @= begin r:=dash_list(h); dash_list(h):=null_dash; while r<>null_dash do begin s:=r; r:=link(r);@/ v:=start_x(s); start_x(s):=stop_x(s); stop_x(s):=v;@/ link(s):=dash_list(h); dash_list(h):=s; end; end @ @= r:=dash_list(h); while r<>null_dash do begin start_x(r):=take_scaled(start_x(r),txx)+tx; stop_x(r):=take_scaled(stop_x(r),txx)+tx;@/ r:=link(r); end @ @= if (txx=0)and(tyy=0) then @ else if (txy<>0)or(tyx<>0) then begin init_bbox(h); goto done1; end; if minx_val(h)<=maxx_val(h) then @; done1: @ @= begin v:=minx_val(h); minx_val(h):=miny_val(h); miny_val(h):=v;@/ v:=maxx_val(h); maxx_val(h):=maxy_val(h); maxy_val(h):=v; end @ The sum ``|txx+txy|'' is whichever of |txx| or |txy| is nonzero. The other sum is similar. @= begin minx_val(h):=take_scaled(minx_val(h),txx+txy)+tx;@/ maxx_val(h):=take_scaled(maxx_val(h),txx+txy)+tx;@/ miny_val(h):=take_scaled(miny_val(h),tyx+tyy)+ty;@/ maxy_val(h):=take_scaled(maxy_val(h),tyx+tyy)+ty;@/ if txx+txy<0 then begin v:=minx_val(h); minx_val(h):=maxx_val(h); maxx_val(h):=v; end; if tyx+tyy<0 then begin v:=miny_val(h); miny_val(h):=maxy_val(h); maxy_val(h):=v; end; end @ Now we ready for the main task of transforming the graphical objects in edge structure~|h|. @= case type(q) of fill_code,stroked_code: begin do_path_trans(path_p(q)); @; end; start_clip_code,start_bounds_code: do_path_trans(path_p(q)); text_code:begin r:=text_tx_loc(q); @; end; stop_clip_code,stop_bounds_code: do_nothing; end {there are no other cases} @ Note that the shift parameters |(tx,ty)| apply only to the path being stroked. The |dash_scale| has to be adjusted to scale the dash lengths in |dash_p(q)| since the \ps\ output procedures will try to compensate for the transformation we are applying to |pen_p(q)|. Since this compensation is based on the square root of the determinant, |sqdet| is the appropriate factor. @= if pen_p(q)<>null then begin sx:=tx; sy:=ty; tx:=0; ty:=0;@/ do_pen_trans(pen_p(q)); if ((type(q)=stroked_code)and(dash_p(q)<>null)) then dash_scale(q):=take_scaled(dash_scale(q),sqdet); if not pen_is_elliptical(pen_p(q)) then if sgndet<0 then pen_p(q):=make_pen(copy_path(pen_p(q)),true); {this unreverses the pen} tx:=sx; ty:=sy; end @ This uses the fact that transformations are stored in the order |(tx,ty,txx,txy,tyx,tyy)|. @^data structure assumptions@> @= trans(r,r+1); sx:=tx; sy:=ty; tx:=0; ty:=0; trans(r+2,r+4); trans(r+3,r+5); tx:=sx; ty:=sy @ The hard cases of transformation occur when big nodes are involved, and when some of their components are unknown. @= @t\4@>@@; procedure big_trans(@!p:pointer;@!c:quarterword); label exit; var @!q,@!r,@!pp,@!qq:pointer; {list manipulation registers} @!s:small_number; {size of a big node} begin s:=big_node_size[type(p)]; q:=value(p); r:=q+s; repeat r:=r-2; if type(r)<>known then @; until r=q; @; exit:end; {node |p| will now be recycled by |do_binary|} @ @= begin set_up_known_trans(c); make_exp_copy(p); r:=value(cur_exp); if cur_type=transform_type then begin bilin1(yy_part_loc(r),tyy,xy_part_loc(q),tyx,0); bilin1(yx_part_loc(r),tyy,xx_part_loc(q),tyx,0); bilin1(xy_part_loc(r),txx,yy_part_loc(q),txy,0); bilin1(xx_part_loc(r),txx,yx_part_loc(q),txy,0); end; bilin1(y_part_loc(r),tyy,x_part_loc(q),tyx,ty); bilin1(x_part_loc(r),txx,y_part_loc(q),txy,tx); return; end @ Let |p| point to a two-word value field inside a big node of |cur_exp|, and let |q| point to a another value field. The |bilin1| procedure replaces |p| by $p\cdot t+q\cdot u+\delta$. @= procedure bilin1(@!p:pointer;@!t:scaled;@!q:pointer;@!u,@!delta:scaled); var @!r:pointer; {list traverser} begin if t<>unity then dep_mult(p,t,true); if u<>0 then if type(q)=known then delta:=delta+take_scaled(value(q),u) else begin @; dep_list(p):=p_plus_fq(dep_list(p),u,dep_list(q),proto_dependent,type(q)); end; if type(p)=known then value(p):=value(p)+delta else begin r:=dep_list(p); while info(r)<>null do r:=link(r); delta:=value(r)+delta; if r<>dep_list(p) then value(r):=delta else begin recycle_value(p); type(p):=known; value(p):=delta; end; end; if fix_needed then fix_dependencies; end; @ @= if type(p)<>proto_dependent then begin if type(p)=known then new_dep(p,const_dependency(value(p))) else dep_list(p):=p_times_v(dep_list(p),unity,dependent,proto_dependent,true); type(p):=proto_dependent; end @ @= set_up_trans(c); if cur_type=known then @ else begin pp:=stash_cur_exp; qq:=value(pp); make_exp_copy(p); r:=value(cur_exp); if cur_type=transform_type then begin bilin2(yy_part_loc(r),yy_part_loc(qq), value(xy_part_loc(q)),yx_part_loc(qq),null); bilin2(yx_part_loc(r),yy_part_loc(qq), value(xx_part_loc(q)),yx_part_loc(qq),null); bilin2(xy_part_loc(r),xx_part_loc(qq), value(yy_part_loc(q)),xy_part_loc(qq),null); bilin2(xx_part_loc(r),xx_part_loc(qq), value(yx_part_loc(q)),xy_part_loc(qq),null); end; bilin2(y_part_loc(r),yy_part_loc(qq), value(x_part_loc(q)),yx_part_loc(qq),y_part_loc(qq)); bilin2(x_part_loc(r),xx_part_loc(qq), value(y_part_loc(q)),xy_part_loc(qq),x_part_loc(qq)); recycle_value(pp); free_node(pp,value_node_size); end; @ Let |p| be a |proto_dependent| value whose dependency list ends at |dep_final|. The following procedure adds |v| times another numeric quantity to~|p|. @= procedure add_mult_dep(@!p:pointer;@!v:scaled;@!r:pointer); begin if type(r)=known then value(dep_final):=value(dep_final)+take_scaled(value(r),v) else begin dep_list(p):= p_plus_fq(dep_list(p),v,dep_list(r),proto_dependent,type(r)); if fix_needed then fix_dependencies; end; end; @ The |bilin2| procedure is something like |bilin1|, but with known and unknown quantities reversed. Parameter |p| points to a value field within the big node for |cur_exp|; and |type(p)=known|. Parameters |t| and~|u| point to value fields elsewhere; so does parameter~|q|, unless it is |null| (which stands for zero). Location~|p| will be replaced by $p\cdot t+v\cdot u+q$. @= procedure bilin2(@!p,@!t:pointer;@!v:scaled;@!u,@!q:pointer); var @!vv:scaled; {temporary storage for |value(p)|} begin vv:=value(p); type(p):=proto_dependent; new_dep(p,const_dependency(0)); {this sets |dep_final|} if vv<>0 then add_mult_dep(p,vv,t); {|dep_final| doesn't change} if v<>0 then add_mult_dep(p,v,u); if q<>null then add_mult_dep(p,unity,q); if dep_list(p)=dep_final then begin vv:=value(dep_final); recycle_value(p); type(p):=known; value(p):=vv; end; end; @ @= begin make_exp_copy(p); r:=value(cur_exp); if cur_type=transform_type then begin bilin3(yy_part_loc(r),tyy,value(xy_part_loc(q)),tyx,0); bilin3(yx_part_loc(r),tyy,value(xx_part_loc(q)),tyx,0); bilin3(xy_part_loc(r),txx,value(yy_part_loc(q)),txy,0); bilin3(xx_part_loc(r),txx,value(yx_part_loc(q)),txy,0); end; bilin3(y_part_loc(r),tyy,value(x_part_loc(q)),tyx,ty); bilin3(x_part_loc(r),txx,value(y_part_loc(q)),txy,tx); end @ Finally, in |bilin3| everything is |known|. @= procedure bilin3(@!p:pointer;@!t,@!v,@!u,@!delta:scaled); begin if t<>unity then delta:=delta+take_scaled(value(p),t) else delta:=delta+value(p); if u<>0 then value(p):=delta+take_scaled(v,u) else value(p):=delta; end; @ @= concatenate: if (cur_type=string_type)and(type(p)=string_type) then cat(p) else bad_binary(p,concatenate); substring_of: if nice_pair(p,type(p))and(cur_type=string_type) then chop_string(value(p)) else bad_binary(p,substring_of); subpath_of: begin if cur_type=pair_type then pair_to_path; if nice_pair(p,type(p))and(cur_type=path_type) then chop_path(value(p)) else bad_binary(p,subpath_of); end; @ @= procedure cat(@!p:pointer); var @!a,@!b:str_number; {the strings being concatenated} @!k:pool_pointer; {index into |str_pool|} begin a:=value(p); b:=cur_exp; str_room(length(a)+length(b)); for k:=str_start[a] to str_stop(a)-1 do append_char(so(str_pool[k])); for k:=str_start[b] to str_stop(b)-1 do append_char(so(str_pool[k])); cur_exp:=make_string; delete_str_ref(b); end; @ @= procedure chop_string(@!p:pointer); var @!a,@!b:integer; {start and stop points} @!l:integer; {length of the original string} @!k:integer; {runs from |a| to |b|} @!s:str_number; {the original string} @!reversed:boolean; {was |a>b|?} begin a:=round_unscaled(value(x_part_loc(p))); b:=round_unscaled(value(y_part_loc(p))); if a<=b then reversed:=false else begin reversed:=true; k:=a; a:=b; b:=k; end; s:=cur_exp; l:=length(s); if a<0 then begin a:=0; if b<0 then b:=0; end; if b>l then begin b:=l; if a>l then a:=l; end; str_room(b-a); if reversed then for k:=str_start[s]+b-1 downto str_start[s]+a do append_char(so(str_pool[k])) else for k:=str_start[s]+a to str_start[s]+b-1 do append_char(so(str_pool[k])); cur_exp:=make_string; delete_str_ref(s); end; @ @= procedure chop_path(@!p:pointer); var @!q:pointer; {a knot in the original path} @!pp,@!qq,@!rr,@!ss:pointer; {link variables for copies of path nodes} @!a,@!b,@!k,@!l:scaled; {indices for chopping} @!reversed:boolean; {was |a>b|?} begin l:=path_length; a:=value(x_part_loc(p)); b:=value(y_part_loc(p)); if a<=b then reversed:=false else begin reversed:=true; k:=a; a:=b; b:=k; end; @l|@>; q:=cur_exp; while a>=unity do begin q:=link(q); a:=a-unity; b:=b-unity; end; if b=a then @ else @; left_type(pp):=endpoint; right_type(qq):=endpoint; link(qq):=pp; toss_knot_list(cur_exp); if reversed then begin cur_exp:=link(htap_ypoc(pp)); toss_knot_list(pp); end else cur_exp:=pp; end; @ @l|@>= if a<0 then if left_type(cur_exp)=endpoint then begin a:=0; if b<0 then b:=0; end else repeat a:=a+l; b:=b+l; until a>=0; {a cycle always has length |l>0|} if b>l then if left_type(cur_exp)=endpoint then begin b:=l; if a>l then a:=l; end else while a>=l do begin a:=a-l; b:=b-l; end @ @= begin pp:=copy_knot(q); qq:=pp; repeat q:=link(q); rr:=qq; qq:=copy_knot(q); link(rr):=qq; b:=b-unity; until b<=0; if a>0 then begin ss:=pp; pp:=link(pp); split_cubic(ss,a*@'10000); pp:=link(ss); free_node(ss,knot_node_size); if rr=ss then begin b:=make_scaled(b,unity-a); rr:=pp; end; end; if b<0 then begin split_cubic(rr,(b+unity)*@'10000); free_node(qq,knot_node_size); qq:=link(rr); end; end @ @= begin if a>0 then begin split_cubic(q,a*@'10000); q:=link(q); end; pp:=copy_knot(q); qq:=pp; end @ @= point_of,precontrol_of,postcontrol_of: begin if cur_type=pair_type then pair_to_path; if (cur_type=path_type)and(type(p)=known) then find_point(value(p),c) else bad_binary(p,c); end; pen_offset_of: if (cur_type=pen_type)and nice_pair(p,type(p)) then set_up_offset(value(p)) else bad_binary(p,pen_offset_of); direction_time_of: begin if cur_type=pair_type then pair_to_path; if (cur_type=path_type)and nice_pair(p,type(p)) then set_up_direction_time(value(p)) else bad_binary(p,direction_time_of); end; @ @= procedure set_up_offset(@!p:pointer); begin find_offset(value(x_part_loc(p)),value(y_part_loc(p)),cur_exp); pair_value(cur_x,cur_y); end; @# procedure set_up_direction_time(@!p:pointer); begin flush_cur_exp(find_direction_time(value(x_part_loc(p)), value(y_part_loc(p)),cur_exp)); end; @ @= procedure find_point(@!v:scaled;@!c:quarterword); var @!p:pointer; {the path} @!n:scaled; {its length} begin p:=cur_exp;@/ if left_type(p)=endpoint then n:=-unity@+else n:=0; repeat p:=link(p); n:=n+unity; until p=cur_exp; if n=0 then v:=0 else if v<0 then if left_type(p)=endpoint then v:=0 else v:=n-1-((-v-1) mod n) else if v>n then if left_type(p)=endpoint then v:=n else v:=v mod n; p:=cur_exp; while v>=unity do begin p:=link(p); v:=v-unity; end; if v<>0 then @; @; end; @ @= begin split_cubic(p,v*@'10000); p:=link(p); end @ @= case c of point_of: pair_value(x_coord(p),y_coord(p)); precontrol_of: if left_type(p)=endpoint then pair_value(x_coord(p),y_coord(p)) else pair_value(left_x(p),left_y(p)); postcontrol_of: if right_type(p)=endpoint then pair_value(x_coord(p),y_coord(p)) else pair_value(right_x(p),right_y(p)); end {there are no other cases} @ @= arc_time_of: begin if cur_type=pair_type then pair_to_path; if (cur_type=path_type)and(type(p)=known) then flush_cur_exp(get_arc_time(cur_exp,value(p))) else bad_binary(p,c); end; @ @= intersect: begin if type(p)=pair_type then begin q:=stash_cur_exp; unstash_cur_exp(p); pair_to_path; p:=stash_cur_exp; unstash_cur_exp(q); end; if cur_type=pair_type then pair_to_path; if (cur_type=path_type)and(type(p)=path_type) then begin path_intersection(value(p),cur_exp); pair_value(cur_t,cur_tt); end else bad_binary(p,intersect); end; @ @= in_font:if (cur_type<>string_type)or(type(p)<>string_type) then bad_binary(p,in_font) else begin do_infont(p); return; end; @ Function |new_text_node| owns the reference count for its second argument (the text string) but not its first (the font name). @= procedure do_infont(@!p:pointer); var @!q:pointer; begin q:=get_node(edge_header_size); init_edges(q); link(obj_tail(q)):=new_text_node(cur_exp,value(p)); obj_tail(q):=link(obj_tail(q)); free_node(p,value_node_size);@/ flush_cur_exp(q); cur_type:=picture_type; end; @* \[40] Statements and commands. The chief executive of \MP\ is the |do_statement| routine, which contains the master switch that causes all the various pieces of \MP\ to do their things, in the right order. In a sense, this is the grand climax of the program: It applies all the tools that we have worked so hard to construct. In another sense, this is the messiest part of the program: It necessarily refers to other pieces of code all over the place, so that a person can't fully understand what is going on without paging back and forth to be reminded of conventions that are defined elsewhere. We are now at the hub of the web. The structure of |do_statement| itself is quite simple. The first token of the statement is fetched using |get_x_next|. If it can be the first token of an expression, we look for an equation, an assignment, or a title. Otherwise we use a \&{case} construction to branch at high speed to the appropriate routine for various and sundry other types of commands, each of which has an ``action procedure'' that does the necessary work. The program uses the fact that $$\hbox{|min_primary_command=max_statement_command=type_name|}$$ to interpret a statement that starts with, e.g., `\&{string}', as a type declaration rather than a boolean expression. @p @@; procedure do_statement; {governs \MP's activities} begin cur_type:=vacuous; get_x_next; if cur_cmd>max_primary_command then @ else if cur_cmd>max_statement_command then @ else @; if cur_cmd; error_count:=0; end; @ The only command codes |>max_primary_command| that can be present at the beginning of a statement are |semicolon| and higher; these occur when the statement is null. @= begin if cur_cmd print_cmd_mod(cur_cmd,cur_mod); print_char("'"); help5("I was looking for the beginning of a new statement.")@/ ("If you just proceed without changing anything, I'll ignore")@/ ("everything up to the next `;'. Please insert a semicolon")@/ ("now in front of anything that you don't want me to delete.")@/ ("(See Chapter 27 of The METAFONTbook for an example.)");@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> back_error; get_x_next; end; end @ The help message printed here says that everything is flushed up to a semicolon, but actually the commands |end_group| and |stop| will also terminate a statement. @= begin print_err("Extra tokens will be flushed"); @.Extra tokens will be flushed@> help6("I've just read as much of that statement as I could fathom,")@/ ("so a semicolon should have been next. It's very puzzling...")@/ ("but I'll try to get myself back together, by ignoring")@/ ("everything up to the next `;'. Please insert a semicolon")@/ ("now in front of anything that you don't want me to delete.")@/ ("(See Chapter 27 of The METAFONTbook for an example.)");@/ @:METAFONTbook}{\sl The {\logos METAFONT\/}book@> back_error; scanner_status:=flushing; repeat get_t_next; @; until end_of_statement; {|cur_cmd=semicolon|, |end_group|, or |stop|} scanner_status:=normal; end @ If |do_statement| ends with |cur_cmd=end_group|, we should have |cur_type=vacuous| unless the statement was simply an expression; in the latter case, |cur_type| and |cur_exp| should represent that expression. @= begin if internal[tracing_commands]>0 then show_cur_cmd_mod; case cur_cmd of type_name:do_type_declaration; macro_def:if cur_mod>var_def then make_op_def else if cur_mod>end_def then scan_def; @t\4@>@@; end; {there are no other cases} cur_type:=vacuous; end @ The most important statements begin with expressions. @= begin var_flag:=assignment; scan_expression; if cur_cmd else if cur_type<>vacuous then begin exp_err("Isolated expression"); @.Isolated expression@> help3("I couldn't find an `=' or `:=' after the")@/ ("expression that is shown above this error message,")@/ ("so I guess I'll just ignore it and carry on."); put_get_error; end; flush_cur_exp(0); cur_type:=vacuous; end; end @ @= begin if internal[tracing_titles]>0 then begin print_nl(""); print(cur_exp); update_terminal; end; end @ Equations and assignments are performed by the pair of mutually recursive @^recursion@> routines |do_equation| and |do_assignment|. These routines are called when |cur_cmd=equals| and when |cur_cmd=assignment|, respectively; the left-hand side is in |cur_type| and |cur_exp|, while the right-hand side is yet to be scanned. After the routines are finished, |cur_type| and |cur_exp| will be equal to the right-hand side (which will normally be equal to the left-hand side). @= @t\4@>@@; @t\4@>@@; procedure@?do_assignment; forward;@t\2@>@/ procedure do_equation; var @!lhs:pointer; {capsule for the left-hand side} @!p:pointer; {temporary register} begin lhs:=stash_cur_exp; get_x_next; var_flag:=assignment; scan_expression; if cur_cmd=equals then do_equation else if cur_cmd=assignment then do_assignment; if internal[tracing_commands]>two then @; if cur_type=unknown_path then if type(lhs)=pair_type then begin p:=stash_cur_exp; unstash_cur_exp(lhs); lhs:=p; end; {in this case |make_eq| will change the pair to a path} make_eq(lhs); {equate |lhs| to |(cur_type,cur_exp)|} end; @ And |do_assignment| is similar to |do_expression|: @= procedure do_assignment; var @!lhs:pointer; {token list for the left-hand side} @!p:pointer; {where the left-hand value is stored} @!q:pointer; {temporary capsule for the right-hand value} begin if cur_type<>token_list then begin exp_err("Improper `:=' will be changed to `='"); @.Improper `:='@> help2("I didn't find a variable name at the left of the `:=',")@/ ("so I'm going to pretend that you said `=' instead.");@/ error; do_equation; end else begin lhs:=cur_exp; cur_type:=vacuous;@/ get_x_next; var_flag:=assignment; scan_expression; if cur_cmd=equals then do_equation else if cur_cmd=assignment then do_assignment; if internal[tracing_commands]>two then @; if info(lhs)>hash_end then @ else @; flush_node_list(lhs); end; end; @ @= begin begin_diagnostic; print_nl("{("); print_exp(lhs,0); print(")=("); print_exp(null,0); print(")}"); end_diagnostic(false); end @ @= begin begin_diagnostic; print_nl("{"); if info(lhs)>hash_end then print(int_name[info(lhs)-(hash_end)]) else show_token_list(lhs,null,1000,0); print(":="); print_exp(null,0); print_char("}"); end_diagnostic(false); end @ @= if cur_type=known then internal[info(lhs)-(hash_end)]:=cur_exp else begin exp_err("Internal quantity `"); @.Internal quantity...@> print(int_name[info(lhs)-(hash_end)]); print("' must receive a known value"); help2("I can't set an internal quantity to anything but a known")@/ ("numeric value, so I'll have to ignore this assignment."); put_get_error; end @ @= begin p:=find_variable(lhs); if p<>null then begin q:=stash_cur_exp; cur_type:=und_type(p); recycle_value(p); type(p):=cur_type; value(p):=null; make_exp_copy(p); p:=stash_cur_exp; unstash_cur_exp(q); make_eq(p); end else begin obliterated(lhs); put_get_error; end; end @ And now we get to the nitty-gritty. The |make_eq| procedure is given a pointer to a capsule that is to be equated to the current expression. @= procedure make_eq(@!lhs:pointer); label restart,done, not_found; var @!t:small_number; {type of the left-hand side} @!v:integer; {value of the left-hand side} @!p,@!q:pointer; {pointers inside of big nodes} begin restart: t:=type(lhs); if t<=pair_type then v:=value(lhs); case t of @t\4@>@@; end; {all cases have been listed} @; done:check_arith; recycle_value(lhs); free_node(lhs,value_node_size); end; @ @= disp_err(lhs,""); exp_err("Equation cannot be performed ("); @.Equation cannot be performed@> if type(lhs)<=pair_type then print_type(type(lhs))@+else print("numeric"); print_char("="); if cur_type<=pair_type then print_type(cur_type)@+else print("numeric"); print_char(")");@/ help2("I'm sorry, but I don't know how to make such things equal.")@/ ("(See the two expressions just above the error message.)"); put_get_error @ @= boolean_type,string_type,pen_type,path_type,picture_type: if cur_type=t+unknown_tag then begin nonlinear_eq(v,cur_exp,false); goto done; end else if cur_type=t then @; unknown_types:if cur_type=t-unknown_tag then begin nonlinear_eq(cur_exp,lhs,true); goto done; end else if cur_type=t then begin ring_merge(lhs,cur_exp); goto done; end else if cur_type=pair_type then if t=unknown_path then begin pair_to_path; goto restart; end; transform_type,color_type,pair_type:if cur_type=t then @; known,dependent,proto_dependent,independent:if cur_type>=known then begin try_eq(lhs,null); goto done; end; vacuous:do_nothing; @ @= begin if cur_type<=string_type then begin if cur_type=string_type then begin if str_vs_str(v,cur_exp)<>0 then goto not_found; end else if v<>cur_exp then goto not_found; @; goto done; end; print_err("Redundant or inconsistent equation"); @.Redundant or inconsistent equation@> help2("An equation between already-known quantities can't help.")@/ ("But don't worry; continue and I'll just ignore it."); put_get_error; goto done; not_found: print_err("Inconsistent equation"); @.Inconsistent equation@> help2("The equation I just read contradicts what was said before.")@/ ("But don't worry; continue and I'll just ignore it."); put_get_error; goto done; end @ @= begin p:=v+big_node_size[t]; q:=value(cur_exp)+big_node_size[t]; repeat p:=p-2; q:=q-2; try_eq(p,q); until p=v; goto done; end @ The first argument to |try_eq| is the location of a value node in a capsule that will soon be recycled. The second argument is either a location within a pair or transform node pointed to by |cur_exp|, or it is |null| (which means that |cur_exp| itself serves as the second argument). The idea is to leave |cur_exp| unchanged, but to equate the two operands. @= procedure try_eq(@!l,@!r:pointer); label done,done1; var @!p:pointer; {dependency list for right operand minus left operand} @!t:known..independent; {the type of list |p|} @!q:pointer; {the constant term of |p| is here} @!pp:pointer; {dependency list for right operand} @!tt:dependent..independent; {the type of list |pp|} @!copied:boolean; {have we copied a list that ought to be recycled?} begin @; @; if info(p)=null then @ else begin linear_eq(p,t); if r=null then if cur_type<>known then if type(cur_exp)=known then begin pp:=cur_exp; cur_exp:=value(cur_exp); cur_type:=known; free_node(pp,value_node_size); end; end; end; @ @= t:=type(l); if t=known then begin t:=dependent; p:=const_dependency(-value(l)); q:=p; end else if t=independent then begin t:=dependent; p:=single_dependency(l); negate(value(p)); q:=dep_final; end else begin p:=dep_list(l); q:=p; loop@+ begin negate(value(q)); if info(q)=null then goto done; q:=link(q); end; done: link(prev_dep(l)):=link(q); prev_dep(link(q)):=prev_dep(l); type(l):=known; end @ @= begin if abs(value(p))>64 then {off by .001 or more} begin print_err("Inconsistent equation");@/ @.Inconsistent equation@> print(" (off by "); print_scaled(value(p)); print_char(")"); help2("The equation I just read contradicts what was said before.")@/ ("But don't worry; continue and I'll just ignore it."); put_get_error; end else if r=null then @; free_node(p,dep_node_size); end @ @= if r=null then if cur_type=known then begin value(q):=value(q)+cur_exp; goto done1; end else begin tt:=cur_type; if tt=independent then pp:=single_dependency(cur_exp) else pp:=dep_list(cur_exp); end else if type(r)=known then begin value(q):=value(q)+value(r); goto done1; end else begin tt:=type(r); if tt=independent then pp:=single_dependency(r) else pp:=dep_list(r); end; if tt<>independent then copied:=false else begin copied:=true; tt:=dependent; end; @; if copied then flush_node_list(pp); done1: @ @= watch_coefs:=false; if t=tt then p:=p_plus_q(p,pp,t) else if t=proto_dependent then p:=p_plus_fq(p,unity,pp,proto_dependent,dependent) else begin q:=p; while info(q)<>null do begin value(q):=round_fraction(value(q)); q:=link(q); end; t:=proto_dependent; p:=p_plus_q(p,pp,t); end; watch_coefs:=true; @ Our next goal is to process type declarations. For this purpose it's convenient to have a procedure that scans a $\langle\,$declared variable$\,\rangle$ and returns the corresponding token list. After the following procedure has acted, the token after the declared variable will have been scanned, so it will appear in |cur_cmd|, |cur_mod|, and~|cur_sym|. @= function scan_declared_variable:pointer; label done; var @!x:pointer; {hash address of the variable's root} @!h,@!t:pointer; {head and tail of the token list to be returned} @!l:pointer; {hash address of left bracket} begin get_symbol; x:=cur_sym; if cur_cmd<>tag_token then clear_symbol(x,false); h:=get_avail; info(h):=x; t:=h;@/ loop@+ begin get_x_next; if cur_sym=0 then goto done; if cur_cmd<>tag_token then if cur_cmd<>internal_quantity then if cur_cmd=left_bracket then @ else goto done; link(t):=get_avail; t:=link(t); info(t):=cur_sym; end; done: if eq_type(x)<>tag_token then clear_symbol(x,false); if equiv(x)=null then new_root(x); scan_declared_variable:=h; end; @ If the subscript isn't collective, we don't accept it as part of the declared variable. @= begin l:=cur_sym; get_x_next; if cur_cmd<>right_bracket then begin back_input; cur_sym:=l; cur_cmd:=left_bracket; goto done; end else cur_sym:=collective_subscript; end @ Type declarations are introduced by the following primitive operations. @= primitive("numeric",type_name,numeric_type);@/ @!@:numeric_}{\&{numeric} primitive@> primitive("string",type_name,string_type);@/ @!@:string_}{\&{string} primitive@> primitive("boolean",type_name,boolean_type);@/ @!@:boolean_}{\&{boolean} primitive@> primitive("path",type_name,path_type);@/ @!@:path_}{\&{path} primitive@> primitive("pen",type_name,pen_type);@/ @!@:pen_}{\&{pen} primitive@> primitive("picture",type_name,picture_type);@/ @!@:picture_}{\&{picture} primitive@> primitive("transform",type_name,transform_type);@/ @!@:transform_}{\&{transform} primitive@> primitive("color",type_name,color_type);@/ @!@:color_}{\&{color} primitive@> primitive("pair",type_name,pair_type);@/ @!@:pair_}{\&{pair} primitive@> @ @= type_name: print_type(m); @ Now we are ready to handle type declarations, assuming that a |type_name| has just been scanned. @= procedure do_type_declaration; var @!t:small_number; {the type being declared} @!p:pointer; {token list for a declared variable} @!q:pointer; {value node for the variable} begin if cur_mod>=transform_type then t:=cur_mod@+else t:=cur_mod+unknown_tag; repeat p:=scan_declared_variable; flush_variable(equiv(info(p)),link(p),false);@/ q:=find_variable(p); if q<>null then begin type(q):=t; value(q):=null; end else begin print_err("Declared variable conflicts with previous vardef"); @.Declared variable conflicts...@> help2("You can't use, e.g., `numeric foo[]' after `vardef foo'.")@/ ("Proceed, and I'll ignore the illegal redeclaration."); put_get_error; end; flush_list(p); if cur_cmd; until end_of_statement; end; @ @= begin print_err("Illegal suffix of declared variable will be flushed"); @.Illegal suffix...flushed@> help5("Variables in declarations must consist entirely of")@/ ("names and collective subscripts, e.g., `x[]a'.")@/ ("Are you trying to use a reserved word in a variable name?")@/ ("I'm going to discard the junk I found here,")@/ ("up to the next comma or the end of the declaration."); if cur_cmd=numeric_token then help_line[2]:="Explicit subscripts like `x15a' aren't permitted."; put_get_error; scanner_status:=flushing; repeat get_t_next; @; until cur_cmd>=comma; {either |end_of_statement| or |cur_cmd=comma|} scanner_status:=normal; end @ \MP's |main_control| procedure just calls |do_statement| repeatedly until coming to the end of the user's program. Each execution of |do_statement| concludes with |cur_cmd=semicolon|, |end_group|, or |stop|. @p procedure main_control; begin repeat do_statement; if cur_cmd=end_group then begin print_err("Extra `endgroup'"); @.Extra `endgroup'@> help2("I'm not currently working on a `begingroup',")@/ ("so I had better not try to end anything."); flush_error(0); end; until cur_cmd=stop; end; @ @= primitive("end",stop,0);@/ @!@:end_}{\&{end} primitive@> primitive("dump",stop,1);@/ @!@:dump_}{\&{dump} primitive@> @ @= stop:if m=0 then print("end")@+else print("dump"); @* \[41] Commands. Let's turn now to statements that are classified as ``commands'' because of their imperative nature. We'll begin with simple ones, so that it will be clear how to hook command processing into the |do_statement| routine; then we'll tackle the tougher commands. Here's one of the simplest: @= random_seed: do_random_seed; @ @= procedure do_random_seed; begin get_x_next; if cur_cmd<>assignment then begin missing_err(":="); @.Missing `:='@> help1("Always say `randomseed:='."); back_error; end; get_x_next; scan_expression; if cur_type<>known then begin exp_err("Unknown value will be ignored"); @.Unknown value...ignored@> help2("Your expression was too random for me to handle,")@/ ("so I won't change the random seed just now.");@/ put_get_flush_error(0); end else @; end; @ @= begin init_randoms(cur_exp); if selector>=log_only then begin old_setting:=selector; selector:=log_only; print_nl("{randomseed:="); print_scaled(cur_exp); print_char("}"); print_nl(""); selector:=old_setting; end; end @ And here's another simple one (somewhat different in flavor): @= mode_command: begin print_ln; interaction:=cur_mod; @; if log_opened then selector:=selector+2; get_x_next; end; @ @= primitive("batchmode",mode_command,batch_mode); @!@:batch_mode_}{\&{batchmode} primitive@> primitive("nonstopmode",mode_command,nonstop_mode); @!@:nonstop_mode_}{\&{nonstopmode} primitive@> primitive("scrollmode",mode_command,scroll_mode); @!@:scroll_mode_}{\&{scrollmode} primitive@> primitive("errorstopmode",mode_command,error_stop_mode); @!@:error_stop_mode_}{\&{errorstopmode} primitive@> @ @= mode_command: case m of batch_mode: print("batchmode"); nonstop_mode: print("nonstopmode"); scroll_mode: print("scrollmode"); othercases print("errorstopmode") endcases; @ The `\&{inner}' and `\&{outer}' commands are only slightly harder. @= protection_command: do_protection; @ @= primitive("inner",protection_command,0);@/ @!@:inner_}{\&{inner} primitive@> primitive("outer",protection_command,1);@/ @!@:outer_}{\&{outer} primitive@> @ @= protection_command: if m=0 then print("inner")@+else print("outer"); @ @= procedure do_protection; var @!m:0..1; {0 to unprotect, 1 to protect} @!t:halfword; {the |eq_type| before we change it} begin m:=cur_mod; repeat get_symbol; t:=eq_type(cur_sym); if m=0 then begin if t>=outer_tag then eq_type(cur_sym):=t-outer_tag; end else if tcomma; end; @ \MP\ never defines the tokens `\.(' and `\.)' to be primitives, but plain \MP\ begins with the declaration `\&{delimiters} \.{()}'. Such a declaration assigns the command code |left_delimiter| to `\.{(}' and |right_delimiter| to `\.{)}'; the |equiv| of each delimiter is the hash address of its mate. @= delimiters: def_delims; @ @= procedure def_delims; var l_delim,r_delim:pointer; {the new delimiter pair} begin get_clear_symbol; l_delim:=cur_sym;@/ get_clear_symbol; r_delim:=cur_sym;@/ eq_type(l_delim):=left_delimiter; equiv(l_delim):=r_delim;@/ eq_type(r_delim):=right_delimiter; equiv(r_delim):=l_delim;@/ get_x_next; end; @ Here is a procedure that is called when \MP\ has reached a point where some right delimiter is mandatory. @= procedure check_delimiter(@!l_delim,@!r_delim:pointer); label exit; begin if cur_cmd=right_delimiter then if cur_mod=l_delim then return; if cur_sym<>r_delim then begin missing_err(text(r_delim));@/ @.Missing `)'@> help2("I found no right delimiter to match a left one. So I've")@/ ("put one in, behind the scenes; this may fix the problem."); back_error; end else begin print_err("The token `"); print(text(r_delim)); @.The token...delimiter@> print("' is no longer a right delimiter"); help3("Strange: This token has lost its former meaning!")@/ ("I'll read it as a right delimiter this time;")@/ ("but watch out, I'll probably miss it later."); error; end; exit:end; @ The next four commands save or change the values associated with tokens. @= save_command: repeat get_symbol; save_variable(cur_sym); get_x_next; until cur_cmd<>comma; interim_command: do_interim; let_command: do_let; new_internal: do_new_internal; @ @= procedure@?do_statement; forward;@t\2@>@/ procedure do_interim; begin get_x_next; if cur_cmd<>internal_quantity then begin print_err("The token `"); @.The token...quantity@> if cur_sym=0 then print("(%CAPSULE)") else print(text(cur_sym)); print("' isn't an internal quantity"); help1("Something like `tracingonline' should follow `interim'."); back_error; end else begin save_internal(cur_mod); back_input; end; do_statement; end; @ The following procedure is careful not to undefine the left-hand symbol too soon, lest commands like `{\tt let x=x}' have a surprising effect. @= procedure do_let; var @!l:pointer; {hash location of the left-hand symbol} begin get_symbol; l:=cur_sym; get_x_next; if cur_cmd<>equals then if cur_cmd<>assignment then begin missing_err("="); @.Missing `='@> help3("You should have said `let symbol = something'.")@/ ("But don't worry; I'll pretend that an equals sign")@/ ("was present. The next token I read will be `something'."); back_error; end; get_symbol; case cur_cmd of defined_macro,secondary_primary_macro,tertiary_secondary_macro, expression_tertiary_macro: add_mac_ref(cur_mod); othercases do_nothing endcases;@/ clear_symbol(l,false); eq_type(l):=cur_cmd; if cur_cmd=tag_token then equiv(l):=null else equiv(l):=cur_mod; get_x_next; end; @ @= procedure do_new_internal; begin repeat if int_ptr=max_internal then overflow("number of internals",max_internal); @:MetaPost capacity exceeded number of int}{\quad number of internals@> get_clear_symbol; incr(int_ptr); eq_type(cur_sym):=internal_quantity; equiv(cur_sym):=int_ptr; int_name[int_ptr]:=text(cur_sym); internal[int_ptr]:=0; get_x_next; until cur_cmd<>comma; end; @ The various `\&{show}' commands are distinguished by modifier fields in the usual way. @d show_token_code=0 {show the meaning of a single token} @d show_stats_code=1 {show current memory and string usage} @d show_code=2 {show a list of expressions} @d show_var_code=3 {show a variable and its descendents} @d show_dependencies_code=4 {show dependent variables in terms of independents} @= primitive("showtoken",show_command,show_token_code);@/ @!@:show_token_}{\&{showtoken} primitive@> primitive("showstats",show_command,show_stats_code);@/ @!@:show_stats_}{\&{showstats} primitive@> primitive("show",show_command,show_code);@/ @!@:show_}{\&{show} primitive@> primitive("showvariable",show_command,show_var_code);@/ @!@:show_var_}{\&{showvariable} primitive@> primitive("showdependencies",show_command,show_dependencies_code);@/ @!@:show_dependencies_}{\&{showdependencies} primitive@> @ @= show_command: case m of show_token_code:print("showtoken"); show_stats_code:print("showstats"); show_code:print("show"); show_var_code:print("showvariable"); othercases print("showdependencies") endcases; @ @= show_command:do_show_whatever; @ The value of |cur_mod| controls the |verbosity| in the |print_exp| routine: if it's |show_code|, complicated structures are abbreviated, otherwise they aren't. @= procedure do_show; begin repeat get_x_next; scan_expression; print_nl(">> "); @.>>@> print_exp(null,2); flush_cur_exp(0); until cur_cmd<>comma; end; @ @= procedure disp_token; begin print_nl("> "); @.>\relax@> if cur_sym=0 then @ else begin print(text(cur_sym)); print_char("="); if eq_type(cur_sym)>=outer_tag then print("(outer) "); print_cmd_mod(cur_cmd,cur_mod); if cur_cmd=defined_macro then begin print_ln; show_macro(cur_mod,null,100000); end; {this avoids recursion between |show_macro| and |print_cmd_mod|} @^recursion@> end; end; @ @= begin if cur_cmd=numeric_token then print_scaled(cur_mod) else if cur_cmd=capsule_token then begin g_pointer:=cur_mod; print_capsule; end else begin print_char(""""); print(cur_mod); print_char(""""); delete_str_ref(cur_mod); end; end @ The following cases of |print_cmd_mod| might arise in connection with |disp_token|, although they don't correspond to any primitive tokens. @= left_delimiter,right_delimiter: begin if c=left_delimiter then print("lef") else print("righ"); print("t delimiter that matches "); print(text(m)); end; tag_token:if m=null then print("tag")@+else print("variable"); defined_macro: print("macro:"); secondary_primary_macro,tertiary_secondary_macro,expression_tertiary_macro: begin print_cmd_mod(macro_def,c); print("'d macro:"); print_ln; show_token_list(link(link(m)),null,1000,0); end; repeat_loop:print("[repeat the loop]"); internal_quantity:print(int_name[m]); @ @= procedure do_show_token; begin repeat get_t_next; disp_token; get_x_next; until cur_cmd<>comma; end; @ @= procedure do_show_stats; begin print_nl("Memory usage "); @.Memory usage...@> @!stat print_int(var_used); print_char("&"); print_int(dyn_used); if false then@+tats@t@>@;@/ print("unknown"); print(" ("); print_int(hi_mem_min-lo_mem_max-1); print(" still untouched)"); print_ln; print_nl("String usage "); stat print_int(strs_in_use-init_str_use); print_char("&"); print_int(pool_in_use-init_pool_ptr); if false then@+tats@t@>@;@/ print("unknown"); print(" ("); print_int(max_strings-1-strs_used_up); print_char("&"); print_int(pool_size-pool_ptr); print(" now untouched)"); print_ln; get_x_next; end; @ Here's a recursive procedure that gives an abbreviated account of a variable, for use by |do_show_var|. @= procedure disp_var(@!p:pointer); var @!q:pointer; {traverses attributes and subscripts} @!n:0..max_print_line; {amount of macro text to show} begin if type(p)=structured then @ else if type(p)>=unsuffixed_macro then @ else if type(p)<>undefined then begin print_nl(""); print_variable_name(p); print_char("="); print_exp(p,0); end; end; @ @= begin q:=attr_head(p); repeat disp_var(q); q:=link(q); until q=end_attr; q:=subscr_head(p); while name_type(q)=subscr do begin disp_var(q); q:=link(q); end; end @ @= begin print_nl(""); print_variable_name(p); if type(p)>unsuffixed_macro then print("@@#"); {|suffixed_macro|} print("=macro:"); if file_offset>=max_print_line-20 then n:=5 else n:=max_print_line-file_offset-15; show_macro(value(p),null,n); end @ @= procedure do_show_var; label done; begin repeat get_t_next; if cur_sym>0 then if cur_sym<=hash_end then if cur_cmd=tag_token then if cur_mod<>null then begin disp_var(cur_mod); goto done; end; disp_token; done:get_x_next; until cur_cmd<>comma; end; @ @= procedure do_show_dependencies; var @!p:pointer; {link that runs through all dependencies} begin p:=link(dep_head); while p<>dep_head do begin if interesting(p) then begin print_nl(""); print_variable_name(p); if type(p)=dependent then print_char("=") else print(" = "); {extra spaces imply proto-dependency} print_dependency(dep_list(p),type(p)); end; p:=dep_list(p); while info(p)<>null do p:=link(p); p:=link(p); end; get_x_next; end; @ Finally we are ready for the procedure that governs all of the show commands. @= procedure do_show_whatever; begin if interaction=error_stop_mode then wake_up_terminal; case cur_mod of show_token_code:do_show_token; show_stats_code:do_show_stats; show_code:do_show; show_var_code:do_show_var; show_dependencies_code:do_show_dependencies; end; {there are no other cases} if internal[showstopping]>0 then begin print_err("OK"); @.OK@> if interaction= primitive("doublepath",thing_to_add,double_path_code);@/ @!@:double_path_}{\&{doublepath} primitive@> primitive("contour",thing_to_add,contour_code);@/ @!@:contour_}{\&{contour} primitive@> primitive("also",thing_to_add,also_code);@/ @!@:also_}{\&{also} primitive@> primitive("withpen",with_option,pen_type);@/ @!@:with_pen_}{\&{withpen} primitive@> primitive("dashed",with_option,picture_type);@/ @!@:dashed_}{\&{dashed} primitive@> primitive("withcolor",with_option,color_type);@/ @!@:with_color_}{\&{withcolor} primitive@> @ @= thing_to_add:if m=contour_code then print("contour") else if m=double_path_code then print("doublepath") else print("also"); with_option:if m=pen_type then print("withpen") else if m=color_type then print("withcolor") else print("dashed"); @ The |scan_with_list| procedure parses a $\langle$with list$\rangle$ and updates the list of graphical objects starting at |p|. Each $\langle$with clause$\rangle$ updates all graphical objects whose |type| is compatible. Other objects are ignored. @= procedure scan_with_list(@!p:pointer); label done, done1, done2; var @!t:small_number; {|cur_mod| of the |with_option| (should match |cur_type|)} @!q:pointer; {for list manipulation} @!cp,@!pp,@!dp:pointer; {objects being updated; |void| initially; |null| to suppress update} begin cp:=void; pp:=void; dp:=void; while cur_cmd=with_option do begin t:=cur_mod; get_x_next; scan_expression; if cur_type<>t then @ else if t=color_type then begin if cp=void then @; if cp<>null then @; flush_cur_exp(0); end else if t=pen_type then begin if pp=void then @; if pp<>null then begin if pen_p(pp)<>null then toss_knot_list(pen_p(pp)); pen_p(pp):=cur_exp; cur_type:=vacuous; end; end else begin if dp=void then @; if dp<>null then begin if dash_p(dp)<>null then delete_edge_ref(dash_p(dp)); dash_p(dp):=make_dashes(cur_exp); dash_scale(dp):=unity; cur_type:=vacuous; end; end; end; @; end; @ @= begin exp_err("Improper type"); @.Improper type@> help2("Next time say `withpen ';")@/ ("I'll ignore the bad `with' clause and look for another."); if t=picture_type then help_line[1]:="Next time say `dashed ';" else if t=color_type then help_line[1]:="Next time say `withcolor ';"; put_get_flush_error(0); end @ Forcing the color to be between |0| and |unity| here guarantees that no picture will ever contain a color outside the legal range for \ps\ graphics. @= begin q:=value(cur_exp); red_val(cp):=value(red_part_loc(q)); green_val(cp):=value(green_part_loc(q)); blue_val(cp):=value(blue_part_loc(q));@/ if red_val(cp)<0 then red_val(cp):=0; if green_val(cp)<0 then green_val(cp):=0; if blue_val(cp)<0 then blue_val(cp):=0; if red_val(cp)>unity then red_val(cp):=unity; if green_val(cp)>unity then green_val(cp):=unity; if blue_val(cp)>unity then blue_val(cp):=unity; end @ @= begin cp:=p; while cp<>null do begin if has_color(cp) then goto done; cp:=link(cp); end; done:do_nothing; end @ @= begin pp:=p; while pp<>null do begin if has_pen(pp) then goto done1; pp:=link(pp); end; done1:do_nothing; end @ @= begin dp:=p; while dp<>null do begin if type(dp)=stroked_code then goto done2; dp:=link(dp); end; done2:do_nothing; end @ @= if cp>void then @; if pp>void then @; if dp>void then @ @ @= begin q:=link(cp); while q<>null do begin if has_color(q) then begin red_val(q):=red_val(cp); green_val(q):=green_val(cp); blue_val(q):=blue_val(cp);@/ end; q:=link(q); end; end @ @= begin q:=link(pp); while q<>null do begin if has_pen(q) then begin if pen_p(q)<>null then toss_knot_list(pen_p(q)); pen_p(q):=copy_pen(pen_p(pp)); end; q:=link(q); end; end @ @= begin q:=link(dp); while q<>null do begin if type(q)=stroked_code then begin if dash_p(q)<>null then delete_edge_ref(dash_p(q)); dash_p(q):=dash_p(dp); dash_scale(q):=unity; if dash_p(q)<>null then add_edge_ref(dash_p(q)); end; q:=link(q); end; end @ One of the things we need to do when we've parsed an \&{addto} or similar command is find the header of a supposed \&{picture} variable, given a token list for that variable. Since the edge structure is about to be updated, we use |private_edges| to make sure that this is possible. @= function find_edges_var(@!t:pointer):pointer; var @!p:pointer; @!cur_edges:pointer; {the return value} begin p:=find_variable(t); cur_edges:=null; if p=null then begin obliterated(t); put_get_error; end else if type(p)<>picture_type then begin print_err("Variable "); show_token_list(t,null,1000,0); @.Variable x is the wrong type@> print(" is the wrong type ("); print_type(type(p)); print_char(")"); help2("I was looking for a ""known"" picture variable.")@/ ("So I'll not change anything just now."); put_get_error; end else begin value(p):=private_edges(value(p)); cur_edges:=value(p); end; flush_node_list(t); find_edges_var:=cur_edges; end; @ @= add_to_command: do_add_to; bounds_command:do_bounds; @ @= primitive("clip",bounds_command,start_clip_code);@/ @!@:clip_}{\&{clip} primitive@> primitive("setbounds",bounds_command,start_bounds_code);@/ @!@:set_bounds_}{\&{setbounds} primitive@> @ @= bounds_command: if m=start_clip_code then print("clip") else print("setbounds"); @ The following function parses the beginning of an \&{addto} or \&{clip} command: it expects a variable name followed by a token with |cur_cmd=sep| and then an expression. The function returns the token list for the variable and stores the command modifier for the separator token in the global variable |last_add_type|. We must be careful because this variable might get overwritten any time we call |get_x_next|. @= @!last_add_type:quarterword; {command modifier that identifies the last \&{addto} command} @ @= function start_draw_cmd(@!sep:quarterword):pointer; var @!lhv:pointer; {variable to add to left} @!add_type:quarterword; {value to be returned in |last_add_type|} begin lhv:=null;@/ get_x_next; var_flag:=sep; scan_primary; if cur_type<>token_list then @ else begin lhv:=cur_exp; add_type:=cur_mod;@/ cur_type:=vacuous; get_x_next; scan_expression; end; last_add_type:=add_type; start_draw_cmd:=lhv; end; @ @= begin exp_err("Not a suitable variable"); @.Not a suitable variable@> help4("At this point I needed to see the name of a picture variable.")@/ ("(Or perhaps you have indeed presented me with one; I might")@/ ("have missed it, if it wasn't followed by the proper token.)")@/ ("So I'll not change anything just now."); put_get_flush_error(0); end @ Here is an example of how to use |start_draw_cmd|. @= procedure do_bounds; var @!lhv,@!lhe:pointer; {variable on left, the corresponding edge structure} @!p:pointer; {for list manipulation} @!m:integer; {initial value of |cur_mod|} begin m:=cur_mod; lhv:=start_draw_cmd(to_token);@/ if lhv<>null then begin lhe:=find_edges_var(lhv); if lhe=null then flush_cur_exp(0) else if cur_type<>path_type then begin exp_err("Improper `clip'"); @.Improper `addto'@> help2("This expression should have specified a known path.")@/ ("So I'll not change anything just now."); put_get_flush_error(0); end else if left_type(cur_exp)=endpoint then @ else @; end; end; @ @= begin print_err("Not a cycle"); @.Not a cycle@> help2("That contour should have ended with `..cycle' or `&cycle'.")@/ ("So I'll not change anything just now."); put_get_error; end @ @= begin p:=new_bounds_node(cur_exp,m); link(p):=link(dummy_loc(lhe)); link(dummy_loc(lhe)):=p;@/ if obj_tail(lhe)=dummy_loc(lhe) then obj_tail(lhe):=p; p:=get_node(gr_object_size[stop_type(m)]); type(p):=stop_type(m); link(obj_tail(lhe)):=p; obj_tail(lhe):=p;@/ init_bbox(lhe); end @ The |do_add_to| procedure is a little like |do_clip| but there are a lot more cases to deal with. @= procedure do_add_to; var @!lhv,@!lhe:pointer; {variable on left, the corresponding edge structure} @!p:pointer; {the graphical object or list for |scan_with_list| to update} @!e:pointer; {an edge structure to be merged} @!add_type:quarterword; {|also_code|, |contour_code|, or |double_path_code|} begin lhv:=start_draw_cmd(thing_to_add); add_type:=last_add_type;@/ if lhv<>null then begin if add_type=also_code then @ else @; scan_with_list(p); @; end; end; @ Setting |p:=null| causes the $\langle$with list$\rangle$ to be ignored; setting |e:=null| prevents anything from being added to |lhe|. @ @= begin p:=null; e:=null; if cur_type<>picture_type then begin exp_err("Improper `addto'"); @.Improper `addto'@> help2("This expression should have specified a known picture.")@/ ("So I'll not change anything just now."); put_get_flush_error(0); end else begin e:=private_edges(cur_exp); cur_type:=vacuous; p:=link(dummy_loc(e)); end; end @ In this case |add_type<>also_code| so setting |p:=null| suppresses future attempts to add to the edge structure. @= begin e:=null; p:=null; if cur_type=pair_type then pair_to_path; if cur_type<>path_type then begin exp_err("Improper `addto'"); @.Improper `addto'@> help2("This expression should have specified a known path.")@/ ("So I'll not change anything just now."); put_get_flush_error(0); end else if add_type=contour_code then if left_type(cur_exp)=endpoint then @ else begin p:=new_fill_node(cur_exp); cur_type:=vacuous; end else begin p:=new_stroked_node(cur_exp); cur_type:=vacuous; end; end @ @= lhe:=find_edges_var(lhv); if lhe=null then begin if (e=null)and(p<>null) then e:=toss_gr_object(p); if e<>null then delete_edge_ref(e); end else if add_type=also_code then if e<>null then @ else do_nothing else if p<>null then begin link(obj_tail(lhe)):=p; obj_tail(lhe):=p; if add_type=double_path_code then if pen_p(p)=null then pen_p(p):=get_pen_circle(0); end @ @= begin if link(dummy_loc(e))<>null then begin link(obj_tail(lhe)):=link(dummy_loc(e)); obj_tail(lhe):=obj_tail(e);@/ obj_tail(e):=dummy_loc(e); link(dummy_loc(e)):=null; flush_dash_list(lhe); end; toss_edges(e); end @ @= ship_out_command: do_ship_out; @ @= @t\4@>@@; @t\4@>@@; procedure do_ship_out; var @!c:integer; {the character code} begin get_x_next; scan_expression; if cur_type<>picture_type then @ else begin c:=round_unscaled(internal[char_code]) mod 256; if c<0 then c:=c+256; @;@/ ship_out(cur_exp); flush_cur_exp(0); end; end; @ @= begin exp_err("Not a known picture"); help1("I can only output known pictures."); put_get_flush_error(0); end @ The \&{everyjob} command simply assigns a nonzero value to the global variable |start_sym|. @= every_job_command: begin get_symbol; start_sym:=cur_sym; get_x_next; end; @ @= @!start_sym:halfword; {a symbolic token to insert at beginning of job} @ @= start_sym:=0; @ Finally, we have only the ``message'' commands remaining. @d message_code=0 @d err_message_code=1 @d err_help_code=2 @= primitive("message",message_command,message_code);@/ @!@:message_}{\&{message} primitive@> primitive("errmessage",message_command,err_message_code);@/ @!@:err_message_}{\&{errmessage} primitive@> primitive("errhelp",message_command,err_help_code);@/ @!@:err_help_}{\&{errhelp} primitive@> @ @= message_command: if m= message_command: do_message; @ @= @@; procedure do_message; var @!m:message_code..err_help_code; {the type of message} begin m:=cur_mod; get_x_next; scan_expression; if cur_type<>string_type then no_string_err("A message should be a known string expression.") else case m of message_code:begin print_nl(""); print(cur_exp); end; err_message_code:@; err_help_code:@; end; {there are no other cases} flush_cur_exp(0); end; @ @= procedure no_string_err(s:str_number); begin exp_err("Not a string"); @.Not a string@> help1(s); put_get_error; end; @ The global variable |err_help| is zero when the user has most recently given an empty help string, or if none has ever been given. @= begin if err_help<>0 then delete_str_ref(err_help); if length(cur_exp)=0 then err_help:=0 else begin err_help:=cur_exp; add_str_ref(err_help); end; end @ If \&{errmessage} occurs often in |scroll_mode|, without user-defined \&{errhelp}, we don't want to give a long help message each time. So we give a verbose explanation only once. @= @!long_help_seen:boolean; {has the long \.{\\errmessage} help been used?} @ @=long_help_seen:=false; @ @= begin print_err(""); print(cur_exp); if err_help<>0 then use_err_help:=true else if long_help_seen then help1("(That was another `errmessage'.)") else begin if interaction ("and deduce the truth by inspired guesses."); end; put_get_error; use_err_help:=false; end @ @= write_command: do_write; @ @= procedure do_write; label continue; var @!t:str_number; {the line of text to be written} @!n,@!n0:write_index; {for searching |wr_fname| and |wr_file| arrays} @!old_setting:0..max_selector; {for saving |selector| during output} begin get_x_next; scan_expression; if cur_type<>string_type then no_string_err("The text to be written should be a known string expression") else if cur_cmd<>to_token then begin print_err("Missing `to' clause"); help1("A write command should end with `to '"); put_get_error; end else begin t:=cur_exp; cur_type:=vacuous; get_x_next; scan_expression; if cur_type<>string_type then no_string_err("I can't write to that file name. It isn't a known string") else @; delete_str_ref(t); end; flush_cur_exp(0); end; @ @= begin @; @; if str_vs_str(t,eof_line)=0 then @ else begin old_setting:=selector; selector:=n; print(t); print_ln; selector := old_setting; end; end @ @= n:=write_files; n0:=write_files; repeat continue:if n=0 then @ else begin decr(n); if wr_fname[n]=0 then begin n0:=n; goto continue; end; end; until str_vs_str(cur_exp,wr_fname[n])=0 @ @= begin if n0=write_files then if write_files= begin a_close(wr_file[n]); delete_str_ref(wr_fname[n]); wr_fname[n]:=0; if n=write_files-1 then write_files:=n; end @* \[42] Writing font metric data. \TeX\ gets its knowledge about fonts from font metric files, also called \.{TFM} files; the `\.T' in `\.{TFM}' stands for \TeX, but other programs know about them too. One of \MP's duties is to write \.{TFM} files so that the user's fonts can readily be applied to typesetting. @:TFM files}{\.{TFM} files@> @^font metric files@> The information in a \.{TFM} file appears in a sequence of 8-bit bytes. Since the number of bytes is always a multiple of~4, we could also regard the file as a sequence of 32-bit words, but \MP\ uses the byte interpretation. The format of \.{TFM} files was designed by Lyle Ramshaw in 1980. The intent is to convey a lot of different kinds @^Ramshaw, Lyle Harold@> of information in a compact but useful form. @= @!tfm_file:byte_file; {the font metric output goes here} @!metric_file_name: str_number; {full name of the font metric file} @ The first 24 bytes (6 words) of a \.{TFM} file contain twelve 16-bit integers that give the lengths of the various subsequent portions of the file. These twelve integers are, in order: $$\vbox{\halign{\hfil#&$\null=\null$#\hfil\cr |lf|&length of the entire file, in words;\cr |lh|&length of the header data, in words;\cr |bc|&smallest character code in the font;\cr |ec|&largest character code in the font;\cr |nw|&number of words in the width table;\cr |nh|&number of words in the height table;\cr |nd|&number of words in the depth table;\cr |ni|&number of words in the italic correction table;\cr |nl|&number of words in the lig/kern table;\cr |nk|&number of words in the kern table;\cr |ne|&number of words in the extensible character table;\cr |np|&number of font parameter words.\cr}}$$ They are all nonnegative and less than $2^{15}$. We must have |bc-1<=ec<=255|, |ne<=256|, and $$\hbox{|lf=6+lh+(ec-bc+1)+nw+nh+nd+ni+nl+nk+ne+np|.}$$ Note that a font may contain as many as 256 characters (if |bc=0| and |ec=255|), and as few as 0 characters (if |bc=ec+1|). Incidentally, when two or more 8-bit bytes are combined to form an integer of 16 or more bits, the most significant bytes appear first in the file. This is called BigEndian order. @!@^BigEndian order@> @ The rest of the \.{TFM} file may be regarded as a sequence of ten data arrays having the informal specification $$\def\arr$[#1]#2${\&{array} $[#1]$ \&{of} #2} \tabskip\centering \halign to\displaywidth{\hfil\\{#}\tabskip=0pt&$\,:\,$\arr#\hfil \tabskip\centering\cr header&|[0..lh-1]@t\\{stuff}@>|\cr char\_info&|[bc..ec]char_info_word|\cr width&|[0..nw-1]fix_word|\cr height&|[0..nh-1]fix_word|\cr depth&|[0..nd-1]fix_word|\cr italic&|[0..ni-1]fix_word|\cr lig\_kern&|[0..nl-1]lig_kern_command|\cr kern&|[0..nk-1]fix_word|\cr exten&|[0..ne-1]extensible_recipe|\cr param&|[1..np]fix_word|\cr}$$ The most important data type used here is a |@!fix_word|, which is a 32-bit representation of a binary fraction. A |fix_word| is a signed quantity, with the two's complement of the entire word used to represent negation. Of the 32 bits in a |fix_word|, exactly 12 are to the left of the binary point; thus, the largest |fix_word| value is $2048-2^{-20}$, and the smallest is $-2048$. We will see below, however, that all but two of the |fix_word| values must lie between $-16$ and $+16$. @ The first data array is a block of header information, which contains general facts about the font. The header must contain at least two words, |header[0]| and |header[1]|, whose meaning is explained below. Additional header information of use to other software routines might also be included, and \MP\ will generate it if the \.{headerbyte} command occurs. For example, 16 more words of header information are in use at the Xerox Palo Alto Research Center; the first ten specify the character coding scheme used (e.g., `\.{XEROX TEXT}' or `\.{TEX MATHSY}'), the next five give the font family name (e.g., `\.{HELVETICA}' or `\.{CMSY}'), and the last gives the ``face byte.'' \yskip\hang|header[0]| is a 32-bit check sum that \MP\ will copy into the \.{GF} output file. This helps ensure consistency between files, since \TeX\ records the check sums from the \.{TFM}'s it reads, and these should match the check sums on actual fonts that are used. The actual relation between this check sum and the rest of the \.{TFM} file is not important; the check sum is simply an identification number with the property that incompatible fonts almost always have distinct check sums. @^check sum@> \yskip\hang|header[1]| is a |fix_word| containing the design size of the font, in units of \TeX\ points. This number must be at least 1.0; it is fairly arbitrary, but usually the design size is 10.0 for a ``10 point'' font, i.e., a font that was designed to look best at a 10-point size, whatever that really means. When a \TeX\ user asks for a font `\.{at} $\delta$ \.{pt}', the effect is to override the design size and replace it by $\delta$, and to multiply the $x$ and~$y$ coordinates of the points in the font image by a factor of $\delta$ divided by the design size. {\sl All other dimensions in the\/ \.{TFM} file are |fix_word|\kern-1pt\ numbers in design-size units.} Thus, for example, the value of |param[6]|, which defines the \.{em} unit, is often the |fix_word| value $2^{20}=1.0$, since many fonts have a design size equal to one em. The other dimensions must be less than 16 design-size units in absolute value; thus, |header[1]| and |param[1]| are the only |fix_word| entries in the whole \.{TFM} file whose first byte might be something besides 0 or 255. @ Next comes the |char_info| array, which contains one |@!char_info_word| per character. Each word in this part of the file contains six fields packed into four bytes as follows. \yskip\hang first byte: |@!width_index| (8 bits)\par \hang second byte: |@!height_index| (4 bits) times 16, plus |@!depth_index| (4~bits)\par \hang third byte: |@!italic_index| (6 bits) times 4, plus |@!tag| (2~bits)\par \hang fourth byte: |@!remainder| (8 bits)\par \yskip\noindent The actual width of a character is \\{width}|[width_index]|, in design-size units; this is a device for compressing information, since many characters have the same width. Since it is quite common for many characters to have the same height, depth, or italic correction, the \.{TFM} format imposes a limit of 16 different heights, 16 different depths, and 64 different italic corrections. Incidentally, the relation $\\{width}[0]=\\{height}[0]=\\{depth}[0]= \\{italic}[0]=0$ should always hold, so that an index of zero implies a value of zero. The |width_index| should never be zero unless the character does not exist in the font, since a character is valid if and only if it lies between |bc| and |ec| and has a nonzero |width_index|. @ The |tag| field in a |char_info_word| has four values that explain how to interpret the |remainder| field. \yskip\hang|tag=0| (|no_tag|) means that |remainder| is unused.\par \hang|tag=1| (|lig_tag|) means that this character has a ligature/kerning program starting at location |remainder| in the |lig_kern| array.\par \hang|tag=2| (|list_tag|) means that this character is part of a chain of characters of ascending sizes, and not the largest in the chain. The |remainder| field gives the character code of the next larger character.\par \hang|tag=3| (|ext_tag|) means that this character code represents an extensible character, i.e., a character that is built up of smaller pieces so that it can be made arbitrarily large. The pieces are specified in |@!exten[remainder]|.\par \yskip\noindent Characters with |tag=2| and |tag=3| are treated as characters with |tag=0| unless they are used in special circumstances in math formulas. For example, \TeX's \.{\\sum} operation looks for a |list_tag|, and the \.{\\left} operation looks for both |list_tag| and |ext_tag|. @d no_tag=0 {vanilla character} @d lig_tag=1 {character has a ligature/kerning program} @d list_tag=2 {character has a successor in a charlist} @d ext_tag=3 {character is extensible} @ The |lig_kern| array contains instructions in a simple programming language that explains what to do for special letter pairs. Each word in this array is a |@!lig_kern_command| of four bytes. \yskip\hang first byte: |skip_byte|, indicates that this is the final program step if the byte is 128 or more, otherwise the next step is obtained by skipping this number of intervening steps.\par \hang second byte: |next_char|, ``if |next_char| follows the current character, then perform the operation and stop, otherwise continue.''\par \hang third byte: |op_byte|, indicates a ligature step if less than~128, a kern step otherwise.\par \hang fourth byte: |remainder|.\par \yskip\noindent In a kern step, an additional space equal to |kern[256*(op_byte-128)+remainder]| is inserted between the current character and |next_char|. This amount is often negative, so that the characters are brought closer together by kerning; but it might be positive. There are eight kinds of ligature steps, having |op_byte| codes $4a+2b+c$ where $0\le a\le b+c$ and $0\le b,c\le1$. The character whose code is |remainder| is inserted between the current character and |next_char|; then the current character is deleted if $b=0$, and |next_char| is deleted if $c=0$; then we pass over $a$~characters to reach the next current character (which may have a ligature/kerning program of its own). If the very first instruction of the |lig_kern| array has |skip_byte=255|, the |next_char| byte is the so-called right boundary character of this font; the value of |next_char| need not lie between |bc| and~|ec|. If the very last instruction of the |lig_kern| array has |skip_byte=255|, there is a special ligature/kerning program for a left boundary character, beginning at location |256*op_byte+remainder|. The interpretation is that \TeX\ puts implicit boundary characters before and after each consecutive string of characters from the same font. These implicit characters do not appear in the output, but they can affect ligatures and kerning. If the very first instruction of a character's |lig_kern| program has |skip_byte>128|, the program actually begins in location |256*op_byte+remainder|. This feature allows access to large |lig_kern| arrays, because the first instruction must otherwise appear in a location |<=255|. Any instruction with |skip_byte>128| in the |lig_kern| array must satisfy the condition $$\hbox{|256*op_byte+remainder=0|, unless $M$ is absent; in the latter case we can have $TR^kB$ for both even and odd values of~|k|. The width of the extensible character is the width of $R$; and the height-plus-depth is the sum of the individual height-plus-depths of the components used, since the pieces are butted together in a vertical list. @d ext_top(#)==exten[#].b0 {|top| piece in a recipe} @d ext_mid(#)==exten[#].b1 {|mid| piece in a recipe} @d ext_bot(#)==exten[#].b2 {|bot| piece in a recipe} @d ext_rep(#)==exten[#].b3 {|rep| piece in a recipe} @ The final portion of a \.{TFM} file is the |param| array, which is another sequence of |fix_word| values. \yskip\hang|param[1]=slant| is the amount of italic slant, which is used to help position accents. For example, |slant=.25| means that when you go up one unit, you also go .25 units to the right. The |slant| is a pure number; it is the only |fix_word| other than the design size itself that is not scaled by the design size. \hang|param[2]=space| is the normal spacing between words in text. Note that character @'40 in the font need not have anything to do with blank spaces. \hang|param[3]=space_stretch| is the amount of glue stretching between words. \hang|param[4]=space_shrink| is the amount of glue shrinking between words. \hang|param[5]=x_height| is the size of one ex in the font; it is also the height of letters for which accents don't have to be raised or lowered. \hang|param[6]=quad| is the size of one em in the font. \hang|param[7]=extra_space| is the amount added to |param[2]| at the ends of sentences. \yskip\noindent If fewer than seven parameters are present, \TeX\ sets the missing parameters to zero. @d slant_code=1 @d space_code=2 @d space_stretch_code=3 @d space_shrink_code=4 @d x_height_code=5 @d quad_code=6 @d extra_space_code=7 @ So that is what \.{TFM} files hold. One of \MP's duties is to output such information, and it does this all at once at the end of a job. In order to prepare for such frenetic activity, it squirrels away the necessary facts in various arrays as information becomes available. Character dimensions (\&{charwd}, \&{charht}, \&{chardp}, and \&{charic}) are stored respectively in |tfm_width|, |tfm_height|, |tfm_depth|, and |tfm_ital_corr|. Other information about a character (e.g., about its ligatures or successors) is accessible via the |char_tag| and |char_remainder| arrays. Other information about the font as a whole is kept in additional arrays called |header_byte|, |lig_kern|, |kern|, |exten|, and |param|. @d undefined_label==lig_table_size {an undefined local label} @= @!bc,@!ec:eight_bits; {smallest and largest character codes shipped out} @!tfm_width:array[eight_bits] of scaled; {\&{charwd} values} @!tfm_height:array[eight_bits] of scaled; {\&{charht} values} @!tfm_depth:array[eight_bits] of scaled; {\&{chardp} values} @!tfm_ital_corr:array[eight_bits] of scaled; {\&{charic} values} @!char_exists:array[eight_bits] of boolean; {has this code been shipped out?} @!char_tag:array[eight_bits] of no_tag..ext_tag; {|remainder| category} @!char_remainder:array[eight_bits] of 0..lig_table_size; {the |remainder| byte} @!header_byte:array[1..header_size] of -1..255; {bytes of the \.{TFM} header, or $-1$ if unset} @!lig_kern:array[0..lig_table_size] of four_quarters; {the ligature/kern table} @!nl:0..32767-256; {the number of ligature/kern steps so far} @!kern:array[0..max_kerns] of scaled; {distinct kerning amounts} @!nk:0..max_kerns; {the number of distinct kerns so far} @!exten:array[eight_bits] of four_quarters; {extensible character recipes} @!ne:0..256; {the number of extensible characters so far} @!param:array[1..max_font_dimen] of scaled; {\&{fontinfo} parameters} @!np:0..max_font_dimen; {the largest \&{fontinfo} parameter specified so far} @!nw,@!nh,@!nd,@!ni:0..256; {sizes of \.{TFM} subtables} @!skip_table:array[eight_bits] of 0..lig_table_size; {local label status} @!lk_started:boolean; {has there been a lig/kern step in this command yet?} @!bchar:integer; {right boundary character} @!bch_label:0..lig_table_size; {left boundary starting location} @!ll,@!lll:0..lig_table_size; {registers used for lig/kern processing} @!label_loc:array[0..256] of -1..lig_table_size; {lig/kern starting addresses} @!label_char:array[1..256] of eight_bits; {characters for |label_loc|} @!label_ptr:0..256; {highest position occupied in |label_loc|} @ @= for k:=0 to 255 do begin tfm_width[k]:=0; tfm_height[k]:=0; tfm_depth[k]:=0; tfm_ital_corr[k]:=0; char_exists[k]:=false; char_tag[k]:=no_tag; char_remainder[k]:=0; skip_table[k]:=undefined_label; end; for k:=1 to header_size do header_byte[k]:=-1; bc:=255; ec:=0; nl:=0; nk:=0; ne:=0; np:=0;@/ internal[boundary_char]:=-unity; bch_label:=undefined_label;@/ label_loc[0]:=-1; label_ptr:=0; @ @= function tfm_check(@!m:small_number):scaled; begin if abs(internal[m])>=fraction_half then begin print_err("Enormous "); print(int_name[m]); @.Enormous charwd...@> @.Enormous chardp...@> @.Enormous charht...@> @.Enormous charic...@> @.Enormous designsize...@> print(" has been reduced"); help1("Font metric dimensions must be less than 2048pt."); put_get_error; if internal[m]>0 then tfm_check:=fraction_half-1 else tfm_check:=1-fraction_half; end else tfm_check:=internal[m]; end; @ @= if cec then ec:=c; char_exists[c]:=true; tfm_width[c]:=tfm_check(char_wd); tfm_height[c]:=tfm_check(char_ht); tfm_depth[c]:=tfm_check(char_dp); tfm_ital_corr[c]:=tfm_check(char_ic) @ Now let's consider \MP's special \.{TFM}-oriented commands. @= tfm_command: do_tfm_command; @ @d char_list_code=0 @d lig_table_code=1 @d extensible_code=2 @d header_byte_code=3 @d font_dimen_code=4 @= primitive("charlist",tfm_command,char_list_code);@/ @!@:char_list_}{\&{charlist} primitive@> primitive("ligtable",tfm_command,lig_table_code);@/ @!@:lig_table_}{\&{ligtable} primitive@> primitive("extensible",tfm_command,extensible_code);@/ @!@:extensible_}{\&{extensible} primitive@> primitive("headerbyte",tfm_command,header_byte_code);@/ @!@:header_byte_}{\&{headerbyte} primitive@> primitive("fontdimen",tfm_command,font_dimen_code);@/ @!@:font_dimen_}{\&{fontdimen} primitive@> @ @= tfm_command: case m of char_list_code:print("charlist"); lig_table_code:print("ligtable"); extensible_code:print("extensible"); header_byte_code:print("headerbyte"); othercases print("fontdimen") endcases; @ @= function get_code:eight_bits; {scans a character code value} label found; var @!c:integer; {the code value found} begin get_x_next; scan_expression; if cur_type=known then begin c:=round_unscaled(cur_exp); if c>=0 then if c<256 then goto found; end else if cur_type=string_type then if length(cur_exp)=1 then begin c:=so(str_pool[str_start[cur_exp]]); goto found; end; exp_err("Invalid code has been replaced by 0"); @.Invalid code...@> help2("I was looking for a number between 0 and 255, or for a")@/ ("string of length 1. Didn't find it; will use 0 instead."); put_get_flush_error(0); c:=0; found: get_code:=c; end; @ @= procedure set_tag(@!c:halfword;@!t:small_number;@!r:halfword); begin if char_tag[c]=no_tag then begin char_tag[c]:=t; char_remainder[c]:=r; if t=lig_tag then begin incr(label_ptr); label_loc[label_ptr]:=r; label_char[label_ptr]:=c; end; end else @; end; @ @= begin print_err("Character "); if (c>" ")and(c<127) then print(c) else if c=256 then print("||") else begin print("code "); print_int(c); end; print(" is already "); @.Character c is already...@> case char_tag[c] of lig_tag: print("in a ligtable"); list_tag: print("in a charlist"); ext_tag: print("extensible"); end; {there are no other cases} help2("It's not legal to label a character more than once.")@/ ("So I'll not change anything just now."); put_get_error; end @ @= procedure do_tfm_command; label continue,done; var @!c,@!cc:0..256; {character codes} @!k:0..max_kerns; {index into the |kern| array} @!j:integer; {index into |header_byte| or |param|} begin case cur_mod of char_list_code: begin c:=get_code; {we will store a list of character successors} while cur_cmd=colon do begin cc:=get_code; set_tag(c,list_tag,cc); c:=cc; end; end; lig_table_code: @; extensible_code: @; header_byte_code, font_dimen_code: begin c:=cur_mod; get_x_next; scan_expression; if (cur_type<>known)or(cur_exp help2("I was looking for a known, positive number.")@/ ("For safety's sake I'll ignore the present command."); put_get_error; end else begin j:=round_unscaled(cur_exp); if cur_cmd<>colon then begin missing_err(":"); @.Missing `:'@> help1("A colon should follow a headerbyte or fontinfo location."); back_error; end; if c=header_byte_code then @ else @; end; end; end; {there are no other cases} end; @ @= begin lk_started:=false; continue: get_x_next; if(cur_cmd=skip_to)and lk_started then @; if cur_cmd=bchar_label then begin c:=256; cur_cmd:=colon;@+end else begin back_input; c:=get_code;@+end; if(cur_cmd=colon)or(cur_cmd=double_colon)then @; if cur_cmd=lig_kern_token then @ else begin print_err("Illegal ligtable step"); @.Illegal ligtable step@> help1("I was looking for `=:' or `kern' here."); back_error; next_char(nl):=qi(0); op_byte(nl):=qi(0); rem_byte(nl):=qi(0);@/ skip_byte(nl):=stop_flag+1; {this specifies an unconditional stop} end; if nl=lig_table_size then overflow("ligtable size",lig_table_size); @:MetaPost capacity exceeded ligtable size}{\quad ligtable size@> incr(nl); if cur_cmd=comma then goto continue; if skip_byte(nl-1)= primitive("=:",lig_kern_token,0); @!@:=:_}{\.{=:} primitive@> primitive("=:|",lig_kern_token,1); @!@:=:/_}{\.{=:\char'174} primitive@> primitive("=:|>",lig_kern_token,5); @!@:=:/>_}{\.{=:\char'174>} primitive@> primitive("|=:",lig_kern_token,2); @!@:=:/_}{\.{\char'174=:} primitive@> primitive("|=:>",lig_kern_token,6); @!@:=:/>_}{\.{\char'174=:>} primitive@> primitive("|=:|",lig_kern_token,3); @!@:=:/_}{\.{\char'174=:\char'174} primitive@> primitive("|=:|>",lig_kern_token,7); @!@:=:/>_}{\.{\char'174=:\char'174>} primitive@> primitive("|=:|>>",lig_kern_token,11); @!@:=:/>_}{\.{\char'174=:\char'174>>} primitive@> primitive("kern",lig_kern_token,128); @!@:kern_}{\&{kern} primitive@> @ @= lig_kern_token: case m of 0:print("=:"); 1:print("=:|"); 2:print("|=:"); 3:print("|=:|"); 5:print("=:|>"); 6:print("|=:>"); 7:print("|=:|>"); 11:print("|=:|>>"); othercases print("kern") endcases; @ Local labels are implemented by maintaining the |skip_table| array, where |skip_table[c]| is either |undefined_label| or the address of the most recent lig/kern instruction that skips to local label~|c|. In the latter case, the |skip_byte| in that instruction will (temporarily) be zero if there were no prior skips to this label, or it will be the distance to the prior skip. We may need to cancel skips that span more than 127 lig/kern steps. @d cancel_skips(#)==ll:=#; repeat lll:=qo(skip_byte(ll)); skip_byte(ll):=stop_flag; ll:=ll-lll; until lll=0 @d skip_error(#)==begin print_err("Too far to skip"); @.Too far to skip@> help1("At most 127 lig/kern steps can separate skipto1 from 1::."); error; cancel_skips(#); end @= begin c:=get_code; if nl-skip_table[c]>128 then {|skip_table[c]<= begin if cur_cmd=colon then if c=256 then bch_label:=nl else set_tag(c,lig_tag,nl) else if skip_table[c]128 then begin skip_error(ll); goto continue; end; skip_byte(ll):=qi(nl-ll-1); ll:=ll-lll; until lll=0; end; goto continue; end @ @= begin next_char(nl):=qi(c); skip_byte(nl):=qi(0); if cur_mod<128 then {ligature op} begin op_byte(nl):=qi(cur_mod); rem_byte(nl):=qi(get_code); end else begin get_x_next; scan_expression; if cur_type<>known then begin exp_err("Improper kern"); @.Improper kern@> help2("The amount of kern should be a known numeric value.")@/ ("I'm zeroing this one. Proceed, with fingers crossed."); put_get_flush_error(0); end; kern[nk]:=cur_exp; k:=0;@+while kern[k]<>cur_exp do incr(k); if k=nk then begin if nk=max_kerns then overflow("kern",max_kerns); @:MetaPost capacity exceeded kern}{\quad kern@> incr(nk); end; op_byte(nl):=kern_flag+(k div 256); rem_byte(nl):=qi((k mod 256)); end; lk_started:=true; end @ @d missing_extensible_punctuation(#)== begin missing_err(#); @.Missing `\char`\#'@> help1("I'm processing `extensible c: t,m,b,r'."); back_error; end @= begin if ne=256 then overflow("extensible",256); @:MetaPost capacity exceeded extensible}{\quad extensible@> c:=get_code; set_tag(c,ext_tag,ne); if cur_cmd<>colon then missing_extensible_punctuation(":"); ext_top(ne):=qi(get_code); if cur_cmd<>comma then missing_extensible_punctuation(","); ext_mid(ne):=qi(get_code); if cur_cmd<>comma then missing_extensible_punctuation(","); ext_bot(ne):=qi(get_code); if cur_cmd<>comma then missing_extensible_punctuation(","); ext_rep(ne):=qi(get_code); incr(ne); end @ @= repeat if j>header_size then overflow("headerbyte",header_size); @:MetaPost capacity exceeded headerbyte}{\quad headerbyte@> header_byte[j]:=get_code; incr(j); until cur_cmd<>comma @ @= repeat if j>max_font_dimen then overflow("fontdimen",max_font_dimen); @:MetaPost capacity exceeded fontdimen}{\quad fontdimen@> while j>np do begin incr(np); param[np]:=0; end; get_x_next; scan_expression; if cur_type<>known then begin exp_err("Improper font parameter"); @.Improper font parameter@> help1("I'm zeroing this one. Proceed, with fingers crossed."); put_get_flush_error(0); end; param[j]:=cur_exp; incr(j); until cur_cmd<>comma @ OK: We've stored all the data that is needed for the \.{TFM} file. All that remains is to output it in the correct format. An interesting problem needs to be solved in this connection, because the \.{TFM} format allows at most 256~widths, 16~heights, 16~depths, and 64~italic corrections. If the data has more distinct values than this, we want to meet the necessary restrictions by perturbing the given values as little as possible. \MP\ solves this problem in two steps. First the values of a given kind (widths, heights, depths, or italic corrections) are sorted; then the list of sorted values is perturbed, if necessary. The sorting operation is facilitated by having a special node of essentially infinite |value| at the end of the current list. @= value(inf_val):=fraction_four; @ Straight linear insertion is good enough for sorting, since the lists are usually not terribly long. As we work on the data, the current list will start at |link(temp_head)| and end at |inf_val|; the nodes in this list will be in increasing order of their |value| fields. Given such a list, the |sort_in| function takes a value and returns a pointer to where that value can be found in the list. The value is inserted in the proper place, if necessary. At the time we need to do these operations, most of \MP's work has been completed, so we will have plenty of memory to play with. The value nodes that are allocated for sorting will never be returned to free storage. @d clear_the_list==link(temp_head):=inf_val @p function sort_in(@!v:scaled):pointer; label found; var @!p,@!q,@!r:pointer; {list manipulation registers} begin p:=temp_head; loop@+ begin q:=link(p); if v<=value(q) then goto found; p:=q; end; found: if vd$ such that the covering found by this algorithm would be different. In particular, |min_cover(0)| returns the number of distinct values in the current list and sets |perturbation| to the minimum distance between adjacent values. @p function min_cover(@!d:scaled):integer; var @!p:pointer; {runs through the current list} @!l:scaled; {the least element covered by the current interval} @!m:integer; {lower bound on the size of the minimum cover} begin m:=0; p:=link(temp_head); perturbation:=el_gordo; while p<>inf_val do begin incr(m); l:=value(p); repeat p:=link(p); until value(p)>l+d; if value(p)-l= @!perturbation:scaled; {quantity related to \.{TFM} rounding} @!excess:integer; {the list is this much too long} @ The smallest |d| such that a given list can be covered with |m| intervals is determined by the |threshold| routine, which is sort of an inverse to |min_cover|. The idea is to increase the interval size rapidly until finding the range, then to go sequentially until the exact borderline has been discovered. @p function threshold(@!m:integer):scaled; var @!d:scaled; {lower bound on the smallest interval size} begin excess:=min_cover(0)-m; if excess<=0 then threshold:=0 else begin repeat d:=perturbation; until min_cover(d+d)<=m; while min_cover(d)>m do d:=perturbation; threshold:=d; end; end; @ The |skimp| procedure reduces the current list to at most |m| entries, by changing values if necessary. It also sets |info(p):=k| if |value(p)| is the |k|th distinct value on the resulting list, and it sets |perturbation| to the maximum amount by which a |value| field has been changed. The size of the resulting list is returned as the value of |skimp|. @p function skimp(@!m:integer):integer; var @!d:scaled; {the size of intervals being coalesced} @!p,@!q,@!r:pointer; {list manipulation registers} @!l:scaled; {the least value in the current interval} @!v:scaled; {a compromise value} begin d:=threshold(m); perturbation:=0; q:=temp_head; m:=0; p:=link(temp_head); while p<>inf_val do begin incr(m); l:=value(p); info(p):=m; if value(link(p))<=l+d then @; q:=p; p:=link(p); end; skimp:=m; end; @ @= begin repeat p:=link(p); info(p):=m; decr(excess);@+if excess=0 then d:=0; until value(link(p))>l+d; v:=l+halfp(value(p)-l); if value(p)-v>perturbation then perturbation:=value(p)-v; r:=q; repeat r:=link(r); value(r):=v; until r=p; link(q):=p; {remove duplicate values from the current list} end @ A warning message is issued whenever something is perturbed by more than 1/16\thinspace pt. @p procedure tfm_warning(@!m:small_number); begin print_nl("(some "); print(int_name[m]); @.some charwds...@> @.some chardps...@> @.some charhts...@> @.some charics...@> print(" values had to be adjusted by as much as "); print_scaled(perturbation); print("pt)"); end; @ Here's an example of how we use these routines. The width data needs to be perturbed only if there are 256 distinct widths, but \MP\ must check for this case even though it is highly unusual. An integer variable |k| will be defined when we use this code. The |dimen_head| array will contain pointers to the sorted lists of dimensions. @= clear_the_list; for k:=bc to ec do if char_exists[k] then tfm_width[k]:=sort_in(tfm_width[k]); nw:=skimp(255)+1; dimen_head[1]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_wd) @ @= @!dimen_head:array[1..4] of pointer; {lists of \.{TFM} dimensions} @ Heights, depths, and italic corrections are different from widths not only because their list length is more severely restricted, but also because zero values do not need to be put into the lists. @= clear_the_list; for k:=bc to ec do if char_exists[k] then if tfm_height[k]=0 then tfm_height[k]:=zero_val else tfm_height[k]:=sort_in(tfm_height[k]); nh:=skimp(15)+1; dimen_head[2]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_ht); clear_the_list; for k:=bc to ec do if char_exists[k] then if tfm_depth[k]=0 then tfm_depth[k]:=zero_val else tfm_depth[k]:=sort_in(tfm_depth[k]); nd:=skimp(15)+1; dimen_head[3]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_dp); clear_the_list; for k:=bc to ec do if char_exists[k] then if tfm_ital_corr[k]=0 then tfm_ital_corr[k]:=zero_val else tfm_ital_corr[k]:=sort_in(tfm_ital_corr[k]); ni:=skimp(63)+1; dimen_head[4]:=link(temp_head); if perturbation>=@'10000 then tfm_warning(char_ic) @ @= value(zero_val):=0; info(zero_val):=0; @ Bytes 5--8 of the header are set to the design size, unless the user has some crazy reason for specifying them differently. Error messages are not allowed at the time this procedure is called, so a warning is printed instead. The value of |max_tfm_dimen| is calculated so that $$\hbox{|make_scaled(16*max_tfm_dimen,internal[design_size])|} < \\{three\_bytes}.$$ @d three_bytes==@'100000000 {$2^{24}$} @p procedure fix_design_size; var @!d:scaled; {the design size} begin d:=internal[design_size]; if (d=fraction_half) then begin if d<>0 then print_nl("(illegal design size has been changed to 128pt)"); @.illegal design size...@> d:=@'40000000; internal[design_size]:=d; end; if header_byte[5]<0 then if header_byte[6]<0 then if header_byte[7]<0 then if header_byte[8]<0 then begin header_byte[5]:=d div @'4000000; header_byte[6]:=(d div 4096) mod 256; header_byte[7]:=(d div 16) mod 256; header_byte[8]:=(d mod 16)*16; end; max_tfm_dimen:=16*internal[design_size]-internal[design_size] div @'10000000; if max_tfm_dimen>=fraction_half then max_tfm_dimen:=fraction_half-1; end; @ The |dimen_out| procedure computes a |fix_word| relative to the design size. If the data was out of range, it is corrected and the global variable |tfm_changed| is increased by~one. @p function dimen_out(@!x:scaled):integer; begin if abs(x)>max_tfm_dimen then begin incr(tfm_changed); if x>0 then x:=three_bytes-1@+else x:=1-three_bytes; end else x:=make_scaled(x*16,internal[design_size]); dimen_out:=x; end; @ @= @!max_tfm_dimen:scaled; {bound on widths, heights, kerns, etc.} @!tfm_changed:integer; {the number of data entries that were out of bounds} @ If the user has not specified any of the first four header bytes, the |fix_check_sum| procedure replaces them by a ``check sum'' computed from the |tfm_width| data relative to the design size. @^check sum@> @p procedure fix_check_sum; label exit; var @!k:eight_bits; {runs through character codes} @!b1,@!b2,@!b3,@!b4:eight_bits; {bytes of the check sum} @!x:integer; {hash value used in check sum computation} begin if header_byte[1]<0 then if header_byte[2]<0 then if header_byte[3]<0 then if header_byte[4]<0 then begin @; header_byte[1]:=b1; header_byte[2]:=b2; header_byte[3]:=b3; header_byte[4]:=b4; return; end; for k:=1 to 4 do if header_byte[k]<0 then header_byte[k]:=0; exit:end; @ @= b1:=bc; b2:=ec; b3:=bc; b4:=ec; tfm_changed:=0; for k:=bc to ec do if char_exists[k] then begin x:=dimen_out(value(tfm_width[k]))+(k+4)*@'20000000; {this is positive} b1:=(b1+b1+x) mod 255; b2:=(b2+b2+x) mod 253; b3:=(b3+b3+x) mod 251; b4:=(b4+b4+x) mod 247; end @ Finally we're ready to actually write the \.{TFM} information. Here are some utility routines for this purpose. @d tfm_out(#)==write(tfm_file,#) {output one byte to |tfm_file|} @p procedure tfm_two(@!x:integer); {output two bytes to |tfm_file|} begin tfm_out(x div 256); tfm_out(x mod 256); end; @# procedure tfm_four(@!x:integer); {output four bytes to |tfm_file|} begin if x>=0 then tfm_out(x div three_bytes) else begin x:=x+@'10000000000; {use two's complement for negative values} x:=x+@'10000000000; tfm_out((x div three_bytes) + 128); end; x:=x mod three_bytes; tfm_out(x div unity); x:=x mod unity; tfm_out(x div @'400); tfm_out(x mod @'400); end; @# procedure tfm_qqqq(@!x:four_quarters); {output four quarterwords to |tfm_file|} begin tfm_out(qo(x.b0)); tfm_out(qo(x.b1)); tfm_out(qo(x.b2)); tfm_out(qo(x.b3)); end; @ @= if job_name=0 then open_log_file; pack_job_name(".tfm"); while not b_open_out(tfm_file) do prompt_file_name("file name for font metrics",".tfm"); metric_file_name:=b_make_name_string(tfm_file); @; @; @; @; @!stat if internal[tracing_stats]>0 then @;@;@+tats@/ print_nl("Font metrics written on "); print(metric_file_name); print_char("."); @.Font metrics written...@> b_close(tfm_file) @ Integer variables |lh|, |k|, and |lk_offset| will be defined when we use this code. @= k:=header_size; while header_byte[k]<0 do decr(k); lh:=(k+3) div 4; {this is the number of header words} if bc>ec then bc:=1; {if there are no characters, |ec=0| and |bc=1|} @; tfm_two(6+lh+(ec-bc+1)+nw+nh+nd+ni+nl+lk_offset+nk+ne+np); {this is the total number of file words that will be output} tfm_two(lh); tfm_two(bc); tfm_two(ec); tfm_two(nw); tfm_two(nh); tfm_two(nd); tfm_two(ni); tfm_two(nl+lk_offset); tfm_two(nk); tfm_two(ne); tfm_two(np); for k:=1 to 4*lh do begin if header_byte[k]<0 then header_byte[k]:=0; tfm_out(header_byte[k]); end @ @= for k:=bc to ec do if not char_exists[k] then tfm_four(0) else begin tfm_out(info(tfm_width[k])); {the width index} tfm_out((info(tfm_height[k]))*16+info(tfm_depth[k])); tfm_out((info(tfm_ital_corr[k]))*4+char_tag[k]); tfm_out(char_remainder[k]); end; tfm_changed:=0; for k:=1 to 4 do begin tfm_four(0); p:=dimen_head[k]; while p<>inf_val do begin tfm_four(dimen_out(value(p))); p:=link(p); end; end @ We need to output special instructions at the beginning of the |lig_kern| array in order to specify the right boundary character and/or to handle starting addresses that exceed 255. The |label_loc| and |label_char| arrays have been set up to record all the starting addresses; we have $-1=|label_loc|[0]<|label_loc|[1]\le\cdots \le|label_loc|[|label_ptr]|$. @= bchar:=round_unscaled(internal[boundary_char]); if(bchar<0)or(bchar>255)then begin bchar:=-1; lk_started:=false; lk_offset:=0;@+end else begin lk_started:=true; lk_offset:=1;@+end; @; if bch_label= k:=label_ptr; {pointer to the largest unallocated label} if label_loc[k]+lk_offset>255 then begin lk_offset:=0; lk_started:=false; {location 0 can do double duty} repeat char_remainder[label_char[k]]:=lk_offset; while label_loc[k-1]=label_loc[k] do begin decr(k); char_remainder[label_char[k]]:=lk_offset; end; incr(lk_offset); decr(k); until lk_offset+label_loc[k]<256; {N.B.: |lk_offset=256| satisfies this when |k=0|} end; if lk_offset>0 then while k>0 do begin char_remainder[label_char[k]] :=char_remainder[label_char[k]]+lk_offset; decr(k); end @ @= for k:=0 to 255 do if skip_table[k] cancel_skips(skip_table[k]); end; if lk_started then {|lk_offset=1| for the special |bchar|} begin tfm_out(255); tfm_out(bchar); tfm_two(0); end else for k:=1 to lk_offset do {output the redirection specs} begin ll:=label_loc[label_ptr]; if bchar<0 then begin tfm_out(254); tfm_out(0); end else begin tfm_out(255); tfm_out(bchar); end; tfm_two(ll+lk_offset); repeat decr(label_ptr); until label_loc[label_ptr]= for k:=0 to ne-1 do tfm_qqqq(exten[k]); for k:=1 to np do if k=1 then if abs(param[1])0 then tfm_four(el_gordo) else tfm_four(-el_gordo); end else tfm_four(dimen_out(param[k])); if tfm_changed>0 then begin if tfm_changed=1 then print_nl("(a font metric dimension") @.a font metric dimension...@> else begin print_nl("("); print_int(tfm_changed); @.font metric dimensions...@> print(" font metric dimensions"); end; print(" had to be decreased)"); end @ @= begin wlog_ln(' '); if bch_label= tfm_infile:byte_file; @ All the width, height, and depth information is stored in an array called |font_info|. This array is allocated sequentially and each font is stored as a series of |char_info| words followed by the width, height, and depth tables. Since |font_name| entries are permanent, their |str_ref| values are set to |max_str_ref|. @= font_number=0..font_max; @ @= font_info:array[0..font_mem_size] of memory_word; {height, width, and depth data} next_fmem:0..font_mem_size; {next unused entry in |font_info|} last_fnum:font_number; {last font number used so far} font_dsize:array[font_number] of scaled; {16 times the ``design'' size in \ps\ points} font_name:array[font_number] of str_number; {name as specified in the \&{infont} command} font_ps_name:array[font_number] of str_number; {PostScript name for use when |internal[prologues]>0|} last_ps_fnum:font_number; {last valid |font_ps_name| index} font_bc,font_ec:array[font_number] of eight_bits; {first and last character code} @ The |font_info| array is indexed via a group directory arrays. For example, the |char_info| data for character~|c| in font~|f| will be in |font_info[char_base[f]+c].qqqq|. @= char_base:array[font_number] of 0..font_mem_size; {base address for |char_info|} width_base:array[font_number] of 0..font_mem_size; {index for zeroth character width} height_base:array[font_number] of 0..font_mem_size; {index for zeroth character height} depth_base:array[font_number] of 0..font_mem_size; {index for zeroth character depth} @ A |null_font| containing no characters is useful for error recovery. Its |font_name| entry starts out empty but is reset each time an erroneous font is found. This helps to cut down on the number of duplicate error messages without wasting a lot of space. @d null_font=0 {the |font_number| for an empty font} @= font_dsize[null_font]:=0; font_name[null_font]:=""; font_ps_name[null_font]:=""; font_bc[null_font]:=1; font_ec[null_font]:=0;@/ char_base[null_font]:=0; width_base[null_font]:=0; height_base[null_font]:=0; depth_base[null_font]:=0;@/ next_fmem:=0; last_fnum:=null_font; last_ps_fnum:=null_font; @ Each |char_info| word is of type |four_quarters|. The |b0| field contains |min_quarter_word| plus the |width index|; the |b1| field contains the height index; the |b2| fields contains the depth index, and the |b3| field used only for temporary storage. (It is used to keep track of which characters occur in an edge structure that is being shipped out.) The corresponding words in the width, height, and depth tables are stored as |scaled| values in units of \ps\ points. With the macros below, the |char_info| word for character~|c| in font~|f| is |char_info(f)(c)| and the width is $$\hbox{|char_width(f)(char_info(f)(c)).sc|.}$$ @d char_info_end(#)==#].qqqq @d char_info(#)==font_info[char_base[#]+char_info_end @d char_width_end(#)==#.b0].sc @d char_width(#)==font_info[width_base[#]+char_width_end @d char_height_end(#)==#.b1].sc @d char_height(#)==font_info[height_base[#]+char_height_end @d char_depth_end(#)==#.b2].sc @d char_depth(#)==font_info[depth_base[#]+char_depth_end @d ichar_exists(#)==(#.b0>min_quarterword) @ The |font_ps_name| for a built-in font should be what PostScript expects. A preliminary name is obtained here from the \.{TFM} name as given in the |fname| argument. This gets updated later from an external table if necessary. @d bad_tfm=11 {go here if the \.{TFM} file is bad} @= @@; function read_font_info(fname:str_number):font_number; label bad_tfm,done; var @!file_opened:boolean; {has |tfm_infile| been opened?} @!n:font_number; {the number to return} @!lf,@!lh,@!bc,@!ec,@!nw,@!nh,@!nd:halfword; {subfile size parameters} @!whd_size:integer; {words needed for heights, widths, and depths} @!i,@!ii:0..font_mem_size; {|font_info| indices} @!jj:0..font_mem_size; {counts bytes to be ignored} @!z:scaled; {used to compute the design size} @!d:fraction; {height, width, or depth as a fraction of design size times $2^{-8}$} @!h_and_d:eight_bits; {height and depth indices being unpacked} begin n:=null_font; @; @; bad_tfm:@; done:if file_opened then b_close(tfm_infile); if n<>null_font then begin font_ps_name[n]:=fname; font_name[n]:=fname; str_ref[fname]:=max_str_ref; end; read_font_info:=n; end; @ \MP\ doesn't bother to check the entire \.{TFM} file for errors or explain precisely what is wrong if it does find a problem. Programs called \.{TFtoPL} @.TFtoPL@> @.PLtoTF@> and \.{PLtoTF} can be used to debug \.{TFM} files. @= print_err("Font "); print(fname); if file_opened then print(" not usable: TFM file is bad") else print(" not usable: TFM file not found"); help3("I wasn't able to read the size data for this font so this")@/ ("`infont' operation won't produce anything. If the font name")@/ ("is right, you might ask an expert to make a TFM file"); if file_opened then help_line[0]:="is right, try asking an expert to fix the TFM file"; error @ @= @; @; @; @ @ A bad \.{TFM} file can be shorter than it claims to be. The code given here might try to read past the end of the file if this happens. Changes will be needed if it causes a system error to refer to |tfm_infile^| or call |get_tfm_infile| when |eof(tfm_infile)| is true. For example, the definition @^system dependencies@> of |tfget| could be changed to ``|begin get(tfm_infile); if eof(tfm_infile) then goto bad_tfm; end|.'' @d tfget==get(tfm_infile) @d tfbyte==tfm_infile^ @d read_two(#)==begin #:=tfbyte; if #>127 then goto bad_tfm; tfget; #:=#*@'400+tfbyte; end @d tf_ignore(#)==for jj:=# downto 1 do tfget @= read_two(lf); tfget; read_two(lh); tfget; read_two(bc); tfget; read_two(ec); if (bc>1+ec)or(ec>255) then goto bad_tfm; tfget; read_two(nw); tfget; read_two(nh); tfget; read_two(nd); whd_size:=(ec+1-bc)+nw+nh+nd; if lf<6+lh+whd_size then goto bad_tfm; tf_ignore(10) @ Offsets are added to |char_base[n]| and |width_base[n]| so that is not necessary to apply the |so| and |qo| macros when looking up the width of a character in the string pool. In order to ensure nonnegative |char_base| values when |bc>0|, it may be necessary to reserve a few unused |font_info| elements. @= if next_fmem=font_mem_size) then @; incr(last_fnum); n:=last_fnum; font_bc[n]:=bc; font_ec[n]:=ec; char_base[n]:=next_fmem-bc-min_pool_ASCII; width_base[n]:=next_fmem+ec-bc+1-min_quarterword; height_base[n]:=width_base[n]+min_quarterword+nw; depth_base[n]:=height_base[n]+nh; next_fmem:=next_fmem+whd_size; @ @= begin print_err("Font "); print(fname); print(" not usable: Not enough space"); help3("This `infont' operation won't produce anything because I")@/ ("don't have enough room to store the character-size data for")@/ ("the font. You may have to ask a wizard to enlarge me."); error; goto done; end @ @= if lh<2 then goto bad_tfm; tf_ignore(4); tfget; read_two(z); tfget; z:=z*@'400+tfbyte; tfget; z:=z*@'400+tfbyte; {now |z| is 16 times the design size} font_dsize[n]:=take_fraction(z,267432584); {times ${72\over72.27}2^{28}$ to convert from \TeX\ points} tf_ignore(4*(lh-2)) @ @= ii:=width_base[n]+min_quarterword; i:=char_base[n]+min_pool_ASCII+bc; while i; if eof(tfm_infile) then goto bad_tfm; goto done @ The raw dimension read into |d| should have magnitude at most $2^{24}$ when interpreted as an integer, and this includes a scale factor of $2^{20}$. Thus we can multiply it by sixteen and think of it as a |fraction| that has been divided by sixteen. This cancels the extra scale factor contained in |font_dsize[n|. @= begin tfget; d:=tfbyte; if d>=@'200 then d:=d-@'400; tfget; d:=d*@'400+tfbyte;@/ tfget; d:=d*@'400+tfbyte;@/ tfget; d:=d*@'400+tfbyte;@/ font_info[i].sc:=take_fraction(d*16,font_dsize[n]); incr(i); end @ @= file_opened:=false; str_scan_file(fname); if cur_area="" then cur_area:=MP_font_area; if cur_ext="" then cur_ext:=".tfm"; pack_cur_name; if not b_open_in(tfm_infile) then goto bad_tfm; file_opened:=true @ When we have a font name and we don't know whether it has been loaded yet, we scan the |font_name| array before calling |read_font_info|. @= function find_font(@!f:str_number):font_number; label exit,found; var @!n:font_number; begin for n:=0 to last_fnum do if str_vs_str(f,font_name[n])=0 then goto found; find_font:=read_font_info(f); return; found:find_font:=n; exit:end; @ One simple application of |find_font| is the implementation of the |font_size| operator that gets the design size for a given font name. @= flush_cur_exp((font_dsize[find_font(cur_exp)]+8) div 16) @ If we discover that the font doesn't have a requested character, we omit it from the bounding box computation and expect the \ps\ interpreter to drop it. This routine issues a warning message if the user has asked for it. @= procedure lost_warning(@!f:font_number;@!k:pool_pointer); begin if internal[tracing_lost_chars]>0 then begin begin_diagnostic; print_nl("Missing character: There is no "); @.Missing character@> print(so(str_pool[k])); print(" in font "); print(font_name[f]); print_char("!"); end_diagnostic(false); end; end; @ The whole purpose of saving the height, width, and depth information is to be able to find the bounding box of an item of text in an edge structure. The |set_text_box| procedure takes a text node and adds this information. @= procedure set_text_box(@!p:pointer); var @!f:font_number; {|font_n(p)|} @!bc,@!ec:pool_ASCII_code; {range of valid characters for font |f|} @!k,kk:pool_pointer; {current character and character to stop at} @!cc:four_quarters; {the |char_info| for the current character} @!h,@!d:scaled; {dimensions of the current character} begin width_val(p):=0; height_val(p):=-el_gordo; depth_val(p):=-el_gordo;@/ f:=font_n(p); bc:=si(font_bc[f]); ec:=si(font_ec[f]);@/ kk:=str_stop(text_p(p)); k:=str_start[text_p(p)]; while k; @; end; @ @= begin if (str_pool[k]ec) then lost_warning(f,k) else begin cc:=char_info(f)(str_pool[k]); if not ichar_exists(cc) then lost_warning(f,k) else begin width_val(p):=width_val(p)+char_width(f)(cc); h:=char_height(f)(cc); d:=char_depth(f)(cc); if h>height_val(p) then height_val(p):=h; if d>depth_val(p) then depth_val(p):=d; end; end; incr(k); end @ Let's hope modern compilers do comparisons correctly when the difference would overflow. @= if height_val(p)<-depth_val(p) then begin height_val(p):=0; depth_val(p):=0; end @ The file |ps_tab_file| gives a table of \TeX\ font names and corresponding PostScript names for fonts that do not have to be downloaded, i.e., fonts that can be used when |internal[prologues]>0|. Each line consists of a \TeX\ name, one or more spaces, a PostScript name, and possibly a space and some other junk. This routine reads the table, updates |font_ps_name| entries starting after |last_ps_fnum|, and sets |last_ps_fnum:=last_fnum|. If the file |ps_tab_file| is missing, we assume that the existing font names are OK and nothing needs to be done. @= procedure read_psname_table; label common_ending, done; var @!k:font_number; {font for possible name match} @!lmax:integer; {upper limit on length of name to match} @!j:integer; {characters left to read before string gets too long} @!c:text_char; {character being read from |ps_tab_file|} @!s:str_number; {possible font name to match} begin name_of_file:=ps_tab_name; if a_open_in(ps_tab_file) then begin @; while not eof(ps_tab_file) do begin @; for k:=last_ps_fnum+1 to last_fnum do if str_vs_str(s,font_name[k])=0 then @<|flush_string(s)|, read in |font_ps_name[k]|, and |goto common_ending|@>; flush_string(s); common_ending:read_ln(ps_tab_file); end; last_ps_fnum:=last_fnum; a_close(ps_tab_file); end; end; @ @= @!ps_tab_file:alpha_file; {file for font name translation table} @ @= lmax:=0; for k:=last_ps_fnum+1 to last_fnum do if length(font_name[k])>lmax then lmax:=length(font_name[k]) @ @= str_room(lmax); j:=lmax; loop @+begin if eoln(ps_tab_file) then fatal_error("The psfont map file is bad!"); read(ps_tab_file,c); if c=' ' then goto done; decr(j); if j>=0 then append_char(xord[c]) else begin flush_cur_string; goto common_ending; end; end; done:s:=make_string @ PostScript font names should be at most 28 characters long but we allow 32 just to be safe. @<|flush_string(s)|, read in |font_ps_name[k]|, and...@>= begin flush_string(s); j:=32; str_room(j); repeat if eoln(ps_tab_file) then fatal_error("The psfont map file is bad!"); read(ps_tab_file,c); until c<>' '; repeat decr(j); if j<0 then fatal_error("The psfont map file is bad!"); append_char(xord[c]); if eoln(ps_tab_file) then c:=' ' @+else read(ps_tab_file,c); until c=' '; delete_str_ref(font_ps_name[k]); font_ps_name[k]:=make_string; goto common_ending; end @* \[44] Shipping pictures out. The |ship_out| procedure, to be described below, is given a pointer to an edge structure. Its mission is to output a file containing the \ps\ description of an edge structure. @ Each time an edge structure is shipped out we write a new \ps\ output file named according to the current \&{charcode}. @:char_code_}{\&{charcode} primitive@> @= procedure open_output_file; var @!c:integer; {\&{charcode} rounded to the nearest integer} @!old_setting:0..max_selector; {previous |selector| setting} @!s:str_number; {a file extension derived from |c|} begin if job_name=0 then open_log_file; c:=round_unscaled(internal[char_code]); if c<0 then s:=".ps" else @; pack_job_name(s); while not a_open_out(ps_file) do prompt_file_name("file name for output",s); delete_str_ref(s); @; @; end; @ The file extension created here could be up to five characters long in extreme cases so it may have to be shortened on some systems. @^system dependencies@> @= begin old_setting:=selector; selector:=new_string; print_char("."); print_int(c); s:=make_string; selector:=old_setting; end @ The user won't want to see all the output file names so we only save the first and last ones and a count of how many there were. For this purpose files are ordered primarily by \&{charcode} and secondarily by order of creation. @:char_code_}{\&{charcode} primitive@> @= if (c=0) then begin first_output_code:=c; delete_str_ref(first_file_name); first_file_name:=a_make_name_string(ps_file); end; if c>=last_output_code then begin last_output_code:=c; delete_str_ref(last_file_name); last_file_name:=a_make_name_string(ps_file); end @ @= @!first_file_name,@!last_file_name:str_number; {full file names} @!first_output_code,@!last_output_code:integer; {rounded \&{charcode} values} @:char_code_}{\&{charcode} primitive@> @!total_shipped:integer; {total number of |ship_out| operations completed} @ @= first_file_name:=""; last_file_name:="";@/ first_output_code:=32768; last_output_code:=-32768;@/ total_shipped:=0; @ @= if term_offset>max_print_line-6 then print_ln else if (term_offset>0)or(file_offset>0) then print_char(" "); print_char("["); if c>=0 then print_int(c) @ @= print_char("]"); update_terminal; incr(total_shipped) @ @= if total_shipped>0 then begin print_nl(""); print_int(total_shipped); print(" output file"); if total_shipped>1 then print_char("s"); print(" written: "); print(first_file_name); if total_shipped>1 then begin if 31+length(first_file_name)+length(last_file_name)>@| max_print_line then print_ln; print(" .. "); print(last_file_name); end; end @ We often need to print a pair of coordinates. @d ps_room(#)==if ps_offset+#>max_print_line then print_ln {optional line break} @= procedure ps_pair_out(@!x,@!y:scaled); begin ps_room(26); print_scaled(x); print_char(" "); print_scaled(y); print_char(" ") end; @ @= procedure ps_print(@!s:str_number); begin ps_room(length(s)); print(s); end; @ The most important output procedure is the one that gives the \ps\ version of a \MP\ path. @= procedure ps_path_out(@!h:pointer); label exit; var @!p,@!q:pointer; {for scanning the path} @!d:scaled; {a temporary value} @!curved:boolean; {|true| unless the cubic is almost straight} begin ps_room(40); if need_newpath then print("newpath "); need_newpath:=true; ps_pair_out(x_coord(h),y_coord(h)); print("moveto");@/ p:=h; repeat if right_type(p)=endpoint then begin if p=h then ps_print(" 0 0 rlineto"); return; end; q:=link(p); @; p:=q; until p=h; ps_print(" closepath"); exit:end; @ @= need_newpath:boolean; {will |ps_path_out| need to issue a \&{newpath} command next time} @:newpath_}{\&{newpath} command@> @ @= curved:=true; @; print_ln; if curved then begin ps_pair_out(right_x(p),right_y(p)); ps_pair_out(left_x(q),left_y(q)); ps_pair_out(x_coord(q),y_coord(q)); ps_print("curveto"); end else if q<>h then begin ps_pair_out(x_coord(q),y_coord(q)); ps_print("lineto"); end @ Two types of straight lines come up often in \MP\ paths: cubics with zero initial and final velocity as created by |make_path| or |make_envelope|, and cubics with control points uniformly spaced on a line as created by |make_choices|. @d bend_tolerance=131 {allow rounding error of $2\cdot10^{-3}$} @= if right_x(p)=x_coord(p) then if right_y(p)=y_coord(p) then if left_x(q)=x_coord(q) then if left_y(q)=y_coord(q) then curved:=false; d:=left_x(q)-right_x(p); if abs(right_x(p)-x_coord(p)-d)<=bend_tolerance then if abs(x_coord(q)-left_x(q)-d)<=bend_tolerance then begin d:=left_y(q)-right_y(p); if abs(right_y(p)-y_coord(p)-d)<=bend_tolerance then if abs(y_coord(q)-left_y(q)-d)<=bend_tolerance then curved:=false; end @ We need to keep track of several parameters from the \ps\ graphics state. @^graphics state@> This allows us to be sure that \ps\ has the correct values when they are needed without wasting time and space setting them unnecessarily. @= @!gs_red,@!gs_green,@!gs_blue:scaled; {color from the last \&{setrgbcolor} or \&{setgray} command} @:setrgbcolor}{\&{setrgbcolor} command@> @:setgray}{\&{setgray} command@> @!gs_ljoin,@!gs_lcap:quarterword; {values from the last \&{setlinejoin} and \&{setlinecap} commands} @:setlinejoin}{\&{setlinejoin} command@> @:setlinecap}{\&{setlinecap} command@> @!gs_miterlim:scaled; {the value from the last \&{setmiterlimit} command} @:setmiterlimit}{\&{setmiterlimit} command@> @!gs_dash_p:pointer; {edge structure for last \&{setdash} command} @:setdash}{\&{setdash} command@> @!gs_dash_sc:scaled; {scale factor used with |gs_dash_p|} @!gs_width:scaled; {width setting or $-1$ if no \&{setlinewidth} command so far} @!gs_adj_wx:boolean; {what resolution-dependent adjustment applies to the width} @:setlinewidth}{\&{setlinewidth} command@> @ To avoid making undue assumptions about the initial graphics state, these parameters are given special values that are guaranteed not to match anything in the edge structure being shipped out. On the other hand, the initial color should be black so that the translation of an all-black picture will have no \&{setcolor} commands. (These would be undesirable in a font application.) Hence we use |c=0| when initializing the graphics state and we use |c<0| to recover from a situation where we have lost track of the graphics state. @= procedure unknown_graphics_state(c:scaled); begin gs_red:=c; gs_green:=c; gs_blue:=c;@/ gs_ljoin:=3; gs_lcap:=3; gs_miterlim:=0;@/ gs_dash_p:=void; gs_dash_sc:=0; gs_width:=-1; end; @ When it is time to output a graphical object, |fix_graphics_state| ensures that \ps's idea of the graphics state agrees with what is stored in the object. @= @@; procedure fix_graphics_state(p:pointer); {get ready to output graphical object |p|} var @!hh,@!pp:pointer; {for list manipulation} @!wx,@!wy,@!ww:scaled; {dimensions of pen bounding box} @!adj_wx:boolean; {whether pixel rounding should be based on |wx| or |wy|} @!tx,@!ty:integer; {temporaries for computing |adj_wx|} @!scf:scaled; {a scale factor for the dash pattern} begin if has_color(p) then @; if (type(p)=fill_code)or(type(p)=stroked_code) then if pen_p(p)<>null then if pen_is_elliptical(pen_p(p)) then begin @; @; @; @; end; if ps_offset>0 then print_ln; end; @ @= if type(p)=stroked_code then if (left_type(path_p(p))=endpoint)or(dash_p(p)<>null) then if gs_lcap<>lcap_val(p) then begin ps_room(13); print_char(" "); print_char("0"+lcap_val(p)); print(" setlinecap"); gs_lcap:=lcap_val(p); end @ @= if gs_ljoin<>ljoin_val(p) then begin ps_room(14); print_char(" "); print_char("0"+ljoin_val(p)); print(" setlinejoin"); gs_ljoin:=ljoin_val(p); end; if gs_miterlim<>miterlim_val(p) then begin ps_room(27); print_char(" "); print_scaled(miterlim_val(p)); print(" setmiterlimit"); gs_miterlim:=miterlim_val(p); end @ @= if (gs_red<>red_val(p))or(gs_green<>green_val(p))or@| (gs_blue<>blue_val(p)) then begin gs_red:=red_val(p); gs_green:=green_val(p); gs_blue:=blue_val(p);@/ if (gs_red=gs_green)and(gs_green=gs_blue) then begin ps_room(16); print_char(" "); print_scaled(gs_red); print(" setgray"); end else begin ps_room(36); print_char(" "); print_scaled(gs_red); print_char(" "); print_scaled(gs_green); print_char(" "); print_scaled(gs_blue); print(" setrgbcolor"); end; end; @ In order to get consistent widths for horizontal and vertical pen strokes, we want \ps\ to use an integer number of pixels for the \&{setwidth} parameter. @:setwidth}{\&{setwidth}command@> We set |gs_width| to the ideal horizontal or vertical stroke width and then generate \ps\ code that computes the rounded value. For non-circular pens, the pen shape will be rescaled so that horizontal or vertical parts of the stroke have the computed width. Rounding the width to whole pixels is not likely to improve the appearance of diagonal or curved strokes, but we do it anyway for consistency. The \&{truncate} command generated here tends to make all the strokes a little @:truncate}{\&{truncate} command@> thinner, but this is appropriate for \ps's scan-conversion rules. Even with truncation, an ideal with of $w$~pixels gets mapped into $\lfloor w\rfloor+1$. It would be better to have $\lceil w\rceil$ but that is ridiculously expensive to compute in \ps. @= @; @; if (ww<>gs_width) or (adj_wx<>gs_adj_wx) then begin if adj_wx then begin ps_room(13); print_char(" "); print_scaled(ww); ps_print(" 0 dtransform exch truncate exch idtransform pop setlinewidth"); end else begin ps_room(15); print(" 0 "); print_scaled(ww); ps_print(" dtransform truncate idtransform setlinewidth pop"); end; gs_width := ww; gs_adj_wx := adj_wx; end @ @= pp:=pen_p(p); if (right_x(pp)=x_coord(pp)) and (left_y(pp)=y_coord(pp)) then begin wx := abs(left_x(pp) - x_coord(pp)); wy := abs(right_y(pp) - y_coord(pp)); end else begin wx := pyth_add(left_x(pp)-x_coord(pp), right_x(pp)-x_coord(pp)); wy := pyth_add(left_y(pp)-y_coord(pp), right_y(pp)-y_coord(pp)); end @ The path is considered ``essentially horizontal'' if its range of $y$~coordinates is less than the $y$~range |wy| for the pen. ``Essentially vertical'' paths are detected similarly. This code ensures that no component of the pen transformation is more that |aspect_bound*(ww+1)|. @d aspect_bound=10 {``less important'' of |wx|, |wy| cannot exceed the other by more than this factor} @= tx:=1; ty:=1; if coord_rangeOK(path_p(p), y_loc(0), wy) then tx:=aspect_bound else if coord_rangeOK(path_p(p), x_loc(0), wx) then ty:=aspect_bound; if wy div ty>=wx div tx then begin ww:=wy; adj_wx:=false; end else begin ww:=wx; adj_wx:=true; end @ This routine quickly tests if path |h| is ``essentially horizontal'' or ``essentially vertical,'' where |zoff| is |x_loc(0)| or |y_loc(0)| and |dz| is allowable range for $x$ or~$y$. We do not need and cannot afford a full bounding-box computation. @= function coord_rangeOK(@!h:pointer; @!zoff:small_number; dz:scaled):boolean; label found, not_found, exit; var @!p:pointer; {for scanning the path form |h|} @!zlo,@!zhi:scaled; {coordinate range so far} @!z:scaled; {coordinate currently being tested} begin zlo:=knot_coord(h+zoff); zhi:=zlo; p:=h; while right_type(p)<>endpoint do begin z:=right_coord(p+zoff);@/ @dz|@>; p:=link(p); z:=left_coord(p+zoff);@/ @dz|@>; z:=knot_coord(p+zoff);@/ @dz|@>; if p=h then goto not_found; end; not_found:coord_rangeOK:=true; return; found:coord_rangeOK:=false; exit:end; @ @dz|@>= if zzhi then zhi:=z; if zhi-zlo>dz then goto found @ Filling with an elliptical pen is implemented via a combination of \&{stroke} and \&{fill} commands and a nontrivial dash pattern would interfere with this. @:stroke}{\&{stroke} command@> @:fill}{\&{fill} command@> Note that we don't use |delete_edge_ref| because |gs_dash_p| is not counted as a reference. @= if type(p)=fill_code then hh:=null else begin hh:=dash_p(p); scf:=get_pen_scale(pen_p(p)); if scf=0 then if gs_width=0 then scf:=dash_scale(p) @+else hh:=null else begin scf:=make_scaled(gs_width,scf); scf:=take_scaled(scf,dash_scale(p)); end; end; if hh=null then begin if gs_dash_p<>null then begin ps_print(" [] 0 setdash"); gs_dash_p:=null; end; end else if (gs_dash_sc<>scf) or not same_dashes(gs_dash_p,hh) then @ @ Translating a dash list into \ps\ is very similar to printing it symbolically in |print_edges|. A dash pattern with |dash_y(hh)=0| has length zero and is ignored. The same fate applies in the bizarre case of a dash pattern that cannot be printed without overflow. @= begin gs_dash_p:=hh; gs_dash_sc:=scf; if (dash_y(hh)=0) or (abs(dash_y(hh)) div unity >= el_gordo div scf) then ps_print(" [] 0 setdash") else begin pp:=dash_list(hh); start_x(null_dash):=start_x(pp)+dash_y(hh);@/ ps_room(28); print(" ["); while pp<>null_dash do begin ps_pair_out(take_scaled(stop_x(pp)-start_x(pp),scf),@| take_scaled(start_x(link(pp))-stop_x(pp),scf)); pp:=link(pp); end; ps_room(22); print("] "); print_scaled(take_scaled(dash_offset(hh),scf)); print(" setdash"); end; end @ @= function same_dashes(@!h,@!hh:pointer):boolean; {do |h| and |hh| represent the same dash pattern?} label done; var @!p,@!pp:pointer; {dash nodes being compared} begin if h=hh then same_dashes:=true else if (h<=void)or(hh<=void) then same_dashes:=false else if dash_y(h)<>dash_y(hh) then same_dashes:=false else @; end; @ @= begin p:=dash_list(h); pp:=dash_list(hh); while (p<>null_dash)and(pp<>null_dash) do if (start_x(p)<>start_x(pp))or(stop_x(p)<>stop_x(pp)) then goto done else begin p:=link(p); pp:=link(pp); end; done:same_dashes:=p=pp; end @ When stroking a path with an elliptical pen, it is necessary to transform the coordinate system so that a unit circular pen will have the desired shape. To keep this transformation local, we enclose it in a $$\&{gsave}\ldots\&{grestore}$$ block. Any translation component must be applied to the path being stroked while the rest of the transformation must apply only to the pen. If |fill_also=true|, the path is to be filled as well as stroked so we must insert commands to do this after giving the path. @= procedure stroke_ellipse(@!h:pointer;@!fill_also:boolean); {generate an elliptical pen stroke from object |h|} var @!txx,@!txy,@!tyx,@!tyy:scaled; {transformation parameters} @!p:pointer; {the pen to stroke with} @!d1,@!det:scaled; {for tweaking transformation parameters} @!s:integer; {also for tweaking transformation paramters} @!transformed:boolean; {keeps track of whether gsave/grestore are needed} begin transformed:=false;@/ @; @; ps_path_out(path_p(h));@/ if fill_also then print_nl("gsave fill grestore"); @; ps_print(" stroke"); if transformed then ps_print(" grestore"); print_ln; end; @ @= p:=pen_p(h); txx:=left_x(p); tyx:=left_y(p);@/ txy:=right_x(p); tyy:=right_y(p); if (x_coord(p)<>0)or(y_coord(p)<>0) then begin print_nl("gsave "); ps_pair_out(x_coord(p),y_coord(p)); ps_print("translate ");@/ txx:=txx-x_coord(p); tyx:=tyx-y_coord(p);@/ txy:=txy-x_coord(p); tyy:=tyy-y_coord(p); transformed:=true; end else print_nl(""); @ @ @= if gs_width<>unity then if gs_width=0 then begin txx:=unity; tyy:=unity; end else begin txx:=make_scaled(txx,gs_width); txy:=make_scaled(txy,gs_width); tyx:=make_scaled(tyx,gs_width); tyy:=make_scaled(tyy,gs_width); end; if (txy<>0)or(tyx<>0)or(txx<>unity)or(tyy<>unity) then if (not transformed) then begin ps_print("gsave "); transformed:=true; end @ @= if (txy<>0)or(tyx<>0) then begin print_ln; print_char("["); ps_pair_out(txx,tyx); ps_pair_out(txy,tyy);@/ ps_print("0 0] concat"); end else if (txx<>unity)or(tyy<>unity) then begin print_ln; ps_pair_out(txx,tyy); print("scale"); end @ The \ps\ interpreter will probably abort if it encounters a singular transformation matrix. The determinant must be large enough to ensure that the printed representation will be nonsingular. Since the printed representation is always within $2^{-17}$ of the internal |scaled| value, the total error is at most $4T_{\rm max}2^{-17}$, where $T_{\rm max}$ is a bound on the magnitudes of |txx/65536|, |txy/65536|, etc. The |aspect_bound*(gs_width+1)| bound on the components of the pen transformation allows $T_{\rm max}$ to be at most |2*aspect_bound|. @= det:=take_scaled(txx,tyy) - take_scaled(txy,tyx); d1:=4*aspect_bound+1; if abs(det)=0 then begin d1:=d1-det; s:=1; @+end else begin d1:=-d1-det; s:=-1; @+end; d1:=d1*unity; if abs(txx)+abs(tyy)>=abs(txy)+abs(tyy) then if abs(txx)>abs(tyy) then tyy:=tyy+(d1+s*abs(txx)) div txx else txx:=txx+(d1+s*abs(tyy)) div tyy else if abs(txy)>abs(tyx) then tyx:=tyx+(d1+s*abs(txy)) div txy else txy:=txy+(d1+s*abs(tyx)) div tyx; end @ Here is a simple routine that just fills a cycle. @= procedure ps_fill_out(@!p:pointer); {fill cyclic path~|p|} begin ps_path_out(p); ps_print(" fill"); print_ln; end; @ Given a cyclic path~|p| and a graphical object~|h|, the |do_outer_envelope| procedure fills the cycle generated by |make_envelope|. It need not do anything unless some region has positive winding number with respect to~|p|, but it does not seem worthwhile to for test this. @= procedure do_outer_envelope(@!p,@!h:pointer); begin p:=make_envelope(p, pen_p(h), ljoin_val(h), 0, miterlim_val(h)); ps_fill_out(p); toss_knot_list(p); end; @ A text node may specify an arbitrary transformation but the usual case involves only shifting, scaling, and occasionally rotation. The purpose of |choose_scale| is to select a scale factor so that the remaining transformation is as ``nice'' as possible. The definition of ``nice'' is somewhat arbitrary but shifting and $90^\circ$ rotation are especially nice because they work out well for bitmap fonts. The code here selects a scale factor equal to $1/\sqrt2$ times the Frobenius norm of the non-shifting part of the transformation matrix. It is careful to avoid additions that might cause undetected overflow. @= function choose_scale(@!p:pointer):scaled; {|p| should point to a text node} var @!a,@!b,@!c,@!d,@!ad,@!bc:scaled; {temporary values} begin a:=txx_val(p); b:=txy_val(p); c:=tyx_val(p); d:=tyy_val(p);@/ if (a<0) then negate(a); if (b<0) then negate(b); if (c<0) then negate(c); if (d<0) then negate(d); ad:=half(a-d); bc:=half(b-c);@/ choose_scale:=pyth_add(pyth_add(d+ad,ad), pyth_add(c+bc,bc)); end; @ @= procedure ps_string_out(s:str_number); var @!i:pool_pointer; {current character code position} @!k:ASCII_code; {bits to be converted to octal} begin print("("); i:=str_start[s]; while imax_print_line then begin print_char("\"); print_ln; end; k:=so(str_pool[i]); if (@) then begin print_char("\"); print_char("0"+(k div 64)); print_char("0"+((k div 8) mod 8)); print_char("0"+(k mod 8)); end else begin if (k="(")or(k=")")or(k="\") then print_char("\"); print_char(k); end; incr(i); end; print(")"); end; @ @= function is_ps_name(@!s:str_number):boolean; label not_found,exit; var @!i:pool_pointer; {current character code position} @!k:ASCII_code; {the character being checked} begin i:=str_start[s]; while i"~") then goto not_found; if (k="(")or(k=")")or(k="<")or(k=">")or@| (k="{")or(k="}")or(k="/")or(k="%") then goto not_found; incr(i); end; is_ps_name:=true; return; not_found:is_ps_name:=false; exit:end; @ @= procedure ps_name_out(@!s:str_number;@!lit:boolean); begin ps_room(length(s)+2); print_char(" "); if is_ps_name(s) then begin if lit then print_char("/"); print(s); end else begin ps_string_out(s); if not lit then ps_print("cvx "); ps_print("cvn"); end; end; @ We also need to keep track of which characters are used in text nodes in the edge structure that is being shipped out. This is done by procedures that use the left-over |b3| field in the |char_info| words; i.e., |char_info(f)(c).b3| gives the status of character |c| in font |f|. @d unused=0 @d used=1 @ @= procedure unmark_font(@!f:font_number); var @!k:0..font_mem_size; {an index into |font_info|} begin for k:= char_base[f]+si(font_bc[f]) to char_base[f]+si(font_ec[f]) do font_info[k].qqqq.b3:=unused; end; @ @= procedure mark_string_chars(@!f:font_number;@!s:str_number); var @!b:integer; {|char_base[f]|} @!bc,@!ec:pool_ASCII_code; {only characters between these bounds are marked} @!k:pool_pointer; {an index into string |s|} begin b:=char_base[f]; bc:=si(font_bc[f]); ec:=si(font_ec[f]);@/ k:=str_stop(s); while k>str_start[s] do begin decr(k); if (str_pool[k]>=bc)and(str_pool[k]<=ec) then font_info[b+str_pool[k]].qqqq.b3:=used; end end; @ @= procedure hex_digit_out(@!d:small_number); begin if d<10 then print_char(d+"0") else print_char(d+"a"-10); end; @ We output the marks as a hexadecimal bit string starting at |c| or |font_bc[f]|, whichever is greater. If the output has to be truncated to avoid exceeding |emergency_line_length| the return value says where to start scanning next time. @= function ps_marks_out(@!f:font_number;@!c:eight_bits):halfword; var @!bc,@!ec:eight_bits; {only encode characters between these bounds} @!lim:integer; {the maximum number of marks to encode before truncating} @!p:0..font_mem_size; {|font_info| index for the current character} @!d,@!b:0..15; {used to construct a hexadecimal digit} begin lim:=4*(emergency_line_length-ps_offset-4); bc:=font_bc[f]; ec:=font_ec[f]; if c>bc then bc:=c; @; @; @; while (ec= p:=char_base[f]+si(bc); while (font_info[p].qqqq.b3=unused)and(bc=bc+lim then ec:=bc+lim-1; p:=char_base[f]+si(ec); while (font_info[p].qqqq.b3=unused)and(bc= print_char(" "); hex_digit_out(bc div 16); hex_digit_out(bc mod 16); print_char(":") @ @= b:=8; d:=0; for p:=char_base[f]+si(bc) to char_base[f]+si(ec) do begin if b=0 then begin hex_digit_out(d); d:=0; b:=8; end; if font_info[p].qqqq.b3<>unused then d:=d+b; b:=halfp(b); end; hex_digit_out(d) @ Here is a simple function that determines whether there are any marked characters in font~|f| with character code at least~|c|. @= function check_ps_marks(@!f:font_number; @!c:integer):boolean; label exit; var @!p:0..font_mem_size; {|font_info| index for the current character} begin for p:=char_base[f]+si(c) to char_base[f]+si(font_ec[f]) do if font_info[p].qqqq.b3=used then begin check_ps_marks:=true; return; end; check_ps_marks:=false; exit: end; @ There may be many sizes of one font and we need to keep track of the characters used for each size. This is done by keeping a linked list of sizes for each font with a counter in each text node giving the appropriate position in the size list for its font. @d sc_factor(#)==mem[#+1].sc {the scale factor stored in a font size node} @d font_size_size=2 {size of a font size node} @= font_sizes:array[font_number] of pointer; @ @d fscale_tolerance==65 {that's $.001\times2^{16}$} @= function size_index(@!f:font_number;@!s:scaled):quarterword; label found; var @!p,@!q:pointer; {the previous and current font size nodes} @!i:quarterword; {the size index for |q|} begin q:=font_sizes[f]; i:=0; while q<>null do begin if abs(s-sc_factor(q))<=fscale_tolerance then goto found else begin p:=q; q:=link(q); incr(i); end; if i=max_quarterword then overflow("sizes per font",max_quarterword); @:MetaPost capacity exceeded sizes per font}{\quad sizes per font@> end; q:=get_node(font_size_size); sc_factor(q):=s; if i=0 then font_sizes[f]:=q @+else link(p):=q; found:size_index:=i; end; @ @= function indexed_size(@!f:font_number;@!j:quarterword):scaled; var @!p:pointer; {a font size node} @!i:quarterword; {the size index for |p|} begin p:=font_sizes[f]; i:=0; if p=null then confusion("size"); while (i<>j) do begin incr(i); p:=link(p); if p=null then confusion("size"); end; indexed_size:=sc_factor(p); end; @ @= procedure clear_sizes; var @!f:font_number; {the font whose size list is being cleared} @!p:pointer; {current font size nodes} begin for f:=null_font+1 to last_fnum do while font_sizes[f]<>null do begin p:=font_sizes[f]; font_sizes[f]:=link(p); free_node(p,font_size_size); end; end; @ The \&{special} command saves up lines of text to be printed during the next |ship_out| operation. The saved items are stored as a list of capsule tokens. @= @!last_pending:pointer; {the last token in a list of pending specials} @ @= last_pending:=spec_head; @ @= special_command:do_special; @ @= procedure do_special; begin get_x_next; scan_expression; if cur_type<>string_type then @ else begin link(last_pending):=stash_cur_exp; last_pending:=link(last_pending); link(last_pending):=null; end; end; @ @= begin exp_err("Unsuitable expression"); help1("Only known strings are allowed for output as specials."); put_get_error; end @ @= t:=link(spec_head); while t<>null do begin if length(value(t))<=emergency_line_length then print(value(t)) else overflow("output line length",emergency_line_length); @:MetaPost capacity exceeded output line length}{\quad output line length@> print_ln; t:=link(t); end; flush_token_list(link(spec_head)); link(spec_head):=null; last_pending:=spec_head @ We are now ready for the main output procedure. Note that the |selector| setting is saved in a global variable so that |begin_diagnostic| can access it. @= procedure ship_out(@!h:pointer); {output edge structure |h|} label done,found; var @!p:pointer; {the current graphical object} @!q:pointer; {something that |p| points to} @!t:integer; {a temporary value} @!f,ff:font_number; {fonts used in a text node or as loop counters} @!ldf:font_number; {the last \.{DocumentFont} listed (otherwise |null_font|)} @!done_fonts:boolean; {have we finished listing the fonts in the header?} @!next_size:quarterword; {the size index for fonts being listed} @!cur_fsize:array[font_number] of pointer; {current positions in |font_sizes|} @!ds,@!scf:scaled; {design size and scale factor for a text node} @!transformed:boolean; {is the coordinate system being transformed?} begin open_output_file; if (internal[prologues]>0) and (last_ps_fnum; if internal[prologues]>0 then @; print("%%EndProlog"); print_nl("%%Page: 1 1"); print_ln; @; unknown_graphics_state(0); need_newpath:=true; p:=link(dummy_loc(h)); while p<>null do begin fix_graphics_state(p); case type(p) of @@; start_bounds_code,stop_bounds_code: do_nothing; end; {all cases are enumerated} p:=link(p); end; print("showpage"); print_ln; print("%%EOF"); print_ln; a_close(ps_file); selector:=non_ps_setting; if internal[prologues]<=0 then clear_sizes; @; if internal[tracing_output]>0 then print_edges(h," (just shipped out)",true); end; @ These special comments described in the {\sl PostScript Language Reference Manual}, 2nd.~edition are understood by some \ps-reading programs. We can't normally output ``conforming'' \ps\ because the structuring conventions don't allow us to say ``Please make sure the following characters are downloaded and define the \.{fshow} macro to access them.'' The exact bounding box is written out if |prologues<0|, although this is not standard \ps, since it allows \TeX\ to calculate the box dimensions accurately. (Overfull boxes are avoided if an illustration is made to match a given \.{\char`\\hsize}.) @= print("%!PS"); if internal[prologues]>0 then print("-Adobe-3.0 EPSF-3.0"); print_nl("%%BoundingBox: "); set_bbox(h,true); if minx_val(h)>maxx_val(h) then print("0 0 0 0") else if internal[prologues]<0 then begin ps_pair_out(minx_val(h),miny_val(h)); ps_pair_out(maxx_val(h),maxy_val(h)); end else begin ps_pair_out(floor_scaled(minx_val(h)),floor_scaled(miny_val(h))); ps_pair_out(-floor_scaled(-maxx_val(h)),-floor_scaled(-maxy_val(h))); end; print_nl("%%Creator: MetaPost"); print_nl("%%CreationDate: "); print_int(round_unscaled(internal[year])); print_char("."); print_dd(round_unscaled(internal[month])); print_char("."); print_dd(round_unscaled(internal[day])); print_char(":");@/ t:=round_unscaled(internal[time]); print_dd(t div 60); print_dd(t mod 60);@/ print_nl("%%Pages: 1");@/ @; print_ln @ @= @; if internal[prologues]>0 then @ else begin next_size:=0; @; repeat done_fonts:=true; for f:=null_font+1 to last_fnum do begin if cur_fsize[f]<>null then @; if cur_fsize[f]<>null then begin unmark_font(f); done_fonts:=false; @+end; end; if not done_fonts then @; until done_fonts; end @ @= for f:=null_font+1 to last_fnum do cur_fsize[f]:=font_sizes[f] @ It's not a good idea to make any assumptions about the |font_ps_name| entries, so we carefully remove duplicates. There is no harm in using a slow, brute-force search. @= begin ldf:=null_font; for f:=null_font+1 to last_fnum do if font_sizes[f]<>null then begin if ldf=null_font then print_nl("%%DocumentFonts:"); for ff:=ldf downto null_font do if font_sizes[ff]<>null then if str_vs_str(font_ps_name[f],font_ps_name[ff])=0 then goto found; if ps_offset+1+length(font_ps_name[f])>max_print_line then print_nl("%%+"); print_char(" "); print(font_ps_name[f]); ldf:=f; found: end; end @ @= for f:=null_font+1 to last_fnum do font_sizes[f]:=null; p:=link(dummy_loc(h)); while p<>null do begin if type(p)=text_code then if font_n(p)<>null_font then begin f:=font_n(p); if internal[prologues]>0 then font_sizes[f]:=void else begin if font_sizes[f]=null then unmark_font(f); name_type(p):=size_index(f,choose_scale(p)); if name_type(p)=0 then mark_string_chars(f,text_p(p)); end; end; p:=link(p); end @ If the file name is so long that it can't be printed without exceeding |emergency_line_length| then there will be missing items in the \.{\%*Font:} line. We might have to repeat line in order to get the character usage information to fit within |emergency_line_length|. @= begin t:=0; while check_ps_marks(f,t) do begin print_nl("%*Font: "); if ps_offset+length(font_name[f])+12>emergency_line_length then goto done; print(font_name[f]); print_char(" "); ds:=(font_dsize[f] + 8) div 16; print_scaled(take_scaled(ds,sc_factor(cur_fsize[f]))); if ps_offset+12>emergency_line_length then goto done; print_char(" "); print_scaled(ds); if ps_offset+5>emergency_line_length then goto done; t:=ps_marks_out(f,t); end; done: cur_fsize[f]:=link(cur_fsize[f]); end @ @= begin incr(next_size); p:=link(dummy_loc(h)); while p<>null do begin if type(p)=text_code then if font_n(p)<>null_font then if name_type(p)=next_size then mark_string_chars(font_n(p),text_p(p)); p:=link(p); end; end @ The prologue defines \.{fshow} and corrects for the fact that \.{fshow} arguments use |font_name| instead of |font_ps_name|. Downloaded bitmap fonts might not have reasonable |font_ps_name| entries, but we just charge ahead anyway. The user should not make \&{prologues} positive if this will cause trouble. @:prologues_}{\&{prologues} primitive@> @= begin if ldf<>null_font then begin for f:=null_font+1 to last_fnum do if font_sizes[f]<>null then begin ps_name_out(font_name[f],true); ps_name_out(font_ps_name[f],true); ps_print(" def"); print_ln; end; print("/fshow {exch findfont exch scalefont setfont show}bind def"); print_ln; end; end @ Since we do not have a stack for the graphics state, it is considered completely unknown after the \.{grestore} from a stop clip object. Procedure |unknown_graphics_state| needs a negative argument in this case. @= start_clip_code:begin print_nl("gsave "); ps_path_out(path_p(p)); ps_print(" clip"); print_ln; end; stop_clip_code:begin print_nl("grestore"); print_ln; unknown_graphics_state(-1); end; @ @= fill_code: if pen_p(p)=null then ps_fill_out(path_p(p)) else if pen_is_elliptical(pen_p(p)) then stroke_ellipse(p,true) else begin do_outer_envelope(copy_path(path_p(p)), p); do_outer_envelope(htap_ypoc(path_p(p)), p); end; stroked_code: if pen_is_elliptical(pen_p(p)) then stroke_ellipse(p,false) else begin q:=copy_path(path_p(p)); t:=lcap_val(p); @; q:=make_envelope(q,pen_p(p),ljoin_val(p),t,miterlim_val(p)); ps_fill_out(q); toss_knot_list(q); end; @ The envelope of a cyclic path~|q| could be computed by calling |make_envelope| once for |q| and once for its reversal. We don't do this because it would fail color regions that are covered by the pen regardless of where it is placed on~|q|. @= if left_type(q)<>endpoint then begin left_type(insert_knot(q,x_coord(q),y_coord(q))):=endpoint; right_type(q):=endpoint; q:=link(q); t:=1; end @ @= text_code: if (font_n(p)<>null_font) and (length(text_p(p))>0) then begin if internal[prologues]>0 then scf:=choose_scale(p) else scf:=indexed_size(font_n(p), name_type(p)); @; ps_string_out(text_p(p)); ps_name_out(font_name[font_n(p)],false); @; print_ln; end; @ @= ps_room(18); print_char(" "); ds:=(font_dsize[font_n(p)]+8) div 16; print_scaled(take_scaled(ds,scf)); print(" fshow"); if transformed then ps_print(" grestore") @ @= transformed:=(txx_val(p)<>scf)or(tyy_val(p)<>scf)or@| (txy_val(p)<>0)or(tyx_val(p)<>0); if transformed then begin print("gsave ["); ps_pair_out(make_scaled(txx_val(p),scf),@|make_scaled(tyx_val(p),scf)); ps_pair_out(make_scaled(txy_val(p),scf),@|make_scaled(tyy_val(p),scf)); ps_pair_out(tx_val(p),ty_val(p));@/ ps_print("] concat 0 0 moveto"); end else begin ps_pair_out(tx_val(p),ty_val(p)); ps_print("moveto"); end; print_ln @ Now that we've finished |ship_out|, let's look at the other commands by which a user can send things to the \.{GF} file. @ @= begin cur_exp:=round_unscaled(cur_exp) mod 256; if cur_exp<0 then cur_exp:=cur_exp+256; boolean_reset(char_exists[cur_exp]); cur_type:=boolean_type; end @* \[45] Dumping and undumping the tables. After \.{INIMP} has seen a collection of macros, it can write all the necessary information on an auxiliary file so that production versions of \MP\ are able to initialize their memory at high speed. The present section of the program takes care of such output and input. We shall consider simultaneously the processes of storing and restoring, so that the inverse relation between them is clear. @.INIMP@> The global variable |mem_ident| is a string that is printed right after the |banner| line when \MP\ is ready to start. For \.{INIMP} this string says simply `\.{(INIMP)}'; for other versions of \MP\ it says, for example, `\.{(preloaded mem=plain 90.4.14)}', showing the year, month, and day that the mem file was created. We have |mem_ident=0| before \MP's tables are loaded. @= @!mem_ident:str_number; @ @= mem_ident:=0; @ @= mem_ident:=" (INIMP)"; @ @= @!init procedure store_mem_file; label done; var @!k:integer; {all-purpose index} @!p,@!q: pointer; {all-purpose pointers} @!x: integer; {something to dump} @!w: four_quarters; {four ASCII codes} @!s: str_number; {all-purpose string} begin @; @; @; @; @; @; @; end; tini @ Corresponding to the procedure that dumps a mem file, we also have a function that reads~one~in. The function returns |false| if the dumped mem is incompatible with the present \MP\ table sizes, etc. @d off_base=6666 {go here if the mem file is unacceptable} @d too_small(#)==begin wake_up_terminal; wterm_ln('---! Must increase the ',#); @.Must increase the x@> goto off_base; end @p @t\4@>@@; function load_mem_file:boolean; label done,off_base,exit; var @!k:integer; {all-purpose index} @!p,@!q: pointer; {all-purpose pointers} @!x: integer; {something undumped} @!s: str_number; {some temporary string} @!w: four_quarters; {four ASCII codes} begin @; @; @; @; @; load_mem_file:=true; return; {it worked!} off_base: wake_up_terminal; wterm_ln('(Fatal mem file error; I''m stymied)'); @.Fatal mem file error@> load_mem_file:=false; exit:end; @ Mem files consist of |memory_word| items, and we use the following macros to dump words of different types: @d dump_wd(#)==begin mem_file^:=#; put(mem_file);@+end @d dump_int(#)==begin mem_file^.int:=#; put(mem_file);@+end @d dump_hh(#)==begin mem_file^.hh:=#; put(mem_file);@+end @d dump_qqqq(#)==begin mem_file^.qqqq:=#; put(mem_file);@+end @= @!mem_file:word_file; {for input or output of mem information} @ The inverse macros are slightly more complicated, since we need to check the range of the values we are reading in. We say `|undump(a)(b)(x)|' to read an integer value |x| that is supposed to be in the range |a<=x<=b|. @d undump_wd(#)==begin get(mem_file); #:=mem_file^;@+end @d undump_int(#)==begin get(mem_file); #:=mem_file^.int;@+end @d undump_hh(#)==begin get(mem_file); #:=mem_file^.hh;@+end @d undump_qqqq(#)==begin get(mem_file); #:=mem_file^.qqqq;@+end @d undump_end_end(#)==#:=x;@+end @d undump_end(#)==(x>#) then goto off_base@+else undump_end_end @d undump(#)==begin undump_int(x); if (x<#) or undump_end @d undump_size_end_end(#)==too_small(#)@+else undump_end_end @d undump_size_end(#)==if x># then undump_size_end_end @d undump_size(#)==begin undump_int(x); if x<# then goto off_base; undump_size_end @ The next few sections of the program should make it clear how we use the dump/undump macros. @= dump_int(@$);@/ dump_int(mem_min);@/ dump_int(mem_top);@/ dump_int(hash_size);@/ dump_int(hash_prime);@/ dump_int(max_in_open) @ Sections of a \.{WEB} program that are ``commented out'' still contribute strings to the string pool; therefore \.{INIMP} and \MP\ will have the same strings. (And it is, of course, a good thing that they do.) @.WEB@> @^string pool@> @= x:=mem_file^.int; if x<>@$ then goto off_base; {check that strings are the same} undump_int(x); if x<>mem_min then goto off_base; undump_int(x); if x<>mem_top then goto off_base; undump_int(x); if x<>hash_size then goto off_base; undump_int(x); if x<>hash_prime then goto off_base; undump_int(x); if x<>max_in_open then goto off_base @ We do string pool compaction to avoid dumping unused strings. @d dump_four_ASCII== w.b0:=qi(so(str_pool[k])); w.b1:=qi(so(str_pool[k+1])); w.b2:=qi(so(str_pool[k+2])); w.b3:=qi(so(str_pool[k+3])); dump_qqqq(w) @= do_compaction(pool_size); dump_int(pool_ptr); dump_int(max_str_ptr); dump_int(str_ptr); k:=0; while (next_str[k]=k+1) and (k<=max_str_ptr) do incr(k); dump_int(k); while k<=max_str_ptr do begin dump_int(next_str[k]); incr(k); end; k:=0; loop @+begin dump_int(str_start[k]); if k=str_ptr then goto done else k:=next_str[k]; end; done:k:=0; while k+4= undump_size(0)(pool_size)('string pool size')(pool_ptr); undump_size(0)(max_strings-1)('max strings')(max_str_ptr); undump(0)(max_str_ptr)(str_ptr); undump(0)(max_str_ptr+1)(s); for k:=0 to s-1 do next_str[k]:=k+1; for k:=s to max_str_ptr do undump(s+1)(max_str_ptr+1)(next_str[k]); fixed_str_use:=0; k:=0; loop @+begin undump(0)(pool_ptr)(str_start[k]); if k=str_ptr then goto done; str_ref[k]:=max_str_ref; incr(fixed_str_use); last_fixed_str:=k; k:=next_str[k]; end; done:k:=0; while k+4= sort_avail; var_used:=0; dump_int(lo_mem_max); dump_int(rover); p:=mem_min; q:=rover; x:=0; repeat for k:=p to q+1 do dump_wd(mem[k]); x:=x+q+2-p; var_used:=var_used+q-p; p:=q+node_size(q); q:=rlink(q); until q=rover; var_used:=var_used+lo_mem_max-p; dyn_used:=mem_end+1-hi_mem_min;@/ for k:=p to lo_mem_max do dump_wd(mem[k]); x:=x+lo_mem_max+1-p; dump_int(hi_mem_min); dump_int(avail); for k:=hi_mem_min to mem_end do dump_wd(mem[k]); x:=x+mem_end+1-hi_mem_min; p:=avail; while p<>null do begin decr(dyn_used); p:=link(p); end; dump_int(var_used); dump_int(dyn_used); print_ln; print_int(x); print(" memory locations dumped; current usage is "); print_int(var_used); print_char("&"); print_int(dyn_used) @ @= undump(lo_mem_stat_max+1000)(hi_mem_stat_min-1)(lo_mem_max); undump(lo_mem_stat_max+1)(lo_mem_max)(rover); p:=mem_min; q:=rover; repeat for k:=p to q+1 do undump_wd(mem[k]); p:=q+node_size(q); if (p>lo_mem_max)or((q>=rlink(q))and(rlink(q)<>rover)) then goto off_base; q:=rlink(q); until q=rover; for k:=p to lo_mem_max do undump_wd(mem[k]); undump(lo_mem_max+1)(hi_mem_stat_min)(hi_mem_min); undump(null)(mem_top)(avail); mem_end:=mem_top; for k:=hi_mem_min to mem_end do undump_wd(mem[k]); undump_int(var_used); undump_int(dyn_used) @ A different scheme is used to compress the hash table, since its lower region is usually sparse. When |text(p)<>0| for |p<=hash_used|, we output three words: |p|, |hash[p]|, and |eqtb[p]|. The hash table is, of course, densely packed for |p>=hash_used|, so the remaining entries are output in~a~block. @= dump_int(hash_used); st_count:=frozen_inaccessible-1-hash_used; for p:=1 to hash_used do if text(p)<>0 then begin dump_int(p); dump_hh(hash[p]); dump_hh(eqtb[p]); incr(st_count); end; for p:=hash_used+1 to hash_end do begin dump_hh(hash[p]); dump_hh(eqtb[p]); end; dump_int(st_count);@/ print_ln; print_int(st_count); print(" symbolic tokens") @ @= undump(1)(frozen_inaccessible)(hash_used); p:=0; repeat undump(p+1)(hash_used)(p); undump_hh(hash[p]); undump_hh(eqtb[p]); until p=hash_used; for p:=hash_used+1 to hash_end do begin undump_hh(hash[p]); undump_hh(eqtb[p]); end; undump_int(st_count) @ We have already printed a lot of statistics, so we set |tracing_stats:=0| to prevent them appearing again. @= dump_int(int_ptr); for k:=1 to int_ptr do begin dump_int(internal[k]); dump_int(int_name[k]); end; dump_int(start_sym); dump_int(interaction); dump_int(mem_ident); dump_int(bg_loc); dump_int(eg_loc); dump_int(serial_no); dump_int(69073); internal[tracing_stats]:=0 @ @= undump(max_given_internal)(max_internal)(int_ptr); for k:=1 to int_ptr do begin undump_int(internal[k]); undump(0)(str_ptr)(int_name[k]); end; undump(0)(frozen_inaccessible)(start_sym); undump(batch_mode)(error_stop_mode)(interaction); undump(0)(str_ptr)(mem_ident); undump(1)(hash_end)(bg_loc); undump(1)(hash_end)(eg_loc); undump_int(serial_no);@/ undump_int(x);@+if (x<>69073)or eof(mem_file) then goto off_base @ @= selector:=new_string; print(" (preloaded mem="); print(job_name); print_char(" "); print_int(round_unscaled(internal[year]) mod 100); print_char("."); print_int(round_unscaled(internal[month])); print_char("."); print_int(round_unscaled(internal[day])); print_char(")"); if interaction=batch_mode then selector:=log_only else selector:=term_and_log; str_room(1); mem_ident:=make_string; str_ref[mem_ident]:=max_str_ref;@/ pack_job_name(mem_extension); while not w_open_out(mem_file) do prompt_file_name("mem file name",mem_extension); print_nl("Beginning to dump on file "); @.Beginning to dump...@> s:=w_make_name_string(mem_file); print(s); flush_string(s); print_nl(mem_ident) @ @= w_close(mem_file) @* \[46] The main program. This is it: the part of \MP\ that executes all those procedures we have written. Well---almost. We haven't put the parsing subroutines into the program yet; and we'd better leave space for a few more routines that may have been forgotten. @p @@; @@; @ @ We've noted that there are two versions of \MP. One, called \.{INIMP}, @.INIMP@> has to be run first; it initializes everything from scratch, without reading a mem file, and it has the capability of dumping a mem file. The other one is called `\.{VIRMP}'; it is a ``virgin'' program that needs @.VIRMP@> to input a mem file in order to get started. \.{VIRMP} typically has a bit more memory capacity than \.{INIMP}, because it does not need the space consumed by the dumping/undumping routines and the numerous calls on |primitive|, etc. The \.{VIRMP} program cannot read a mem file instantaneously, of course; the best implementations therefore allow for production versions of \MP\ that not only avoid the loading routine for \PASCAL\ object code, they also have a mem file pre-loaded. This is impossible to do if we stick to standard \PASCAL; but there is a simple way to fool many systems into avoiding the initialization, as follows:\quad(1)~We declare a global integer variable called |ready_already|. The probability is negligible that this variable holds any particular value like 314159 when \.{VIRMP} is first loaded.\quad(2)~After we have read in a mem file and initialized everything, we set |ready_already:=314159|.\quad(3)~Soon \.{VIRMP} will print `\.*', waiting for more input; and at this point we interrupt the program and save its core image in some form that the operating system can reload speedily.\quad(4)~When that core image is activated, the program starts again at the beginning; but now |ready_already=314159| and all the other global variables have their initial values too. The former chastity has vanished! In other words, if we allow ourselves to test the condition |ready_already=314159|, before |ready_already| has been assigned a value, we can avoid the lengthy initialization. Dirty tricks rarely pay off so handsomely. @^dirty \PASCAL@> @^system dependencies@> @= @!ready_already:integer; {a sacrifice of purity for economy} @ Now this is really it: \MP\ starts and ends here. The initial test involving |ready_already| should be deleted if the \PASCAL\ runtime system is smart enough to detect such a ``mistake.'' @^system dependencies@> @p begin @!{|start_here|} history:=fatal_error_stop; {in case we quit during initialization} t_open_out; {open the terminal for output} if ready_already=314159 then goto start_of_MP; @@; if bad>0 then begin wterm_ln('Ouch---my internal constants have been clobbered!', '---case ',bad:1); @.Ouch...clobbered@> goto final_end; end; initialize; {set global variables to their starting values} @!init if not get_strings_started then goto final_end; init_tab; {initialize the tables} init_prim; {call |primitive| for each primitive} init_str_use:=str_ptr; init_pool_ptr:=pool_ptr;@/ max_str_ptr:=str_ptr; max_pool_ptr:=pool_ptr; fix_date_and_time; tini@/ ready_already:=314159; start_of_MP: @; @; history:=spotless; {ready to go!} if start_sym>0 then {insert the `\&{everyjob}' symbol} begin cur_sym:=start_sym; back_input; end; main_control; {come to life} final_cleanup; {prepare for death} end_of_MP: close_files_and_terminate; final_end: ready_already:=0; end. @ Here we do whatever is needed to complete \MP's job gracefully on the local operating system. The code here might come into play after a fatal error; it must therefore consist entirely of ``safe'' operations that cannot produce error messages. For example, it would be a mistake to call |str_room| or |make_string| at this time, because a call on |overflow| might lead to an infinite loop. @^system dependencies@> This program doesn't bother to close the input files that may still be open. @= procedure close_files_and_terminate; var @!k:integer; {all-purpose index} @!lh:integer; {the length of the \.{TFM} header, in words} @!lk_offset:0..256; {extra words inserted at beginning of |lig_kern| array} @!p:pointer; {runs through a list of \.{TFM} dimensions} begin @; @!stat if internal[tracing_stats]>0 then @;@;@+tats@/ wake_up_terminal; @; @; if log_opened then begin wlog_cr; a_close(log_file); selector:=selector-2; if selector=term_only then begin print_nl("Transcript written on "); @.Transcript written...@> print(log_name); print_char("."); end; end; end; @ @= for k:=0 to read_files-1 do if rd_fname[k]<>0 then a_close(rd_file[k]); for k:=0 to write_files-1 do if wr_fname[k]<>0 then a_close(wr_file[k]) @ We want to produce a \.{TFM} file if and only if |fontmaking| is positive. We reclaim all of the variable-size memory at this point, so that there is no chance of another memory overflow after the memory capacity has already been exceeded. @= if internal[fontmaking]>0 then begin @; @; fix_design_size; fix_check_sum; @; internal[fontmaking]:=0; {avoid loop in case of fatal error} @; end @ @= rover:=lo_mem_stat_max+1; link(rover):=empty_flag; lo_mem_max:=hi_mem_min-1; if lo_mem_max-rover>max_halfword then lo_mem_max:=max_halfword+rover; node_size(rover):=lo_mem_max-rover; llink(rover):=rover; rlink(rover):=rover; link(lo_mem_max):=null; info(lo_mem_max):=null @ The present section goes directly to the log file instead of using |print| commands, because there's no need for these strings to take up |str_pool| memory when a non-{\bf stat} version of \MP\ is being used. @= if log_opened then begin wlog_ln(' '); wlog_ln('Here is how much of MetaPost''s memory',' you used:'); @.Here is how much...@> wlog(' ',max_strs_used-init_str_use:1,' string'); if max_strs_used<>init_str_use+1 then wlog('s'); wlog_ln(' out of ', max_strings-1-init_str_use:1);@/ wlog_ln(' ',max_pl_used-init_pool_ptr:1,' string characters out of ', pool_size-init_pool_ptr:1);@/ wlog_ln(' ',lo_mem_max-mem_min+mem_end-hi_mem_min+2:1,@| ' words of memory out of ',mem_end+1-mem_min:1);@/ wlog_ln(' ',st_count:1,' symbolic tokens out of ', hash_size:1);@/ wlog_ln(' ',max_in_stack:1,'i,',@| int_ptr:1,'n,',@| max_param_stack:1,'p,',@| max_buf_stack+1:1,'b stack positions out of ',@| stack_size:1,'i,', max_internal:1,'n,', param_size:1,'p,', buf_size:1,'b'); wlog_ln(' ',pact_count:1,' string compactions (moved ', pact_chars:1,' characters, ', pact_strs:1,' strings)'); end @ We get to the |final_cleanup| routine when \&{end} or \&{dump} has been scanned. @= procedure final_cleanup; label exit; var c:small_number; {0 for \&{end}, 1 for \&{dump}} begin c:=cur_mod; if job_name=0 then open_log_file; while input_ptr>0 do if token_state then end_token_list@+else end_file_reading; while loop_ptr<>null do stop_iteration; while open_parens>0 do begin print(" )"); decr(open_parens); end; while cond_ptr<>null do begin print_nl("(end occurred when ");@/ @.end occurred...@> print_cmd_mod(fi_or_else,cur_if); {`\.{if}' or `\.{elseif}' or `\.{else}'} if if_line<>0 then begin print(" on line "); print_int(if_line); end; print(" was incomplete)"); if_line:=if_line_field(cond_ptr); cur_if:=name_type(cond_ptr); cond_ptr:=link(cond_ptr); end; if history<>spotless then if ((history=warning_issued)or(interaction selector:=term_and_log; end; if c=1 then begin @!init store_mem_file; return;@+tini@/ print_nl("(dump is performed only by INIMP)"); return; @.dump...only by INIMP@> end; exit:end; @ @= @!init procedure init_prim; {initialize all the primitives} begin @; end; @# procedure init_tab; {initialize other tables} var @!k:integer; {all-purpose index} begin @@; end; tini @ When we begin the following code, \MP's tables may still contain garbage; the strings might not even be present. Thus we must proceed cautiously to get bootstrapped in. But when we finish this part of the program, \MP\ is ready to call on the |main_control| routine to do its work. @= begin @; if (mem_ident=0)or(buffer[loc]="&") then begin if mem_ident<>0 then initialize; {erase preloaded mem} if not open_mem_file then goto final_end; if not load_mem_file then begin w_close(mem_file); goto final_end; end; w_close(mem_file); while (loc; if loc"\" then start_input; {\&{input} assumed} end @* \[47] Debugging. Once \MP\ is working, you should be able to diagnose most errors with the \.{show} commands and other diagnostic features. But for the initial stages of debugging, and for the revelation of really deep mysteries, you can compile \MP\ with a few more aids, including the \PASCAL\ runtime checks and its debugger. An additional routine called |debug_help| will also come into play when you type `\.D' after an error message; |debug_help| also occurs just before a fatal error causes \MP\ to succumb. @^debugging@> @^system dependencies@> The interface to |debug_help| is primitive, but it is good enough when used with a \PASCAL\ debugger that allows you to set breakpoints and to read variables and change their values. After getting the prompt `\.{debug \#}', you type either a negative number (this exits |debug_help|), or zero (this goes to a location where you can set a breakpoint, thereby entering into dialog with the \PASCAL\ debugger), or a positive number |m| followed by an argument |n|. The meaning of |m| and |n| will be clear from the program below. (If |m=13|, there is an additional argument, |l|.) @.debug \#@> @d breakpoint=888 {place where a breakpoint is desirable} @= @!debug procedure debug_help; {routine to display various things} label breakpoint,exit; var @!k,@!l,@!m,@!n:integer; begin loop begin wake_up_terminal; print_nl("debug # (-1 to exit):"); update_terminal; @.debug \#@> read(term_in,m); if m<0 then return else if m=0 then begin goto breakpoint;@\ {go to every label at least once} breakpoint: m:=0; @{'BREAKPOINT'@}@\ end else begin read(term_in,n); case m of @t\4@>@@; othercases print("?") endcases; end; end; exit:end; gubed @ @= 1: print_word(mem[n]); {display |mem[n]| in all forms} 2: print_int(info(n)); 3: print_int(link(n)); 4: begin print_int(eq_type(n)); print_char(":"); print_int(equiv(n)); end; 5: print_variable_name(n); 6: print_int(internal[n]); 7: do_show_dependencies; 9: show_token_list(n,null,100000,0); 10: print(n); 11: check_mem(n>0); {check wellformedness; print new busy locations if |n>0|} 12: search_mem(n); {look for pointers to |n|} 13: begin read(term_in,l); print_cmd_mod(n,l); end; 14: for k:=0 to n do print(buffer[k]); 15: panicking:=not panicking; @* \[48] System-dependent changes. This section should be replaced, if necessary, by any special modification of the program that are necessary to make \MP\ work at a particular installation. It is usually best to design your change file so that all changes to previous sections preserve the section numbering; then everybody's version will be consistent with the published program. More extensive changes, which introduce new sections, can be inserted here; then only the index itself will get a new section number. @^system dependencies@> @* \[49] Index. Here is where you can find all uses of each identifier in the program, with underlined entries pointing to where the identifier was defined. If the identifier is only one letter long, however, you get to see only the underlined entries. {\sl All references are to section numbers instead of page numbers.} This index also lists error messages and other aspects of the program that you might want to look up some day. For example, the entry for ``system dependencies'' lists all sections that should receive special attention from people who are installing \MP\ in a new operating environment. A list of various things that can't happen appears under ``this can't happen''. Approximately 25 sections are listed under ``inner loop''; these account for more than 60\pct! of \MP's running time, exclusive of input and output.