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authorKarl Berry <karl@freefriends.org>2021-02-25 19:22:25 +0000
committerKarl Berry <karl@freefriends.org>2021-02-25 19:22:25 +0000
commitad547a6b5986815fda458221149728d9d9ab1d87 (patch)
tree16296910eb3eca724371474ea9aea3994dc69614 /Build/source/texk/web2c/mf.web
parent947b43de3dd21d58ccc2ffadefc4441ea1c2a813 (diff)
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+% This program is copyright (C) 1984 by D. E. Knuth; all rights are reserved.
+% Unlimited copying and redistribution of this file are permitted as long
+% as this file is not modified. Modifications are permitted, but only if
+% the resulting file is not named mf.web. (The WEB system provides
+% for alterations via an auxiliary file; the master file should stay intact.)
+% In other words, METAFONT is under essentially the same ground rules as TeX.
+
+% TeX is a trademark of the American Mathematical Society.
+% METAFONT is a trademark of Addison-Wesley Publishing Company.
+
+% Version 0 was completed on July 28, 1984.
+% Version 1 was completed on January 4, 1986; it corresponds to "Volume D".
+% Version 1.1 trivially corrected the punctuation in one message (June 1986).
+% Version 1.2 corrected an arithmetic overflow problem (July 1986).
+% Version 1.3 improved rounding when elliptical pens are made (November 1986).
+% Version 1.4 corrected scan_declared_variable timing (May 1988).
+% Version 1.5 fixed negative halving in allocator when mem_min<0 (June 1988).
+% Version 1.6 kept open_log_file from calling fatal_error (November 1988).
+% Version 1.7 solved that problem a better way (December 1988).
+% Version 1.8 introduced major changes for 8-bit extensions (September 1989).
+% Version 1.9 improved skimping and was edited for style (December 1989).
+% Version 2.0 fixed bug in addto; released with TeX version 3.0 (March 1990).
+% Version 2.7 made consistent with TeX version 3.1 (September 1990).
+% Version 2.71 fixed bug in draw, allowed unprintable filenames (March 1992).
+% Version 2.718 fixed bug in <Choose a dependent...> (March 1995).
+% Version 2.7182 fixed bugs related to "<unprintable char>" (August 1996).
+% Version 2.71828 suppressed autorounding in dangerous cases (June 2003).
+% Version 2.718281 was a general cleanup with minor fixes (February 2008).
+% Version 2.7182818 was similar (January 2014).
+% Version 2.71828182 was similar (January 2021).
+
+% A reward of $327.68 will be paid to the first finder of any remaining bug.
+
+% Although considerable effort has been expended to make the METAFONT program
+% correct and reliable, no warranty is implied; the author disclaims any
+% obligation or liability for damages, including but not limited to
+% special, indirect, or consequential damages arising out of or in
+% connection with the use or performance of this software. This work has
+% been a ``labor of love'' and the author hopes that users enjoy it.
+
+% Here is TeX material that gets inserted after \input webmac
+\def\hang{\hangindent 3em\noindent\ignorespaces}
+\def\textindent#1{\hangindent2.5em\noindent\hbox to2.5em{\hss#1 }\ignorespaces}
+\font\ninerm=cmr9
+\let\mc=\ninerm % medium caps for names like SAIL
+\def\PASCAL{Pascal}
+\def\ph{\hbox{Pascal-H}}
+\def\psqrt#1{\sqrt{\mathstrut#1}}
+\def\k{_{k+1}}
+\def\pct!{{\char`\%}} % percent sign in ordinary text
+\font\tenlogo=logo10 % font used for the METAFONT logo
+\font\logos=logosl10
+\font\eightlogo=logo8
+\def\MF{{\tenlogo META}\-{\tenlogo FONT}}
+\def\<#1>{$\langle#1\rangle$}
+\def\section{\mathhexbox278}
+\let\swap=\leftrightarrow
+\def\round{\mathop{\rm round}\nolimits}
+
+\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{{\eightlogo METAFONT}}
+\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 "<Global...>"
+\def\gglob{20, 26} % this should be the next two sections of "<Global...>"
+
+@* \[1] Introduction.
+This is \MF, a font compiler intended to produce typefaces of high quality.
+The \PASCAL\ program that follows is the definition of \MF84, a standard
+@:PASCAL}{\PASCAL@>
+@!@:METAFONT84}{\MF84@>
+version of \MF\ that is designed to be highly portable so that identical output
+will be obtainable on a great variety of computers. The conventions
+of \MF84 are the same as those of \TeX82.
+
+The main purpose of the following program is to explain the algorithms of \MF\
+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 \MF\ 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
+\MF\ language itself, since the reader is supposed to be familiar with
+{\sl The {\logos METAFONT\/}book}.
+@.WEB@>
+@:METAFONTbook}{\sl The {\logos METAFONT\/}book@>
+
+@ The present implementation has a long ancestry, beginning in the spring
+of~1977, when its author wrote a prototype set of subroutines and macros
+@^Knuth, Donald Ervin@>
+that were used to develop the first Computer Modern fonts.
+This original proto-\MF\ required the user to recompile a {\mc SAIL} program
+whenever any character was changed, because it was not a ``language'' for
+font design; the language was {\mc SAIL}. After several hundred characters
+had been designed in that way, the author developed an interpretable language
+called \MF, in which it was possible to express the Computer Modern programs
+less cryptically. A complete \MF\ processor was designed and coded by the
+author in 1979. This program, written in {\mc SAIL}, was adapted for use
+with a variety of typesetting equipment and display terminals by Leo Guibas,
+Lyle Ramshaw, and David Fuchs.
+@^Guibas, Leonidas Ioannis@>
+@^Ramshaw, Lyle Harold@>
+@^Fuchs, David Raymond@>
+Major improvements to the design of Computer Modern fonts were made in the
+spring of 1982, after which it became clear that a new language would
+better express the needs of letterform designers. Therefore an entirely
+new \MF\ language and system were developed in 1984; the present system
+retains the name and some of the spirit of \MF79, but all of the details
+have changed.
+
+No doubt there still is plenty of room for improvement, but the author
+is firmly committed to keeping \MF84 ``frozen'' from now on; stability
+and reliability are to be its main virtues.
+
+On the other hand, the \.{WEB} description can be extended without changing
+the core of \MF84 itself, and the program has been designed so that such
+extensions are not extremely difficult to make.
+The |banner| string defined here should be changed whenever \MF\
+undergoes any modifications, so that it will be clear which version of
+\MF\ might be the guilty party when a problem arises.
+@^extensions to \MF@>
+@^system dependencies@>
+
+If this program is changed, the resulting system should not be called
+`\MF\kern.5pt'; the official name `\MF\kern.5pt' by itself is reserved
+for software systems that are fully compatible with each other.
+A special test suite called the ``\.{TRAP} test'' is available for
+helping to determine whether an implementation deserves to be
+known as `\MF\kern.5pt' [cf.~Stanford Computer Science report CS1095,
+January 1986].
+
+@d banner=='This is METAFONT, Version 2.71828182' {printed when \MF\ starts}
+
+@ Different \PASCAL s have slightly different conventions, and the present
+@!@:PASCAL H}{\ph@>
+program expresses \MF\ in terms of the \PASCAL\ that was
+available to the author in 1984. 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 and Experience \bf6} (1976),
+29--42. The \MF\ program below is intended to be adaptable, without
+extensive changes, to most other versions of \PASCAL, so it does not fully
+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
+or in the system-dependent |paint_row| procedure;
+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 \MF\ user
+to specify a file name if |output| were specified here.
+@:PASCAL H}{\ph@>
+@^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@>@<Compiler directives@>@/
+program MF; {all file names are defined dynamically}
+label @<Labels in the outer block@>@/
+const @<Constants in the outer block@>@/
+mtype @<Types in the outer block@>@/
+var @<Global variables@>@/
+@#
+procedure initialize; {this procedure gets things started properly}
+ var @<Local variables for initialization@>@/
+ begin @<Set initial values of key variables@>@/
+ end;@#
+@t\4@>@<Basic printing procedures@>@/
+@t\4@>@<Error handling procedures@>@/
+
+@ The overall \MF\ 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 \MF'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_MF=1 {go here when \MF's variables are initialized}
+@d end_of_MF=9998 {go here to close files and terminate gracefully}
+@d final_end=9999 {this label marks the ending of the program}
+
+@<Labels in the out...@>=
+start_of_MF@t\hskip-2pt@>, end_of_MF@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 \MF\ is being installed or
+when system wizards are fooling around with \MF\ 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 \MF's memory usage. The |stat| $\ldots$ |tats| code also
+implements special diagnostic information that is printed when
+$\\{tracingedges}>1$.
+@^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 \.{INIMF}, which does the extra calculations needed to
+@.INIMF@>
+initialize \MF'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 \MF\ 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.
+@:PASCAL H}{\ph@>
+@^system dependencies@>
+@^overflow in arithmetic@>
+
+@<Compiler directives@>=
+@{@&$C-,A+,D-@} {no range check, catch arithmetic overflow, no debug overhead}
+@!debug @{@&$C+,D+@}@+ gubed {but turn everything on when debugging}
+
+@ This \MF\ 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 \MF\ 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!)
+@:PASCAL H}{\ph@>
+
+@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 \MF's capacity. They may have different values in \.{INIMF} and
+in production versions of \MF.
+@.INIMF@>
+@^system dependencies@>
+
+@<Constants...@>=
+@!mem_max=30000; {greatest index in \MF's internal |mem| array;
+ must be strictly less than |max_halfword|;
+ must be equal to |mem_top| in \.{INIMF}, 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}
+@!screen_width=768; {number of pixels in each row of screen display}
+@!screen_depth=1024; {number of pixels in each column of screen display}
+@!stack_size=30; {maximum number of simultaneous input sources}
+@!max_strings=2000; {maximum number of strings; must not exceed |max_halfword|}
+@!string_vacancies=8000; {the minimum number of characters that should be
+ available for the user's identifier names and strings,
+ after \MF's own error messages are stored}
+@!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 \MF's own strings, which is currently about 22000}
+@!move_size=5000; {space for storing moves in a single octant}
+@!max_wiggle=300; {number of autorounded points per cycle}
+@!gf_buf_size=800; {size of the output buffer, must be a multiple of 8}
+@!file_name_size=40; {file names shouldn't be longer than this}
+@!pool_name='MFbases:MF.POOL ';
+ {string of length |file_name_size|; tells where the string pool appears}
+@.MFbases@>
+@!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 \MF's capacity. But if they are changed,
+it is necessary to rerun the initialization program \.{INIMF}
+@.INIMF@>
+to generate new tables for the production \MF\ 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 \.{INIMF};
+ 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}
+@^system dependencies@>
+
+@ In case somebody has inadvertently made bad settings of the ``constants,''
+\MF\ checks them using a global variable called |bad|.
+
+This is the first of many sections of \MF\ where global variables are
+defined.
+
+@<Glob...@>=
+@!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.)
+
+@<Check the ``constant'' values for consistency@>=
+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 gf_buf_size mod 8<>0 then bad:=3;
+if mem_min+1100>mem_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 \MF\ 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 \MF\ only with respect to
+the \&{char} and \&{ASCII} operations, and the comparison of strings.
+
+@ Characters of text that have been converted to \MF's internal form
+are said to be of type |ASCII_code|, which is a subrange of the integers.
+
+@<Types...@>=
+@!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 \MF\ 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|}
+
+@<Local variables for init...@>=
+@!i:integer;
+
+@ The \MF\ 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.
+
+@<Glob...@>=
+@!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 \MF\ with less complete character
+sets, and in such cases it will be necessary to change something here.
+@^system dependencies@>
+
+@<Set init...@>=
+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 \MF\ 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 \MF\ 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 \MF\ are allowed to have in their input files.
+Appropriate changes to \MF's |char_class| table should then be made.
+(Unlike \TeX, each installation of \MF\ 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@>
+
+@<Set init...@>=
+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<j<@'177|, the value of |xord[xchr[i]]| will turn out to be
+|j| or more; hence, standard ASCII code numbers will be used instead of
+codes below @'40 in case there is a coincidence.
+
+@<Set init...@>=
+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.
+
+\MF\ 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 base 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.
+
+@<Types...@>=
+@!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
+\MF; 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 \MF\ can open a file whose external name is specified by
+|name_of_file|.
+@^system dependencies@>
+
+@<Glob...@>=
+@!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 present version of \MF\ was prepared has
+extended 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
+\MF\ to undertake appropriate corrective action.
+@:PASCAL H}{\ph@>
+@^system dependencies@>
+
+\MF'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_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
+@:PASCAL H}{\ph@>
+@^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.
+\MF'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.
+
+@<Glob...@>=
+@!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
+file 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 \MF\
+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<buf_size| will always hold, so that there is always room for an
+``empty'' line.
+
+The variable |max_buf_stack|, which is used to keep track of how large
+the |buf_size| parameter must be to accommodate the present job, is
+also kept up to date by |input_ln|.
+
+If the |bypass_eoln| parameter is |true|, |input_ln| will do a |get|
+before looking at the first character of the line; this skips over
+an |eoln| that was in |f^|. The procedure does not do a |get| when it
+reaches the end of the line; therefore it can be used to acquire input
+from the user's terminal as well as from ordinary text files.
+
+Standard \PASCAL\ says that a file should have |eoln| immediately
+before |eof|, but \MF\ needs only a weaker restriction: If |eof|
+occurs in the middle of a line, the system function |eoln| should return
+a |true| result (even though |f^| will be undefined).
+
+@p function input_ln(var @!f:alpha_file;@!bypass_eoln:boolean):boolean;
+ {inputs the next line or returns |false|}
+var @!last_nonblank:0..buf_size; {|last| with trailing blanks removed}
+begin if bypass_eoln then if not eof(f) then get(f);
+ {input the first character of the line into |f^|}
+last:=first; {cf.\ Matthew 19\thinspace:\thinspace30}
+if eof(f) then input_ln:=false
+else begin last_nonblank:=first;
+ while not eoln(f) do
+ begin if last>=max_buf_stack then
+ begin max_buf_stack:=last+1;
+ if max_buf_stack=buf_size then
+ @<Report overflow of the input buffer, and abort@>;
+ 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@>
+
+@<Glob...@>=
+@!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|.
+@:PASCAL H}{\ph@>
+@^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:
+@:PASCAL H}{\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 \MF\ 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 `\.{mfput.log}'. (The transcript file
+will not contain error messages generated by the first line before the
+first \.{input} command.)
+@.mfput@>
+
+The first line is even more special if we are lucky enough to have an operating
+system that treats \MF\ differently from a run-of-the-mill \PASCAL\ object
+program. It's nice to let the user start running a \MF\ job by typing
+a command line like `\.{MF cmr10}'; in such a case, \MF\ 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 \MF.
+
+The first line is special also because it may be read before \MF\ has
+input a base file. In such cases, normal error messages cannot yet
+be given. The following code uses concepts that will be explained later.
+(If the \PASCAL\ compiler does not support non-local |@!goto|\unskip, the
+@^system dependencies@>
+statement `|goto final_end|' should be replaced by something that
+quietly terminates the program.)
+
+@<Report overflow of the input buffer, and abort@>=
+if base_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);
+@:METAFONT 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 \MF\ is in |buffer[loc]|. This
+ character should not be blank, and we should have |loc<last|.
+
+\yskip\noindent(It may be necessary to prompt the user several times
+before a non-blank line comes in. The prompt is `\.{**}' instead of the
+later `\.*' because the meaning is slightly different: `\.{input}' need
+not be typed immediately after~`\.{**}'.)
+
+@d loc==cur_input.loc_field {location of first unread character in |buffer|}
+
+@ The following program does the required initialization
+without retrieving a possible command line.
+It should be clear how to modify this routine to deal with command lines,
+if the system permits them.
+@^system dependencies@>
+
+@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<last)and(buffer[loc]=" ") do incr(loc);
+ if loc<last then
+ begin init_terminal:=true;
+ return; {return unless the line was all blank}
+ end;
+ write_ln(term_out,'Please type the name of your input file.');
+ end;
+exit:end;
+
+@* \[4] String handling.
+Symbolic token names and diagnostic messages are variable-length strings
+of eight-bit characters. Since \PASCAL\ does not have a well-developed string
+mechanism, \MF\ does all of its string processing by homegrown methods.
+
+Elaborate facilities for dynamic strings are not needed, so all of the
+necessary operations can be handled with a simple data structure.
+The array |str_pool| contains all of the (eight-bit) ASCII codes in all
+of the strings, and the array |str_start| contains indices of the starting
+points of each string. Strings are referred to by integer numbers, so that
+string number |s| comprises the characters |str_pool[j]| for
+|str_start[s]<=j<str_start[s+1]|. Additional integer variables
+|pool_ptr| and |str_ptr| indicate the number of entries used so far
+in |str_pool| and |str_start|, respectively; locations
+|str_pool[pool_ptr]| and |str_start[str_ptr]| are
+ready for the next string to be allocated.
+
+String numbers 0 to 255 are reserved for strings that correspond to single
+ASCII characters. This is in accordance with the conventions of \.{WEB},
+@.WEB@>
+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 \MF\ 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 only via macros that can easily be redefined.
+@^system dependencies@>
+
+@d si(#) == # {convert from |ASCII_code| to |packed_ASCII_code|}
+@d so(#) == # {convert from |packed_ASCII_code| to |ASCII_code|}
+
+@<Types...@>=
+@!pool_pointer = 0..pool_size; {for variables that point into |str_pool|}
+@!str_number = 0..max_strings; {for variables that point into |str_start|}
+@!packed_ASCII_code = 0..255; {elements of |str_pool| array}
+
+@ @<Glob...@>=
+@!str_pool:packed array[pool_pointer] of packed_ASCII_code; {the characters}
+@!str_start : array[str_number] of pool_pointer; {the starting pointers}
+@!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_ptr : str_number; {the starting value of |str_ptr|}
+@!max_pool_ptr : pool_pointer; {the maximum so far of |pool_ptr|}
+@!max_str_ptr : str_number; {the maximum so far of |str_ptr|}
+
+@ 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 length(#)==(str_start[#+1]-str_start[#]) {the number of characters
+ in string number \#}
+
+@ The length of the current string is called |cur_length|:
+
+@d cur_length == (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 aborts \MF\ and gives an
+apologetic error message if there isn't enough room.
+
+@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
+ begin if pool_ptr+# > pool_size then
+ overflow("pool size",pool_size-init_pool_ptr);
+@:METAFONT capacity exceeded pool size}{\quad pool size@>
+ max_pool_ptr:=pool_ptr+#;
+ end;
+ end
+
+@ \MF'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.
+
+Such a scheme can be justified because string expressions aren't a big
+deal in \MF\ applications; strings rarely need to be saved from one
+statement to the next. But it would waste space needlessly if we didn't
+try to reclaim the space of strings that are going to be used only once.
+
+Therefore a simple reference count mechanism is provided: If there are
+@^reference counts@>
+no references to a certain string from elsewhere in the program, and
+if there are no references to any strings created subsequent to it,
+then the string space will be reclaimed.
+
+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[#]<max_str_ref then incr(str_ref[#]);
+ end
+
+@<Glob...@>=
+@!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[#]<max_str_ref then
+ if str_ref[#]>1 then decr(str_ref[#])@+else flush_string(#);
+ end
+
+@<Declare the procedure called |flush_string|@>=
+procedure flush_string(@!s:str_number);
+begin if s<str_ptr-1 then str_ref[s]:=0
+else repeat decr(str_ptr);
+ until str_ref[str_ptr-1]<>0;
+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.
+
+@p function make_string : str_number; {current string enters the pool}
+begin if str_ptr=max_str_ptr then
+ begin if str_ptr=max_strings then
+ overflow("number of strings",max_strings-init_str_ptr);
+@:METAFONT capacity exceeded number of strings}{\quad number of strings@>
+ incr(max_str_ptr);
+ end;
+str_ref[str_ptr]:=1; incr(str_ptr); str_start[str_ptr]:=pool_ptr;
+make_string:=str_ptr-1;
+end;
+
+@ 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 j<str_start[s+1] do
+ begin if so(str_pool[j])<>buffer[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 \.{INIMF} program, based in part
+on the information that \.{WEB} has output while processing \MF.
+@.INIMF@>
+@^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; {the string just created}
+@!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;
+@<Make the first 256 strings@>;
+@<Read the other strings from the \.{MF.POOL} file and return |true|,
+ or give an error message and return |false|@>;
+exit:end;
+tini
+
+@ @d app_lc_hex(#)==l:=#;
+ if l<10 then append_char(l+"0")@+else append_char(l-10+"a")
+
+@<Make the first 256...@>=
+for k:=0 to 255 do
+ begin if (@<Character |k| cannot be printed@>) then
+ begin append_char("^"); append_char("^");
+ if k<@'100 then append_char(k+@'100)
+ else if k<@'200 then append_char(k-@'100)
+ else begin app_lc_hex(k div 16); app_lc_hex(k mod 16);
+ end;
+ end
+ else 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 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 \MF\ 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, @'136, @'141..@'146]|
+must be printable.
+@^character set dependencies@>
+@^system dependencies@>
+
+@<Character |k| cannot be printed@>=
+ (k<" ")or(k>"~")
+
+@ When the \.{WEB} system program called \.{TANGLE} processes the \.{MF.WEB}
+description that you are now reading, it outputs the \PASCAL\ program
+\.{MF.PAS} and also a string pool file called \.{MF.POOL}. The \.{INIMF}
+@.WEB@>@.INIMF@>
+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
+\MF's string memory.
+
+@<Glob...@>=
+@!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
+@<Read the other strings...@>=
+name_of_file:=pool_name; {we needn't set |name_length|}
+if a_open_in(pool_file) then
+ begin c:=false;
+ repeat @<Read one string, but return |false| if the
+ string memory space is getting too tight for comfort@>;
+ until c;
+ a_close(pool_file); get_strings_started:=true;
+ end
+else bad_pool('! I can''t read MF.POOL.')
+@.I can't read MF.POOL@>
+
+@ @<Read one string...@>=
+begin if eof(pool_file) then bad_pool('! MF.POOL has no check sum.');
+@.MF.POOL has no check sum@>
+read(pool_file,m,n); {read two digits of string length}
+if m='*' then @<Check the pool check sum@>
+else begin if (xord[m]<"0")or(xord[m]>"9")or@|
+ (xord[n]<"0")or(xord[n]>"9") then
+ bad_pool('! MF.POOL line doesn''t begin with two digits.');
+@.MF.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@>
+ 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 \.{MF.POOL} file; any other value means that the wrong pool
+file has been loaded.
+@^check sum@>
+
+@<Check the pool check sum@>=
+begin a:=0; k:=1;
+loop@+ begin if (xord[n]<"0")or(xord[n]>"9") then
+ bad_pool('! MF.POOL check sum doesn''t have nine digits.');
+@.MF.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('! MF.POOL doesn''t match; TANGLE me again.');
+@.MF.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 |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.
+
+\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|.
+
+Three additional global variables, |tally| and |term_offset| and
+|file_offset|, record the number of characters that have been printed
+since they were most recently cleared to zero. We use |tally| to record
+the length of (possibly very long) stretches of printing; |term_offset|
+and |file_offset|, on the other hand, keep track of how many characters
+have appeared so far on the current line that has been output to the
+terminal or to the transcript file, respectively.
+
+@d no_print=0 {|selector| setting that makes data disappear}
+@d term_only=1 {printing is destined for the terminal only}
+@d log_only=2 {printing is destined for the transcript file only}
+@d term_and_log=3 {normal |selector| setting}
+@d pseudo=4 {special |selector| setting for |show_context|}
+@d new_string=5 {printing is deflected to the string pool}
+@d max_selector=5 {highest selector setting}
+
+@<Glob...@>=
+@!log_file : alpha_file; {transcript of \MF\ session}
+@!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}
+@!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}
+
+@ @<Initialize the output routines@>=
+selector:=term_only; tally:=0; term_offset:=0; file_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)
+
+@ To end a line of text output, we call |print_ln|.
+
+@<Basic print...@>=
+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;
+no_print,pseudo,new_string: do_nothing;
+end; {there are no other cases}
+end; {note that |tally| is not affected}
+
+@ The |print_char| procedure sends one character to the desired destination,
+using the |xchr| array to map it into an external character compatible with
+|input_ln|. All printing comes through |print_ln| or |print_char|.
+
+@<Basic printing...@>=
+procedure print_char(@!s:ASCII_code); {prints a single character}
+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;
+no_print: do_nothing;
+pseudo: if tally<trick_count then trick_buf[tally mod error_line]:=s;
+new_string: begin if pool_ptr<pool_size then append_char(s);
+ end; {we drop characters if the string space is full}
+end; {there are no other cases}
+incr(tally);
+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 \MF\ 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@>
+
+@<Basic print...@>=
+procedure print(@!s:integer); {prints string |s|}
+var @!j:pool_pointer; {current character code position}
+begin if (s<0)or(s>=str_ptr) then s:="???"; {this can't happen}
+@.???@>
+if (s<256)and(selector>pseudo) then print_char(s)
+else begin j:=str_start[s];
+ while j<str_start[s+1] do
+ begin print_char(so(str_pool[j])); incr(j);
+ end;
+ end;
+end;
+
+@ Sometimes it's necessary to print a string whose characters
+may not be visible ASCII codes. In that case |slow_print| is used.
+
+@<Basic print...@>=
+procedure slow_print(@!s:integer); {prints string |s|}
+var @!j:pool_pointer; {current character code position}
+begin if (s<0)or(s>=str_ptr) then s:="???"; {this can't happen}
+@.???@>
+if (s<256)and(selector>pseudo) then print_char(s)
+else begin j:=str_start[s];
+ while j<str_start[s+1] do
+ begin print(so(str_pool[j])); incr(j);
+ end;
+ end;
+end;
+
+@ Here is the very first thing that \MF\ prints: a headline that identifies
+the version number and base name. The |term_offset| variable is temporarily
+incorrect, but the discrepancy is not serious since we assume that this
+part of the program is system dependent.
+@^system dependencies@>
+
+@<Initialize the output...@>=
+wterm(banner);
+if base_ident=0 then wterm_ln(' (no base preloaded)')
+else begin slow_print(base_ident); print_ln;
+ end;
+update_terminal;
+
+@ The procedure |print_nl| is like |print|, but it makes sure that the
+string appears at the beginning of a new line.
+
+@<Basic print...@>=
+procedure print_nl(@!s:str_number); {prints string |s| at beginning of line}
+begin if ((term_offset>0)and(odd(selector)))or@|
+ ((file_offset>0)and(selector>=log_only)) then print_ln;
+print(s);
+end;
+
+@ An array of digits in the range |0..9| is printed by |print_the_digs|.
+
+@<Basic print...@>=
+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.
+
+@<Basic print...@>=
+procedure print_int(@!n:integer); {prints an integer in decimal form}
+var k:0..23; {index to current digit; we assume that $\vert n\vert<10^{23}$}
+@!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;
+
+@ \MF\ 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<scroll_mode|.
+
+@d prompt_input(#)==begin wake_up_terminal; print(#); term_input;
+ end {prints a string and gets a line of input}
+
+@p procedure term_input; {gets a line from the terminal}
+var @!k:0..buf_size; {index into |buffer|}
+begin update_terminal; {now the user sees the prompt for sure}
+if not input_ln(term_in,true) then fatal_error("End of file on the terminal!");
+@.End of file on the terminal@>
+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, \MF\ 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
+
+@<Glob...@>=
+@!interaction:batch_mode..error_stop_mode; {current level of interaction}
+
+@ @<Set init...@>=interaction:=error_stop_mode;
+
+@ \MF\ 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).
+
+@<Initialize the print |selector| based on |interaction|@>=
+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, \MF\ 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}
+
+@<Glob...@>=
+@!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 \MF\ survives the initialization process.
+
+@<Set init...@>=
+deletions_allowed:=true; error_count:=0; {|history| is initialized elsewhere}
+
+@ Since errors can be detected almost anywhere in \MF, 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 \MF\ is trying to fix a non-fatal one. But such recursion
+@^recursion@>
+is never more than two levels deep.
+
+@<Error handling...@>=
+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@>@<Declare the procedure called |flush_string|@>
+
+@ 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}
+
+@<Glob...@>=
+@!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}}
+
+@ @<Set init...@>=
+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_MF|. This is the only nontrivial |@!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.
+
+@<Error hand...@>=
+procedure jump_out;
+begin goto end_of_MF;
+end;
+
+@ Here now is the general |error| routine.
+
+@<Error hand...@>=
+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<error_message_issued then history:=error_message_issued;
+print_char("."); show_context;
+if interaction=error_stop_mode then @<Get user's advice and |return|@>;
+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;
+@<Put help message on the transcript file@>;
+exit:end;
+
+@ @<Get user's advice...@>=
+loop@+begin continue: if interaction<>error_stop_mode then return;
+ 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}
+ @<Interpret code |c| and |return| if done@>;
+ end
+
+@ It is desirable to provide an `\.E' option here that gives the user
+an easy way to return from \MF\ 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@>
+
+@<Interpret code |c| and |return| if done@>=
+case c of
+"0","1","2","3","4","5","6","7","8","9": if deletions_allowed then
+ @<Delete |c-"0"| tokens and |goto continue|@>;
+@t\4\4@>@;@+@!debug "D":begin debug_help;goto continue;@+end;@+gubed@/
+"E": if file_ptr>0 then if input_stack[file_ptr].name_field>=256 then
+ begin print_nl("You want to edit file ");
+@.You want to edit file x@>
+ slow_print(input_stack[file_ptr].name_field);
+ print(" at line "); print_int(line);@/
+ interaction:=scroll_mode; jump_out;
+ end;
+"H": @<Print the help information and |goto continue|@>;
+"I":@<Introduce new material from the terminal and |return|@>;
+"Q","R","S":@<Change the interaction level and |return|@>;
+"X":begin interaction:=scroll_mode; jump_out;
+ end;
+othercases do_nothing
+endcases;@/
+@<Print the menu of available options@>
+
+@ @<Print the menu...@>=
+begin print("Type <return> to proceed, S to scroll future error messages,");@/
+@.Type <return> to proceed...@>
+print_nl("R to run without stopping, Q to run quietly,");@/
+print_nl("I to insert something, ");
+if file_ptr>0 then if input_stack[file_ptr].name_field>=256 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 \MF\ 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@>
+
+@<Change the interaction...@>=
+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 \MF's input stacks.
+
+@<Introduce new material...@>=
+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.
+
+@<Delete |c-"0"| tokens...@>=
+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}
+ @<Decrease the string reference count, if the current token is a string@>;
+ 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
+
+@ @<Print the help info...@>=
+begin if use_err_help then
+ begin @<Print the string |err_help|, possibly on several lines@>;
+ 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
+
+@ @<Print the string |err_help|, possibly on several lines@>=
+j:=str_start[err_help];
+while j<str_start[err_help+1] do
+ begin if str_pool[j]<>si("%") then print(so(str_pool[j]))
+ else if j+1=str_start[err_help+1] then print_ln
+ else if str_pool[j+1]<>si("%") then print_ln
+ else begin incr(j); print_char("%");
+ end;
+ incr(j);
+ end
+
+@ @<Put help message on the transcript file@>=
+if interaction>batch_mode then decr(selector); {avoid terminal output}
+if use_err_help then
+ begin print_nl("");
+ @<Print the string |err_help|, possibly on several lines@>;
+ 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 \MF'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
+
+@<Error hand...@>=
+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.
+
+@<Error hand...@>=
+procedure overflow(@!s:str_number;@!n:integer); {stop due to finiteness}
+begin normalize_selector;
+print_err("METAFONT capacity exceeded, sorry [");
+@.METAFONT 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 \MF\
+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@>
+
+@<Error hand...@>=
+procedure confusion(@!s:str_number);
+ {consistency check violated; |s| tells where}
+begin normalize_selector;
+if history<error_message_issued then
+ begin print_err("This can't happen ("); print(s); print_char(")");
+@.This can't happen@>
+ 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 \MF\ 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 signalled. 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
+
+@<Global...@>=
+@!interrupt:integer; {should \MF\ pause for instructions?}
+@!OK_to_interrupt:boolean; {should interrupts be observed?}
+
+@ @<Set init...@>=
+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. \MF\ 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 an instruction 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 \MF'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 \MF\ 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 \MF 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 \MF\ in a similar way.
+
+@d el_gordo == @'17777777777 {$2^{31}-1$, the largest value that \MF\ likes}
+
+@ One of \MF'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 \MF\ is being implemented with languages that permit
+binary shifting, the |half| macro should be changed to make this operation
+as efficient as possible.
+
+@d half(#)==(#) 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.
+@^overflow in arithmetic@>
+
+@<Glob...@>=
+@!arith_error:boolean; {has arithmetic overflow occurred recently?}
+
+@ @<Set init...@>=
+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}
+
+@<Types...@>=
+@!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:=half(a+1);
+end;
+
+@ Conversely, here is a procedure analogous to |print_int|. If the output
+of this procedure is subsequently read by \MF\ 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 $s-\delta\L10\cdot2^{16}f<s$.
+We can stop if and only if $f=0$ satisfies this condition; the loop will
+terminate before $s$ can possibly become zero.
+
+@<Basic printing...@>=
+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.
+
+@<Basic printing...@>=
+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 \MF\ programs are generally supposed to be
+less than $2^{12}$ in absolute value, so \MF\ 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}
+
+@<Types...@>=
+@!fraction=integer; {this type is used for scaled fractions}
+
+@ In fact, the two sorts of scaling discussed above aren't quite
+sufficient; \MF\ 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 $360^\circ$}
+
+@<Types...@>=
+@!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 \MF's internal computations have
+been designed to avoid this sort of error.
+
+Notice that if 64-bit integer arithmetic were available,
+we could simply compute |@t$(2^{29}$@>*p+q)div (2*q)|.
+But when we are restricted to \PASCAL's 32-bit arithmetic we
+must either resort to multiple-precision maneuvering
+or 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 \MF'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 \MF\ 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;
+ @<Compute $f=\lfloor 2^{28}(1+p/q)+{1\over2}\rfloor$@>;
+ 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<q|; (ii)~$fq+p=2^k(q+p_0)$, where $k$ is an integer and
+$p_0$ is the original value of~$p$.
+
+Notice that the computation specifies
+|(p-q)+p| instead of |(p+p)-q|, because the latter could overflow.
+Let us hope that optimizing compilers do not miss this point; a
+special variable |be_careful| is used to emphasize the necessary
+order of computation. Optimizing compilers should keep |be_careful|
+in a register, not store it in memory.
+@^inner loop@>
+
+@<Compute $f=\lfloor 2^{28}(1+p/q)+{1\over2}\rfloor$@>=
+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 \MF's computation
+time during typical jobs, so a machine-language or 64-bit
+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 @<Reduce to the case that |f>=0| and |q>=0|@>;
+if f<fraction_one then n:=0
+else begin n:=f div fraction_one; f:=f mod fraction_one;
+ if q<=el_gordo div n then n:=n*q
+ else begin arith_error:=true; n:=el_gordo;
+ end;
+ end;
+f:=f+fraction_one;
+@<Compute $p=\lfloor qf/2^{28}+{1\over2}\rfloor-q$@>;
+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;
+
+@ @<Reduce to the case that |f>=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@>
+
+@<Compute $p=\lfloor qf/2^{28}+{1\over2}\rfloor-q$@>=
+p:=fraction_half; {that's $2^{27}$; the invariants hold now with $k=28$}
+if q<fraction_four then
+ repeat if odd(f) then p:=half(p+q)@+else p:=half(p);
+ f:=half(f);
+ until f=1
+else repeat if odd(f) then p:=p+half(q-p)@+else p:=half(p);
+ f:=half(f);
+ until f=1
+
+
+@ When we want to multiply something by a |scaled| quantity, we use a scheme
+analogous to |take_fraction| but with a different scaling.
+Given positive operands, |take_scaled|
+computes the quantity $p=\lfloor qf/2^{16}+{1\over2}\rfloor$.
+
+Once again it is a good idea to use 64-bit arithmetic if
+possible; otherwise |take_scaled| will use more than 2\pct! of the running time
+when the Computer Modern fonts are being generated.
+@^inner loop@>
+
+@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 @<Reduce to the case that |f>=0| and |q>=0|@>;
+if f<unity then n:=0
+else begin n:=f div unity; f:=f mod unity;
+ if q<=el_gordo div n then n:=n*q
+ else begin arith_error:=true; n:=el_gordo;
+ end;
+ end;
+f:=f+unity;
+@<Compute $p=\lfloor qf/2^{16}+{1\over2}\rfloor-q$@>;
+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;
+
+@ @<Compute $p=\lfloor qf/2^{16}+{1\over2}\rfloor-q$@>=
+p:=half_unit; {that's $2^{15}$; the invariants hold now with $k=16$}
+@^inner loop@>
+if q<fraction_four then
+ repeat if odd(f) then p:=half(p+q)@+else p:=half(p);
+ f:=half(f);
+ until f=1
+else repeat if odd(f) then p:=p+half(q-p)@+else p:=half(p);
+ f:=half(f);
+ until f=1
+
+@ For completeness, there's also |make_scaled|, which computes a
+quotient as a |scaled| number instead of as a |fraction|.
+In other words, the result is $\lfloor2^{16}p/q+{1\over2}\rfloor$, if the
+operands are positive. \ (This procedure is not used especially often,
+so it is not part of \MF's inner loop.)
+
+@p function make_scaled(@!p,@!q:integer):scaled;
+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>=@'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;
+ @<Compute $f=\lfloor 2^{16}(1+p/q)+{1\over2}\rfloor$@>;
+ if negative then make_scaled:=-(f+n)@+else make_scaled:=f+n;
+ end;
+end;
+
+@ @<Compute $f=\lfloor 2^{16}(1+p/q)+{1\over2}\rfloor$@>=
+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 \MF'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, \MF\ 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 @<Reduce to the case that |a,c>=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;
+
+@ @<Reduce to the case that |a...@>=
+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 floor_unscaled(@!x:scaled):integer;
+ {$\lfloor x/2^{16}\rfloor$}
+var @!be_careful:integer; {temporary register}
+begin if x>=0 then floor_unscaled:=x div unity
+else begin be_careful:=x+1; floor_unscaled:=-(1+((-be_careful) div 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.
+\MF\ 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 $x=2^{46-2k}x_0\bmod 2^{30}$, $0<y=\lfloor 2^{16-2k}x_0\rfloor
+-s^2+s\L q=2s$, where $x_0$ is the initial value of $x$. The value of~$y$
+might, however, be zero at the start of the first iteration.
+
+@p function square_rt(@!x:scaled):scaled;
+var @!k:small_number; {iteration control counter}
+@!y,@!q:integer; {registers for intermediate calculations}
+begin if x<=0 then @<Handle square root of zero or negative argument@>
+else begin k:=23; q:=2;
+ while x<fraction_two do {i.e., |while x<@t$2^{29}$@>|\unskip}
+ begin decr(k); x:=x+x+x+x;
+ end;
+ if x<fraction_four then y:=0
+ else begin x:=x-fraction_four; y:=1;
+ end;
+ repeat @<Decrease |k| by 1, maintaining the invariant
+ relations between |x|, |y|, and~|q|@>;
+ until k=0;
+ square_rt:=half(q);
+ end;
+end;
+
+@ @<Handle square root of zero...@>=
+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
+
+@ @<Decrease |k| by 1, maintaining...@>=
+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 a<b then
+ begin r:=b; b:=a; a:=r;
+ end; {now |0<=b<=a|}
+if b>0 then
+ begin if a<fraction_two then big:=false
+ else begin a:=a div 4; b:=b div 4; big:=true;
+ end; {we reduced the precision to avoid arithmetic overflow}
+ @<Replace |a| by an approximation to $\psqrt{a^2+b^2}$@>;
+ if big then
+ if a<fraction_two then a:=a+a+a+a
+ else begin arith_error:=true; a:=el_gordo;
+ end;
+ end;
+pyth_add:=a;
+end;
+
+@ The key idea here is to reflect the vector $(a,b)$ about the
+line through $(a,b/2)$.
+
+@<Replace |a| by an approximation to $\psqrt{a^2+b^2}$@>=
+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 @<Handle erroneous |pyth_sub| and set |a:=0|@>
+else begin if a<fraction_four then big:=false
+ else begin a:=half(a); b:=half(b); big:=true;
+ end;
+ @<Replace |a| by an approximation to $\psqrt{a^2-b^2}$@>;
+ if big then a:=a+a;
+ end;
+pyth_sub:=a;
+end;
+
+@ @<Replace |a| by an approximation to $\psqrt{a^2-b^2}$@>=
+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:
+
+@ @<Handle erroneous |pyth_sub| and set |a:=0|@>=
+begin if a<b then
+ begin print_err("Pythagorean subtraction "); print_scaled(a);
+ print("+-+"); print_scaled(b); print(" has been replaced by 0");
+@.Pythagorean...@>
+ 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.
+
+@<Glob...@>=
+@!two_to_the:array[0..30] of integer; {powers of two}
+@!spec_log:array[1..28] of integer; {special logarithms}
+
+@ @<Local variables for initialization@>=
+@!k:integer; {all-purpose loop index}
+
+@ @<Set init...@>=
+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 @<Handle non-positive logarithm@>
+else begin y:=1302456956+4-100; {$14\times2^{27}\ln2\approx1302456956.421063$}
+ z:=27595+6553600; {and $2^{16}\times .421063\approx 27595$}
+ while x<fraction_four do
+ begin double(x); y:=y-93032639; z:=z-48782;
+ end; {$2^{27}\ln2\approx 93032639.74436163$
+ and $2^{16}\times.74436163\approx 48782$}
+ y:=y+(z div unity); k:=2;
+ while x>fraction_four+4 do
+ @<Increase |k| until |x| can be multiplied by a
+ factor of $2^{-k}$, and adjust $y$ accordingly@>;
+ m_log:=y div 8;
+ end;
+end;
+
+@ @<Increase |k| until |x| can...@>=
+begin z:=((x-1) div two_to_the[k])+1; {$z=\lceil x/2^k\rceil$}
+while x<fraction_four+z do
+ begin z:=half(z+1); k:=k+1;
+ end;
+y:=y+spec_log[k]; x:=x-z;
+end
+
+@ @<Handle non-positive logarithm@>=
+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;
+ @<Multiply |y| by $\exp(-z/2^{27})$@>;
+ 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|.
+
+@<Multiply |y| by...@>=
+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~$1/2^k$.
+
+@<Glob...@>=
+@!spec_atan:array[1..26] of angle; {$\arctan2^{-k}$ times $2^{20}\cdot180/\pi$}
+
+@ @<Set init...@>=
+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<y then
+ begin t:=y; y:=x; x:=t; octant:=octant+switch_x_and_y;
+ end;
+if x=0 then @<Handle undefined arg@>
+else begin @<Set variable |z| to the arg of $(x,y)$@>;
+ @<Return an appropriate answer based on |z| and |octant|@>;
+ end;
+end;
+
+@ @<Handle undefined arg@>=
+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
+
+@ @<Return an appropriate answer...@>=
+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.
+
+@<Set variable |z| to the arg...@>=
+while x>=fraction_two do
+ begin x:=half(x); y:=half(y);
+ end;
+z:=0;
+if y>0 then
+ begin while x<fraction_one do
+ begin double(x); double(y);
+ end;
+ @<Increase |z| to the arg of $(x,y)$@>;
+ 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.
+
+@<Increase |z|...@>=
+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|.
+
+@<Glob...@>=
+@!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<three_sixty_deg|}
+q:=z div forty_five_deg; z:=z mod forty_five_deg;
+x:=fraction_one; y:=x;
+if not odd(q) then z:=forty_five_deg-z;
+@<Subtract angle |z| from |(x,y)|@>;
+@<Convert |(x,y)| to the octant determined by~|q|@>;
+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.
+
+@<Convert |(x,...@>=
+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.
+
+@<Subtract angle |z|...@>=
+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.
+\MF\ 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 $x_n=(x_{n-55}-x_{n-24})\bmod 2^{28}$,
+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.
+
+@<Glob...@>=
+@!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]|. The |next_random| macro
+actually accesses the numbers backwards; blocks of 55~$x$'s are
+essentially being ``flipped.'' But that doesn't make them less 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:=half(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<x| or |0>=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)<u;
+x:=make_fraction(x,u);
+l:=139548960-m_log(u); {$2^{24}\cdot12\ln2\approx139548959.6165$}
+until ab_vs_cd(1024,l,x,x)>=0;
+norm_rand:=x;
+end;
+
+@* \[9] Packed data.
+In order to make efficient use of storage space, \MF\ 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. \MF\ 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 \MF\ 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):
+
+@<Check the ``constant''...@>=
+init if mem_max<>mem_top then bad:=10;@+tini@;@/
+if mem_max<mem_top then bad:=10;
+if (min_quarterword>0)or(max_quarterword<127) then bad:=11;
+if (min_halfword>0)or(max_halfword<32767) then bad:=12;
+if (min_quarterword<min_halfword)or@|
+ (max_quarterword>max_halfword) then bad:=13;
+if (mem_min<min_halfword)or(mem_max>=max_halfword) then bad:=14;
+if max_strings>max_halfword then bad:=15;
+if buf_size>max_halfword then bad:=16;
+if (max_quarterword-min_quarterword<255)or@|
+ (max_halfword-min_halfword<65535) then bad:=17;
+
+@ The operation of subtracting |min_halfword| occurs rather frequently in
+\MF, so it is convenient to abbreviate this operation by using the macro
+|ho| defined here. \MF\ 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|}
+
+@<Types...@>=
+@!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 \MF\ 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 \MF\ are handled by providing a large array |mem| in
+which consecutive blocks of words are used as nodes by the \MF\ 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 base files, by the \.{INIMF} preprocessor.
+@.INIMF@>
+Production versions of \MF\ 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:
+$$\hbox{|null=mem_min<lo_mem_max<hi_mem_min<mem_top<=mem_end<=mem_max|.}$$
+
+@<Glob...@>=
+@!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 specific applications 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,'' \MF\ will run a bit slower but it will
+report these statistics when |tracing_stats| is positive.
+
+@<Glob...@>=
+@!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}
+
+@<Glob...@>=
+@!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@>@<Declare the procedure called |show_token_list|@>@;
+@t\4@>@<Declare the procedure called |runaway|@>
+
+@ The function |get_avail| returns a pointer to a new one-word node whose
+|link| field is null. However, \MF\ 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<mem_max then {or go into virgin territory}
+ begin incr(mem_end); p:=mem_end;
+ end
+else begin decr(hi_mem_min); p:=hi_mem_min;
+ if hi_mem_min<=lo_mem_max then
+ begin runaway; {if memory is exhausted, display possible runaway text}
+ overflow("main memory size",mem_max+1-mem_min);
+ {quit; all one-word nodes are busy}
+@:METAFONT capacity exceeded main memory size}{\quad main memory size@>
+ 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<max_halfword| because terrible things can happen
+when |max_halfword| appears in the |link| field of a nonempty node.)
+
+@d empty_flag == max_halfword {the |link| of an empty variable-size node}
+@d is_empty(#) == (link(#)=empty_flag) {tests for empty node}
+@d node_size == info {the size field in empty variable-size nodes}
+@d llink(#) == info(#+1) {left link in doubly-linked list of empty nodes}
+@d rlink(#) == link(#+1) {right link in doubly-linked list of empty nodes}
+
+@<Glob...@>=
+@!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 @<Try to allocate within node |p| and its physical successors,
+ and |goto found| if allocation was possible@>;
+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<hi_mem_min then if lo_mem_max+2<=mem_min+max_halfword then
+ @<Grow more variable-size memory and |goto restart|@>;
+overflow("main memory size",mem_max+1-mem_min);
+ {sorry, nothing satisfactory is left}
+@:METAFONT 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 \MF\ is
+implemented on ``virtual memory'' systems.
+@^virtual memory@>
+
+@<Grow more variable-size memory and |goto restart|@>=
+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<=t<hi_mem_min|}
+if t>mem_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
+
+@ @<Try to allocate...@>=
+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 @<Allocate from the top of node |p| and |goto found|@>;
+if r=p then if rlink(p)<>p then
+ @<Allocate entire node |p| and |goto found|@>;
+node_size(p):=q-p {reset the size in case it grew}
+
+@ @<Allocate from the top...@>=
+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.
+
+@<Allocate entire...@>=
+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 \.{INIMF} 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 @<Sort |p| into the list starting at |rover|
+ and advance |p| to |rlink(p)|@>;
+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.
+
+@<Sort |p|...@>=
+if p<rover then
+ begin q:=p; p:=rlink(q); rlink(q):=rover; rover:=q;
+ end
+else begin q:=rover;
+ while rlink(q)<p do q:=rlink(q);
+ r:=rlink(p); rlink(p):=rlink(q); rlink(q):=p; p:=r;
+ end
+
+@* \[11] Memory layout.
+Some areas of |mem| are dedicated to fixed usage, since static allocation is
+more efficient than dynamic allocation when we can get away with it. For
+example, locations |mem_min| to |mem_min+2| are always used to store the
+specification for null pen coordinates that are `$(0,0)$'. The
+following macro definitions accomplish the static allocation by giving
+symbolic names to the fixed positions. Static variable-size nodes appear
+in locations |mem_min| through |lo_mem_stat_max|, and static single-word nodes
+appear in locations |hi_mem_stat_min| through |mem_top|, inclusive.
+
+@d null_coords==mem_min {specification for pen offsets of $(0,0)$}
+@d null_pen==null_coords+3 {we will define |coord_node_size=3|}
+@d dep_head==null_pen+10 {and |pen_node_size=10|}
+@d zero_val==dep_head+2 {two words for a permanently zero value}
+@d temp_val==zero_val+2 {two words for a temporary value node}
+@d end_attr==temp_val {we use |end_attr+2| only}
+@d inf_val==end_attr+2 {and |inf_val+1| only}
+@d bad_vardef==inf_val+2 {two words for \&{vardef} error recovery}
+@d lo_mem_stat_max==bad_vardef+1 {largest statically
+ allocated word in the variable-size |mem|}
+@#
+@d sentinel==mem_top {end of sorted lists}
+@d temp_head==mem_top-1 {head of a temporary list of some kind}
+@d hold_head==mem_top-2 {head of a temporary list of another kind}
+@d hi_mem_stat_min==mem_top-2 {smallest statically allocated word in
+ the one-word |mem|}
+
+@ The following code gets the dynamic part of |mem| off to a good start,
+when \MF\ is initializing itself the slow way.
+
+@<Initialize table entries (done by \.{INIMF} only)@>=
+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_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 r<hi_mem_min then goto done;
+ until r=sentinel;
+ done: {now |q| is the last node on the list}
+ link(q):=avail; avail:=p;
+ end;
+end;
+@#
+procedure flush_node_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_node(q,2)@+else free_avail(q);
+ end;
+end;
+
+@ If \MF\ is extended improperly, the |mem| array might get screwed up.
+For example, some pointers might be wrong, or some ``dead'' nodes might not
+have been freed when the last reference to them disappeared. Procedures
+|check_mem| and |search_mem| are available to help diagnose such
+problems. These procedures make use of two arrays called |free| and
+|was_free| that are present only if \MF's debugging routines have
+been included. (You may want to decrease the size of |mem| while you
+@^debugging@>
+are debugging.)
+
+@<Glob...@>=
+@!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
+
+@ @<Set initial...@>=
+@!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; {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}
+@<Check single-word |avail| list@>;
+@<Check variable-size |avail| list@>;
+@<Check flags of unavailable nodes@>;
+@<Check the list of linear dependencies@>;
+if print_locs then @<Print newly busy locations@>;
+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
+
+@ @<Check single-word...@>=
+p:=avail; q:=null; clobbered:=false;
+while p<>null do
+ begin if (p>mem_end)or(p<hi_mem_min) then clobbered:=true
+ else if free[p] then clobbered:=true;
+ if clobbered then
+ begin print_nl("AVAIL list clobbered at ");
+@.AVAIL list clobbered...@>
+ print_int(q); goto done1;
+ end;
+ free[p]:=true; q:=p; p:=link(q);
+ end;
+done1:
+
+@ @<Check variable-size...@>=
+p:=rover; q:=null; clobbered:=false;
+repeat if (p>=lo_mem_max)or(p<mem_min) then clobbered:=true
+ else if (rlink(p)>=lo_mem_max)or(rlink(p)<mem_min) then clobbered:=true
+ else if not(is_empty(p))or(node_size(p)<2)or@|
+ (p+node_size(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:
+
+@ @<Check flags...@>=
+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
+
+@ @<Print newly busy...@>=
+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
+ begin print_char(" "); print_int(p);
+ end;
+for p:=hi_mem_min to mem_end do
+ if not free[p] and
+ ((p<was_hi_min) or (p>was_mem_end) or was_free[p]) then
+ begin print_char(" "); print_int(p);
+ end;
+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;
+@<Search |eqtb| for equivalents equal to |p|@>;
+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 \MF. 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. The ordering of the highest-numbered
+commands (|comma<semicolon<end_group<stop|) is crucial for the parsing
+and error-recovery methods of this program.
+
+At any rate, here is the list, for future reference.
+
+@d if_test=1 {conditional text (\&{if})}
+@d fi_or_else=2 {delimiters for conditionals (\&{elseif}, \&{else}, \&{fi})}
+@d input=3 {input a source file (\&{input}, \&{endinput})}
+@d iteration=4 {iterate (\&{for}, \&{forsuffixes}, \&{forever}, \&{endfor})}
+@d repeat_loop=5 {special command substituted for \&{endfor}}
+@d exit_test=6 {premature exit from a loop (\&{exitif})}
+@d relax=7 {do nothing (\.{\char`\\})}
+@d scan_tokens=8 {put a string into the input buffer}
+@d expand_after=9 {look ahead one token}
+@d defined_macro=10 {a macro defined by the user}
+@d min_command=defined_macro+1
+@d display_command=11 {online graphic output (\&{display})}
+@d save_command=12 {save a list of tokens (\&{save})}
+@d interim_command=13 {save an internal quantity (\&{interim})}
+@d let_command=14 {redefine a symbolic token (\&{let})}
+@d new_internal=15 {define a new internal quantity (\&{newinternal})}
+@d macro_def=16 {define a macro (\&{def}, \&{vardef}, etc.)}
+@d ship_out_command=17 {output a character (\&{shipout})}
+@d add_to_command=18 {add to edges (\&{addto})}
+@d cull_command=19 {cull and normalize edges (\&{cull})}
+@d tfm_command=20 {command for font metric info (\&{ligtable}, etc.)}
+@d protection_command=21 {set protection flag (\&{outer}, \&{inner})}
+@d show_command=22 {diagnostic output (\&{show}, \&{showvariable}, etc.)}
+@d mode_command=23 {set interaction level (\&{batchmode}, etc.)}
+@d random_seed=24 {initialize random number generator (\&{randomseed})}
+@d message_command=25 {communicate to user (\&{message}, \&{errmessage})}
+@d every_job_command=26 {designate a starting token (\&{everyjob})}
+@d delimiters=27 {define a pair of delimiters (\&{delimiters})}
+@d open_window=28 {define a window on the screen (\&{openwindow})}
+@d special_command=29 {output special info (\&{special}, \&{numspecial})}
+@d type_name=30 {declare a type (\&{numeric}, \&{pair}, etc.)}
+@d max_statement_command=type_name
+@d min_primary_command=type_name
+@d left_delimiter=31 {the left delimiter of a matching pair}
+@d begin_group=32 {beginning of a group (\&{begingroup})}
+@d nullary=33 {an operator without arguments (e.g., \&{normaldeviate})}
+@d unary=34 {an operator with one argument (e.g., \&{sqrt})}
+@d str_op=35 {convert a suffix to a string (\&{str})}
+@d cycle=36 {close a cyclic path (\&{cycle})}
+@d primary_binary=37 {binary operation taking `\&{of}' (e.g., \&{point})}
+@d capsule_token=38 {a value that has been put into a token list}
+@d string_token=39 {a string constant (e.g., |"hello"|)}
+@d internal_quantity=40 {internal numeric parameter (e.g., \&{pausing})}
+@d min_suffix_token=internal_quantity
+@d tag_token=41 {a symbolic token without a primitive meaning}
+@d numeric_token=42 {a numeric constant (e.g., \.{3.14159})}
+@d max_suffix_token=numeric_token
+@d plus_or_minus=43 {either `\.+' or `\.-'}
+@d max_primary_command=plus_or_minus {should also be |numeric_token+1|}
+@d min_tertiary_command=plus_or_minus
+@d tertiary_secondary_macro=44 {a macro defined by \&{secondarydef}}
+@d tertiary_binary=45 {an operator at the tertiary level (e.g., `\.{++}')}
+@d max_tertiary_command=tertiary_binary
+@d left_brace=46 {the operator `\.{\char`\{}'}
+@d min_expression_command=left_brace
+@d path_join=47 {the operator `\.{..}'}
+@d ampersand=48 {the operator `\.\&'}
+@d expression_tertiary_macro=49 {a macro defined by \&{tertiarydef}}
+@d expression_binary=50 {an operator at the expression level (e.g., `\.<')}
+@d equals=51 {the operator `\.='}
+@d max_expression_command=equals
+@d and_command=52 {the operator `\&{and}'}
+@d min_secondary_command=and_command
+@d secondary_primary_macro=53 {a macro defined by \&{primarydef}}
+@d slash=54 {the operator `\./'}
+@d secondary_binary=55 {an operator at the binary level (e.g., \&{shifted})}
+@d max_secondary_command=secondary_binary
+@d param_type=56 {type of parameter (\&{primary}, \&{expr}, \&{suffix}, etc.)}
+@d controls=57 {specify control points explicitly (\&{controls})}
+@d tension=58 {specify tension between knots (\&{tension})}
+@d at_least=59 {bounded tension value (\&{atleast})}
+@d curl_command=60 {specify curl at an end knot (\&{curl})}
+@d macro_special=61 {special macro operators (\&{quote}, \.{\#\AT!}, etc.)}
+@d right_delimiter=62 {the right delimiter of a matching pair}
+@d left_bracket=63 {the operator `\.['}
+@d right_bracket=64 {the operator `\.]'}
+@d right_brace=65 {the operator `\.{\char`\}}'}
+@d with_option=66 {option for filling (\&{withpen}, \&{withweight})}
+@d cull_op=67 {the operator `\&{keeping}' or `\&{dropping}'}
+@d thing_to_add=68
+ {variant of \&{addto} (\&{contour}, \&{doublepath}, \&{also})}
+@d of_token=69 {the operator `\&{of}'}
+@d from_token=70 {the operator `\&{from}'}
+@d to_token=71 {the operator `\&{to}'}
+@d at_token=72 {the operator `\&{at}'}
+@d in_window=73 {the operator `\&{inwindow}'}
+@d step_token=74 {the operator `\&{step}'}
+@d until_token=75 {the operator `\&{until}'}
+@d lig_kern_token=76
+ {the operators `\&{kern}' and `\.{=:}' and `\.{=:\char'174}', etc.}
+@d assignment=77 {the operator `\.{:=}'}
+@d skip_to=78 {the operation `\&{skipto}'}
+@d bchar_label=79 {the operator `\.{\char'174\char'174:}'}
+@d double_colon=80 {the operator `\.{::}'}
+@d colon=81 {the operator `\.:'}
+@#
+@d comma=82 {the operator `\.,', must be |colon+1|}
+@d end_of_statement==cur_cmd>comma
+@d semicolon=83 {the operator `\.;', must be |comma+1|}
+@d end_group=84 {end a group (\&{endgroup}), must be |semicolon+1|}
+@d stop=85 {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}
+
+@<Types...@>=
+@!command_code=1..max_command_code;
+
+@ Variables and capsules in \MF\ have a variety of ``types,''
+distinguished by the following code numbers:
+
+@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 future_pen=8 {subexpression that will become a \&{pen} at a higher level}
+@d path_type=9 {\&{path} with a known value}
+@d unknown_path=path_type+unknown_tag
+@d picture_type=11 {\&{picture} with a known value}
+@d unknown_picture=picture_type+unknown_tag
+@d transform_type=13 {\&{transform} 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
+
+@<Basic printing procedures@>=
+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");
+future_pen:print("future pen");
+path_type:print("path");
+unknown_path:print("unknown path");
+picture_type:print("picture");
+unknown_picture:print("unknown picture");
+transform_type:print("transform");
+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 \MF\ 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 capsule=11 {|name_type| in stashed-away subexpressions}
+@d token=12 {|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.
+
+@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 odd_op=38 {operation code for \.{odd}}
+@d known_op=39 {operation code for \.{known}}
+@d unknown_op=40 {operation code for \.{unknown}}
+@d not_op=41 {operation code for \.{not}}
+@d decimal=42 {operation code for \.{decimal}}
+@d reverse=43 {operation code for \.{reverse}}
+@d make_path_op=44 {operation code for \.{makepath}}
+@d make_pen_op=45 {operation code for \.{makepen}}
+@d total_weight_op=46 {operation code for \.{totalweight}}
+@d oct_op=47 {operation code for \.{oct}}
+@d hex_op=48 {operation code for \.{hex}}
+@d ASCII_op=49 {operation code for \.{ASCII}}
+@d char_op=50 {operation code for \.{char}}
+@d length_op=51 {operation code for \.{length}}
+@d turning_op=52 {operation code for \.{turningnumber}}
+@d x_part=53 {operation code for \.{xpart}}
+@d y_part=54 {operation code for \.{ypart}}
+@d xx_part=55 {operation code for \.{xxpart}}
+@d xy_part=56 {operation code for \.{xypart}}
+@d yx_part=57 {operation code for \.{yxpart}}
+@d yy_part=58 {operation code for \.{yypart}}
+@d sqrt_op=59 {operation code for \.{sqrt}}
+@d m_exp_op=60 {operation code for \.{mexp}}
+@d m_log_op=61 {operation code for \.{mlog}}
+@d sin_d_op=62 {operation code for \.{sind}}
+@d cos_d_op=63 {operation code for \.{cosd}}
+@d floor_op=64 {operation code for \.{floor}}
+@d uniform_deviate=65 {operation code for \.{uniformdeviate}}
+@d char_exists_op=66 {operation code for \.{charexists}}
+@d angle_op=67 {operation code for \.{angle}}
+@d cycle_op=68 {operation code for \.{cycle}}
+@d plus=69 {operation code for \.+}
+@d minus=70 {operation code for \.-}
+@d times=71 {operation code for \.*}
+@d over=72 {operation code for \./}
+@d pythag_add=73 {operation code for \.{++}}
+@d pythag_sub=74 {operation code for \.{+-+}}
+@d or_op=75 {operation code for \.{or}}
+@d and_op=76 {operation code for \.{and}}
+@d less_than=77 {operation code for \.<}
+@d less_or_equal=78 {operation code for \.{<=}}
+@d greater_than=79 {operation code for \.>}
+@d greater_or_equal=80 {operation code for \.{>=}}
+@d equal_to=81 {operation code for \.=}
+@d unequal_to=82 {operation code for \.{<>}}
+@d concatenate=83 {operation code for \.\&}
+@d rotated_by=84 {operation code for \.{rotated}}
+@d slanted_by=85 {operation code for \.{slanted}}
+@d scaled_by=86 {operation code for \.{scaled}}
+@d shifted_by=87 {operation code for \.{shifted}}
+@d transformed_by=88 {operation code for \.{transformed}}
+@d x_scaled=89 {operation code for \.{xscaled}}
+@d y_scaled=90 {operation code for \.{yscaled}}
+@d z_scaled=91 {operation code for \.{zscaled}}
+@d intersect=92 {operation code for \.{intersectiontimes}}
+@d double_dot=93 {operation code for improper \.{..}}
+@d substring_of=94 {operation code for \.{substring}}
+@d min_of=substring_of
+@d subpath_of=95 {operation code for \.{subpath}}
+@d direction_time_of=96 {operation code for \.{directiontime}}
+@d point_of=97 {operation code for \.{point}}
+@d precontrol_of=98 {operation code for \.{precontrol}}
+@d postcontrol_of=99 {operation code for \.{postcontrol}}
+@d pen_offset_of=100 {operation code for \.{penoffset}}
+
+@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");
+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");
+total_weight_op:print("totalweight");
+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");
+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");
+angle_op:print("angle");
+cycle_op:print("cycle");
+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");
+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");
+othercases print("..")
+endcases;
+end;
+
+@ \MF\ 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 subdivision of paths into octants before digitizing}
+@d tracing_pens=6 {show details of pens that are made}
+@d tracing_commands=7 {show commands and operations before they are performed}
+@d tracing_restores=8 {show when a variable or internal is restored}
+@d tracing_macros=9 {show macros before they are expanded}
+@d tracing_edges=10 {show digitized edges as they are computed}
+@d tracing_output=11 {show digitized edges as they are output}
+@d tracing_stats=12 {show memory usage at end of job}
+@d tracing_online=13 {show long diagnostics on terminal and in the log file}
+@d year=14 {the current year (e.g., 1984)}
+@d month=15 {the current month (e.g., 3 $\equiv$ March)}
+@d day=16 {the current day of the month}
+@d time=17 {the number of minutes past midnight when this job started}
+@d char_code=18 {the number of the next character to be output}
+@d char_ext=19 {the extension code of the next character to be output}
+@d char_wd=20 {the width of the next character to be output}
+@d char_ht=21 {the height of the next character to be output}
+@d char_dp=22 {the depth of the next character to be output}
+@d char_ic=23 {the italic correction of the next character to be output}
+@d char_dx=24 {the device's $x$ movement for the next character, in pixels}
+@d char_dy=25 {the device's $y$ movement for the next character, in pixels}
+@d design_size=26 {the unit of measure used for |char_wd..char_ic|, in points}
+@d hppp=27 {the number of horizontal pixels per point}
+@d vppp=28 {the number of vertical pixels per point}
+@d x_offset=29 {horizontal displacement of shipped-out characters}
+@d y_offset=30 {vertical displacement of shipped-out characters}
+@d pausing=31 {positive to display lines on the terminal before they are read}
+@d showstopping=32 {positive to stop after each \&{show} command}
+@d fontmaking=33 {positive if font metric output is to be produced}
+@d proofing=34 {positive for proof mode, negative to suppress output}
+@d smoothing=35 {positive if moves are to be ``smoothed''}
+@d autorounding=36 {controls path modification to ``good'' points}
+@d granularity=37 {autorounding uses this pixel size}
+@d fillin=38 {extra darkness of diagonal lines}
+@d turning_check=39 {controls reorientation of clockwise paths}
+@d warning_check=40 {controls error message when variable value is large}
+@d boundary_char=41 {the boundary character for ligatures}
+@d max_given_internal=41
+
+@<Glob...@>=
+@!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}
+
+@ @<Set init...@>=
+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 \MF'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.
+
+@<Put each of \MF's primitives into the hash table@>=
+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("tracingpens",internal_quantity,tracing_pens);@/
+@!@:tracing_pens_}{\&{tracingpens} 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("tracingedges",internal_quantity,tracing_edges);@/
+@!@:tracing_edges_}{\&{tracingedges} primitive@>
+primitive("tracingoutput",internal_quantity,tracing_output);@/
+@!@:tracing_output_}{\&{tracingoutput} primitive@>
+primitive("tracingstats",internal_quantity,tracing_stats);@/
+@!@:tracing_stats_}{\&{tracingstats} 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("chardx",internal_quantity,char_dx);@/
+@!@:char_dx_}{\&{chardx} primitive@>
+primitive("chardy",internal_quantity,char_dy);@/
+@!@:char_dy_}{\&{chardy} primitive@>
+primitive("designsize",internal_quantity,design_size);@/
+@!@:design_size_}{\&{designsize} primitive@>
+primitive("hppp",internal_quantity,hppp);@/
+@!@:hppp_}{\&{hppp} primitive@>
+primitive("vppp",internal_quantity,vppp);@/
+@!@:vppp_}{\&{vppp} primitive@>
+primitive("xoffset",internal_quantity,x_offset);@/
+@!@:x_offset_}{\&{xoffset} primitive@>
+primitive("yoffset",internal_quantity,y_offset);@/
+@!@:y_offset_}{\&{yoffset} 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("proofing",internal_quantity,proofing);@/
+@!@:proofing_}{\&{proofing} primitive@>
+primitive("smoothing",internal_quantity,smoothing);@/
+@!@:smoothing_}{\&{smoothing} primitive@>
+primitive("autorounding",internal_quantity,autorounding);@/
+@!@:autorounding_}{\&{autorounding} primitive@>
+primitive("granularity",internal_quantity,granularity);@/
+@!@:granularity_}{\&{granularity} primitive@>
+primitive("fillin",internal_quantity,fillin);@/
+@!@:fillin_}{\&{fillin} primitive@>
+primitive("turningcheck",internal_quantity,turning_check);@/
+@!@:turning_check_}{\&{turningcheck} primitive@>
+primitive("warningcheck",internal_quantity,warning_check);@/
+@!@:warning_check_}{\&{warningcheck} primitive@>
+primitive("boundarychar",internal_quantity,boundary_char);@/
+@!@:boundary_char_}{\&{boundarychar} primitive@>
+
+@ Well, we do have to list the names one more time, for use in symbolic
+printouts.
+
+@<Initialize table...@>=
+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_pens]:="tracingpens";
+int_name[tracing_commands]:="tracingcommands";
+int_name[tracing_restores]:="tracingrestores";
+int_name[tracing_macros]:="tracingmacros";
+int_name[tracing_edges]:="tracingedges";
+int_name[tracing_output]:="tracingoutput";
+int_name[tracing_stats]:="tracingstats";
+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[char_dx]:="chardx";
+int_name[char_dy]:="chardy";
+int_name[design_size]:="designsize";
+int_name[hppp]:="hppp";
+int_name[vppp]:="vppp";
+int_name[x_offset]:="xoffset";
+int_name[y_offset]:="yoffset";
+int_name[pausing]:="pausing";
+int_name[showstopping]:="showstopping";
+int_name[fontmaking]:="fontmaking";
+int_name[proofing]:="proofing";
+int_name[smoothing]:="smoothing";
+int_name[autorounding]:="autorounding";
+int_name[granularity]:="granularity";
+int_name[fillin]:="fillin";
+int_name[turning_check]:="turningcheck";
+int_name[warning_check]:="warningcheck";
+int_name[boundary_char]:="boundarychar";
+
+@ The following procedure, which is called just before \MF\ 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 assumes that suitable values appear in
+the global variables \\{sys\_time}, \\{sys\_day}, \\{sys\_month}, and
+\\{sys\_year} (which are initialized to noon on 4 July 1776,
+in case the implementor is careless).
+
+Note that the values are |scaled| integers. Hence \MF\ can no longer
+be used after the year 32767.
+
+@p procedure fix_date_and_time;
+begin sys_time:=12*60;
+sys_day:=4; sys_month:=7; sys_year:=1776; {self-evident truths}
+internal[time]:=sys_time*unity; {minutes since midnight}
+internal[day]:=sys_day*unity; {day of the month}
+internal[month]:=sys_month*unity; {month of the year}
+internal[year]:=sys_year*unity; {Anno Domini}
+end;
+
+@ \MF\ 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:
+
+@<Basic printing...@>=
+procedure begin_diagnostic; {prepare to do some tracing}
+begin old_setting:=selector;
+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;
+
+@ Of course we had better declare a few more global variables, if the previous
+routines are going to work.
+
+@<Glob...@>=
+@!old_setting:0..max_selector;
+@!sys_time,@!sys_day,@!sys_month,@!sys_year:integer;
+ {date and time supplied by external system}
+
+@ We will occasionally use |begin_diagnostic| in connection with line-number
+printing, as follows. (The parameter |s| is typically |"Path"| or
+|"Cycle spec"|, etc.)
+
+@<Basic printing...@>=
+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(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 \MF.
+
+@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}
+
+@<Glob...@>=
+@!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@>
+
+@<Set init...@>=
+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?}
+
+@<Glob...@>=
+@!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_bad_vardef==hash_top+11 {|hash| location of `\.{a bad variable}'}
+@d frozen_undefined==hash_top+12 {|hash| location that never gets defined}
+@d hash_end==hash_top+12 {the actual size of the |hash| and |eqtb| arrays}
+
+@<Glob...@>=
+@!hash: array[1..hash_end] of two_halves; {the hash table}
+@!eqtb: array[1..hash_end] of two_halves; {the equivalents}
+
+@ @<Set init...@>=
+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;
+
+@ @<Initialize table entries...@>=
+hash_used:=frozen_inaccessible; {nothing is used}
+st_count:=0;@/
+text(frozen_bad_vardef):="a bad variable";
+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;
+
+@ @<Check the ``constant'' values...@>=
+if hash_end+max_internal>max_halfword then bad:=21;
+
+@ 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 @<Treat special case of length 1 and |goto found|@>;
+@<Compute the hash code |h|@>;
+p:=h+hash_base; {we start searching here; note that |0<=h<hash_prime|}
+loop@+ begin if text(p)>0 then if length(text(p))=l then
+ if str_eq_buf(text(p),j) then goto found;
+ if next(p)=0 then
+ @<Insert a new symbolic token after |p|, then
+ make |p| point to it and |goto found|@>;
+ p:=next(p);
+ end;
+found: id_lookup:=p;
+end;
+
+@ @<Treat special case of length 1...@>=
+begin p:=buffer[j]+1; text(p):=p-1; goto found;
+end
+
+@ @<Insert a new symbolic...@>=
+begin if text(p)>0 then
+ begin repeat if hash_is_full then
+ overflow("hash size",hash_size);
+@:METAFONT 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@>
+
+@<Compute the hash code |h|@>=
+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
+
+@ @<Search |eqtb| for equivalents equal to |p|@>=
+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 \MF'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 \MF\ 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_start[s+1]-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(str_ptr-1); text(cur_sym):=s;
+ end;
+eq_type(cur_sym):=c; equiv(cur_sym):=o;
+end;
+tini
+
+@ Many of \MF's primitives need no |equiv|, since they are identifiable
+by their |eq_type| alone. These primitives are loaded into the hash table
+as follows:
+
+@<Put each of \MF's primitives into the hash table@>=
+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("at",at_token,0);@/
+@!@:at_}{\&{at} 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("cull",cull_command,0);@/
+@!@:cull_}{\&{cull} primitive@>
+primitive("curl",curl_command,0);@/
+@!@:curl_}{\&{curl} primitive@>
+primitive("delimiters",delimiters,0);@/
+@!@:delimiters_}{\&{delimiters} primitive@>
+primitive("display",display_command,0);@/
+@!@:display_}{\&{display} 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("from",from_token,0);@/
+@!@:from_}{\&{from} primitive@>
+primitive("inwindow",in_window,0);@/
+@!@:in_window_}{\&{inwindow} 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("openwindow",open_window,0);@/
+@!@:open_window_}{\&{openwindow} 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("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@>
+
+@ 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.
+
+@<Cases of |print_cmd_mod| for symbolic printing of primitives@>=
+add_to_command:print("addto");
+assignment:print(":=");
+at_least:print("atleast");
+at_token:print("at");
+bchar_label:print("||:");
+begin_group:print("begingroup");
+colon:print(":");
+comma:print(",");
+controls:print("controls");
+cull_command:print("cull");
+curl_command:print("curl");
+delimiters:print("delimiters");
+display_command:print("display");
+double_colon:print("::");
+end_group:print("endgroup");
+every_job_command:print("everyjob");
+exit_test:print("exitif");
+expand_after:print("expandafter");
+from_token:print("from");
+in_window:print("inwindow");
+interim_command:print("interim");
+left_brace:print("{");
+left_bracket:print("[");
+let_command:print("let");
+new_internal:print("newinternal");
+of_token:print("of");
+open_window:print("openwindow");
+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");
+step_token:print("step");
+str_op:print("str");
+tension:print("tension");
+to_token:print("to");
+until_token:print("until");
+
+@ 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 \MF\ 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<param_size|.)
+Actual values of these parameters are kept in a separate stack, as we will
+see later. The constants |expr_base|, |suffix_base|, and |text_base| are,
+of course, chosen so that there will be no confusion between symbolic
+tokens and parameters of various types.
+
+It turns out that |value(null)=0|, because |null=null_coords|;
+we will make use of this coincidence later.
+
+Incidentally, while we're speaking of coincidences, we might note that
+the `\\{type}' field of a node has nothing to do with ``type'' in a
+printer's sense. It's curious that the same word is used in such different ways.
+
+@d type(#) == mem[#].hh.b0 {identifies what kind of value this is}
+@d name_type(#) == mem[#].hh.b1 {a clue to the name of this value}
+@d token_node_size=2 {the number of words in a large token node}
+@d value_loc(#)==#+1 {the word that contains the |value| field}
+@d value(#)==mem[value_loc(#)].int {the value stored in a large token node}
+@d expr_base==hash_end+1 {code for the zeroth \&{expr} parameter}
+@d suffix_base==expr_base+param_size {code for the zeroth \&{suffix} parameter}
+@d text_base==suffix_base+param_size {code for the zeroth \&{text} parameter}
+
+@<Check the ``constant''...@>=
+if text_base+param_size>max_halfword then bad:=22;
+
+@ 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,future_pen,picture_type,
+ pair_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}'.
+
+@<Declare the procedure called |show_token_list|@>=
+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<l) do
+ begin if p=q then @<Do magic computation@>;
+ @<Display token |p| and set |c| to its class;
+ but |return| if there are problems@>;
+ class:=c; p:=link(p);
+ end;
+if p<>null then print(" ETC.");
+@.ETC@>
+exit:
+end;
+
+@ @<Display token |p| and set |c| to its class...@>=
+c:=letter_class; {the default}
+if (p<mem_min)or(p>mem_end) then
+ begin print(" CLOBBERED"); return;
+@.CLOBBERED@>
+ end;
+if p<hi_mem_min then @<Display two-word token@>
+else begin r:=info(p);
+ if r>=expr_base then @<Display a parameter token@>
+ else if r<1 then
+ if r=0 then @<Display a collective subscript@>
+ else print(" IMPOSSIBLE")
+@.IMPOSSIBLE@>
+ else begin r:=text(r);
+ if (r<0)or(r>=str_ptr) then print(" NONEXISTENT")
+@.NONEXISTENT@>
+ else @<Print string |r| as a symbolic token
+ and set |c| to its class@>;
+ end;
+ end
+
+@ @<Display two-word token@>=
+if name_type(p)=token then
+ if type(p)=known then @<Display a numeric token@>
+ else if type(p)<>string_type then print(" BAD")
+@.BAD@>
+ else begin print_char(""""); slow_print(value(p)); print_char("""");
+ c:=string_class;
+ end
+else if (name_type(p)<>capsule)or(type(p)<vacuous)or(type(p)>independent) then
+ print(" BAD")
+else begin g_pointer:=p; print_capsule; c:=right_paren_class;
+ end
+
+@ @<Display a numeric token@>=
+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 definition of attribute nodes) that
+it is convenient to let |info(p)=0| stand for `\.{[]}'.
+
+@<Display a collective subscript@>=
+begin if class=left_bracket_class then print_char(" ");
+print("[]"); c:=right_bracket_class;
+end
+
+@ @<Display a parameter token@>=
+begin if r<suffix_base then
+ begin print("(EXPR"); r:=r-(expr_base);
+@.EXPR@>
+ end
+else if r<text_base then
+ begin print("(SUFFIX"); r:=r-(suffix_base);
+@.SUFFIX@>
+ end
+else begin print("(TEXT"); r:=r-(text_base);
+@.TEXT@>
+ end;
+print_int(r); print_char(")"); c:=right_paren_class;
+end
+
+@ @<Print string |r| as a symbolic token...@>=
+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;
+slow_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@>
+
+@<Declare miscellaneous procedures that were declared |forward|@>=
+procedure print_capsule;
+begin print_char("("); print_exp(g_pointer,0); print_char(")");
+end;
+@#
+procedure token_recycle;
+begin recycle_value(g_pointer);
+end;
+
+@ @<Glob...@>=
+@!g_pointer:pointer; {(global) parameter to the |forward| procedures}
+
+@ Macro definitions are kept in \MF'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 consist of zero or more parameter tokens followed
+by a code for the type of macro.
+
+@d ref_count==info {reference count preceding a macro definition or pen 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@>@<Declare the procedure called |print_cmd_mod|@>@;
+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("<expr>->");
+of_macro:print("<expr>of<primary>->");
+suffix_macro:print("<suffix>->");
+text_macro:print("<text>->");
+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 \MF\ programs can be simple, like `\.x', or they can
+combine the structural properties of arrays and records, like `\.{x20a.b}'.
+A \MF\ user assigns a type to a variable like \.{x20a.b} by saying, for
+example, `\.{boolean} \.{x[]a.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 \MF\ 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 \MF\ 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
+has the fields of 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))=tag_token| 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 three-word ``attribute-as-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|,
+|attr_loc(qq)=0|, |parent(qq)=p|, 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=link(qqq1)|, as you can well imagine.
+Similarly, the value of `\.{x.a}' appears in node |q2=link(q1)|, where
+|attr_loc(q2)=h(a)| and |parent(q2)=p|.
+
+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 `\.{[]}'}
+
+@<Initialize table...@>=
+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|.
+
+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 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}
+
+@<Glob...@>=
+@!big_node_size:array[transform_type..pair_type] of small_number;
+
+@ The |big_node_size| array simply contains two constants that \MF\
+occasionally needs to know.
+
+@<Set init...@>=
+big_node_size[transform_type]:=transform_node_size;
+big_node_size[pair_type]:=pair_node_size;
+
+@ 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; @<Make variable |q+s| newly independent@>;
+name_type(q+s):=half(s)+x_part_sector; 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
+ @<Preface the output with a part specifier; |return| in the
+ case of a capsule@>;
+q:=null;
+while name_type(p)>saved_root do
+ @<Ascend one level, pushing a token onto list |q|
+ and replacing |p| by its parent@>;
+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;
+
+@ @<Ascend one level, pushing a token onto list |q|...@>=
+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
+
+@ @<Preface the output with a part specifier...@>=
+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");
+capsule: begin print("%CAPSULE"); print_int(p-null); return;
+@.CAPSULE@>
+ end;
+end; {there are no other cases}
+print("part "); p:=link(p-2*(name_type(p)-x_part_sector));
+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-2*(t-x_part_sector)));
+ 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: @<Link a new subscript node |r| in place of node |p|@>;
+attr: @<Link a new attribute node |r| in place of node |p|@>;
+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;
+
+@ @<Link a new subscript node |r| in place of node |p|@>=
+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.
+
+@<Link a new attribute node |r| in place of node |p|@>=
+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<type(p)<structured|.
+
+@d abort_find==begin find_variable:=null; return;@+end
+
+@p function find_variable(@!t:pointer):pointer;
+label exit;
+var @!p,@!q,@!r,@!s:pointer; {nodes in the ``value'' line}
+@!pp,@!qq,@!rr,@!ss:pointer; {nodes in the ``collective'' line}
+@!n:integer; {subscript or attribute}
+@!save_word:memory_word; {temporary storage for a word of |mem|}
+@^inner loop@>
+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 @<Make sure that both nodes |p| and |pp| are of |structured| type@>;
+ if t<hi_mem_min then
+ @<Descend one level for the subscript |value(t)|@>
+ else @<Descend one level for the attribute |info(t)|@>;
+ 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.
+
+@<Make sure that both nodes |p| and |pp|...@>=
+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.
+
+@<Descend one level for the subscript |value(t)|@>=
+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
+
+@ @<Descend one level for the attribute |info(t)|@>=
+begin n:=info(t);
+ss:=attr_head(pp);
+repeat rr:=ss; ss:=link(ss);
+until n<=attr_loc(ss);
+if n<attr_loc(ss) then
+ begin qq:=get_node(attr_node_size); link(rr):=qq; link(qq):=ss;
+ attr_loc(qq):=n; name_type(qq):=attr; type(qq):=undefined;
+ parent(qq):=pp; ss:=qq;
+ end;
+if p=pp then
+ begin p:=ss; pp:=ss;
+ end
+else begin pp:=ss; s:=attr_head(p);
+ repeat r:=s; s:=link(s);
+ until n<=attr_loc(s);
+ if n=attr_loc(s) then p:=s
+ else begin q:=get_node(attr_node_size); link(r):=q; link(q):=s;
+ attr_loc(q):=n; name_type(q):=attr; type(q):=undefined;
+ parent(q):=p; p:=q;
+ end;
+ end;
+end
+
+@ Variables lose their former values when they appear in a type declaration,
+or when they are defined to be macros or \&{let} equal to something else.
+A subroutine will be defined later that recycles the storage associated
+with any particular |type| or |value|; our goal now is to study a higher
+level process called |flush_variable|, which selectively frees parts of a
+variable structure.
+
+This routine has some complexity because of examples such as
+`\hbox{\tt numeric x[]a[]b}',
+which recycles all variables of the form \.{x[i]a[j]b} (and no others), while
+`\hbox{\tt vardef x[]a[]=...}'
+discards all variables of the form \.{x[i]a[j]} followed by an arbitrary
+suffix, except for the collective node \.{x[]a[]} itself. The obvious way
+to handle such examples is to use recursion; so that's what we~do.
+@^recursion@>
+
+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@>@<Declare subroutines for printing expressions@>@;@/
+@t\4@>@<Declare basic dependency-list subroutines@>@;
+@t\4@>@<Declare the recycling subroutines@>@;
+@t\4@>@<Declare the procedure called |flush_cur_exp|@>@;
+@t\4@>@<Declare the procedure called |flush_below_variable|@>@;
+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.
+
+@<Declare the procedure called |flush_below_variable|@>=
+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,future_pen:und_type:=unknown_pen;
+path_type,unknown_path:und_type:=unknown_path;
+picture_type,unknown_picture:und_type:=unknown_picture;
+transform_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 provided 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, \MF\ 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.
+
+\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
+
+@<Glob...@>=@!save_ptr:pointer; {the most recently saved item}
+
+@ @<Set init...@>=save_ptr:=null;
+
+@ The |save_variable| routine is given a hash address |q|; it salts this
+address away 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 ");
+ slow_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 ");
+ slow_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 \MF\ user specifies a path, \MF\ 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 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; \MF\ 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 \MF\ 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 \MF'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 \MF\ should choose the direction}
+
+@ Here is a diagnostic routine that prints a given knot list
+in symbolic form. It illustrates the conventions discussed above,
+and checks for anomalies that might arise while \MF\ is being debugged.
+
+@<Declare subroutines for printing expressions@>=
+procedure print_path(@!h:pointer;@!s:str_number;@!nuline:boolean);
+label done,done1;
+var @!p,@!q:pointer; {for list traversal}
+begin print_diagnostic("Path",s,nuline); print_ln;
+@.Path at line...@>
+p:=h;
+repeat q:=link(p);
+if (p=null)or(q=null) then
+ begin print_nl("???"); goto done; {this won't happen}
+@.???@>
+ end;
+@<Print information for adjacent knots |p| and |q|@>;
+p:=q;
+if (p<>h)or(left_type(h)<>endpoint) then
+ @<Print two dots, followed by |given| or |curl| if present@>;
+until p=h;
+if left_type(h)<>endpoint then print("cycle");
+done:end_diagnostic(true);
+end;
+
+@ @<Print information for adjacent knots...@>=
+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: @<Print control points between |p| and |q|, then |goto done1|@>;
+open: @<Print information for a curve that begins |open|@>;
+curl,given: @<Print information for a curve that begins |curl| or |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
+ @<Print tension between |p| and |q|@>;
+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.
+
+@<Print two dots...@>=
+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
+
+@ @<Print tension between |p| and |q|@>=
+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
+
+@ @<Print control points between |p| and |q|, then |goto done1|@>=
+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
+
+@ @<Print information for a curve that begins |open|@>=
+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
+\MF's default curl is present.
+
+@<Print information for a curve that begins |curl|...@>=
+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
+
+@ 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;
+label exit;
+var @!q,@!pp,@!qq:pointer; {for list manipulation}
+begin q:=get_node(knot_node_size); {this will correspond to |p|}
+qq:=q; pp:=p;
+loop@+ begin left_type(qq):=left_type(pp);
+ right_type(qq):=right_type(pp);@/
+ x_coord(qq):=x_coord(pp); y_coord(qq):=y_coord(pp);@/
+ left_x(qq):=left_x(pp); left_y(qq):=left_y(pp);@/
+ right_x(qq):=right_x(pp); right_y(qq):=right_y(pp);@/
+ if link(pp)=p then
+ begin link(qq):=q; copy_path:=q; return;
+ end;
+ link(qq):=get_node(knot_node_size); qq:=link(qq); pp:=link(pp);
+ end;
+exit: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;
+
+@ @<Glob...@>=
+@!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.
+
+@<Declare the recycling subroutines@>=
+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 \MF'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@>@<Declare the procedure called |solve_choices|@>@;
+procedure make_choices(@!knots:pointer);
+label done;
+var @!h:pointer; {the first breakpoint}
+@!p,@!q:pointer; {consecutive breakpoints being processed}
+@<Other local variables for |make_choices|@>@;
+begin check_arith; {make sure that |arith_error=false|}
+if internal[tracing_choices]>0 then
+ print_path(knots,", before choices",true);
+@<If consecutive knots are equal, join them explicitly@>;
+@<Find the first breakpoint, |h|, on the path;
+ insert an artificial breakpoint if the path is an unbroken cycle@>;
+p:=h;
+repeat @<Fill in the control points between |p| and the next breakpoint,
+ then advance |p| to that breakpoint@>;
+until p=h;
+if internal[tracing_choices]>0 then
+ print_path(knots,", after choices",true);
+if arith_error then @<Report an unexpected problem during the choice-making@>;
+end;
+
+@ @<Report an unexpected problem during the choice...@>=
+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.
+
+@<If consecutive knots are equal, join them explicitly@>=
+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
+
+@<Find the first breakpoint, |h|, on the path...@>=
+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)<given| and |q=link(p)|, we must have
+|right_type(p)=left_type(q)=explicit| or |endpoint|.
+
+@<Fill in the control points between |p| and the next breakpoint...@>=
+q:=link(p);
+if right_type(p)>=given then
+ begin while (left_type(q)=open)and(right_type(q)=open) do q:=link(q);
+ @<Fill in the control information between
+ consecutive breakpoints |p| and |q|@>;
+ end;
+p:=q
+
+@ 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 \MF'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 $0<k<n$, and it will be convenient to introduce
+equations of the same form for $k=0$ and $k=n$, where
+$$A_0=B_0=C_n=D_n=0.$$
+If $\theta_0$ is supposed to have a given value $E_0$, we simply
+define $C_0=1$, $D_0=0$, and $R_0=E_0$. Otherwise a curl
+parameter, $\gamma_0$, has been specified at~$z_0$; this means
+that the mock curvature at $z_0$ should be $\gamma_0$ times the
+mock curvature at $z_1$; i.e.,
+$${2\beta_1(\theta_0+\phi_1)-6\theta_0\over\alpha_0^2d_{01}}
+=\gamma_0{2\alpha_0(\theta_0+\phi_1)-6\phi_1\over\beta_1^2d_{01}}.$$
+This equation simplifies to
+$$(\alpha_0\chi_0+3-\beta_1)\theta_0+
+ \bigl((3-\alpha_0)\chi_0+\beta_1\bigr)\theta_1=
+ -\bigl((3-\alpha_0)\chi_0+\beta_1\bigr)\psi_1,$$
+where $\chi_0=\alpha_0^2\gamma_0/\beta_1^2$; so we can set $C_0=
+\chi_0\alpha_0+3-\beta_1$, $D_0=(3-\alpha_0)\chi_0+\beta_1$, $R_0=-D_0\psi_1$.
+It can be shown that $C_0>0$ 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.
+
+@<Fill in the control information between...@>=
+@<Calculate the turning angles $\psi_k$ and the distances $d_{k,k+1}$;
+ set $n$ to the length of the path@>;
+@<Remove |open| types at the breakpoints@>;
+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]|.
+
+@<Glob...@>=
+@!delta_x,@!delta_y,@!delta:array[0..path_size] of scaled; {knot differences}
+@!psi:array[1..path_size] of angle; {turning angles}
+
+@ @<Other local variables for |make_choices|@>=
+@!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}
+
+@ @<Calculate the turning 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;
+@:METAFONT 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|.
+
+@<Remove |open| types at the breakpoints@>=
+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 tri-diagonal
+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|.
+
+@<Glob...@>=
+@!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.
+
+@<Declare the procedure called |solve_choices|@>=
+@t\4@>@<Declare subroutines needed by |solve_choices|@>@;
+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}
+@<Other local variables for |solve_choices|@>@;
+begin k:=0; s:=p;
+loop@+ begin t:=link(s);
+ if k=0 then @<Get the linear equations started; or |return|
+ with the control points in place, if linear equations
+ needn't be solved@>
+ else case left_type(s) of
+ end_cycle,open:@<Set up equation to match mock curvatures
+ at $z_k$; then |goto found| with $\theta_n$
+ adjusted to equal $\theta_0$, if a cycle has ended@>;
+ curl:@<Set up equation for a curl at $\theta_n$
+ and |goto found|@>;
+ given:@<Calculate the given value of $\theta_n$
+ and |goto found|@>;
+ end; {there are no other cases}
+ r:=s; s:=t; incr(k);
+ end;
+found:@<Finish choosing angles and assigning control points@>;
+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 $A_0=B_0=0$.
+
+@<Get the linear equations started...@>=
+case right_type(s) of
+given: if left_type(t)=given then @<Reduce to simple case of two givens
+ and |return|@>
+ else @<Set up the equation for a given value of $\theta_0$@>;
+curl: if left_type(t)=curl then @<Reduce to simple case of straight line
+ and |return|@>
+ else @<Set up the equation for a curl at $\theta_0$@>;
+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.
+
+@<Other local variables for |solve_choices|@>=
+@!aa,@!bb,@!cc,@!ff,@!acc:fraction; {temporary registers}
+@!dd,@!ee:scaled; {likewise, but |scaled|}
+@!lt,@!rt:scaled; {tension values}
+
+@ @<Set up equation to match mock curvatures...@>=
+begin @<Calculate the values $\\{aa}=A_k/B_k$, $\\{bb}=D_k/C_k$,
+ $\\{dd}=(3-\alpha_{k-1})d_{k,k+1}$, $\\{ee}=(3-\beta\k)d_{k-1,k}$,
+ and $\\{cc}=(B_k-u_{k-1}A_k)/B_k$@>;
+@<Calculate the ratio $\\{ff}=C_k/(C_k+B_k-u_{k-1}A_k)$@>;
+uu[k]:=take_fraction(ff,bb);
+@<Calculate the values of $v_k$ and $w_k$@>;
+if left_type(s)=end_cycle then
+ @<Adjust $\theta_n$ to equal $\theta_0$ and |goto found|@>;
+end
+
+@ Since tension values are never less than 3/4, the values |aa| and
+|bb| computed here are never more than 4/5.
+
+@<Calculate the values $\\{aa}=...@>=
+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.
+
+@<Calculate the ratio $\\{ff}=C_k/(C_k+B_k-u_{k-1}A_k)$@>=
+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<rt then
+ begin ff:=make_fraction(lt,rt);
+ ff:=take_fraction(ff,ff); {$\alpha_k^2/\beta_k^2$}
+ dd:=take_fraction(dd,ff);
+ end
+ else begin ff:=make_fraction(rt,lt);
+ ff:=take_fraction(ff,ff); {$\beta_k^2/\alpha_k^2$}
+ ee:=take_fraction(ee,ff);
+ end;
+ff:=make_fraction(ee,ee+dd)
+
+@ The value of $u_{k-1}$ will be |<=1| except when $k=1$ and the previous
+equation was specified by a curl. In that case we must use a special
+method of computation to prevent overflow.
+
+Fortunately, the calculations turn out to be even simpler in this ``hard''
+case. The curl equation makes $w_0=0$ and $v_0=-u_0\psi_1$, hence
+$-B_1\psi_1-A_1v_0=-(B_1-u_0A_1)\psi_1=-\\{cc}\cdot B_1\psi_1$.
+
+@<Calculate the values of $v_k$ and $w_k$@>=
+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<n|, so that the cyclic case can be finished up just as if there
+were no cycle.
+
+The idea in the following code is to observe that
+$$\eqalign{\theta_n&=v_n+w_n\theta_0-u_n\theta_1=\cdots\cr
+&=v_n+w_n\theta_0-u_n\bigl(v_1+w_1\theta_0-u_1(v_2+\cdots
+ -u_{n-2}(v_{n-1}+w_{n-1}\theta_0-u_{n-1}\theta_0)\ldots{})\bigr),\cr}$$
+so we can solve for $\theta_n=\theta_0$.
+
+@<Adjust $\theta_n$ to equal $\theta_0$ and |goto found|@>=
+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
+
+@<Calculate the given value of $\theta_n$...@>=
+begin theta[n]:=left_given(s)-n_arg(delta_x[n-1],delta_y[n-1]);
+reduce_angle(theta[n]);
+goto found;
+end
+
+@ @<Set up the equation for a given value of $\theta_0$@>=
+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
+
+@ @<Set up the equation for a curl at $\theta_0$@>=
+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
+
+@ @<Set up equation for a curl at $\theta_n$...@>=
+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.
+
+@<Declare subroutines needed by |solve_choices|@>=
+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.
+
+@<Finish choosing angles and assigning control points@>=
+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.
+
+@<Glob...@>=
+@!st,@!ct,@!sf,@!cf:fraction; {sines and cosines}
+
+@ @<Declare subroutines needed by |solve_choices|@>=
+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 @<Decrease the velocities,
+ if necessary, to stay inside the bounding triangle@>;
+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.''
+
+@<Decrease the velocities, if necessary...@>=
+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.
+
+@<Reduce to simple case of two givens and |return|@>=
+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
+
+@ @<Reduce to simple case of straight line and |return|@>=
+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] Generating discrete moves.
+The purpose of the next part of \MF\ is to compute discrete approximations
+to curves described as parametric polynomial functions $z(t)$.
+We shall start with the low level first, because an efficient ``engine''
+is needed to support the high-level constructions.
+
+Most of the subroutines are based on variations of a single theme,
+namely the idea of {\sl bisection}. 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<n-j|, for |0<=j<n|.
+
+\smallskip\noindent
+Then
+$$B(z_0,z_1,\ldots,z_n;t)=B(z_0^{(0)},z_0^{(1)},\ldots,z_0^{(n)};2t)
+ =B(z_0^{(n)},z_1^{(n-1)},\ldots,z_n^{(0)};2t-1).$$
+This formula gives us the coefficients of polynomials to use over the ranges
+$0\L t\L{1\over2}$ and ${1\over2}\L t\L1$.
+
+In our applications it will usually be possible to work indirectly with
+numbers that allow us to deduce relevant properties of the polynomials
+without actually computing the polynomial values. We will deal with
+coefficients $Z_k=2^l(z_k-z_{k-1})$ for |1<=k<=n|, instead of
+the actual numbers $z_0$, $z_1$, \dots,~$z_n$, and the value of~|l| will
+increase by~1 at each bisection step. This technique reduces the
+amount of calculation needed for bisection and also increases the
+accuracy of evaluation (since one bit of precision is gained at each
+bisection). Indeed, the bisection process now becomes one level shorter:
+
+\smallskip
+\textindent{$1'$)} Let $Z_k^{(1)}=Z_k$, for |1<=k<=n|.
+
+\smallskip
+\textindent{$2'$)} Let $Z_k^{(j+1)}={1\over2}(Z_k^{(j)}+Z\k^{(j)})$, for
+|1<=k<=n-j|, for |1<=j<n|.
+
+\smallskip\noindent
+The relevant coefficients $(Z'_1,\ldots,Z'_n)$ and $(Z''_1,\ldots,Z''_n)$
+for the two subintervals after bisection are respectively
+$(Z_1^{(1)},Z_1^{(2)},\ldots,Z_1^{(n)})$ and
+$(Z_1^{(n)},Z_2^{(n-1)},\ldots,Z_n^{(1)})$.
+And the values of $z_0$ appropriate for the bisected interval are $z'_0=z_0$
+and $z''_0=z_0+(Z'_1+Z'_2+\cdots+Z'_n)/2^{l+1}$.
+
+Step $2'$ involves division by~2, which introduces computational errors
+of at most $1\over2$ at each step; thus after $l$~levels of bisection the
+integers $Z_k$ will differ from their true values by at most $(n-1)l/2$.
+This error rate is quite acceptable, considering that we have $l$~more
+bits of precision in the $Z$'s by comparison with the~$z$'s. Note also
+that the $Z$'s remain bounded; there's no danger of integer overflow, even
+though we have the identity $Z_k=2^l(z_k-z_{k-1})$ for arbitrarily large~$l$.
+
+In fact, we can show not only that the $Z$'s remain bounded, but also that
+they become nearly equal, since they are control points for a polynomial
+of one less degree. If $\vert Z\k-Z_k\vert\L M$ initially, it is possible
+to prove that $\vert Z\k-Z_k\vert\L\lceil M/2^l\rceil$ after $l$~levels
+of bisection, even in the presence of rounding errors. Here's the
+proof [cf.~Lane and Riesenfeld, {\sl IEEE Trans.\ on Pattern Analysis
+@^Lane, Jeffrey Michael@>
+@^Riesenfeld, Richard Franklin@>
+and Machine Intelligence\/ \bf PAMI-2} (1980), 35--46]: Assuming that
+$\vert Z\k-Z_k\vert\L M$ before bisection, we want to prove that
+$\vert Z\k-Z_k\vert\L\lceil M/2\rceil$ afterward. First we show that
+$\vert Z\k^{(j)}-Z_k^{(j)}\vert\L M$ for all $j$ and~$k$, by induction
+on~$j$; this follows from the fact that
+$$\bigl\vert\\{half}(a+b)-\\{half}(b+c)\bigr\vert\L
+ \max\bigl(\vert a-b\vert,\vert b-c\vert\bigr)$$
+holds for both of the rounding rules $\\{half}(x)=\lfloor x/2\rfloor$
+and $\\{half}(x)={\rm sign}(x)\lfloor\vert x/2\vert\rfloor$.
+(If $\vert a-b\vert$ and $\vert b-c\vert$ are equal, then
+$a+b$ and $b+c$ are both even or both odd. The rounding errors either
+cancel or round the numbers toward each other; hence
+$$\eqalign{\bigl\vert\\{half}(a+b)-\\{half}(b+c)\bigr\vert
+&\L\textstyle\bigl\vert{1\over2}(a+b)-{1\over2}(b+c)\bigr\vert\cr
+&=\textstyle\bigl\vert{1\over2}(a-b)+{1\over2}(b-c)\bigr\vert
+\L\max\bigl(\vert a-b\vert,\vert b-c\vert\bigr),\cr}$$
+as required. A simpler argument applies if $\vert a-b\vert$ and
+$\vert b-c\vert$ are unequal.) Now it is easy to see that
+$\vert Z_1^{(j+1)}-Z_1^{(j)}\vert\L\bigl\lfloor{1\over2}
+\vert Z_2^{(j)}-Z_1^{(j)}\vert+{1\over2}\bigr\rfloor
+\L\bigl\lfloor{1\over2}(M+1)\bigr\rfloor=\lceil M/2\rceil$.
+
+Another interesting fact about bisection is the identity
+$$Z_1'+\cdots+Z_n'+Z_1''+\cdots+Z_n''=2(Z_1+\cdots+Z_n+E),$$
+where $E$ is the sum of the rounding errors in all of the halving
+operations ($\vert E\vert\L n(n-1)/4$).
+
+@ We will later reduce the problem of digitizing a complex cubic
+$z(t)=B(z_0,z_1,z_2,z_3;t)$ to the following simpler problem:
+Given two real cubics
+$x(t)=B(x_0,x_1,x_2,x_3;t)$
+and $y(t)=B(y_0,y_1,y_2,y_3;t)$ that are monotone nondecreasing,
+determine the set of integer points
+$$P=\bigl\{\bigl(\lfloor x(t)\rfloor,\lfloor y(t)\rfloor\bigr)
+\bigm\vert 0\L t\L 1\bigr\}.$$
+Well, the problem isn't actually quite so clean as this; when the path
+goes very near an integer point $(a,b)$, computational errors may
+make us think that $P$ contains $(a-1,b)$ while in reality it should
+contain $(a,b-1)$. Furthermore, if the path goes {\sl exactly\/}
+through the integer points $(a-1,b-1)$ and
+$(a,b)$, we will want $P$ to contain one
+of the two points $(a-1,b)$ or $(a,b-1)$, so that $P$ can be described
+entirely by ``rook moves'' upwards or to the right; no diagonal
+moves from $(a-1,b-1)$ to~$(a,b)$ will be allowed.
+
+Thus, the set $P$ we wish to compute will merely be an approximation
+to the set described in the formula above. It will consist of
+$\lfloor x(1)\rfloor-\lfloor x(0)\rfloor$ rightward moves and
+$\lfloor y(1)\rfloor-\lfloor y(0)\rfloor$ upward moves, intermixed
+in some order. Our job will be to figure out a suitable order.
+
+The following recursive strategy suggests itself, when we recall that
+$x(0)=x_0$, $x(1)=x_3$, $y(0)=y_0$, and $y(1)=y_3$:
+
+\smallskip
+If $\lfloor x_0\rfloor=\lfloor x_3\rfloor$ then take
+$\lfloor y_3\rfloor-\lfloor y_0\rfloor$ steps up.
+
+Otherwise if $\lfloor y_0\rfloor=\lfloor y_3\rfloor$ then take
+$\lfloor x_3\rfloor-\lfloor x_0\rfloor$ steps to the right.
+
+Otherwise bisect the current cubics and repeat the process on both halves.
+
+\yskip\noindent
+This intuitively appealing formulation does not quite solve the problem,
+because it may never terminate. For example, it's not hard to see that
+no steps will {\sl ever\/} be taken if $(x_0,x_1,x_2,x_3)=(y_0,y_1,y_2,y_3)$!
+However, we can surmount this difficulty with a bit of care; so let's
+proceed to flesh out the algorithm as stated, before worrying about
+such details.
+
+The bisect-and-double strategy discussed above suggests that we represent
+$(x_0,x_1,x_2,x_3)$ by $(X_1,X_2,X_3)$, where $X_k=2^l(x_k-x_{k-1})$
+for some~$l$. Initially $l=16$, since the $x$'s are |scaled|.
+In order to deal with other aspects of the algorithm we will want to
+maintain also the quantities $m=\lfloor x_3\rfloor-\lfloor x_0\rfloor$
+and $R=2^l(x_0\bmod 1)$. Similarly,
+$(y_0,y_1,y_2,y_3)$ will be represented by $(Y_1,Y_2,Y_3)$,
+$n=\lfloor y_3\rfloor-\lfloor y_0\rfloor$,
+and $S=2^l(y_0\bmod 1)$. The algorithm now takes the following form:
+
+\smallskip
+If $m=0$ then take $n$ steps up.
+
+Otherwise if $n=0$ then take $m$ steps to the right.
+
+Otherwise bisect the current cubics and repeat the process on both halves.
+
+\smallskip\noindent
+The bisection process for $(X_1,X_2,X_3,m,R,l)$ reduces, in essence,
+to the following formulas:
+$$\vbox{\halign{$#\hfil$\cr
+X_2'=\\{half}(X_1+X_2),\quad
+X_2''=\\{half}(X_2+X_3),\quad
+X_3'=\\{half}(X_2'+X_2''),\cr
+X_1'=X_1,\quad
+X_1''=X_3',\quad
+X_3''=X_3,\cr
+R'=2R,\quad
+T=X_1'+X_2'+X_3'+R',\quad
+R''=T\bmod 2^{l+1},\cr
+m'=\lfloor T/2^{l+1}\rfloor,\quad
+m''=m-m'.\cr}}$$
+
+@ When $m=n=1$, the computation can be speeded up because we simply
+need to decide between two alternatives, (up,\thinspace right)
+versus (right,\thinspace up). There appears to be no simple, direct
+way to make the correct decision by looking at the values of
+$(X_1,X_2,X_3,R)$ and
+$(Y_1,Y_2,Y_3,S)$; but we can streamline the bisection process, and
+we can use the fact that only one of the two descendants needs to
+be examined after each bisection. Furthermore, we observed earlier
+that after several levels of bisection the $X$'s and $Y$'s will be nearly
+equal; so we will be justified in assuming that the curve is essentially a
+straight line. (This, incidentally, solves the problem of infinite
+recursion mentioned earlier.)
+
+It is possible to show that
+$$m=\bigl\lfloor(X_1+X_2+X_3+R+E)\,/\,2^l\bigr\rfloor,$$
+where $E$ is an accumulated rounding error that is at most
+$3\cdot(2^{l-16}-1)$ in absolute value. We will make sure that
+the $X$'s are less than $2^{28}$; hence when $l=30$ we must
+have |m<=1|. This proves that the special case $m=n=1$ is
+bound to be reached by the time $l=30$. Furthermore $l=30$ is
+a suitable time to make the straight line approximation,
+if the recursion hasn't already died out, because the maximum
+difference between $X$'s will then be $<2^{14}$; this corresponds
+to an error of $<1$ with respect to the original scaling.
+(Stating this another way, each bisection makes the curve two bits
+closer to a straight line, hence 14 bisections are sufficient for
+28-bit accuracy.)
+
+In the case of a straight line, the curve goes first right, then up,
+if and only if $(T-2^l)(2^l-S)>(U-2^l)(2^l-R)$, where
+$T=X_1+X_2+X_3+R$ and $U=Y_1+Y_2+Y_3+S$. For the actual curve
+essentially runs from $(R/2^l,S/2^l)$ to $(T/2^l,U/2^l)$, and
+we are testing whether or not $(1,1)$ is above the straight
+line connecting these two points. (This formula assumes that $(1,1)$
+is not exactly on the line.)
+
+@ We have glossed over the problem of tie-breaking in ambiguous
+cases when the cubic curve passes exactly through integer points.
+\MF\ finesses this problem by assuming that coordinates
+$(x,y)$ actually stand for slightly perturbed values $(x+\xi,y+\eta)$,
+where $\xi$ and~$\eta$ are infinitesimals whose signs will determine
+what to do when $x$ and/or~$y$ are exact integers. The quantities
+$\lfloor x\rfloor$ and~$\lfloor y\rfloor$ in the formulas above
+should actually read $\lfloor x+\xi\rfloor$ and $\lfloor y+\eta\rfloor$.
+
+If $x$ is a |scaled| value, we have $\lfloor x+\xi\rfloor=\lfloor x\rfloor$
+if $\xi>0$, and $\lfloor x+\xi\rfloor=\lfloor x-2^{-16}\rfloor$ if
+$\xi<0$. It is convenient to represent $\xi$ by the integer |xi_corr|,
+defined to be 0~if $\xi>0$ and 1~if $\xi<0$; then, for example, the
+integer $\lfloor x+\xi\rfloor$ can be computed as
+|floor_unscaled(x-xi_corr)|. Similarly, $\eta$ is conveniently
+represented by~|eta_corr|.
+
+In our applications the sign of $\xi-\eta$ will always be the same as
+the sign of $\xi$. Therefore it turns out that the rule for straight
+lines, as stated above, should be modified as follows in the case of
+ties: The line goes first right, then up, if and only if
+$(T-2^l)(2^l-S)+\xi>(U-2^l)(2^l-R)$. And this relation holds iff
+$|ab_vs_cd|(T-2^l,2^l-S,U-2^l,2^l-R)-|xi_corr|\ge0$.
+
+These conventions for rounding are symmetrical, in the sense that the
+digitized moves obtained from $(x_0,x_1,x_2,x_3,y_0,y_1,y_2,y_3,\xi,\eta)$
+will be exactly complementary to the moves that would be obtained from
+$(-x_3,-x_2,-x_1,-x_0,-y_3,-y_2,-y_1,-y_0,-\xi,-\eta)$, if arithmetic
+is exact. However, truncation errors in the bisection process might
+upset the symmetry. We can restore much of the lost symmetry by adding
+|xi_corr| or |eta_corr| when halving the data.
+
+@ One further possibility needs to be mentioned: The algorithm
+will be applied only to cubic polynomials $B(x_0,x_1,x_2,x_3;t)$ that
+are nondecreasing as $t$~varies from 0 to~1; this condition turns
+out to hold if and only if $x_0\L x_1$ and $x_2\L x_3$, and either
+$x_1\L x_2$ or $(x_1-x_2)^2\L(x_1-x_0)(x_3-x_2)$. If bisection were
+carried out with perfect accuracy, these relations would remain
+invariant. But rounding errors can creep in, hence the bisection
+algorithm can produce non-monotonic subproblems from monotonic
+initial conditions. This leads to the potential danger that $m$ or~$n$
+could become negative in the algorithm described above.
+
+For example, if we start with $(x_1-x_0,x_2-x_1,x_3-x_2)=
+(X_1,X_2,X_3)=(7,-16,39)$, the corresponding polynomial is
+monotonic, because $16^2<7\cdot39$. But the bisection algorithm
+produces the left descendant $(7,-5,3)$, which is nonmonotonic;
+its right descendant is~$(0,-1,3)$.
+
+\def\xt{{\tilde x}}
+Fortunately we can prove that such rounding errors will never cause
+the algorithm to make a tragic mistake. At every stage we are working
+with numbers corresponding to a cubic polynomial $B(\xt_0,
+\xt_1,\xt_2,\xt_3)$ that approximates some
+monotonic polynomial $B(x_0,x_1,x_2,x_3)$. The accumulated errors are
+controlled so that $\vert x_k-\xt_k\vert<\epsilon=3\cdot2^{-16}$.
+If bisection is done at some stage of the recursion, we have
+$m=\lfloor\xt_3\rfloor-\lfloor\xt_0\rfloor>0$, and the algorithm
+computes a bisection value $\bar x$ such that $m'=\lfloor\bar x\rfloor-
+\lfloor\xt_0\rfloor$
+and $m''=\lfloor\xt_3\rfloor-\lfloor\bar x\rfloor$. We want to prove
+that neither $m'$ nor $m''$ can be negative. Since $\bar x$ is an
+approximation to a value in the interval $[x_0,x_3]$, we have
+$\bar x>x_0-\epsilon$ and $\bar x<x_3+\epsilon$, hence $\bar x>
+\xt_0-2\epsilon$ and $\bar x<\xt_3+2\epsilon$.
+If $m'$ is negative we must have $\xt_0\bmod 1<2\epsilon$;
+if $m''$ is negative we must have $\xt_3\bmod 1>1-2\epsilon$.
+In either case the condition $\lfloor\xt_3\rfloor-\lfloor\xt_0\rfloor>0$
+implies that $\xt_3-\xt_0>1-2\epsilon$, hence $x_3-x_0>1-4\epsilon$.
+But it can be shown that if $B(x_0,x_1,x_2,x_3;t)$ is a monotonic
+cubic, then $B(x_0,x_1,x_2,x_3;{1\over2})$ is always between
+$.06[x_0,x_3]$ and $.94[x_0,x_3]$; and it is impossible for $\bar x$
+to be within~$\epsilon$ of such a number. Contradiction!
+(The constant .06 is actually $(2-\sqrt3\,)/4$; the worst case
+occurs for polynomials like $B(0,2-\sqrt3,1-\sqrt3,3;t)$.)
+
+@ OK, now that a long theoretical preamble has justified the
+bisection-and-doubling algorithm, we are ready to proceed with
+its actual coding. But we still haven't discussed the
+form of the output.
+
+For reasons to be discussed later, we shall find it convenient to
+record the output as follows: Moving one step up is represented by
+appending a `1' to a list; moving one step right is represented by
+adding unity to the element at the end of the list. Thus, for example,
+the net effect of ``(up, right, right, up, right)'' is to append
+$(3,2)$.
+
+The list is kept in a global array called |move|. Before starting the
+algorithm, \MF\ should check that $\\{move\_ptr}+\lfloor y_3\rfloor
+-\lfloor y_0\rfloor\L\\{move\_size}$, so that the list won't exceed
+the bounds of this array.
+
+@<Glob...@>=
+@!move:array[0..move_size] of integer; {the recorded moves}
+@!move_ptr:0..move_size; {the number of items in the |move| list}
+
+@ When bisection occurs, we ``push'' the subproblem corresponding
+to the right-hand subinterval onto the |bisect_stack| while
+we continue to work on the left-hand subinterval. Thus, the |bisect_stack|
+will hold $(X_1,X_2,X_3,R,m,Y_1,Y_2,Y_3,S,n,l)$ values for
+subproblems yet to be tackled.
+
+At most 15 subproblems will be on the stack at once (namely, for
+$l=15$,~16, \dots,~29); but the stack is bigger than this, because
+it is used also for more complicated bisection algorithms.
+
+@d stack_x1==bisect_stack[bisect_ptr] {stacked value of $X_1$}
+@d stack_x2==bisect_stack[bisect_ptr+1] {stacked value of $X_2$}
+@d stack_x3==bisect_stack[bisect_ptr+2] {stacked value of $X_3$}
+@d stack_r==bisect_stack[bisect_ptr+3] {stacked value of $R$}
+@d stack_m==bisect_stack[bisect_ptr+4] {stacked value of $m$}
+@d stack_y1==bisect_stack[bisect_ptr+5] {stacked value of $Y_1$}
+@d stack_y2==bisect_stack[bisect_ptr+6] {stacked value of $Y_2$}
+@d stack_y3==bisect_stack[bisect_ptr+7] {stacked value of $Y_3$}
+@d stack_s==bisect_stack[bisect_ptr+8] {stacked value of $S$}
+@d stack_n==bisect_stack[bisect_ptr+9] {stacked value of $n$}
+@d stack_l==bisect_stack[bisect_ptr+10] {stacked value of $l$}
+@d move_increment=11 {number of items pushed by |make_moves|}
+
+@<Glob...@>=
+@!bisect_stack:array[0..bistack_size] of integer;
+@!bisect_ptr:0..bistack_size;
+
+@ @<Check the ``constant'' values...@>=
+if 15*move_increment>bistack_size then bad:=31;
+
+@ The |make_moves| subroutine is given |scaled| values $(x_0,x_1,x_2,x_3)$
+and $(y_0,y_1,y_2,y_3)$ that represent monotone-nondecreasing polynomials;
+it makes $\lfloor x_3+\xi\rfloor-\lfloor x_0+\xi\rfloor$ rightward moves
+and $\lfloor y_3+\eta\rfloor-\lfloor y_0+\eta\rfloor$ upward moves, as
+explained earlier. (Here $\lfloor x+\xi\rfloor$ actually stands for
+$\lfloor x/2^{16}-|xi_corr|\rfloor$, if $x$ is regarded as an integer
+without scaling.) The unscaled integers $x_k$ and~$y_k$ should be less
+than $2^{28}$ in magnitude.
+
+It is assumed that $|move_ptr| + \lfloor y_3+\eta\rfloor -
+\lfloor y_0+\eta\rfloor < |move_size|$ when this procedure is called,
+so that the capacity of the |move| array will not be exceeded.
+
+The variables |r| and |s| in this procedure stand respectively for
+$R-|xi_corr|$ and $S-|eta_corr|$ in the theory discussed above.
+
+@p procedure make_moves(@!xx0,@!xx1,@!xx2,@!xx3,@!yy0,@!yy1,@!yy2,@!yy3:
+ scaled;@!xi_corr,@!eta_corr:small_number);
+label continue, done, exit;
+var @!x1,@!x2,@!x3,@!m,@!r,@!y1,@!y2,@!y3,@!n,@!s,@!l:integer;
+ {bisection variables explained above}
+@!q,@!t,@!u,@!x2a,@!x3a,@!y2a,@!y3a:integer; {additional temporary registers}
+begin if (xx3<xx0)or(yy3<yy0) then confusion("m");
+@:this can't happen m}{\quad m@>
+l:=16; bisect_ptr:=0;@/
+x1:=xx1-xx0; x2:=xx2-xx1; x3:=xx3-xx2;
+if xx0>=xi_corr then r:=(xx0-xi_corr) mod unity
+else r:=unity-1-((-xx0+xi_corr-1) mod unity);
+m:=(xx3-xx0+r) div unity;@/
+y1:=yy1-yy0; y2:=yy2-yy1; y3:=yy3-yy2;
+if yy0>=eta_corr then s:=(yy0-eta_corr) mod unity
+else s:=unity-1-((-yy0+eta_corr-1) mod unity);
+n:=(yy3-yy0+s) div unity;@/
+if (xx3-xx0>=fraction_one)or(yy3-yy0>=fraction_one) then
+ @<Divide the variables by two, to avoid overflow problems@>;
+loop@+ begin continue:@<Make moves for current subinterval;
+ if bisection is necessary, push the second subinterval
+ onto the stack, and |goto continue| in order to handle
+ the first subinterval@>;
+ if bisect_ptr=0 then return;
+ @<Remove a subproblem for |make_moves| from the stack@>;
+ end;
+exit: end;
+
+@ @<Remove a subproblem for |make_moves| from the stack@>=
+bisect_ptr:=bisect_ptr-move_increment;@/
+x1:=stack_x1; x2:=stack_x2; x3:=stack_x3; r:=stack_r; m:=stack_m;@/
+y1:=stack_y1; y2:=stack_y2; y3:=stack_y3; s:=stack_s; n:=stack_n;@/
+l:=stack_l
+
+@ Our variables |(x1,x2,x3)| correspond to $(X_1,X_2,X_3)$ in the notation
+of the theory developed above. We need to keep them less than $2^{28}$
+in order to avoid integer overflow in weird circumstances.
+For example, data like $x_0=-2^{28}+2^{16}-1$ and $x_1=x_2=x_3=2^{28}-1$
+would otherwise be problematical. Hence this part of the code is
+needed, if only to thwart malicious users.
+
+@<Divide the variables by two, to avoid overflow problems@>=
+begin x1:=half(x1+xi_corr); x2:=half(x2+xi_corr); x3:=half(x3+xi_corr);
+r:=half(r+xi_corr);@/
+y1:=half(y1+eta_corr); y2:=half(y2+eta_corr); y3:=half(y3+eta_corr);
+s:=half(s+eta_corr);@/
+l:=15;
+end
+
+@ @<Make moves...@>=
+if m=0 then @<Move upward |n| steps@>
+else if n=0 then @<Move to the right |m| steps@>
+else if m+n=2 then @<Make one move of each kind@>
+else begin incr(l); stack_l:=l;@/
+ stack_x3:=x3; stack_x2:=half(x2+x3+xi_corr); x2:=half(x1+x2+xi_corr);
+ x3:=half(x2+stack_x2+xi_corr); stack_x1:=x3;@/
+ r:=r+r+xi_corr; t:=x1+x2+x3+r;@/
+ q:=t div two_to_the[l]; stack_r:=t mod two_to_the[l];@/
+ stack_m:=m-q; m:=q;@/
+ stack_y3:=y3; stack_y2:=half(y2+y3+eta_corr); y2:=half(y1+y2+eta_corr);
+ y3:=half(y2+stack_y2+eta_corr); stack_y1:=y3;@/
+ s:=s+s+eta_corr; u:=y1+y2+y3+s;@/
+ q:=u div two_to_the[l]; stack_s:=u mod two_to_the[l];@/
+ stack_n:=n-q; n:=q;@/
+ bisect_ptr:=bisect_ptr+move_increment; goto continue;
+ end
+
+@ @<Move upward |n| steps@>=
+while n>0 do
+ begin incr(move_ptr); move[move_ptr]:=1; decr(n);
+ end
+
+@ @<Move to the right |m| steps@>=
+move[move_ptr]:=move[move_ptr]+m
+
+@ @<Make one move of each kind@>=
+begin r:=two_to_the[l]-r; s:=two_to_the[l]-s;@/
+while l<30 do
+ begin x3a:=x3; x2a:=half(x2+x3+xi_corr); x2:=half(x1+x2+xi_corr);
+ x3:=half(x2+x2a+xi_corr);
+ t:=x1+x2+x3; r:=r+r-xi_corr;@/
+ y3a:=y3; y2a:=half(y2+y3+eta_corr); y2:=half(y1+y2+eta_corr);
+ y3:=half(y2+y2a+eta_corr);
+ u:=y1+y2+y3; s:=s+s-eta_corr;@/
+ if t<r then if u<s then @<Switch to the right subinterval@>
+ else begin @<Move up then right@>; goto done;
+ end
+ else if u<s then
+ begin @<Move right then up@>; goto done;
+ end;
+ incr(l);
+ end;
+r:=r-xi_corr; s:=s-eta_corr;
+if ab_vs_cd(x1+x2+x3,s,y1+y2+y3,r)-xi_corr>=0 then @<Move right then up@>
+ else @<Move up then right@>;
+done:
+end
+
+@ @<Switch to the right subinterval@>=
+begin x1:=x3; x2:=x2a; x3:=x3a; r:=r-t;
+y1:=y3; y2:=y2a; y3:=y3a; s:=s-u;
+end
+
+@ @<Move right then up@>=
+begin incr(move[move_ptr]); incr(move_ptr); move[move_ptr]:=1;
+end
+
+@ @<Move up then right@>=
+begin incr(move_ptr); move[move_ptr]:=2;
+end
+
+@ After |make_moves| has acted, possibly for several curves that move toward
+the same octant, a ``smoothing'' operation might be done on the |move| array.
+This removes optical glitches that can arise even when the curve has been
+digitized without rounding errors.
+
+The smoothing process replaces the integers $a_0\ldots a_n$ in
+|move[b..t]| by ``smoothed'' integers $a_0'\ldots a_n'$ defined as
+follows:
+$$a_k'=a_k+\delta\k-\delta_k;\qquad
+\delta_k=\cases{+1,&if $1<k<n$ and $a_{k-2}\G a_{k-1}\ll a_k\G a\k$;\cr
+-1,&if $1<k<n$ and $a_{k-2}\L a_{k-1}\gg a_k\L a\k$;\cr
+0,&otherwise.\cr}$$
+Here $a\ll b$ means that $a\L b-2$, and $a\gg b$ means that $a\G b+2$.
+
+The smoothing operation is symmetric in the sense that, if $a_0\ldots a_n$
+smooths to $a_0'\ldots a_n'$, then the reverse sequence $a_n\ldots a_0$
+smooths to $a_n'\ldots a_0'$; also the complementary sequence
+$(m-a_0)\ldots(m-a_n)$ smooths to $(m-a_0')\ldots(m-a_n')$.
+We have $a_0'+\cdots+a_n'=a_0+\cdots+a_n$ because $\delta_0=\delta_{n+1}=0$.
+
+@p procedure smooth_moves(@!b,@!t:integer);
+var@!k:1..move_size; {index into |move|}
+@!a,@!aa,@!aaa:integer; {original values of |move[k],move[k-1],move[k-2]|}
+begin if t-b>=3 then
+ begin k:=b+2; aa:=move[k-1]; aaa:=move[k-2];
+ repeat a:=move[k];
+ if abs(a-aa)>1 then
+ @<Increase and decrease |move[k-1]| and |move[k]| by $\delta_k$@>;
+ incr(k); aaa:=aa; aa:=a;
+ until k=t;
+ end;
+end;
+
+@ @<Increase and decrease |move[k-1]| and |move[k]| by $\delta_k$@>=
+if a>aa then
+ begin if aaa>=aa then if a>=move[k+1] then
+ begin incr(move[k-1]); move[k]:=a-1;
+ end;
+ end
+else begin if aaa<=aa then if a<=move[k+1] then
+ begin decr(move[k-1]); move[k]:=a+1;
+ end;
+ end
+
+@* \[20] Edge structures.
+Now we come to \MF's internal scheme for representing what the user can
+actually ``see,'' the edges between pixels. Each pixel has an integer
+weight, obtained by summing the weights on all edges to its left. \MF\
+represents only the nonzero edge weights, since most of the edges are
+weightless; in this way, the data storage requirements grow only linearly
+with respect to the number of pixels per point, even though two-dimensional
+data is being represented. (Well, the actual dependence on the underlying
+resolution is order $n\log n$, but the $\log n$ factor is buried in our
+implicit restriction on the maximum raster size.) The sum of all edge
+weights in each row should be zero.
+
+The data structure for edge weights must be compact and flexible,
+yet it should support efficient updating and display operations. We
+want to be able to have many different edge structures in memory at
+once, and we want the computer to be able to translate them, reflect them,
+and/or merge them together with relative ease.
+
+\MF's solution to this problem requires one single-word node per
+nonzero edge weight, plus one two-word node for each row in a contiguous
+set of rows. There's also a header node that provides global information
+about the entire structure.
+
+@ Let's consider the edge-weight nodes first. The |info| field of such
+nodes contains both an $m$~value and a weight~$w$, in the form
+$8m+w+c$, where $c$ is a constant that depends on data found in the header.
+We shall consider $c$ in detail later; for now, it's best just to think
+of it as a way to compensate for the fact that $m$ and~$w$ can be negative,
+together with the fact that an |info| field must have a value between
+|min_halfword| and |max_halfword|. The $m$ value is an unscaled $x$~coordinate,
+so it satisfies $\vert m\vert<
+4096$; the $w$ value is always in the range $1\L\vert w\vert\L3$. We can
+unpack the data in the |info| field by fetching |ho(info(p))=
+info(p)-min_halfword| and dividing this nonnegative number by~8;
+the constant~$c$ will be chosen so that the remainder of this division
+is $4+w$. Thus, for example, a remainder of~3 will correspond to
+the edge weight $w=-1$.
+
+Every row of an edge structure contains two lists of such edge-weight
+nodes, called the |sorted| and |unsorted| lists, linked together by their
+|link| fields in the normal way. The difference between them is that we
+always have |info(p)<=info(link(p))| in the |sorted| list, but there's no
+such restriction on the elements of the |unsorted| list. The reason for
+this distinction is that it would take unnecessarily long to maintain
+edge-weight lists in sorted order while they're being updated; but when we
+need to process an entire row from left to right in order of the
+$m$~values, it's fairly easy and quick to sort a short list of unsorted
+elements and to merge them into place among their sorted cohorts.
+Furthermore, the fact that the |unsorted| list is empty can sometimes be
+used to good advantage, because it allows us to conclude that a particular
+row has not changed since the last time we sorted it.
+
+The final |link| of the |sorted| list will be |sentinel|, which points to
+a special one-word node whose |info| field is essentially infinite; this
+facilitates the sorting and merging operations. The final |link| of the
+|unsorted| list will be either |null| or |void|, where |void=null+1|
+is used to avoid redisplaying data that has not changed:
+A |void| value is stored at the head of the
+unsorted list whenever the corresponding row has been displayed.
+
+@d zero_w=4
+@d void==null+1
+
+@<Initialize table entries...@>=
+info(sentinel):=max_halfword; {|link(sentinel)=null|}
+
+@ The rows themselves are represented by row header nodes that
+contain four link fields. Two of these four, |sorted| and |unsorted|,
+point to the first items of the edge-weight lists just mentioned.
+The other two, |link| and |knil|, point to the headers of the two
+adjacent rows. If |p| points to the header for row number~|n|, then
+|link(p)| points up to the header for row~|n+1|, and |knil(p)| points
+down to the header for row~|n-1|. This double linking makes it
+convenient to move through consecutive rows either upward or downward;
+as usual, we have |link(knil(p))=knil(link(p))=p| for all row headers~|p|.
+
+The row associated with a given value of |n| contains weights for
+edges that run between the lattice points |(m,n)| and |(m,n+1)|.
+
+@d knil==info {inverse of the |link| field, in a doubly linked list}
+@d sorted_loc(#)==#+1 {where the |sorted| link field resides}
+@d sorted(#)==link(sorted_loc(#)) {beginning of the list of sorted edge weights}
+@d unsorted(#)==info(#+1) {beginning of the list of unsorted edge weights}
+@d row_node_size=2 {number of words in a row header node}
+
+@ The main header node |h| for an edge structure has |link| and |knil|
+fields that link it above the topmost row and below the bottommost row.
+It also has fields called |m_min|, |m_max|, |n_min|, and |n_max| that
+bound the current extent of the edge data: All |m| values in edge-weight
+nodes should lie between |m_min(h)-4096| and |m_max(h)-4096|, inclusive.
+Furthermore the topmost row header, pointed to by |knil(h)|,
+is for row number |n_max(h)-4096|; the bottommost row header, pointed to by
+|link(h)|, is for row number |n_min(h)-4096|.
+
+The offset constant |c| that's used in all of the edge-weight data is
+represented implicitly in |m_offset(h)|; its actual value is
+$$\hbox{|c=min_halfword+zero_w+8*m_offset(h)|.}$$
+Notice that it's possible to shift an entire edge structure by an
+amount $(\Delta m,\Delta n)$ by adding $\Delta n$ to |n_min(h)| and |n_max(h)|,
+adding $\Delta m$ to |m_min(h)| and |m_max(h)|, and subtracting
+$\Delta m$ from |m_offset(h)|;
+none of the other edge data needs to be modified. Initially the |m_offset|
+field is~4096, but it will change if the user requests such a shift.
+The contents of these five fields should always be positive and less than
+8192; |n_max| should, in fact, be less than 8191. Furthermore
+|m_min+m_offset-4096| and |m_max+m_offset-4096| must also lie strictly
+between 0 and 8192, so that the |info| fields of edge-weight nodes will
+fit in a halfword.
+
+The header node of an edge structure also contains two somewhat unusual
+fields that are called |last_window(h)| and |last_window_time(h)|. When this
+structure is displayed in window~|k| of the user's screen, after that
+window has been updated |t| times, \MF\ sets |last_window(h):=k| and
+|last_window_time(h):=t|; it also sets |unsorted(p):=void| for all row
+headers~|p|, after merging any existing unsorted weights with the sorted
+ones. A subsequent display in the same window will be able to avoid
+redisplaying rows whose |unsorted| list is still |void|, if the window
+hasn't been used for something else in the meantime.
+
+A pointer to the row header of row |n_pos(h)-4096| is provided in
+|n_rover(h)|. Most of the algorithms that update an edge structure
+are able to get by without random row references; they usually
+access rows that are neighbors of each other or of the current |n_pos| row.
+Exception: If |link(h)=h| (so that the edge structure contains
+no rows), we have |n_rover(h)=h|, and |n_pos(h)| is irrelevant.
+
+@d zero_field=4096 {amount added to coordinates to make them positive}
+@d n_min(#)==info(#+1) {minimum row number present, plus |zero_field|}
+@d n_max(#)==link(#+1) {maximum row number present, plus |zero_field|}
+@d m_min(#)==info(#+2) {minimum column number present, plus |zero_field|}
+@d m_max(#)==link(#+2) {maximum column number present, plus |zero_field|}
+@d m_offset(#)==info(#+3) {translation of $m$ data in edge-weight nodes}
+@d last_window(#)==link(#+3) {the last display went into this window}
+@d last_window_time(#)==mem[#+4].int {after this many window updates}
+@d n_pos(#)==info(#+5) {the row currently in |n_rover|, plus |zero_field|}
+@d n_rover(#)==link(#+5) {a row recently referenced}
+@d edge_header_size=6 {number of words in an edge-structure header}
+@d valid_range(#)==(abs(#-4096)<4096) {is |#| strictly between 0 and 8192?}
+@d empty_edges(#)==link(#)=# {are there no rows in this edge header?}
+
+@p procedure init_edges(@!h:pointer); {initialize an edge header to null values}
+begin knil(h):=h; link(h):=h;@/
+n_min(h):=zero_field+4095; n_max(h):=zero_field-4095;
+m_min(h):=zero_field+4095; m_max(h):=zero_field-4095;
+m_offset(h):=zero_field;@/
+last_window(h):=0; last_window_time(h):=0;@/
+n_rover(h):=h; n_pos(h):=0;@/
+end;
+
+@ When a lot of work is being done on a particular edge structure, we plant
+a pointer to its main header in the global variable |cur_edges|.
+This saves us from having to pass this pointer as a parameter over and
+over again between subroutines.
+
+Similarly, |cur_wt| is a global weight that is being used by several
+procedures at once.
+
+@<Glob...@>=
+@!cur_edges:pointer; {the edge structure of current interest}
+@!cur_wt:integer; {the edge weight of current interest}
+
+@ The |fix_offset| routine goes through all the edge-weight nodes of
+|cur_edges| and adds a constant to their |info| fields, so that
+|m_offset(cur_edges)| can be brought back to |zero_field|. (This
+is necessary only in unusual cases when the offset has gotten too
+large or too small.)
+
+@p procedure fix_offset;
+var @!p,@!q:pointer; {list traversers}
+@!delta:integer; {the amount of change}
+begin delta:=8*(m_offset(cur_edges)-zero_field);
+m_offset(cur_edges):=zero_field;
+q:=link(cur_edges);
+while q<>cur_edges do
+ begin p:=sorted(q);
+ while p<>sentinel do
+ begin info(p):=info(p)-delta; p:=link(p);
+ end;
+ p:=unsorted(q);
+ while p>void do
+ begin info(p):=info(p)-delta; p:=link(p);
+ end;
+ q:=link(q);
+ end;
+end;
+
+@ The |edge_prep| routine makes the |cur_edges| structure ready to
+accept new data whose coordinates satisfy |ml<=m<=mr| and |nl<=n<=nr-1|,
+assuming that |-4096<ml<=mr<4096| and |-4096<nl<=nr<4096|. It makes
+appropriate adjustments to |m_min|, |m_max|, |n_min|, and |n_max|,
+adding new empty rows if necessary.
+
+@p procedure edge_prep(@!ml,@!mr,@!nl,@!nr:integer);
+var @!delta:halfword; {amount of change}
+@!p,@!q:pointer; {for list manipulation}
+begin ml:=ml+zero_field; mr:=mr+zero_field;
+nl:=nl+zero_field; nr:=nr-1+zero_field;@/
+if ml<m_min(cur_edges) then m_min(cur_edges):=ml;
+if mr>m_max(cur_edges) then m_max(cur_edges):=mr;
+if not valid_range(m_min(cur_edges)+m_offset(cur_edges)-zero_field) or@|
+ not valid_range(m_max(cur_edges)+m_offset(cur_edges)-zero_field) then
+ fix_offset;
+if empty_edges(cur_edges) then {there are no rows}
+ begin n_min(cur_edges):=nr+1; n_max(cur_edges):=nr;
+ end;
+if nl<n_min(cur_edges) then
+ @<Insert exactly |n_min(cur_edges)-nl| empty rows at the bottom@>;
+if nr>n_max(cur_edges) then
+ @<Insert exactly |nr-n_max(cur_edges)| empty rows at the top@>;
+end;
+
+@ @<Insert exactly |n_min(cur_edges)-nl| empty rows at the bottom@>=
+begin delta:=n_min(cur_edges)-nl; n_min(cur_edges):=nl;
+p:=link(cur_edges);
+repeat q:=get_node(row_node_size); sorted(q):=sentinel; unsorted(q):=void;
+knil(p):=q; link(q):=p; p:=q; decr(delta);
+until delta=0;
+knil(p):=cur_edges; link(cur_edges):=p;
+if n_rover(cur_edges)=cur_edges then n_pos(cur_edges):=nl-1;
+end
+
+@ @<Insert exactly |nr-n_max(cur_edges)| empty rows at the top@>=
+begin delta:=nr-n_max(cur_edges); n_max(cur_edges):=nr;
+p:=knil(cur_edges);
+repeat q:=get_node(row_node_size); sorted(q):=sentinel; unsorted(q):=void;
+link(p):=q; knil(q):=p; p:=q; decr(delta);
+until delta=0;
+link(p):=cur_edges; knil(cur_edges):=p;
+if n_rover(cur_edges)=cur_edges then n_pos(cur_edges):=nr+1;
+end
+
+@ The |print_edges| subroutine gives a symbolic rendition of an edge
+structure, for use in `\&{show}' commands. A rather terse output
+format has been chosen since edge structures can grow quite large.
+
+@<Declare subroutines for printing expressions@>=
+@t\4@>@<Declare the procedure called |print_weight|@>@;@/
+procedure print_edges(@!s:str_number;@!nuline:boolean;@!x_off,@!y_off:integer);
+var @!p,@!q,@!r:pointer; {for list traversal}
+@!n:integer; {row number}
+begin print_diagnostic("Edge structure",s,nuline);
+p:=knil(cur_edges); n:=n_max(cur_edges)-zero_field;
+while p<>cur_edges do
+ begin q:=unsorted(p); r:=sorted(p);
+ if(q>void)or(r<>sentinel) then
+ begin print_nl("row "); print_int(n+y_off); print_char(":");
+ while q>void do
+ begin print_weight(q,x_off); q:=link(q);
+ end;
+ print(" |");
+ while r<>sentinel do
+ begin print_weight(r,x_off); r:=link(r);
+ end;
+ end;
+ p:=knil(p); decr(n);
+ end;
+end_diagnostic(true);
+end;
+
+@ @<Declare the procedure called |print_weight|@>=
+procedure print_weight(@!q:pointer;@!x_off:integer);
+var @!w,@!m:integer; {unpacked weight and coordinate}
+@!d:integer; {temporary data register}
+begin d:=ho(info(q)); w:=d mod 8; m:=(d div 8)-m_offset(cur_edges);
+if file_offset>max_print_line-9 then print_nl(" ")
+else print_char(" ");
+print_int(m+x_off);
+while w>zero_w do
+ begin print_char("+"); decr(w);
+ end;
+while w<zero_w do
+ begin print_char("-"); incr(w);
+ end;
+end;
+
+@ Here's a trivial subroutine that copies an edge structure. (Let's hope
+that the given structure isn't too gigantic.)
+
+@p function copy_edges(@!h:pointer):pointer;
+var @!p,@!r:pointer; {variables that traverse the given structure}
+@!hh,@!pp,@!qq,@!rr,@!ss:pointer; {variables that traverse the new structure}
+begin hh:=get_node(edge_header_size);
+mem[hh+1]:=mem[h+1]; mem[hh+2]:=mem[h+2];
+mem[hh+3]:=mem[h+3]; mem[hh+4]:=mem[h+4]; {we've now copied |n_min|, |n_max|,
+ |m_min|, |m_max|, |m_offset|, |last_window|, and |last_window_time|}
+n_pos(hh):=n_max(hh)+1;n_rover(hh):=hh;@/
+p:=link(h); qq:=hh;
+while p<>h do
+ begin pp:=get_node(row_node_size); link(qq):=pp; knil(pp):=qq;
+ @<Copy both |sorted| and |unsorted| lists of |p| to |pp|@>;
+ p:=link(p); qq:=pp;
+ end;
+link(qq):=hh; knil(hh):=qq;
+copy_edges:=hh;
+end;
+
+@ @<Copy both |sorted| and |unsorted|...@>=
+r:=sorted(p); rr:=sorted_loc(pp); {|link(rr)=sorted(pp)|}
+while r<>sentinel do
+ begin ss:=get_avail; link(rr):=ss; rr:=ss; info(rr):=info(r);@/
+ r:=link(r);
+ end;
+link(rr):=sentinel;@/
+r:=unsorted(p); rr:=temp_head;
+while r>void do
+ begin ss:=get_avail; link(rr):=ss; rr:=ss; info(rr):=info(r);@/
+ r:=link(r);
+ end;
+link(rr):=r; unsorted(pp):=link(temp_head)
+
+@ Another trivial routine flips |cur_edges| about the |x|-axis
+(i.e., negates all the |y| coordinates), assuming that at least
+one row is present.
+
+@p procedure y_reflect_edges;
+var @!p,@!q,@!r:pointer; {list manipulation registers}
+begin p:=n_min(cur_edges);
+n_min(cur_edges):=zero_field+zero_field-1-n_max(cur_edges);
+n_max(cur_edges):=zero_field+zero_field-1-p;
+n_pos(cur_edges):=zero_field+zero_field-1-n_pos(cur_edges);@/
+p:=link(cur_edges); q:=cur_edges; {we assume that |p<>q|}
+repeat r:=link(p); link(p):=q; knil(q):=p; q:=p; p:=r;
+until q=cur_edges;
+last_window_time(cur_edges):=0;
+end;
+
+@ It's somewhat more difficult, yet not too hard, to reflect about the |y|-axis.
+
+@p procedure x_reflect_edges;
+var @!p,@!q,@!r,@!s:pointer; {list manipulation registers}
+@!m:integer; {|info| fields will be reflected with respect to this number}
+begin p:=m_min(cur_edges);
+m_min(cur_edges):=zero_field+zero_field-m_max(cur_edges);
+m_max(cur_edges):=zero_field+zero_field-p;
+m:=(zero_field+m_offset(cur_edges))*8+zero_w+min_halfword+zero_w+min_halfword;
+m_offset(cur_edges):=zero_field;
+p:=link(cur_edges);
+repeat @<Reflect the edge-and-weight data in |sorted(p)|@>;
+@<Reflect the edge-and-weight data in |unsorted(p)|@>;
+p:=link(p);
+until p=cur_edges;
+last_window_time(cur_edges):=0;
+end;
+
+@ We want to change the sign of the weight as we change the sign of the
+|x|~coordinate. Fortunately, it's easier to do this than to negate
+one without the other.
+
+@<Reflect the edge-and-weight data in |unsorted(p)|@>=
+q:=unsorted(p);
+while q>void do
+ begin info(q):=m-info(q); q:=link(q);
+ end
+
+@ Reversing the order of a linked list is best thought of as the process of
+popping nodes off one stack and pushing them on another. In this case we
+pop from stack~|q| and push to stack~|r|.
+
+@<Reflect the edge-and-weight data in |sorted(p)|@>=
+q:=sorted(p); r:=sentinel;
+while q<>sentinel do
+ begin s:=link(q); link(q):=r; r:=q; info(r):=m-info(q); q:=s;
+ end;
+sorted(p):=r
+
+@ Now let's multiply all the $y$~coordinates of a nonempty edge structure
+by a small integer $s>1$:
+
+@p procedure y_scale_edges(@!s:integer);
+var @!p,@!q,@!pp,@!r,@!rr,@!ss:pointer; {list manipulation registers}
+@!t:integer; {replication counter}
+begin if (s*(n_max(cur_edges)+1-zero_field)>=4096) or@|
+ (s*(n_min(cur_edges)-zero_field)<=-4096) then
+ begin print_err("Scaled picture would be too big");
+@.Scaled picture...big@>
+ help3("I can't yscale the picture as requested---it would")@/
+ ("make some coordinates too large or too small.")@/
+ ("Proceed, and I'll omit the transformation.");
+ put_get_error;
+ end
+else begin n_max(cur_edges):=s*(n_max(cur_edges)+1-zero_field)-1+zero_field;
+ n_min(cur_edges):=s*(n_min(cur_edges)-zero_field)+zero_field;
+ @<Replicate every row exactly $s$ times@>;
+ last_window_time(cur_edges):=0;
+ end;
+end;
+
+@ @<Replicate...@>=
+p:=cur_edges;
+repeat q:=p; p:=link(p);
+for t:=2 to s do
+ begin pp:=get_node(row_node_size); link(q):=pp; knil(p):=pp;
+ link(pp):=p; knil(pp):=q; q:=pp;
+ @<Copy both |sorted| and |unsorted|...@>;
+ end;
+until link(p)=cur_edges
+
+@ Scaling the $x$~coordinates is, of course, our next task.
+
+@p procedure x_scale_edges(@!s:integer);
+var @!p,@!q:pointer; {list manipulation registers}
+@!t:0..65535; {unpacked |info| field}
+@!w:0..7; {unpacked weight}
+@!delta:integer; {amount added to scaled |info|}
+begin if (s*(m_max(cur_edges)-zero_field)>=4096) or@|
+ (s*(m_min(cur_edges)-zero_field)<=-4096) then
+ begin print_err("Scaled picture would be too big");
+@.Scaled picture...big@>
+ help3("I can't xscale the picture as requested---it would")@/
+ ("make some coordinates too large or too small.")@/
+ ("Proceed, and I'll omit the transformation.");
+ put_get_error;
+ end
+else if (m_max(cur_edges)<>zero_field)or(m_min(cur_edges)<>zero_field) then
+ begin m_max(cur_edges):=s*(m_max(cur_edges)-zero_field)+zero_field;
+ m_min(cur_edges):=s*(m_min(cur_edges)-zero_field)+zero_field;
+ delta:=8*(zero_field-s*m_offset(cur_edges))+min_halfword;
+ m_offset(cur_edges):=zero_field;@/
+ @<Scale the $x$~coordinates of each row by $s$@>;
+ last_window_time(cur_edges):=0;
+ end;
+end;
+
+@ The multiplications cannot overflow because we know that |s<4096|.
+
+@<Scale the $x$~coordinates of each row by $s$@>=
+q:=link(cur_edges);
+repeat p:=sorted(q);
+while p<>sentinel do
+ begin t:=ho(info(p)); w:=t mod 8; info(p):=(t-w)*s+w+delta; p:=link(p);
+ end;
+p:=unsorted(q);
+while p>void do
+ begin t:=ho(info(p)); w:=t mod 8; info(p):=(t-w)*s+w+delta; p:=link(p);
+ end;
+q:=link(q);
+until q=cur_edges
+
+@ Here is a routine that changes the signs of all the weights, without
+changing anything else.
+
+@p procedure negate_edges(@!h:pointer);
+label done;
+var @!p,@!q,@!r,@!s,@!t,@!u:pointer; {structure traversers}
+begin p:=link(h);
+while p<>h do
+ begin q:=unsorted(p);
+ while q>void do
+ begin info(q):=8-2*((ho(info(q))) mod 8)+info(q); q:=link(q);
+ end;
+ q:=sorted(p);
+ if q<>sentinel then
+ begin repeat info(q):=8-2*((ho(info(q))) mod 8)+info(q); q:=link(q);
+ until q=sentinel;
+ @<Put the list |sorted(p)| back into sort@>;
+ end;
+ p:=link(p);
+ end;
+last_window_time(h):=0;
+end;
+
+@ \MF\ would work even if the code in this section were omitted, because
+a list of edge-and-weight data that is sorted only by
+|m| but not~|w| turns out to be good enough for correct operation.
+However, the author decided not to make the program even trickier than
+it is already, since |negate_edges| isn't needed very often.
+The simpler-to-state condition, ``keep the |sorted| list fully sorted,''
+is therefore being preserved at the cost of extra computation.
+
+@<Put the list |sorted(p)|...@>=
+u:=sorted_loc(p); q:=link(u); r:=q; s:=link(r); {|q=sorted(p)|}
+loop@+ if info(s)>info(r) then
+ begin link(u):=q;
+ if s=sentinel then goto done;
+ u:=r; q:=s; r:=q; s:=link(r);
+ end
+ else begin t:=s; s:=link(t); link(t):=q; q:=t;
+ end;
+done: link(r):=sentinel
+
+@ The |unsorted| edges of a row are merged into the |sorted| ones by
+a subroutine called |sort_edges|. It uses simple insertion sort,
+followed by a merge, because the unsorted list is supposedly quite short.
+However, the unsorted list is assumed to be nonempty.
+
+@p procedure sort_edges(@!h:pointer); {|h| is a row header}
+label done;
+var @!k:halfword; {key register that we compare to |info(q)|}
+@!p,@!q,@!r,@!s:pointer;
+begin r:=unsorted(h); unsorted(h):=null;
+p:=link(r); link(r):=sentinel; link(temp_head):=r;
+while p>void do {sort node |p| into the list that starts at |temp_head|}
+ begin k:=info(p); q:=temp_head;
+ repeat r:=q; q:=link(r);
+ until k<=info(q);
+ link(r):=p; r:=link(p); link(p):=q; p:=r;
+ end;
+@<Merge the |temp_head| list into |sorted(h)|@>;
+end;
+
+@ In this step we use the fact that |sorted(h)=link(sorted_loc(h))|.
+
+@<Merge the |temp_head| list into |sorted(h)|@>=
+begin r:=sorted_loc(h); q:=link(r); p:=link(temp_head);
+loop@+ begin k:=info(p);
+ while k>info(q) do
+ begin r:=q; q:=link(r);
+ end;
+ link(r):=p; s:=link(p); link(p):=q;
+ if s=sentinel then goto done;
+ r:=p; p:=s;
+ end;
+done:end
+
+@ The |cull_edges| procedure ``optimizes'' an edge structure by making all
+the pixel weights either |w_out| or~|w_in|. The weight will be~|w_in| after the
+operation if and only if it was in the closed interval |[w_lo,w_hi]|
+before, where |w_lo<=w_hi|. Either |w_out| or |w_in| is zero, while the other is
+$\pm1$, $\pm2$, or $\pm3$. The parameters will be such that zero-weight
+pixels will remain of weight zero. (This is fortunate,
+because there are infinitely many of them.)
+
+The procedure also computes the tightest possible bounds on the resulting
+data, by updating |m_min|, |m_max|, |n_min|, and~|n_max|.
+
+@p procedure cull_edges(@!w_lo,@!w_hi,@!w_out,@!w_in:integer);
+label done;
+var @!p,@!q,@!r,@!s:pointer; {for list manipulation}
+@!w:integer; {new weight after culling}
+@!d:integer; {data register for unpacking}
+@!m:integer; {the previous column number, including |m_offset|}
+@!mm:integer; {the next column number, including |m_offset|}
+@!ww:integer; {accumulated weight before culling}
+@!prev_w:integer; {value of |w| before column |m|}
+@!n,@!min_n,@!max_n:pointer; {current and extreme row numbers}
+@!min_d,@!max_d:pointer; {extremes of the new edge-and-weight data}
+begin min_d:=max_halfword; max_d:=min_halfword;
+min_n:=max_halfword; max_n:=min_halfword;@/
+p:=link(cur_edges); n:=n_min(cur_edges);
+while p<>cur_edges do
+ begin if unsorted(p)>void then sort_edges(p);
+ if sorted(p)<>sentinel then
+ @<Cull superfluous edge-weight entries from |sorted(p)|@>;
+ p:=link(p); incr(n);
+ end;
+@<Delete empty rows at the top and/or bottom;
+ update the boundary values in the header@>;
+last_window_time(cur_edges):=0;
+end;
+
+@ The entire |sorted| list is returned to available memory in this step;
+a new list is built starting (temporarily) at |temp_head|.
+Since several edges can occur at the same column, we need to be looking
+ahead of where the actual culling takes place. This means that it's
+slightly tricky to get the iteration started and stopped.
+
+@<Cull superfluous...@>=
+begin r:=temp_head; q:=sorted(p); ww:=0; m:=1000000; prev_w:=0;
+loop@+ begin if q=sentinel then mm:=1000000
+ else begin d:=ho(info(q)); mm:=d div 8; ww:=ww+(d mod 8)-zero_w;
+ end;
+ if mm>m then
+ begin @<Insert an edge-weight for edge |m|, if the new pixel
+ weight has changed@>;
+ if q=sentinel then goto done;
+ end;
+ m:=mm;
+ if ww>=w_lo then if ww<=w_hi then w:=w_in
+ else w:=w_out
+ else w:=w_out;
+ s:=link(q); free_avail(q); q:=s;
+ end;
+done: link(r):=sentinel; sorted(p):=link(temp_head);
+if r<>temp_head then @<Update the max/min amounts@>;
+end
+
+@ @<Insert an edge-weight for edge |m|, if...@>=
+if w<>prev_w then
+ begin s:=get_avail; link(r):=s;
+ info(s):=8*m+min_halfword+zero_w+w-prev_w;
+ r:=s; prev_w:=w;
+ end
+
+@ @<Update the max/min amounts@>=
+begin if min_n=max_halfword then min_n:=n;
+max_n:=n;
+if min_d>info(link(temp_head)) then min_d:=info(link(temp_head));
+if max_d<info(r) then max_d:=info(r);
+end
+
+@ @<Delete empty rows at the top and/or bottom...@>=
+if min_n>max_n then @<Delete all the row headers@>
+else begin n:=n_min(cur_edges); n_min(cur_edges):=min_n;
+ while min_n>n do
+ begin p:=link(cur_edges); link(cur_edges):=link(p);
+ knil(link(p)):=cur_edges;
+ free_node(p,row_node_size); incr(n);
+ end;
+ n:=n_max(cur_edges); n_max(cur_edges):=max_n;
+ n_pos(cur_edges):=max_n+1; n_rover(cur_edges):=cur_edges;
+ while max_n<n do
+ begin p:=knil(cur_edges); knil(cur_edges):=knil(p);
+ link(knil(p)):=cur_edges;
+ free_node(p,row_node_size); decr(n);
+ end;
+ m_min(cur_edges):=((ho(min_d)) div 8)-m_offset(cur_edges)+zero_field;
+ m_max(cur_edges):=((ho(max_d)) div 8)-m_offset(cur_edges)+zero_field;
+ end
+
+@ We get here if the edges have been entirely culled away.
+
+@<Delete all the row headers@>=
+begin p:=link(cur_edges);
+while p<>cur_edges do
+ begin q:=link(p); free_node(p,row_node_size); p:=q;
+ end;
+init_edges(cur_edges);
+end
+
+
+@ The last and most difficult routine for transforming an edge structure---and
+the most interesting one!---is |xy_swap_edges|, which interchanges the
+r\^^Doles of rows and columns. Its task can be viewed as the job of
+creating an edge structure that contains only horizontal edges, linked
+together in columns, given an edge structure that contains only
+vertical edges linked together in rows; we must do this without changing
+the implied pixel weights.
+
+Given any two adjacent rows of an edge structure, it is not difficult to
+determine the horizontal edges that lie ``between'' them: We simply look
+for vertically adjacent pixels that have different weight, and insert
+a horizontal edge containing the difference in weights. Every horizontal
+edge determined in this way should be put into an appropriate linked
+list. Since random access to these linked lists is desirable, we use
+the |move| array to hold the list heads. If we work through the given
+edge structure from top to bottom, the constructed lists will not need
+to be sorted, since they will already be in order.
+
+The following algorithm makes use of some ideas suggested by John Hobby.
+@^Hobby, John Douglas@>
+It assumes that the edge structure is non-null, i.e., that |link(cur_edges)
+<>cur_edges|, hence |m_max(cur_edges)>=m_min(cur_edges)|.
+
+@p procedure xy_swap_edges; {interchange |x| and |y| in |cur_edges|}
+label done;
+var @!m_magic,@!n_magic:integer; {special values that account for offsets}
+@!p,@!q,@!r,@!s:pointer; {pointers that traverse the given structure}
+@<Other local variables for |xy_swap_edges|@>@;
+begin @<Initialize the array of new edge list heads@>;
+@<Insert blank rows at the top and bottom, and set |p| to the new top row@>;
+@<Compute the magic offset values@>;
+repeat q:=knil(p);@+if unsorted(q)>void then sort_edges(q);
+@<Insert the horizontal edges defined by adjacent rows |p,q|,
+ and destroy row~|p|@>;
+p:=q; n_magic:=n_magic-8;
+until knil(p)=cur_edges;
+free_node(p,row_node_size); {now all original rows have been recycled}
+@<Adjust the header to reflect the new edges@>;
+end;
+
+@ Here we don't bother to keep the |link| entries up to date, since the
+procedure looks only at the |knil| fields as it destroys the former
+edge structure.
+
+@<Insert blank rows at the top and bottom...@>=
+p:=get_node(row_node_size); sorted(p):=sentinel; unsorted(p):=null;@/
+knil(p):=cur_edges; knil(link(cur_edges)):=p; {the new bottom row}
+p:=get_node(row_node_size); sorted(p):=sentinel;
+knil(p):=knil(cur_edges); {the new top row}
+
+@ The new lists will become |sorted| lists later, so we initialize
+empty lists to |sentinel|.
+
+@<Initialize the array of new edge list heads@>=
+m_spread:=m_max(cur_edges)-m_min(cur_edges); {this is |>=0| by assumption}
+if m_spread>move_size then overflow("move table size",move_size);
+@:METAFONT capacity exceeded move table size}{\quad move table size@>
+for j:=0 to m_spread do move[j]:=sentinel
+
+@ @<Other local variables for |xy_swap_edges|@>=
+@!m_spread:integer; {the difference between |m_max| and |m_min|}
+@!j,@!jj:0..move_size; {indices into |move|}
+@!m,@!mm:integer; {|m| values at vertical edges}
+@!pd,@!rd:integer; {data fields from edge-and-weight nodes}
+@!pm,@!rm:integer; {|m| values from edge-and-weight nodes}
+@!w:integer; {the difference in accumulated weight}
+@!ww:integer; {as much of |w| that can be stored in a single node}
+@!dw:integer; {an increment to be added to |w|}
+
+@ At the point where we test |w<>0|, variable |w| contains
+the accumulated weight from edges already passed in
+row~|p| minus the accumulated weight from edges already passed in row~|q|.
+
+@<Insert the horizontal edges defined by adjacent rows |p,q|...@>=
+r:=sorted(p); free_node(p,row_node_size); p:=r;@/
+pd:=ho(info(p)); pm:=pd div 8;@/
+r:=sorted(q); rd:=ho(info(r)); rm:=rd div 8; w:=0;
+loop@+ begin if pm<rm then mm:=pm@+else mm:=rm;
+ if w<>0 then
+ @<Insert horizontal edges of weight |w| between |m| and~|mm|@>;
+ if pd<rd then
+ begin dw:=(pd mod 8)-zero_w;
+ @<Advance pointer |p| to the next vertical edge,
+ after destroying the previous one@>;
+ end
+ else begin if r=sentinel then goto done; {|rd=pd=ho(max_halfword)|}
+ dw:=-((rd mod 8)-zero_w);
+ @<Advance pointer |r| to the next vertical edge@>;
+ end;
+ m:=mm; w:=w+dw;
+ end;
+done:
+
+@ @<Advance pointer |r| to the next vertical edge@>=
+r:=link(r); rd:=ho(info(r)); rm:=rd div 8
+
+@ @<Advance pointer |p| to the next vertical edge...@>=
+s:=link(p); free_avail(p); p:=s; pd:=ho(info(p)); pm:=pd div 8
+
+@ Certain ``magic'' values are needed to make the following code work,
+because of the various offsets in our data structure. For now, let's not
+worry about their precise values; we shall compute |m_magic| and |n_magic|
+later, after we see what the code looks like.
+
+@ @<Insert horizontal edges of weight |w| between |m| and~|mm|@>=
+if m<>mm then
+ begin if mm-m_magic>=move_size then confusion("xy");
+@:this can't happen xy}{\quad xy@>
+ extras:=(abs(w)-1) div 3;
+ if extras>0 then
+ begin if w>0 then xw:=+3@+else xw:=-3;
+ ww:=w-extras*xw;
+ end
+ else ww:=w;
+ repeat j:=m-m_magic;
+ for k:=1 to extras do
+ begin s:=get_avail; info(s):=n_magic+xw;
+ link(s):=move[j]; move[j]:=s;
+ end;
+ s:=get_avail; info(s):=n_magic+ww;
+ link(s):=move[j]; move[j]:=s;@/
+ incr(m);
+ until m=mm;
+ end
+
+@ @<Other local variables for |xy...@>=
+@!extras:integer; {the number of additional nodes to make weights |>3|}
+@!xw:-3..3; {the additional weight in extra nodes}
+@!k:integer; {loop counter for inserting extra nodes}
+
+@ At the beginning of this step, |move[m_spread]=sentinel|, because no
+horizontal edges will extend to the right of column |m_max(cur_edges)|.
+
+@<Adjust the header to reflect the new edges@>=
+move[m_spread]:=0; j:=0;
+while move[j]=sentinel do incr(j);
+if j=m_spread then init_edges(cur_edges) {all edge weights are zero}
+else begin mm:=m_min(cur_edges);
+ m_min(cur_edges):=n_min(cur_edges);
+ m_max(cur_edges):=n_max(cur_edges)+1;
+ m_offset(cur_edges):=zero_field;
+ jj:=m_spread-1;
+ while move[jj]=sentinel do decr(jj);
+ n_min(cur_edges):=j+mm; n_max(cur_edges):=jj+mm; q:=cur_edges;
+ repeat p:=get_node(row_node_size); link(q):=p; knil(p):=q;
+ sorted(p):=move[j]; unsorted(p):=null; incr(j); q:=p;
+ until j>jj;
+ link(q):=cur_edges; knil(cur_edges):=q;
+ n_pos(cur_edges):=n_max(cur_edges)+1; n_rover(cur_edges):=cur_edges;
+ last_window_time(cur_edges):=0;
+ end;
+
+@ The values of |m_magic| and |n_magic| can be worked out by trying the
+code above on a small example; if they work correctly in simple cases,
+they should work in general.
+
+@<Compute the magic offset values@>=
+m_magic:=m_min(cur_edges)+m_offset(cur_edges)-zero_field;
+n_magic:=8*n_max(cur_edges)+8+zero_w+min_halfword
+
+@ Now let's look at the subroutine that merges the edges from a given
+edge structure into |cur_edges|. The given edge structure loses all its
+edges.
+
+@p procedure merge_edges(@!h:pointer);
+label done;
+var @!p,@!q,@!r,@!pp,@!qq,@!rr:pointer; {list manipulation registers}
+@!n:integer; {row number}
+@!k:halfword; {key register that we compare to |info(q)|}
+@!delta:integer; {change to the edge/weight data}
+begin if link(h)<>h then
+ begin if (m_min(h)<m_min(cur_edges))or(m_max(h)>m_max(cur_edges))or@|
+ (n_min(h)<n_min(cur_edges))or(n_max(h)>n_max(cur_edges)) then
+ edge_prep(m_min(h)-zero_field,m_max(h)-zero_field,
+ n_min(h)-zero_field,n_max(h)-zero_field+1);
+ if m_offset(h)<>m_offset(cur_edges) then
+ @<Adjust the data of |h| to account for a difference of offsets@>;
+ n:=n_min(cur_edges); p:=link(cur_edges); pp:=link(h);
+ while n<n_min(h) do
+ begin incr(n); p:=link(p);
+ end;
+ repeat @<Merge row |pp| into row |p|@>;
+ pp:=link(pp); p:=link(p);
+ until pp=h;
+ end;
+end;
+
+@ @<Adjust the data of |h| to account for a difference of offsets@>=
+begin pp:=link(h); delta:=8*(m_offset(cur_edges)-m_offset(h));
+repeat qq:=sorted(pp);
+while qq<>sentinel do
+ begin info(qq):=info(qq)+delta; qq:=link(qq);
+ end;
+qq:=unsorted(pp);
+while qq>void do
+ begin info(qq):=info(qq)+delta; qq:=link(qq);
+ end;
+pp:=link(pp);
+until pp=h;
+end
+
+@ The |sorted| and |unsorted| lists are merged separately. After this
+step, row~|pp| will have no edges remaining, since they will all have
+been merged into row~|p|.
+
+@<Merge row |pp|...@>=
+qq:=unsorted(pp);
+if qq>void then
+ if unsorted(p)<=void then unsorted(p):=qq
+ else begin while link(qq)>void do qq:=link(qq);
+ link(qq):=unsorted(p); unsorted(p):=unsorted(pp);
+ end;
+unsorted(pp):=null; qq:=sorted(pp);
+if qq<>sentinel then
+ begin if unsorted(p)=void then unsorted(p):=null;
+ sorted(pp):=sentinel; r:=sorted_loc(p); q:=link(r); {|q=sorted(p)|}
+ if q=sentinel then sorted(p):=qq
+ else loop@+begin k:=info(qq);
+ while k>info(q) do
+ begin r:=q; q:=link(r);
+ end;
+ link(r):=qq; rr:=link(qq); link(qq):=q;
+ if rr=sentinel then goto done;
+ r:=qq; qq:=rr;
+ end;
+ end;
+done:
+
+@ The |total_weight| routine computes the total of all pixel weights
+in a given edge structure. It's not difficult to prove that this is
+the sum of $(-w)$ times $x$ taken over all edges,
+where $w$ and~$x$ are the weight and $x$~coordinates stored in an edge.
+It's not necessary to worry that this quantity will overflow the
+size of an |integer| register, because it will be less than~$2^{31}$
+unless the edge structure has more than 174,762 edges. However, we had
+better not try to compute it as a |scaled| integer, because a total
+weight of almost $12\times 2^{12}$ can be produced by only four edges.
+
+@p function total_weight(@!h:pointer):integer; {|h| is an edge header}
+var @!p,@!q:pointer; {variables that traverse the given structure}
+@!n:integer; {accumulated total so far}
+@!m:0..65535; {packed $x$ and $w$ values, including offsets}
+begin n:=0; p:=link(h);
+while p<>h do
+ begin q:=sorted(p);
+ while q<>sentinel do
+ @<Add the contribution of node |q| to the total weight,
+ and set |q:=link(q)|@>;
+ q:=unsorted(p);
+ while q>void do
+ @<Add the contribution of node |q| to the total weight,
+ and set |q:=link(q)|@>;
+ p:=link(p);
+ end;
+total_weight:=n;
+end;
+
+@ It's not necessary to add the offsets to the $x$ coordinates, because
+an entire edge structure can be shifted without affecting its total weight.
+Similarly, we don't need to subtract |zero_field|.
+
+@<Add the contribution of node |q| to the total weight...@>=
+begin m:=ho(info(q)); n:=n-((m mod 8)-zero_w)*(m div 8);
+q:=link(q);
+end
+
+@ So far we've done lots of things to edge structures assuming that
+edges are actually present, but we haven't seen how edges get created
+in the first place. Let's turn now to the problem of generating new edges.
+
+\MF\ will display new edges as they are being computed, if |tracing_edges|
+is positive. In order to keep such data reasonably compact, only the
+points at which the path makes a $90^\circ$ or $180^\circ$ turn are listed.
+
+The tracing algorithm must remember some past history in order to suppress
+unnecessary data. Three variables |trace_x|, |trace_y|, and |trace_yy|
+provide this history: The last coordinates printed were |(trace_x,trace_y)|,
+and the previous edge traced ended at |(trace_x,trace_yy)|. Before anything
+at all has been traced, |trace_x=-4096|.
+
+@<Glob...@>=
+@!trace_x:integer; {$x$~coordinate most recently shown in a trace}
+@!trace_y:integer; {$y$~coordinate most recently shown in a trace}
+@!trace_yy:integer; {$y$~coordinate most recently encountered}
+
+@ Edge tracing is initiated by the |begin_edge_tracing| routine,
+continued by the |trace_a_corner| routine, and terminated by the
+|end_edge_tracing| routine.
+
+@p procedure begin_edge_tracing;
+begin print_diagnostic("Tracing edges","",true);
+print(" (weight "); print_int(cur_wt); print_char(")"); trace_x:=-4096;
+end;
+@#
+procedure trace_a_corner;
+begin if file_offset>max_print_line-13 then print_nl("");
+print_char("("); print_int(trace_x); print_char(","); print_int(trace_yy);
+print_char(")"); trace_y:=trace_yy;
+end;
+@#
+procedure end_edge_tracing;
+begin if trace_x=-4096 then print_nl("(No new edges added.)")
+@.No new edges added@>
+else begin trace_a_corner; print_char(".");
+ end;
+end_diagnostic(true);
+end;
+
+@ Just after a new edge weight has been put into the |info| field of
+node~|r|, in row~|n|, the following routine continues an ongoing trace.
+
+@p procedure trace_new_edge(@!r:pointer;@!n:integer);
+var @!d:integer; {temporary data register}
+@!w:-3..3; {weight associated with an edge transition}
+@!m,@!n0,@!n1:integer; {column and row numbers}
+begin d:=ho(info(r)); w:=(d mod 8)-zero_w; m:=(d div 8)-m_offset(cur_edges);
+if w=cur_wt then
+ begin n0:=n+1; n1:=n;
+ end
+else begin n0:=n; n1:=n+1;
+ end; {the edges run from |(m,n0)| to |(m,n1)|}
+if m<>trace_x then
+ begin if trace_x=-4096 then
+ begin print_nl(""); trace_yy:=n0;
+ end
+ else if trace_yy<>n0 then print_char("?") {shouldn't happen}
+ else trace_a_corner;
+ trace_x:=m; trace_a_corner;
+ end
+else begin if n0<>trace_yy then print_char("!"); {shouldn't happen}
+ if ((n0<n1)and(trace_y>trace_yy))or((n0>n1)and(trace_y<trace_yy)) then
+ trace_a_corner;
+ end;
+trace_yy:=n1;
+end;
+
+@ One way to put new edge weights into an edge structure is to use the
+following routine, which simply draws a straight line from |(x0,y0)| to
+|(x1,y1)|. More precisely, it introduces weights for the edges of the
+discrete path $\bigl(\lfloor t[x_0,x_1]+{1\over2}+\epsilon\rfloor,
+\lfloor t[y_0,y_1]+{1\over2}+\epsilon\delta\rfloor\bigr)$,
+as $t$ varies from 0 to~1, where $\epsilon$ and $\delta$ are extremely small
+positive numbers.
+
+The structure header is assumed to be |cur_edges|; downward edge weights
+will be |cur_wt|, while upward ones will be |-cur_wt|.
+
+Of course, this subroutine will be called only in connection with others
+that eventually draw a complete cycle, so that the sum of the edge weights
+in each row will be zero whenever the row is displayed.
+
+@p procedure line_edges(@!x0,@!y0,@!x1,@!y1:scaled);
+label done,done1;
+var @!m0,@!n0,@!m1,@!n1:integer; {rounded and unscaled coordinates}
+@!delx,@!dely:scaled; {the coordinate differences of the line}
+@!yt:scaled; {smallest |y| coordinate that rounds the same as |y0|}
+@!tx:scaled; {tentative change in |x|}
+@!p,@!r:pointer; {list manipulation registers}
+@!base:integer; {amount added to edge-and-weight data}
+@!n:integer; {current row number}
+begin n0:=round_unscaled(y0);
+n1:=round_unscaled(y1);
+if n0<>n1 then
+ begin m0:=round_unscaled(x0); m1:=round_unscaled(x1);
+ delx:=x1-x0; dely:=y1-y0;
+ yt:=n0*unity-half_unit; y0:=y0-yt; y1:=y1-yt;
+ if n0<n1 then @<Insert upward edges for a line@>
+ else @<Insert downward edges for a line@>;
+ n_rover(cur_edges):=p; n_pos(cur_edges):=n+zero_field;
+ end;
+end;
+
+@ Here we are careful to cancel any effect of rounding error.
+
+@<Insert upward edges for a line@>=
+begin base:=8*m_offset(cur_edges)+min_halfword+zero_w-cur_wt;
+if m0<=m1 then edge_prep(m0,m1,n0,n1)@+else edge_prep(m1,m0,n0,n1);
+@<Move to row |n0|, pointed to by |p|@>;
+y0:=unity-y0;
+loop@+ begin r:=get_avail; link(r):=unsorted(p); unsorted(p):=r;@/
+ tx:=take_fraction(delx,make_fraction(y0,dely));
+ if ab_vs_cd(delx,y0,dely,tx)<0 then decr(tx);
+ {now $|tx|=\lfloor|y0|\cdot|delx|/|dely|\rfloor$}
+ info(r):=8*round_unscaled(x0+tx)+base;@/
+ y1:=y1-unity;
+ if internal[tracing_edges]>0 then trace_new_edge(r,n);
+ if y1<unity then goto done;
+ p:=link(p); y0:=y0+unity; incr(n);
+ end;
+done: end
+
+@ @<Insert downward edges for a line@>=
+begin base:=8*m_offset(cur_edges)+min_halfword+zero_w+cur_wt;
+if m0<=m1 then edge_prep(m0,m1,n1,n0)@+else edge_prep(m1,m0,n1,n0);
+decr(n0); @<Move to row |n0|, pointed to by |p|@>;
+loop@+ begin r:=get_avail; link(r):=unsorted(p); unsorted(p):=r;@/
+ tx:=take_fraction(delx,make_fraction(y0,dely));
+ if ab_vs_cd(delx,y0,dely,tx)<0 then incr(tx);
+ {now $|tx|=\lceil|y0|\cdot|delx|/|dely|\rceil$, since |dely<0|}
+ info(r):=8*round_unscaled(x0-tx)+base;@/
+ y1:=y1+unity;
+ if internal[tracing_edges]>0 then trace_new_edge(r,n);
+ if y1>=0 then goto done1;
+ p:=knil(p); y0:=y0+unity; decr(n);
+ end;
+done1: end
+
+@ @<Move to row |n0|, pointed to by |p|@>=
+n:=n_pos(cur_edges)-zero_field; p:=n_rover(cur_edges);
+if n<>n0 then
+ if n<n0 then
+ repeat incr(n); p:=link(p);
+ until n=n0
+ else repeat decr(n); p:=knil(p);
+ until n=n0
+
+@ \MF\ inserts most of its edges into edge structures via the
+|move_to_edges| subroutine, which uses the data stored in the |move| array
+to specify a sequence of ``rook moves.'' The starting point |(m0,n0)|
+and finishing point |(m1,n1)| of these moves, as seen from the standpoint
+of the first octant, are supplied as parameters; the moves should, however,
+be rotated into a given octant. (We're going to study octant
+transformations in great detail later; the reader may wish to come back to
+this part of the program after mastering the mysteries of octants.)
+
+The rook moves themselves are defined as follows, from a |first_octant|
+point of view: ``Go right |move[k]| steps, then go up one, for |0<=k<n1-n0|;
+then go right |move[n1-n0]| steps and stop.'' The sum of |move[k]|
+for |0<=k<=n1-n0| will be equal to |m1-m0|.
+
+As in the |line_edges| routine, we use |+cur_wt| as the weight of
+all downward edges and |-cur_wt| as the weight of all upward edges,
+after the moves have been rotated to the proper octant direction.
+
+There are two main cases to consider: \\{fast\_case} is for moves that
+travel in the direction of octants 1, 4, 5, and~8, while \\{slow\_case}
+is for moves that travel toward octants 2, 3, 6, and~7. The latter directions
+are comparatively cumbersome because they generate more upward or downward
+edges; a curve that travels horizontally doesn't produce any edges at all,
+but a curve that travels vertically touches lots of rows.
+
+@d fast_case_up=60 {for octants 1 and 4}
+@d fast_case_down=61 {for octants 5 and 8}
+@d slow_case_up=62 {for octants 2 and 3}
+@d slow_case_down=63 {for octants 6 and 7}
+
+@p procedure move_to_edges(@!m0,@!n0,@!m1,@!n1:integer);
+label fast_case_up,fast_case_down,slow_case_up,slow_case_down,done;
+var @!delta:0..move_size; {extent of |move| data}
+@!k:0..move_size; {index into |move|}
+@!p,@!r:pointer; {list manipulation registers}
+@!dx:integer; {change in edge-weight |info| when |x| changes by 1}
+@!edge_and_weight:integer; {|info| to insert}
+@!j:integer; {number of consecutive vertical moves}
+@!n:integer; {the current row pointed to by |p|}
+debug @!sum:integer;@+gubed@;@/
+begin delta:=n1-n0;
+debug sum:=move[0]; for k:=1 to delta do sum:=sum+abs(move[k]);
+if sum<>m1-m0 then confusion("0");@+gubed@;@/
+@:this can't happen 0}{\quad 0@>
+@<Prepare for and switch to the appropriate case, based on |octant|@>;
+fast_case_up:@<Add edges for first or fourth octants, then |goto done|@>;
+fast_case_down:@<Add edges for fifth or eighth octants, then |goto done|@>;
+slow_case_up:@<Add edges for second or third octants, then |goto done|@>;
+slow_case_down:@<Add edges for sixth or seventh octants, then |goto done|@>;
+done: n_pos(cur_edges):=n+zero_field; n_rover(cur_edges):=p;
+end;
+
+@ The current octant code appears in a global variable. If, for example,
+we have |octant=third_octant|, it means that a curve traveling in a north to
+north-westerly direction has been rotated for the purposes of internal
+calculations so that the |move| data travels in an east to north-easterly
+direction. We want to unrotate as we update the edge structure.
+
+@<Glob...@>=
+@!octant:first_octant..sixth_octant; {the current octant of interest}
+
+@ @<Prepare for and switch to the appropriate case, based on |octant|@>=
+case octant of
+first_octant:begin dx:=8; edge_prep(m0,m1,n0,n1); goto fast_case_up;
+ end;
+second_octant:begin dx:=8; edge_prep(n0,n1,m0,m1); goto slow_case_up;
+ end;
+third_octant:begin dx:=-8; edge_prep(-n1,-n0,m0,m1); negate(n0);
+ goto slow_case_up;
+ end;
+fourth_octant:begin dx:=-8; edge_prep(-m1,-m0,n0,n1); negate(m0);
+ goto fast_case_up;
+ end;
+fifth_octant:begin dx:=-8; edge_prep(-m1,-m0,-n1,-n0); negate(m0);
+ goto fast_case_down;
+ end;
+sixth_octant:begin dx:=-8; edge_prep(-n1,-n0,-m1,-m0); negate(n0);
+ goto slow_case_down;
+ end;
+seventh_octant:begin dx:=8; edge_prep(n0,n1,-m1,-m0); goto slow_case_down;
+ end;
+eighth_octant:begin dx:=8; edge_prep(m0,m1,-n1,-n0); goto fast_case_down;
+ end;
+end; {there are only eight octants}
+
+@ @<Add edges for first or fourth octants, then |goto done|@>=
+@<Move to row |n0|, pointed to by |p|@>;
+if delta>0 then
+ begin k:=0;
+ edge_and_weight:=8*(m0+m_offset(cur_edges))+min_halfword+zero_w-cur_wt;
+ repeat edge_and_weight:=edge_and_weight+dx*move[k];
+ fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight;
+ if internal[tracing_edges]>0 then trace_new_edge(r,n);
+ unsorted(p):=r; p:=link(p); incr(k); incr(n);
+ until k=delta;
+ end;
+goto done
+
+@ @<Add edges for fifth or eighth octants, then |goto done|@>=
+n0:=-n0-1; @<Move to row |n0|, pointed to by |p|@>;
+if delta>0 then
+ begin k:=0;
+ edge_and_weight:=8*(m0+m_offset(cur_edges))+min_halfword+zero_w+cur_wt;
+ repeat edge_and_weight:=edge_and_weight+dx*move[k];
+ fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight;
+ if internal[tracing_edges]>0 then trace_new_edge(r,n);
+ unsorted(p):=r; p:=knil(p); incr(k); decr(n);
+ until k=delta;
+ end;
+goto done
+
+@ @<Add edges for second or third octants, then |goto done|@>=
+edge_and_weight:=8*(n0+m_offset(cur_edges))+min_halfword+zero_w-cur_wt;
+n0:=m0; k:=0; @<Move to row |n0|, pointed to by |p|@>;
+repeat j:=move[k];
+while j>0 do
+ begin fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight;
+ if internal[tracing_edges]>0 then trace_new_edge(r,n);
+ unsorted(p):=r; p:=link(p); decr(j); incr(n);
+ end;
+edge_and_weight:=edge_and_weight+dx; incr(k);
+until k>delta;
+goto done
+
+@ @<Add edges for sixth or seventh octants, then |goto done|@>=
+edge_and_weight:=8*(n0+m_offset(cur_edges))+min_halfword+zero_w+cur_wt;
+n0:=-m0-1; k:=0; @<Move to row |n0|, pointed to by |p|@>;
+repeat j:=move[k];
+while j>0 do
+ begin fast_get_avail(r); link(r):=unsorted(p); info(r):=edge_and_weight;
+ if internal[tracing_edges]>0 then trace_new_edge(r,n);
+ unsorted(p):=r; p:=knil(p); decr(j); decr(n);
+ end;
+edge_and_weight:=edge_and_weight+dx; incr(k);
+until k>delta;
+goto done
+
+@ All the hard work of building an edge structure is undone by the following
+subroutine.
+
+@<Declare the recycling subroutines@>=
+procedure toss_edges(@!h:pointer);
+var @!p,@!q:pointer; {for list manipulation}
+begin q:=link(h);
+while q<>h do
+ begin flush_list(sorted(q));
+ if unsorted(q)>void then flush_list(unsorted(q));
+ p:=q; q:=link(q); free_node(p,row_node_size);
+ end;
+free_node(h,edge_header_size);
+end;
+
+@* \[21] Subdivision into octants.
+When \MF\ digitizes a path, it reduces the problem to the special
+case of paths that travel in ``first octant'' directions; i.e.,
+each cubic $z(t)=\bigl(x(t),y(t)\bigr)$ being digitized will have the property
+that $0\L y'(t)\L x'(t)$. This assumption makes digitizing simpler
+and faster than if the direction of motion has to be tested repeatedly.
+
+When $z(t)$ is cubic, $x'(t)$ and $y'(t)$ are quadratic, hence the four
+polynomials $x'(t)$, $y'(t)$, $x'(t)-y'(t)$, and $x'(t)+y'(t)$ cross
+through~0 at most twice each. If we subdivide the given cubic at these
+places, we get at most nine subintervals in each of which
+$x'(t)$, $y'(t)$, $x'(t)-y'(t)$, and $x'(t)+y'(t)$ all have a constant
+sign. The curve can be transformed in each of these subintervals so that
+it travels entirely in first octant directions, if we reflect $x\swap-x$,
+$y\swap-y$, and/or $x\swap y$ as necessary. (Incidentally, it can be
+shown that a cubic such that $x'(t)=16(2t-1)^2+2(2t-1)-1$ and
+$y'(t)=8(2t-1)^2+4(2t-1)$ does indeed split into nine subintervals.)
+
+@ The transformation that rotates coordinates, so that first octant motion
+can be assumed, is defined by the |skew| subroutine, which sets global
+variables |cur_x| and |cur_y| to the values that are appropriate in a
+given octant. (Octants are encoded as they were in the |n_arg| subroutine.)
+
+This transformation is ``skewed'' by replacing |(x,y)| by |(x-y,y)|,
+once first octant motion has been established. It turns out that
+skewed coordinates are somewhat better to work with when curves are
+actually digitized.
+
+@d set_two_end(#)==cur_y:=#;@+end
+@d set_two(#)==begin cur_x:=#; set_two_end
+
+@p procedure skew(@!x,@!y:scaled;@!octant:small_number);
+begin case octant of
+first_octant: set_two(x-y)(y);
+second_octant: set_two(y-x)(x);
+third_octant: set_two(y+x)(-x);
+fourth_octant: set_two(-x-y)(y);
+fifth_octant: set_two(-x+y)(-y);
+sixth_octant: set_two(-y+x)(-x);
+seventh_octant: set_two(-y-x)(x);
+eighth_octant: set_two(x+y)(-y);
+end; {there are no other cases}
+end;
+
+@ Conversely, the following subroutine sets |cur_x| and
+|cur_y| to the original coordinate values of a point, given an octant
+code and the point's coordinates |(x,y)| after they have been mapped into
+the first octant and skewed.
+
+@<Declare subroutines for printing expressions@>=
+procedure unskew(@!x,@!y:scaled;@!octant:small_number);
+begin case octant of
+first_octant: set_two(x+y)(y);
+second_octant: set_two(y)(x+y);
+third_octant: set_two(-y)(x+y);
+fourth_octant: set_two(-x-y)(y);
+fifth_octant: set_two(-x-y)(-y);
+sixth_octant: set_two(-y)(-x-y);
+seventh_octant: set_two(y)(-x-y);
+eighth_octant: set_two(x+y)(-y);
+end; {there are no other cases}
+end;
+
+@ @<Glob...@>=
+@!cur_x,@!cur_y:scaled;
+ {outputs of |skew|, |unskew|, and a few other routines}
+
+@ The conversion to skewed and rotated coordinates takes place in
+stages, and at one point in the transformation we will have negated the
+$x$ and/or $y$ coordinates so as to make curves travel in the first
+{\sl quadrant}. At this point the relevant ``octant'' code will be
+either |first_octant| (when no transformation has been done),
+or |fourth_octant=first_octant+negate_x| (when $x$ has been negated),
+or |fifth_octant=first_octant+negate_x+negate_y| (when both have been
+negated), or |eighth_octant=first_octant+negate_y| (when $y$ has been
+negated). The |abnegate| routine is sometimes needed to convert
+from one of these transformations to another.
+
+@p procedure abnegate(@!x,@!y:scaled;
+ @!octant_before,@!octant_after:small_number);
+begin if odd(octant_before)=odd(octant_after) then cur_x:=x
+ else cur_x:=-x;
+if (octant_before>negate_y)=(octant_after>negate_y) then cur_y:=y
+ else cur_y:=-y;
+end;
+
+@ Now here's a subroutine that's handy for subdivision: Given a
+quadratic polynomial $B(a,b,c;t)$, the |crossing_point| function
+returns the unique |fraction| value |t| between 0 and~1 at which
+$B(a,b,c;t)$ changes from positive to negative, or returns
+|t=fraction_one+1| if no such value exists. If |a<0| (so that $B(a,b,c;t)$
+is already negative at |t=0|), |crossing_point| returns the value zero.
+
+@d no_crossing==begin crossing_point:=fraction_one+1; return;
+ end
+@d one_crossing==begin crossing_point:=fraction_one; return;
+ end
+@d zero_crossing==begin crossing_point:=0; return;
+ end
+
+@p function crossing_point(@!a,@!b,@!c:integer):fraction;
+label exit;
+var @!d:integer; {recursive counter}
+@!x,@!xx,@!x0,@!x1,@!x2:integer; {temporary registers for bisection}
+begin if a<0 then zero_crossing;
+if c>=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;
+@<Use bisection to find the crossing point, if one exists@>;
+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 $\vert b-c\vert<2^{30}$.
+
+@<Use bisection to find the crossing point...@>=
+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
+
+@ Octant subdivision is applied only to cycles, i.e., to closed paths.
+A ``cycle spec'' is a data structure that contains specifications of
+@!@^cycle spec@>
+cubic curves and octant mappings for the cycle that has been subdivided
+into segments belonging to single octants. It is composed entirely of
+knot nodes, similar to those in the representation of paths; but the
+|explicit| type indications have been replaced by positive numbers
+that give further information. Additional |endpoint| data is also
+inserted at the octant boundaries.
+
+Recall that a cubic polynomial is represented by four control points
+that appear in adjacent nodes |p| and~|q| of a knot list. The |x|~coordinates
+are |x_coord(p)|, |right_x(p)|, |left_x(q)|, and |x_coord(q)|; the
+|y|~coordinates are similar. We shall call this ``the cubic following~|p|''
+or ``the cubic between |p| and~|q|'' or ``the cubic preceding~|q|.''
+
+Cycle specs are circular lists of cubic curves mixed with octant
+boundaries. Like cubics, the octant boundaries are represented in
+consecutive knot nodes |p| and~|q|. In such cases |right_type(p)=
+left_type(q)=endpoint|, and the fields |right_x(p)|, |right_y(p)|,
+|left_x(q)|, and |left_y(q)| are replaced by other fields called
+|right_octant(p)|, |right_transition(p)|, |left_octant(q)|, and
+|left_transition(q)|, respectively. For example, when the curve direction
+moves from the third octant to the fourth octant, the boundary nodes say
+|right_octant(p)=third_octant|, |left_octant(q)=fourth_octant|,
+and |right_transition(p)=left_transition(q)=diagonal|. A |diagonal|
+transition occurs when moving between octants 1~\AM~2, 3~\AM~4, 5~\AM~6, or
+7~\AM~8; an |axis| transition occurs when moving between octants 8~\AM~1,
+2~\AM~3, 4~\AM~5, 6~\AM~7. (Such transition information is redundant
+but convenient.) Fields |x_coord(p)| and |y_coord(p)| will contain
+coordinates of the transition point after rotation from third octant
+to first octant; i.e., if the true coordinates are $(x,y)$, the
+coordinates $(y,-x)$ will appear in node~|p|. Similarly, a fourth-octant
+transformation will have been applied after the transition, so
+we will have |x_coord(q)=@t$-x$@>| and |y_coord(q)=y|.
+
+The cubic between |p| and |q| will contain positive numbers in the
+fields |right_type(p)| and |left_type(q)|; this makes cubics
+distinguishable from octant boundaries, because |endpoint=0|.
+The value of |right_type(p)| will be the current octant code,
+during the time that cycle specs are being constructed; it will
+refer later to a pen offset position, if the envelope of a cycle is
+being computed. A cubic that comes from some subinterval of the $k$th
+step in the original cyclic path will have |left_type(q)=k|.
+
+@d right_octant==right_x {the octant code before a transition}
+@d left_octant==left_x {the octant after a transition}
+@d right_transition==right_y {the type of transition}
+@d left_transition==left_y {ditto, either |axis| or |diagonal|}
+@d axis=0 {a transition across the $x'$- or $y'$-axis}
+@d diagonal=1 {a transition where $y'=\pm x'$}
+
+@ Here's a routine that prints a cycle spec in symbolic form, so that it
+is possible to see what subdivision has been made. The point coordinates
+are converted back from \MF's internal ``rotated'' form to the external
+``true'' form. The global variable~|cur_spec| should point to a knot just
+after the beginning of an octant boundary, i.e., such that
+|left_type(cur_spec)=endpoint|.
+
+@d print_two_true(#)==unskew(#,octant); print_two(cur_x,cur_y)
+
+@p procedure print_spec(@!s:str_number);
+label not_found,done;
+var @!p,@!q:pointer; {for list traversal}
+@!octant:small_number; {the current octant code}
+begin print_diagnostic("Cycle spec",s,true);
+@.Cycle spec at line...@>
+p:=cur_spec; octant:=left_octant(p); print_ln;
+print_two_true(x_coord(cur_spec),y_coord(cur_spec));
+print(" % beginning in octant `");
+loop@+ begin print(octant_dir[octant]); print_char("'");
+ loop@+ begin q:=link(p);
+ if right_type(p)=endpoint then goto not_found;
+ @<Print the cubic between |p| and |q|@>;
+ p:=q;
+ end;
+not_found: if q=cur_spec then goto done;
+ p:=q; octant:=left_octant(p); print_nl("% entering octant `");
+ end;
+@.entering the nth octant@>
+done: print_nl(" & cycle"); end_diagnostic(true);
+end;
+
+@ Symbolic octant direction names are kept in the |octant_dir| array.
+
+@<Glob...@>=
+@!octant_dir:array[first_octant..sixth_octant] of str_number;
+
+@ @<Set init...@>=
+octant_dir[first_octant]:="ENE";
+octant_dir[second_octant]:="NNE";
+octant_dir[third_octant]:="NNW";
+octant_dir[fourth_octant]:="WNW";
+octant_dir[fifth_octant]:="WSW";
+octant_dir[sixth_octant]:="SSW";
+octant_dir[seventh_octant]:="SSE";
+octant_dir[eighth_octant]:="ESE";
+
+@ @<Print the cubic between...@>=
+begin print_nl(" ..controls ");
+print_two_true(right_x(p),right_y(p));
+print(" and ");
+print_two_true(left_x(q),left_y(q));
+print_nl(" ..");
+print_two_true(x_coord(q),y_coord(q));
+print(" % segment "); print_int(left_type(q)-1);
+end
+
+@ A much more compact version of a spec is printed to help users identify
+``strange paths.''
+
+@p procedure print_strange(@!s:str_number);
+var @!p:pointer; {for list traversal}
+@!f:pointer; {starting point in the cycle}
+@!q:pointer; {octant boundary to be printed}
+@!t:integer; {segment number, plus 1}
+begin if interaction=error_stop_mode then wake_up_terminal;
+print_nl(">");
+@.>\relax@>
+@<Find the starting point, |f|@>;
+@<Determine the octant boundary |q| that precedes |f|@>;
+t:=0;
+repeat if left_type(p)<>endpoint then
+ begin if left_type(p)<>t then
+ begin t:=left_type(p); print_char(" "); print_int(t-1);
+ end;
+ if q<>null then
+ begin @<Print the turns, if any, that start at |q|, and advance |q|@>;
+ print_char(" "); print(octant_dir[left_octant(q)]); q:=null;
+ end;
+ end
+else if q=null then q:=p;
+p:=link(p);
+until p=f;
+print_char(" "); print_int(left_type(p)-1);
+if q<>null then @<Print the turns...@>;
+print_err(s);
+end;
+
+@ If the segment numbers on the cycle are $t_1$, $t_2$, \dots, $t_m$,
+and if |m<=max_quarterword|,
+we have $t_{k-1}\L t_k$ except for at most one value of~$k$. If there are
+no exceptions, $f$ will point to $t_1$; otherwise it will point to the
+exceptional~$t_k$.
+
+There is at least one segment number (i.e., we always have $m>0$), because
+|print_strange| is never called upon to display an entirely ``dead'' cycle.
+
+@<Find the starting point, |f|@>=
+p:=cur_spec; t:=max_quarterword+1;
+repeat p:=link(p);
+if left_type(p)<>endpoint then
+ begin if left_type(p)<t then f:=p;
+ t:=left_type(p);
+ end;
+until p=cur_spec
+
+@ @<Determine the octant boundary...@>=
+p:=cur_spec; q:=p;
+repeat p:=link(p);
+if left_type(p)=endpoint then q:=p;
+until p=f
+
+@ When two octant boundaries are adjacent, the path is simply changing direction
+without moving. Such octant directions are shown in parentheses.
+
+@<Print the turns...@>=
+if left_type(link(q))=endpoint then
+ begin print(" ("); print(octant_dir[left_octant(q)]); q:=link(q);
+ while left_type(link(q))=endpoint do
+ begin print_char(" "); print(octant_dir[left_octant(q)]); q:=link(q);
+ end;
+ print_char(")");
+ end
+
+@ The |make_spec| routine is what subdivides paths into octants:
+Given a pointer |cur_spec| to a cyclic path, |make_spec| mungs the path data
+and returns a pointer to the corresponding cyclic spec.
+All ``dead'' cubics (i.e., cubics that don't move at all from
+their starting points) will have been removed from the result.
+@!@^dead cubics@>
+
+The idea of |make_spec| is fairly simple: Each cubic is first
+subdivided, if necessary, into pieces belonging to single octants;
+then the octant boundaries are inserted. But some of the details of
+this transformation are not quite obvious.
+
+If |autorounding>0|, the path will be adjusted so that critical tangent
+directions occur at ``good'' points with respect to the pen called |cur_pen|.
+
+The resulting spec will have all |x| and |y| coordinates at most
+$2^{28}-|half_unit|-1-|safety_margin|$ in absolute value. The pointer
+that is returned will start some octant, as required by |print_spec|.
+
+@p @t\4@>@<Declare subroutines needed by |make_spec|@>@;
+function make_spec(@!h:pointer;
+ @!safety_margin:scaled;@!tracing:integer):pointer;
+ {converts a path to a cycle spec}
+label continue,done;
+var @!p,@!q,@!r,@!s:pointer; {for traversing the lists}
+@!k:integer; {serial number of path segment, or octant code}
+@!chopped:integer; {positive if data truncated,
+ negative if data dangerously large}
+@<Other local variables for |make_spec|@>@;
+begin cur_spec:=h;
+if tracing>0 then
+ print_path(cur_spec,", before subdivision into octants",true);
+max_allowed:=fraction_one-half_unit-1-safety_margin;
+@<Truncate the values of all coordinates that exceed |max_allowed|, and stamp
+ segment numbers in each |left_type| field@>;
+quadrant_subdivide; {subdivide each cubic into pieces belonging to quadrants}
+if (internal[autorounding]>0)and(chopped=0) then xy_round;
+octant_subdivide; {complete the subdivision}
+if (internal[autorounding]>unity)and(chopped=0) then diag_round;
+@<Remove dead cubics@>;
+@<Insert octant boundaries and compute the turning number@>;
+while left_type(cur_spec)<>endpoint do cur_spec:=link(cur_spec);
+if tracing>0 then
+ if (internal[autorounding]<=0)or(chopped<>0) then
+ print_spec(", after subdivision")
+ else if internal[autorounding]>unity then
+ print_spec(", after subdivision and double autorounding")
+ else print_spec(", after subdivision and autorounding");
+make_spec:=cur_spec;
+end;
+
+@ The |make_spec| routine has an interesting side effect, namely to set
+the global variable |turning_number| to the number of times the tangent
+vector of the given cyclic path winds around the origin.
+
+Another global variable |cur_spec| points to the specification as it is
+being made, since several subroutines must go to work on it.
+
+And there are two global variables that affect the rounding
+decisions, as we'll see later; they are called |cur_pen| and |cur_path_type|.
+The latter will be |double_path_code| if |make_spec| is being
+applied to a double path.
+
+@d double_path_code=0 {command modifier for `\&{doublepath}'}
+@d contour_code=1 {command modifier for `\&{contour}'}
+@d also_code=2 {command modifier for `\&{also}'}
+
+@<Glob...@>=
+@!cur_spec:pointer; {the principal output of |make_spec|}
+@!turning_number:integer; {another output of |make_spec|}
+@!cur_pen:pointer; {an implicit input of |make_spec|, used in autorounding}
+@!cur_path_type:double_path_code..contour_code; {likewise}
+@!max_allowed:scaled; {coordinates must be at most this big}
+
+@ First we do a simple preprocessing step. The segment numbers inserted
+here will propagate to all descendants of cubics that are split into
+subintervals. These numbers must be nonzero, but otherwise they are
+present merely for diagnostic purposes. The cubic from |p| to~|q|
+that represents ``time interval'' |(t-1)..t| usually has |left_type(q)=t|,
+except when |t| is too large to be stored in a quarterword.
+
+@d procrustes(#)==@+if abs(#)>=dmax then
+ if abs(#)>max_allowed then
+ begin chopped:=1;
+ if #>0 then #:=max_allowed@+else #:=-max_allowed;
+ end
+ else if chopped=0 then chopped:=-1
+
+@<Truncate the values of all coordinates that exceed...@>=
+p:=cur_spec; k:=1; chopped:=0; dmax:=half(max_allowed);
+repeat procrustes(left_x(p)); procrustes(left_y(p));
+procrustes(x_coord(p)); procrustes(y_coord(p));
+procrustes(right_x(p)); procrustes(right_y(p));@/
+p:=link(p); left_type(p):=k;
+if k<max_quarterword then incr(k)@+else k:=1;
+until p=cur_spec;
+if chopped>0 then
+ begin print_err("Curve out of range");
+@.Curve out of range@>
+ help4("At least one of the coordinates in the path I'm about to")@/
+ ("digitize was really huge (potentially bigger than 4095).")@/
+ ("So I've cut it back to the maximum size.")@/
+ ("The results will probably be pretty wild.");
+ put_get_error;
+ end
+
+@ We may need to get rid of constant ``dead'' cubics that clutter up
+the data structure and interfere with autorounding.
+
+@<Declare subroutines needed by |make_spec|@>=
+procedure remove_cubic(@!p:pointer); {removes the cubic following~|p|}
+var @!q:pointer; {the node that disappears}
+begin q:=link(p); right_type(p):=right_type(q); link(p):=link(q);@/
+x_coord(p):=x_coord(q); y_coord(p):=y_coord(q);@/
+right_x(p):=right_x(q); right_y(p):=right_y(q);@/
+free_node(q,knot_node_size);
+end;
+
+@ The subdivision process proceeds by first swapping $x\swap-x$, if
+necessary, to ensure that $x'\G0$; then swapping $y\swap-y$, if necessary,
+to ensure that $y'\G0$; and finally swapping $x\swap y$, if necessary,
+to ensure that $x'\G y'$.
+
+Recall that the octant codes have been defined in such a way that, for
+example, |third_octant=first_octant+negate_x+switch_x_and_y|. The program
+uses the fact that |negate_x<negate_y<switch_x_and_y| to handle ``double
+negation'': If |c| is an octant code that possibly involves |negate_x|
+and/or |negate_y|, but not |switch_x_and_y|, then negating~|y| changes~|c|
+either to |c+negate_y| or |c-negate_y|, depending on whether
+|c<=negate_y| or |c>negate_y|. Octant codes are always greater than zero.
+
+The first step is to subdivide on |x| and |y| only, so that horizontal
+and vertical autorounding can be done before we compare $x'$ to $y'$.
+
+@<Declare subroutines needed by |make_spec|@>=
+@t\4@>@<Declare the procedure called |split_cubic|@>@;
+procedure quadrant_subdivide;
+label continue,exit;
+var @!p,@!q,@!r,@!s,@!pp,@!qq:pointer; {for traversing the lists}
+@!first_x,@!first_y:scaled; {unnegated coordinates of node |cur_spec|}
+@!del1,@!del2,@!del3,@!del,@!dmax:scaled; {proportional to the control
+ points of a quadratic derived from a cubic}
+@!t:fraction; {where a quadratic crosses zero}
+@!dest_x,@!dest_y:scaled; {final values of |x| and |y| in the current cubic}
+@!constant_x:boolean; {is |x| constant between |p| and |q|?}
+begin p:=cur_spec; first_x:=x_coord(cur_spec); first_y:=y_coord(cur_spec);
+repeat continue: q:=link(p);
+@<Subdivide the cubic between |p| and |q| so that the results travel
+ toward the right halfplane@>;
+@<Subdivide all cubics between |p| and |q| so that the results travel
+ toward the first quadrant; but |return| or |goto continue| if the
+ cubic from |p| to |q| was dead@>;
+p:=q;
+until p=cur_spec;
+exit:end;
+
+@ All three subdivision processes are similar, so it's possible to
+get the general idea by studying the first one (which is the simplest).
+The calculation makes use of the fact that the derivatives of
+Bernshte{\u\i}n polynomials satisfy
+$B'(z_0,z_1,\ldots,z_n;t)=nB(z_1-z_0,\ldots,z_n-z_{n-1};t)$.
+
+When this routine begins, |right_type(p)| is |explicit|; we should
+set |right_type(p):=first_octant|. However, no assignment is made,
+because |explicit=first_octant|. The author apologizes for using
+such trickery here; it is really hard to do redundant computations
+just for the sake of purity.
+
+@<Subdivide the cubic between |p| and |q| so that the results travel
+ toward the right halfplane...@>=
+if q=cur_spec then
+ begin dest_x:=first_x; dest_y:=first_y;
+ end
+else begin dest_x:=x_coord(q); dest_y:=y_coord(q);
+ end;
+del1:=right_x(p)-x_coord(p); del2:=left_x(q)-right_x(p);
+del3:=dest_x-left_x(q);
+@<Scale up |del1|, |del2|, and |del3| for greater accuracy;
+ also set |del| to the first nonzero element of |(del1,del2,del3)|@>;
+if del=0 then constant_x:=true
+else begin constant_x:=false;
+ if del<0 then @<Complement the |x| coordinates of the
+ cubic between |p| and~|q|@>;
+ t:=crossing_point(del1,del2,del3);
+ if t<fraction_one then
+ @<Subdivide the cubic with respect to $x'$, possibly twice@>;
+ end
+
+@ If |del1=del2=del3=0|, it's impossible to obey the title of this
+section. We just set |del=0| in that case.
+@^inner loop@>
+
+@<Scale up |del1|, |del2|, and |del3| for greater accuracy...@>=
+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<fraction_half do
+ begin double(dmax); double(del1); double(del2); double(del3);
+ end;
+ end
+
+@ During the subdivision phases of |make_spec|, the |x_coord| and |y_coord|
+fields of node~|q| are not transformed to agree with the octant
+stated in |right_type(p)|; they remain consistent with |right_type(q)|.
+But |left_x(q)| and |left_y(q)| are governed by |right_type(p)|.
+
+@<Complement the |x| coordinates...@>=
+begin negate(x_coord(p)); negate(right_x(p));
+negate(left_x(q));@/
+negate(del1); negate(del2); negate(del3);@/
+negate(dest_x);
+right_type(p):=first_octant+negate_x;
+end
+
+@ When a cubic is split at a |fraction| value |t|, we obtain two cubics
+whose B\'ezier control points are obtained by a generalization of the
+bisection process: The formula
+`$z_k^{(j+1)}={1\over2}(z_k^{(j)}+z\k^{(j)})$' becomes
+`$z_k^{(j+1)}=t[z_k^{(j)},z\k^{(j)}]$'.
+
+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]|.
+
+If |0<=t<=1|, the quantity |t[a,b]| is always between |a| and~|b|, even in
+the presence of rounding errors. Our subroutines
+also obey the identity |t[a,b]+t[b,a]=a+b|.
+
+@d t_of_the_way_end(#)==#,t@=)@>
+@d t_of_the_way(#)==#-take_fraction@=(@>#-t_of_the_way_end
+
+@<Declare the procedure called |split_cubic|@>=
+procedure split_cubic(@!p:pointer;@!t:fraction;
+ @!xq,@!yq:scaled); {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):=left_type(q); right_type(r):=right_type(p);@#
+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))(xq);
+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))(yq);
+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;
+
+@ Since $x'(t)$ is a quadratic equation, it can cross through zero
+at~most twice. When it does cross zero, we make doubly sure that the
+derivative is really zero at the splitting point, in case rounding errors
+have caused the split cubic to have an apparently nonzero derivative.
+We also make sure that the split cubic is monotonic.
+
+@<Subdivide the cubic with respect to $x'$, possibly twice@>=
+begin split_cubic(p,t,dest_x,dest_y); r:=link(p);
+if right_type(r)>negate_x then right_type(r):=first_octant
+else right_type(r):=first_octant+negate_x;
+if x_coord(r)<x_coord(p) then x_coord(r):=x_coord(p);
+left_x(r):=x_coord(r);
+if right_x(p)>x_coord(r) then right_x(p):=x_coord(r);
+ {we always have |x_coord(p)<=right_x(p)|}
+negate(x_coord(r)); right_x(r):=x_coord(r);
+negate(left_x(q)); negate(dest_x);@/
+del2:=t_of_the_way(del2)(del3);
+ {now |0,del2,del3| represent $x'$ on the remaining interval}
+if del2>0 then del2:=0;
+t:=crossing_point(0,-del2,-del3);
+if t<fraction_one then @<Subdivide the cubic a second time
+ with respect to $x'$@>
+else begin if x_coord(r)>dest_x then
+ begin x_coord(r):=dest_x; left_x(r):=-x_coord(r); right_x(r):=x_coord(r);
+ end;
+ if left_x(q)>dest_x then left_x(q):=dest_x
+ else if left_x(q)<x_coord(r) then left_x(q):=x_coord(r);
+ end;
+end
+
+@ @<Subdivide the cubic a second time with respect to $x'$@>=
+begin split_cubic(r,t,dest_x,dest_y); s:=link(r);
+if x_coord(s)<dest_x then x_coord(s):=dest_x;
+if x_coord(s)<x_coord(r) then x_coord(s):=x_coord(r);
+right_type(s):=right_type(p);
+left_x(s):=x_coord(s); {now |x_coord(r)=right_x(r)<=left_x(s)|}
+if left_x(q)<dest_x then left_x(q):=-dest_x
+else if left_x(q)>x_coord(s) then left_x(q):=-x_coord(s)
+else negate(left_x(q));
+negate(x_coord(s)); right_x(s):=x_coord(s);
+end
+
+@ The process of subdivision with respect to $y'$ is like that with respect
+to~$x'$, with the slight additional complication that two or three cubics
+might now appear between |p| and~|q|.
+
+@<Subdivide all cubics between |p| and |q| so that the results travel
+ toward the first quadrant...@>=
+pp:=p;
+repeat qq:=link(pp);
+abnegate(x_coord(qq),y_coord(qq),right_type(qq),right_type(pp));
+dest_x:=cur_x; dest_y:=cur_y;@/
+del1:=right_y(pp)-y_coord(pp); del2:=left_y(qq)-right_y(pp);
+del3:=dest_y-left_y(qq);
+@<Scale up |del1|, |del2|, and |del3| for greater accuracy;
+ also set |del| to the first nonzero element of |(del1,del2,del3)|@>;
+if del<>0 then {they weren't all zero}
+ begin if del<0 then @<Complement the |y| coordinates of the
+ cubic between |pp| and~|qq|@>;
+ t:=crossing_point(del1,del2,del3);
+ if t<fraction_one then
+ @<Subdivide the cubic with respect to $y'$, possibly twice@>;
+ end
+else @<Do any special actions needed when |y| is constant;
+ |return| or |goto continue| if a dead cubic from |p| to |q| is removed@>;
+pp:=qq;
+until pp=q;
+if constant_x then @<Correct the octant code in segments with decreasing |y|@>
+
+@ @<Complement the |y| coordinates...@>=
+begin negate(y_coord(pp)); negate(right_y(pp));
+negate(left_y(qq));@/
+negate(del1); negate(del2); negate(del3);@/
+negate(dest_y);
+right_type(pp):=right_type(pp)+negate_y;
+end
+
+@ @<Subdivide the cubic with respect to $y'$, possibly twice@>=
+begin split_cubic(pp,t,dest_x,dest_y); r:=link(pp);
+if right_type(r)>negate_y then right_type(r):=right_type(r)-negate_y
+else right_type(r):=right_type(r)+negate_y;
+if y_coord(r)<y_coord(pp) then y_coord(r):=y_coord(pp);
+left_y(r):=y_coord(r);
+if right_y(pp)>y_coord(r) then right_y(pp):=y_coord(r);
+ {we always have |y_coord(pp)<=right_y(pp)|}
+negate(y_coord(r)); right_y(r):=y_coord(r);
+negate(left_y(qq)); negate(dest_y);@/
+if x_coord(r)<x_coord(pp) then x_coord(r):=x_coord(pp)
+else if x_coord(r)>dest_x then x_coord(r):=dest_x;
+if left_x(r)>x_coord(r) then
+ begin left_x(r):=x_coord(r);
+ if right_x(pp)>x_coord(r) then right_x(pp):=x_coord(r);
+ end;
+if right_x(r)<x_coord(r) then
+ begin right_x(r):=x_coord(r);
+ if left_x(qq)<x_coord(r) then left_x(qq):=x_coord(r);
+ end;
+del2:=t_of_the_way(del2)(del3);
+ {now |0,del2,del3| represent $y'$ on the remaining interval}
+if del2>0 then del2:=0;
+t:=crossing_point(0,-del2,-del3);
+if t<fraction_one then @<Subdivide the cubic a second time
+ with respect to $y'$@>
+else begin if y_coord(r)>dest_y then
+ begin y_coord(r):=dest_y; left_y(r):=-y_coord(r); right_y(r):=y_coord(r);
+ end;
+ if left_y(qq)>dest_y then left_y(qq):=dest_y
+ else if left_y(qq)<y_coord(r) then left_y(qq):=y_coord(r);
+ end;
+end
+
+@ @<Subdivide the cubic a second time with respect to $y'$@>=
+begin split_cubic(r,t,dest_x,dest_y); s:=link(r);@/
+if y_coord(s)<dest_y then y_coord(s):=dest_y;
+if y_coord(s)<y_coord(r) then y_coord(s):=y_coord(r);
+right_type(s):=right_type(pp);
+left_y(s):=y_coord(s); {now |y_coord(r)=right_y(r)<=left_y(s)|}
+if left_y(qq)<dest_y then left_y(qq):=-dest_y
+else if left_y(qq)>y_coord(s) then left_y(qq):=-y_coord(s)
+else negate(left_y(qq));
+negate(y_coord(s)); right_y(s):=y_coord(s);
+if x_coord(s)<x_coord(r) then x_coord(s):=x_coord(r)
+else if x_coord(s)>dest_x then x_coord(s):=dest_x;
+if left_x(s)>x_coord(s) then
+ begin left_x(s):=x_coord(s);
+ if right_x(r)>x_coord(s) then right_x(r):=x_coord(s);
+ end;
+if right_x(s)<x_coord(s) then
+ begin right_x(s):=x_coord(s);
+ if left_x(qq)<x_coord(s) then left_x(qq):=x_coord(s);
+ end;
+end
+
+@ If the cubic is constant in $y$ and increasing in $x$, we have classified
+it as traveling in the first octant. If the cubic is constant
+in~$y$ and decreasing in~$x$, it is desirable to classify it as traveling
+in the fifth octant (not the fourth), because autorounding will be consistent
+with respect to doublepaths only if the octant number changes by four when
+the path is reversed. Therefore we negate the $y$~coordinates
+when they are constant but the curve is decreasing in~$x$; this gives
+the desired result except in pathological paths.
+
+If the cubic is ``dead,'' i.e., constant in both |x| and |y|, we remove
+it unless it is the only cubic in the entire path. We |goto continue|
+if it wasn't the final cubic, so that the test |p=cur_spec| does not
+falsely imply that all cubics have been processed.
+
+@<Do any special actions needed when |y| is constant...@>=
+if constant_x then {|p=pp|, |q=qq|, and the cubic is dead}
+ begin if q<>p then
+ begin remove_cubic(p); {remove the dead cycle and recycle node |q|}
+ if cur_spec<>q then goto continue
+ else begin cur_spec:=p; return;
+ end; {the final cubic was dead and is gone}
+ end;
+ end
+else if not odd(right_type(pp)) then {the $x$ coordinates were negated}
+ @<Complement the |y| coordinates...@>
+
+@ A similar correction to octant codes deserves to be made when |x| is
+constant and |y| is decreasing.
+
+@<Correct the octant code in segments with decreasing |y|@>=
+begin pp:=p;
+repeat qq:=link(pp);
+if right_type(pp)>negate_y then {the $y$ coordinates were negated}
+ begin right_type(pp):=right_type(pp)+negate_x;
+ negate(x_coord(pp)); negate(right_x(pp)); negate(left_x(qq));
+ end;
+pp:=qq;
+until pp=q;
+end
+
+@ Finally, the process of subdividing to make $x'\G y'$ is like the other
+two subdivisions, with a few new twists. We skew the coordinates at this time.
+
+@<Declare subroutines needed by |make_spec|@>=
+procedure octant_subdivide;
+var @!p,@!q,@!r,@!s:pointer; {for traversing the lists}
+@!del1,@!del2,@!del3,@!del,@!dmax:scaled; {proportional to the control
+ points of a quadratic derived from a cubic}
+@!t:fraction; {where a quadratic crosses zero}
+@!dest_x,@!dest_y:scaled; {final values of |x| and |y| in the current cubic}
+begin p:=cur_spec;
+repeat q:=link(p);@/
+x_coord(p):=x_coord(p)-y_coord(p);
+right_x(p):=right_x(p)-right_y(p);
+left_x(q):=left_x(q)-left_y(q);@/
+@<Subdivide the cubic between |p| and |q| so that the results travel
+ toward the first octant@>;
+p:=q;
+until p=cur_spec;
+end;
+
+@ @<Subdivide the cubic between |p| and |q| so that the results travel
+ toward the first octant@>=
+@<Set up the variables |(del1,del2,del3)| to represent $x'-y'$@>;
+@<Scale up |del1|, |del2|, and |del3| for greater accuracy;
+ also set |del| to the first nonzero element of |(del1,del2,del3)|@>;
+if del<>0 then {they weren't all zero}
+ begin if del<0 then @<Swap the |x| and |y| coordinates of the
+ cubic between |p| and~|q|@>;
+ t:=crossing_point(del1,del2,del3);
+ if t<fraction_one then
+ @<Subdivide the cubic with respect to $x'-y'$, possibly twice@>;
+ end
+
+@ @<Set up the variables |(del1,del2,del3)| to represent $x'-y'$@>=
+if q=cur_spec then
+ begin unskew(x_coord(q),y_coord(q),right_type(q));
+ skew(cur_x,cur_y,right_type(p)); dest_x:=cur_x; dest_y:=cur_y;
+ end
+else begin abnegate(x_coord(q),y_coord(q),right_type(q),right_type(p));
+ dest_x:=cur_x-cur_y; dest_y:=cur_y;
+ end;
+del1:=right_x(p)-x_coord(p); del2:=left_x(q)-right_x(p);
+del3:=dest_x-left_x(q)
+
+@ The swapping here doesn't simply interchange |x| and |y| values,
+because the coordinates are skewed. It turns out that this is easier
+than ordinary swapping, because it can be done in two assignment statements
+rather than three.
+
+@ @<Swap the |x| and |y| coordinates...@>=
+begin y_coord(p):=x_coord(p)+y_coord(p); negate(x_coord(p));@/
+right_y(p):=right_x(p)+right_y(p); negate(right_x(p));@/
+left_y(q):=left_x(q)+left_y(q); negate(left_x(q));@/
+negate(del1); negate(del2); negate(del3);@/
+dest_y:=dest_x+dest_y; negate(dest_x);@/
+right_type(p):=right_type(p)+switch_x_and_y;
+end
+
+@ A somewhat tedious case analysis is carried out here to make sure that
+nasty rounding errors don't destroy our assumptions of monotonicity.
+
+@<Subdivide the cubic with respect to $x'-y'$, possibly twice@>=
+begin split_cubic(p,t,dest_x,dest_y); r:=link(p);
+if right_type(r)>switch_x_and_y then right_type(r):=right_type(r)-switch_x_and_y
+else right_type(r):=right_type(r)+switch_x_and_y;
+if y_coord(r)<y_coord(p) then y_coord(r):=y_coord(p)
+else if y_coord(r)>dest_y then y_coord(r):=dest_y;
+if x_coord(p)+y_coord(r)>dest_x+dest_y then
+ y_coord(r):=dest_x+dest_y-x_coord(p);
+if left_y(r)>y_coord(r) then
+ begin left_y(r):=y_coord(r);
+ if right_y(p)>y_coord(r) then right_y(p):=y_coord(r);
+ end;
+if right_y(r)<y_coord(r) then
+ begin right_y(r):=y_coord(r);
+ if left_y(q)<y_coord(r) then left_y(q):=y_coord(r);
+ end;
+if x_coord(r)<x_coord(p) then x_coord(r):=x_coord(p)
+else if x_coord(r)+y_coord(r)>dest_x+dest_y then
+ x_coord(r):=dest_x+dest_y-y_coord(r);
+left_x(r):=x_coord(r);
+if right_x(p)>x_coord(r) then right_x(p):=x_coord(r);
+ {we always have |x_coord(p)<=right_x(p)|}
+y_coord(r):=y_coord(r)+x_coord(r); right_y(r):=right_y(r)+x_coord(r);@/
+negate(x_coord(r)); right_x(r):=x_coord(r);@/
+left_y(q):=left_y(q)+left_x(q); negate(left_x(q));@/
+dest_y:=dest_y+dest_x; negate(dest_x);
+if right_y(r)<y_coord(r) then
+ begin right_y(r):=y_coord(r);
+ if left_y(q)<y_coord(r) then left_y(q):=y_coord(r);
+ end;
+del2:=t_of_the_way(del2)(del3);
+ {now |0,del2,del3| represent $x'-y'$ on the remaining interval}
+if del2>0 then del2:=0;
+t:=crossing_point(0,-del2,-del3);
+if t<fraction_one then
+ @<Subdivide the cubic a second time with respect to $x'-y'$@>
+else begin if x_coord(r)>dest_x then
+ begin x_coord(r):=dest_x; left_x(r):=-x_coord(r); right_x(r):=x_coord(r);
+ end;
+ if left_x(q)>dest_x then left_x(q):=dest_x
+ else if left_x(q)<x_coord(r) then left_x(q):=x_coord(r);
+ end;
+end
+
+@ @<Subdivide the cubic a second time with respect to $x'-y'$@>=
+begin split_cubic(r,t,dest_x,dest_y); s:=link(r);@/
+if y_coord(s)<y_coord(r) then y_coord(s):=y_coord(r)
+else if y_coord(s)>dest_y then y_coord(s):=dest_y;
+if x_coord(r)+y_coord(s)>dest_x+dest_y then
+ y_coord(s):=dest_x+dest_y-x_coord(r);
+if left_y(s)>y_coord(s) then
+ begin left_y(s):=y_coord(s);
+ if right_y(r)>y_coord(s) then right_y(r):=y_coord(s);
+ end;
+if right_y(s)<y_coord(s) then
+ begin right_y(s):=y_coord(s);
+ if left_y(q)<y_coord(s) then left_y(q):=y_coord(s);
+ end;
+if x_coord(s)+y_coord(s)>dest_x+dest_y then x_coord(s):=dest_x+dest_y-y_coord(s)
+else begin if x_coord(s)<dest_x then x_coord(s):=dest_x;
+ if x_coord(s)<x_coord(r) then x_coord(s):=x_coord(r);
+ end;
+right_type(s):=right_type(p);
+left_x(s):=x_coord(s); {now |x_coord(r)=right_x(r)<=left_x(s)|}
+if left_x(q)<dest_x then
+ begin left_y(q):=left_y(q)+dest_x; left_x(q):=-dest_x;@+end
+else if left_x(q)>x_coord(s) then
+ begin left_y(q):=left_y(q)+x_coord(s); left_x(q):=-x_coord(s);@+end
+else begin left_y(q):=left_y(q)+left_x(q); negate(left_x(q));@+end;
+y_coord(s):=y_coord(s)+x_coord(s); right_y(s):=right_y(s)+x_coord(s);@/
+negate(x_coord(s)); right_x(s):=x_coord(s);@/
+if right_y(s)<y_coord(s) then
+ begin right_y(s):=y_coord(s);
+ if left_y(q)<y_coord(s) then left_y(q):=y_coord(s);
+ end;
+end
+
+@ It's time now to consider ``autorounding,'' which tries to make horizontal,
+vertical, and diagonal tangents occur at places that will produce appropriate
+images after the curve is digitized.
+
+The first job is to fix things so that |x(t)| plus the horizontal pen offset
+is an integer multiple of the
+current ``granularity'' when the derivative $x'(t)$ crosses through zero.
+The given cyclic path contains regions where $x'(t)\G0$ and regions
+where $x'(t)\L0$. The |quadrant_subdivide| routine is called into action
+before any of the path coordinates have been skewed, but some of them
+may have been negated. In regions where $x'(t)\G0$ we have |right_type=
+first_octant| or |right_type=eighth_octant|; in regions where $x'(t)\L0$,
+we have |right_type=fifth_octant| or |right_type=fourth_octant|.
+
+Within any such region the transformed $x$ values increase monotonically
+from, say, $x_0$ to~$x_1$. We want to modify things by applying a linear
+transformation to all $x$ coordinates in the region, after which
+the $x$ values will increase monotonically from round$(x_0)$ to round$(x_1)$.
+
+This rounding scheme sounds quite simple, and it usually is. But several
+complications can arise that might make the task more difficult. In the
+first place, autorounding is inappropriate at cusps where $x'$ jumps
+discontinuously past zero without ever being zero. In the second place,
+the current pen might be unsymmetric in such a way that $x$ coordinates
+should round differently in different parts of the curve.
+These considerations imply that round$(x_0)$ might be greater
+than round$(x_1)$, even though $x_0\L x_1$; in such cases we do not want
+to carry out the linear transformation. Furthermore, it's possible to have
+round$(x_1)-\hbox{round} (x_0)$ positive but much greater than $x_1-x_0$;
+then the transformation might distort the curve drastically, and again we
+want to avoid it. Finally, the rounded points must be consistent between
+adjacent regions, hence we can't transform one region without knowing
+about its neighbors.
+
+To handle all these complications, we must first look at the whole
+cycle and choose rounded $x$ values that are ``safe.'' The following
+procedure does this: Given $m$~values $(b_0,b_1,\ldots,b_{m-1})$ before
+rounding and $m$~corresponding values $(a_0,a_1,\ldots,a_{m-1})$ that would
+be desirable after rounding, the |make_safe| routine sets $a$'s to $b$'s
+if necessary so that $0\L(a\k-a_k)/(b\k-b_k)\L2$ afterwards. It is
+symmetric under cyclic permutation, reversal, and/or negation of the inputs.
+(Instead of |a|, |b|, and~|m|, the program uses the names |after|,
+|before|, and |cur_rounding_ptr|.)
+
+@<Declare subroutines needed by |make_spec|@>=
+procedure make_safe;
+var @!k:0..max_wiggle; {runs through the list of inputs}
+@!all_safe:boolean; {does everything look OK so far?}
+@!next_a:scaled; {|after[k]| before it might have changed}
+@!delta_a,@!delta_b:scaled; {|after[k+1]-after[k]| and |before[k+1]-before[k]|}
+begin before[cur_rounding_ptr]:=before[0]; {wrap around}
+node_to_round[cur_rounding_ptr]:=node_to_round[0];
+repeat after[cur_rounding_ptr]:=after[0]; all_safe:=true; next_a:=after[0];
+for k:=0 to cur_rounding_ptr-1 do
+ begin delta_b:=before[k+1]-before[k];
+ if delta_b>=0 then delta_a:=after[k+1]-next_a
+ else delta_a:=next_a-after[k+1];
+ next_a:=after[k+1];
+ if (delta_a<0)or(delta_a>abs(delta_b+delta_b)) then
+ begin all_safe:=false; after[k]:=before[k];
+ if k=cur_rounding_ptr-1 then after[0]:=before[0]
+ else after[k+1]:=before[k+1];
+ end;
+ end;
+until all_safe;
+end;
+
+@ The global arrays used by |make_safe| are accompanied by an array of
+pointers into the current knot list.
+
+@<Glob...@>=
+@!before,@!after:array[0..max_wiggle] of scaled; {data for |make_safe|}
+@!node_to_round:array[0..max_wiggle] of pointer; {reference back to the path}
+@!cur_rounding_ptr:0..max_wiggle; {how many are being used}
+@!max_rounding_ptr:0..max_wiggle; {how many have been used}
+
+@ @<Set init...@>=
+max_rounding_ptr:=0;
+
+@ New entries go into the tables via the |before_and_after| routine:
+
+@<Declare subroutines needed by |make_spec|@>=
+procedure before_and_after(@!b,@!a:scaled;@!p:pointer);
+begin if cur_rounding_ptr=max_rounding_ptr then
+ if max_rounding_ptr<max_wiggle then incr(max_rounding_ptr)
+ else overflow("rounding table size",max_wiggle);
+@:METAFONT capacity exceeded rounding table size}{\quad rounding table size@>
+after[cur_rounding_ptr]:=a; before[cur_rounding_ptr]:=b;
+node_to_round[cur_rounding_ptr]:=p; incr(cur_rounding_ptr);
+end;
+
+@ A global variable called |cur_gran| is used instead of |internal[
+granularity]|, because we want to work with a number that's guaranteed to
+be positive.
+
+@<Glob...@>=
+@!cur_gran:scaled; {the current granularity (which normally is |unity|)}
+
+@ The |good_val| function computes a number |a| that's as close as
+possible to~|b|, with the property that |a+o| is a multiple of
+|cur_gran|.
+
+If we assume that |cur_gran| is even (since it will in fact be a multiple
+of |unity| in all reasonable applications), we have the identity
+|good_val(-b-1,-o)=-good_val(b,o)|.
+
+@<Declare subroutines needed by |make_spec|@>=
+function good_val(@!b,@!o:scaled):scaled;
+var @!a:scaled; {accumulator}
+begin a:=b+o;
+if a>=0 then a:=a-(a mod cur_gran)-o
+else a:=a+((-(a+1)) mod cur_gran)-cur_gran+1-o;
+if b-a<a+cur_gran-b then good_val:=a
+else good_val:=a+cur_gran;
+end;
+
+@ When we're rounding a doublepath, we might need to compromise between
+two opposing tendencies, if the pen thickness is not a multiple of the
+granularity. The following ``compromise'' adjustment, suggested by
+John Hobby, finds the best way out of the dilemma. (Only the value
+@^Hobby, John Douglas@>
+modulo |cur_gran| is relevant in our applications, so the result turns
+out to be essentially symmetric in |u| and~|v|.)
+
+@<Declare subroutines needed by |make_spec|@>=
+function compromise(@!u,@!v:scaled):scaled;
+begin compromise:=half(good_val(u+u,-u-v));
+end;
+
+@ Here, then, is the procedure that rounds $x$ coordinates as described;
+it does the same for $y$ coordinates too, independently.
+
+@<Declare subroutines needed by |make_spec|@>=
+procedure xy_round;
+var @!p,@!q:pointer; {list manipulation registers}
+@!b,@!a:scaled; {before and after values}
+@!pen_edge:scaled; {offset that governs rounding}
+@!alpha:fraction; {coefficient of linear transformation}
+begin cur_gran:=abs(internal[granularity]);
+if cur_gran=0 then cur_gran:=unity;
+p:=cur_spec; cur_rounding_ptr:=0;
+repeat q:=link(p);
+@<If node |q| is a transition point for |x| coordinates,
+ compute and save its before-and-after coordinates@>;
+p:=q;
+until p=cur_spec;
+if cur_rounding_ptr>0 then @<Transform the |x| coordinates@>;
+p:=cur_spec; cur_rounding_ptr:=0;
+repeat q:=link(p);
+@<If node |q| is a transition point for |y| coordinates,
+ compute and save its before-and-after coordinates@>;
+p:=q;
+until p=cur_spec;
+if cur_rounding_ptr>0 then @<Transform the |y| coordinates@>;
+end;
+
+@ When |x| has been negated, the |octant| codes are even. We allow
+for an error of up to .01 pixel (i.e., 655 |scaled| units) in the
+derivative calculations at transition nodes.
+
+@<If node |q| is a transition point for |x| coordinates...@>=
+if odd(right_type(p))<>odd(right_type(q)) then
+ begin if odd(right_type(q)) then b:=x_coord(q)@+else b:=-x_coord(q);
+ if (abs(x_coord(q)-right_x(q))<655)or@|
+ (abs(x_coord(q)+left_x(q))<655) then
+ @<Compute before-and-after |x| values based on the current pen@>
+ else a:=b;
+ if abs(a)>max_allowed then
+ if a>0 then a:=max_allowed@+else a:=-max_allowed;
+ before_and_after(b,a,q);
+ end
+
+@ When we study the data representation for pens, we'll learn that the
+|x|~coordinate of the current pen's west edge is
+$$\hbox{|y_coord(link(cur_pen+seventh_octant))|},$$
+and that there are similar ways to address other important offsets.
+
+@d north_edge(#)==y_coord(link(#+fourth_octant))
+@d south_edge(#)==y_coord(link(#+first_octant))
+@d east_edge(#)==y_coord(link(#+second_octant))
+@d west_edge(#)==y_coord(link(#+seventh_octant))
+
+@<Compute before-and-after |x| values based on the current pen@>=
+begin if cur_pen=null_pen then pen_edge:=0
+else if cur_path_type=double_path_code then
+ pen_edge:=compromise(east_edge(cur_pen),west_edge(cur_pen))
+else if odd(right_type(q)) then pen_edge:=west_edge(cur_pen)
+else pen_edge:=east_edge(cur_pen);
+a:=good_val(b,pen_edge);
+end
+
+@ The monotone transformation computed here with fixed-point arithmetic is
+guaranteed to take consecutive |before| values $(b,b')$ into consecutive
+|after| values $(a,a')$, even in the presence of rounding errors,
+as long as $\vert b-b'\vert<2^{28}$.
+
+@<Transform the |x| coordinates@>=
+begin make_safe;
+repeat decr(cur_rounding_ptr);
+if (after[cur_rounding_ptr]<>before[cur_rounding_ptr])or@|
+ (after[cur_rounding_ptr+1]<>before[cur_rounding_ptr+1]) then
+ begin p:=node_to_round[cur_rounding_ptr];
+ if odd(right_type(p)) then
+ begin b:=before[cur_rounding_ptr]; a:=after[cur_rounding_ptr];
+ end
+ else begin b:=-before[cur_rounding_ptr]; a:=-after[cur_rounding_ptr];
+ end;
+ if before[cur_rounding_ptr]=before[cur_rounding_ptr+1] then
+ alpha:=fraction_one
+ else alpha:=make_fraction(after[cur_rounding_ptr+1]-after[cur_rounding_ptr],@|
+ before[cur_rounding_ptr+1]-before[cur_rounding_ptr]);
+ repeat x_coord(p):=take_fraction(alpha,x_coord(p)-b)+a;
+ right_x(p):=take_fraction(alpha,right_x(p)-b)+a;
+ p:=link(p); left_x(p):=take_fraction(alpha,left_x(p)-b)+a;
+ until p=node_to_round[cur_rounding_ptr+1];
+ end;
+until cur_rounding_ptr=0;
+end
+
+@ When |y| has been negated, the |octant| codes are |>negate_y|. Otherwise
+these routines are essentially identical to the routines for |x| coordinates
+that we have just seen.
+
+@<If node |q| is a transition point for |y| coordinates...@>=
+if (right_type(p)>negate_y)<>(right_type(q)>negate_y) then
+ begin if right_type(q)<=negate_y then b:=y_coord(q)@+else b:=-y_coord(q);
+ if (abs(y_coord(q)-right_y(q))<655)or@|
+ (abs(y_coord(q)+left_y(q))<655) then
+ @<Compute before-and-after |y| values based on the current pen@>
+ else a:=b;
+ if abs(a)>max_allowed then
+ if a>0 then a:=max_allowed@+else a:=-max_allowed;
+ before_and_after(b,a,q);
+ end
+
+@ @<Compute before-and-after |y| values based on the current pen@>=
+begin if cur_pen=null_pen then pen_edge:=0
+else if cur_path_type=double_path_code then
+ pen_edge:=compromise(north_edge(cur_pen),south_edge(cur_pen))
+else if right_type(q)<=negate_y then pen_edge:=south_edge(cur_pen)
+else pen_edge:=north_edge(cur_pen);
+a:=good_val(b,pen_edge);
+end
+
+@ @<Transform the |y| coordinates@>=
+begin make_safe;
+repeat decr(cur_rounding_ptr);
+if (after[cur_rounding_ptr]<>before[cur_rounding_ptr])or@|
+ (after[cur_rounding_ptr+1]<>before[cur_rounding_ptr+1]) then
+ begin p:=node_to_round[cur_rounding_ptr];
+ if right_type(p)<=negate_y then
+ begin b:=before[cur_rounding_ptr]; a:=after[cur_rounding_ptr];
+ end
+ else begin b:=-before[cur_rounding_ptr]; a:=-after[cur_rounding_ptr];
+ end;
+ if before[cur_rounding_ptr]=before[cur_rounding_ptr+1] then
+ alpha:=fraction_one
+ else alpha:=make_fraction(after[cur_rounding_ptr+1]-after[cur_rounding_ptr],@|
+ before[cur_rounding_ptr+1]-before[cur_rounding_ptr]);
+ repeat y_coord(p):=take_fraction(alpha,y_coord(p)-b)+a;
+ right_y(p):=take_fraction(alpha,right_y(p)-b)+a;
+ p:=link(p); left_y(p):=take_fraction(alpha,left_y(p)-b)+a;
+ until p=node_to_round[cur_rounding_ptr+1];
+ end;
+until cur_rounding_ptr=0;
+end
+
+@ Rounding at diagonal tangents takes place after the subdivision into
+octants is complete, hence after the coordinates have been skewed.
+The details are somewhat tricky, because we want to round to points
+whose skewed coordinates are halfway between integer multiples of
+the granularity. Furthermore, both coordinates change when they are
+rounded; this means we need a generalization of the |make_safe| routine,
+ensuring safety in both |x| and |y|.
+
+In spite of these extra complications, we can take comfort in the fact
+that the basic structure of the routine is the same as before.
+
+@<Declare subroutines needed by |make_spec|@>=
+procedure diag_round;
+var @!p,@!q,@!pp:pointer; {list manipulation registers}
+@!b,@!a,@!bb,@!aa,@!d,@!c,@!dd,@!cc:scaled; {before and after values}
+@!pen_edge:scaled; {offset that governs rounding}
+@!alpha,@!beta:fraction; {coefficients of linear transformation}
+@!next_a:scaled; {|after[k]| before it might have changed}
+@!all_safe:boolean; {does everything look OK so far?}
+@!k:0..max_wiggle; {runs through before-and-after values}
+@!first_x,@!first_y:scaled; {coordinates before rounding}
+begin p:=cur_spec; cur_rounding_ptr:=0;
+repeat q:=link(p);
+@<If node |q| is a transition point between octants,
+ compute and save its before-and-after coordinates@>;
+p:=q;
+until p=cur_spec;
+if cur_rounding_ptr>0 then @<Transform the skewed coordinates@>;
+end;
+
+@ We negate the skewed |x| coordinates in the before-and-after table when
+the octant code is greater than |switch_x_and_y|.
+
+@<If node |q| is a transition point between octants...@>=
+if right_type(p)<>right_type(q) then
+ begin if right_type(q)>switch_x_and_y then b:=-x_coord(q)
+ else b:=x_coord(q);
+ if abs(right_type(q)-right_type(p))=switch_x_and_y then
+ if (abs(x_coord(q)-right_x(q))<655)or(abs(x_coord(q)+left_x(q))<655) then
+ @<Compute a good coordinate at a diagonal transition@>
+ else a:=b
+ else a:=b;
+ before_and_after(b,a,q);
+ end
+
+@ In octants whose code number is even, $x$~has been
+negated; we want to round ambiguous cases downward instead of upward,
+so that the rounding will be consistent with octants whose code
+number is odd. This downward bias can be achieved by
+subtracting~1 from the first argument of |good_val|.
+
+@d diag_offset(#)==x_coord(knil(link(cur_pen+#)))
+
+@<Compute a good coordinate at a diagonal transition@>=
+begin if cur_pen=null_pen then pen_edge:=0
+else if cur_path_type=double_path_code then @<Compute a compromise |pen_edge|@>
+else if right_type(q)<=switch_x_and_y then pen_edge:=diag_offset(right_type(q))
+else pen_edge:=-diag_offset(right_type(q));
+if odd(right_type(q)) then a:=good_val(b,pen_edge+half(cur_gran))
+else a:=good_val(b-1,pen_edge+half(cur_gran));
+end
+
+@ (It seems a shame to compute these compromise offsets repeatedly. The
+author would have stored them directly in the pen data structure, if the
+granularity had been constant.)
+
+@<Compute a compromise...@>=
+case right_type(q) of
+first_octant,second_octant:pen_edge:=compromise(diag_offset(first_octant),@|
+ -diag_offset(fifth_octant));
+fifth_octant,sixth_octant:pen_edge:=-compromise(diag_offset(first_octant),@|
+ -diag_offset(fifth_octant));
+third_octant,fourth_octant:pen_edge:=compromise(diag_offset(fourth_octant),@|
+ -diag_offset(eighth_octant));
+seventh_octant,eighth_octant:pen_edge:=-compromise(diag_offset(fourth_octant),@|
+ -diag_offset(eighth_octant));
+end {there are no other cases}
+
+@ @<Transform the skewed coordinates@>=
+begin p:=node_to_round[0]; first_x:=x_coord(p); first_y:=y_coord(p);
+@<Make sure that all the diagonal roundings are safe@>;
+for k:=0 to cur_rounding_ptr-1 do
+ begin a:=after[k]; b:=before[k];
+ aa:=after[k+1]; bb:=before[k+1];
+ if (a<>b)or(aa<>bb) then
+ begin p:=node_to_round[k]; pp:=node_to_round[k+1];
+ @<Determine the before-and-after values of both coordinates@>;
+ if b=bb then alpha:=fraction_one
+ else alpha:=make_fraction(aa-a,bb-b);
+ if d=dd then beta:=fraction_one
+ else beta:=make_fraction(cc-c,dd-d);
+ repeat x_coord(p):=take_fraction(alpha,x_coord(p)-b)+a;
+ y_coord(p):=take_fraction(beta,y_coord(p)-d)+c;
+ right_x(p):=take_fraction(alpha,right_x(p)-b)+a;
+ right_y(p):=take_fraction(beta,right_y(p)-d)+c;
+ p:=link(p); left_x(p):=take_fraction(alpha,left_x(p)-b)+a;
+ left_y(p):=take_fraction(beta,left_y(p)-d)+c;
+ until p=pp;
+ end;
+ end;
+end
+
+@ In node |p|, the coordinates |(b,d)| will be rounded to |(a,c)|;
+in node |pp|, the coordinates |(bb,dd)| will be rounded to |(aa,cc)|.
+(We transform the values from node |pp| so that they agree with the
+conventions of node |p|.)
+
+If |aa<>bb|, we know that |abs(right_type(p)-right_type(pp))=switch_x_and_y|.
+
+@<Determine the before-and-after values of both coordinates@>=
+if aa=bb then
+ begin if pp=node_to_round[0] then
+ unskew(first_x,first_y,right_type(pp))
+ else unskew(x_coord(pp),y_coord(pp),right_type(pp));
+ skew(cur_x,cur_y,right_type(p));
+ bb:=cur_x; aa:=bb; dd:=cur_y; cc:=dd;
+ if right_type(p)>switch_x_and_y then
+ begin b:=-b; a:=-a;
+ end;
+ end
+else begin if right_type(p)>switch_x_and_y then
+ begin bb:=-bb; aa:=-aa; b:=-b; a:=-a;
+ end;
+ if pp=node_to_round[0] then dd:=first_y-bb@+else dd:=y_coord(pp)-bb;
+ if odd(aa-bb) then
+ if right_type(p)>switch_x_and_y then cc:=dd-half(aa-bb+1)
+ else cc:=dd-half(aa-bb-1)
+ else cc:=dd-half(aa-bb);
+ end;
+d:=y_coord(p);
+if odd(a-b) then
+ if right_type(p)>switch_x_and_y then c:=d-half(a-b-1)
+ else c:=d-half(a-b+1)
+else c:=d-half(a-b)
+
+@ @<Make sure that all the diagonal roundings are safe@>=
+before[cur_rounding_ptr]:=before[0]; {cf.~|make_safe|}
+node_to_round[cur_rounding_ptr]:=node_to_round[0];
+repeat after[cur_rounding_ptr]:=after[0]; all_safe:=true; next_a:=after[0];
+for k:=0 to cur_rounding_ptr-1 do
+ begin a:=next_a; b:=before[k]; next_a:=after[k+1];
+ aa:=next_a; bb:=before[k+1];
+ if (a<>b)or(aa<>bb) then
+ begin p:=node_to_round[k]; pp:=node_to_round[k+1];
+ @<Determine the before-and-after values of both coordinates@>;
+ if (aa<a)or(cc<c)or(aa-a>2*(bb-b))or(cc-c>2*(dd-d)) then
+ begin all_safe:=false; after[k]:=before[k];
+ if k=cur_rounding_ptr-1 then after[0]:=before[0]
+ else after[k+1]:=before[k+1];
+ end;
+ end;
+ end;
+until all_safe
+
+@ Here we get rid of ``dead'' cubics, i.e., polynomials that don't move at
+all when |t|~changes, since the subdivision process might have introduced
+such things. If the cycle reduces to a single point, however, we are left
+with a single dead cubic that will not be removed until later.
+
+@<Remove dead cubics@>=
+p:=cur_spec;
+repeat continue: q:=link(p);
+if p<>q then
+ begin if x_coord(p)=right_x(p) then
+ if y_coord(p)=right_y(p) then
+ if x_coord(p)=left_x(q) then
+ if y_coord(p)=left_y(q) then
+ begin unskew(x_coord(q),y_coord(q),right_type(q));
+ skew(cur_x,cur_y,right_type(p));
+ if x_coord(p)=cur_x then if y_coord(p)=cur_y then
+ begin remove_cubic(p); {remove the cubic following |p|}
+ if q<>cur_spec then goto continue;
+ cur_spec:=p; q:=p;
+ end;
+ end;
+ end;
+p:=q;
+until p=cur_spec;
+
+@ Finally we come to the last steps of |make_spec|, when boundary nodes
+are inserted between cubics that move in different octants. The main
+complication remaining arises from consecutive cubics whose octants
+are not adjacent; we should insert more than one octant boundary
+at such sharp turns, so that the envelope-forming routine will work.
+
+For this purpose, conversion tables between numeric and Gray codes for
+octants are desirable.
+
+@<Glob...@>=
+@!octant_number:array[first_octant..sixth_octant] of 1..8;
+@!octant_code:array[1..8] of first_octant..sixth_octant;
+
+@ @<Set init...@>=
+octant_code[1]:=first_octant;
+octant_code[2]:=second_octant;
+octant_code[3]:=third_octant;
+octant_code[4]:=fourth_octant;
+octant_code[5]:=fifth_octant;
+octant_code[6]:=sixth_octant;
+octant_code[7]:=seventh_octant;
+octant_code[8]:=eighth_octant;
+for k:=1 to 8 do octant_number[octant_code[k]]:=k;
+
+@ The main loop for boundary insertion deals with three consecutive
+nodes |p,q,r|.
+
+@<Insert octant boundaries and compute the turning number@>=
+turning_number:=0;
+p:=cur_spec; q:=link(p);
+repeat r:=link(q);
+if (right_type(p)<>right_type(q))or(q=r) then
+ @<Insert one or more octant boundary nodes just before~|q|@>;
+p:=q; q:=r;
+until p=cur_spec;
+
+@ The |new_boundary| subroutine comes in handy at this point. It inserts
+a new boundary node just after a given node |p|, using a given octant code
+to transform the new node's coordinates. The ``transition'' fields are
+not computed here.
+
+@<Declare subroutines needed by |make_spec|@>=
+procedure new_boundary(@!p:pointer;@!octant:small_number);
+var @!q,@!r:pointer; {for list manipulation}
+begin q:=link(p); {we assume that |right_type(q)<>endpoint|}
+r:=get_node(knot_node_size); link(r):=q; link(p):=r;
+left_type(r):=left_type(q); {but possibly |left_type(q)=endpoint|}
+left_x(r):=left_x(q); left_y(r):=left_y(q);
+right_type(r):=endpoint; left_type(q):=endpoint;
+right_octant(r):=octant; left_octant(q):=right_type(q);
+unskew(x_coord(q),y_coord(q),right_type(q));
+skew(cur_x,cur_y,octant); x_coord(r):=cur_x; y_coord(r):=cur_y;
+end;
+
+@ The case |q=r| occurs if and only if |p=q=r=cur_spec|, when we want to turn
+$360^\circ$ in eight steps and then remove a solitary dead cubic.
+The program below happens to work in that case, but the reader isn't
+expected to understand why.
+
+@<Insert one or more octant boundary nodes just before~|q|@>=
+begin new_boundary(p,right_type(p)); s:=link(p);
+o1:=octant_number[right_type(p)]; o2:=octant_number[right_type(q)];
+case o2-o1 of
+1,-7,7,-1: goto done;
+2,-6: clockwise:=false;
+3,-5,4,-4,5,-3: @<Decide whether or not to go clockwise@>;
+6,-2: clockwise:=true;
+0:clockwise:=rev_turns;
+end; {there are no other cases}
+@<Insert additional boundary nodes, then |goto done|@>;
+done: if q=r then
+ begin q:=link(q); r:=q; p:=s; link(s):=q; left_octant(q):=right_octant(q);
+ left_type(q):=endpoint; free_node(cur_spec,knot_node_size); cur_spec:=q;
+ end;
+@<Fix up the transition fields and adjust the turning number@>;
+end
+
+@ @<Other local variables for |make_spec|@>=
+@!o1,@!o2:small_number; {octant numbers}
+@!clockwise:boolean; {should we turn clockwise?}
+@!dx1,@!dy1,@!dx2,@!dy2:integer; {directions of travel at a cusp}
+@!dmax,@!del:integer; {temporary registers}
+
+@ A tricky question arises when a path jumps four octants. We want the
+direction of turning to be counterclockwise if the curve has changed
+direction by $180^\circ$, or by something so close to $180^\circ$ that
+the difference is probably due to rounding errors; otherwise we want to
+turn through an angle of less than $180^\circ$. This decision needs to
+be made even when a curve seems to have jumped only three octants, since
+a curve may approach direction $(-1,0)$ from the fourth octant, then
+it might leave from direction $(+1,0)$ into the first.
+
+The following code solves the problem by analyzing the incoming
+direction |(dx1,dy1)| and the outgoing direction |(dx2,dy2)|.
+
+@<Decide whether or not to go clockwise@>=
+begin @<Compute the incoming and outgoing directions@>;
+unskew(dx1,dy1,right_type(p)); del:=pyth_add(cur_x,cur_y);@/
+dx1:=make_fraction(cur_x,del); dy1:=make_fraction(cur_y,del);
+ {$\cos\theta_1$ and $\sin\theta_1$}
+unskew(dx2,dy2,right_type(q)); del:=pyth_add(cur_x,cur_y);@/
+dx2:=make_fraction(cur_x,del); dy2:=make_fraction(cur_y,del);
+ {$\cos\theta_2$ and $\sin\theta_2$}
+del:=take_fraction(dx1,dy2)-take_fraction(dx2,dy1); {$\sin(\theta_2-\theta_1)$}
+if del>4684844 then clockwise:=false
+else if del<-4684844 then clockwise:=true
+ {$2^{28}\cdot\sin 1^\circ\approx4684844.68$}
+else clockwise:=rev_turns;
+end
+
+@ Actually the turnarounds just computed will be clockwise,
+not counterclockwise, if
+the global variable |rev_turns| is |true|; it is usually |false|.
+
+@<Glob...@>=
+@!rev_turns:boolean; {should we make U-turns in the English manner?}
+
+@ @<Set init...@>=
+rev_turns:=false;
+
+@ @<Compute the incoming and outgoing directions@>=
+dx1:=x_coord(s)-left_x(s); dy1:=y_coord(s)-left_y(s);
+if dx1=0 then if dy1=0 then
+ begin dx1:=x_coord(s)-right_x(p); dy1:=y_coord(s)-right_y(p);
+ if dx1=0 then if dy1=0 then
+ begin dx1:=x_coord(s)-x_coord(p); dy1:=y_coord(s)-y_coord(p);
+ end; {and they {\sl can't} both be zero}
+ end;
+dmax:=abs(dx1);@+if abs(dy1)>dmax then dmax:=abs(dy1);
+while dmax<fraction_one do
+ begin double(dmax); double(dx1); double(dy1);
+ end;
+dx2:=right_x(q)-x_coord(q); dy2:=right_y(q)-y_coord(q);
+if dx2=0 then if dy2=0 then
+ begin dx2:=left_x(r)-x_coord(q); dy2:=left_y(r)-y_coord(q);
+ if dx2=0 then if dy2=0 then
+ begin if right_type(r)=endpoint then
+ begin cur_x:=x_coord(r); cur_y:=y_coord(r);
+ end
+ else begin unskew(x_coord(r),y_coord(r),right_type(r));
+ skew(cur_x,cur_y,right_type(q));
+ end;
+ dx2:=cur_x-x_coord(q); dy2:=cur_y-y_coord(q);
+ end; {and they {\sl can't} both be zero}
+ end;
+dmax:=abs(dx2);@+if abs(dy2)>dmax then dmax:=abs(dy2);
+while dmax<fraction_one do
+ begin double(dmax); double(dx2); double(dy2);
+ end
+
+@ @<Insert additional boundary nodes...@>=
+loop@+ begin if clockwise then
+ if o1=1 then o1:=8@+else decr(o1)
+ else if o1=8 then o1:=1@+else incr(o1);
+ if o1=o2 then goto done;
+ new_boundary(s,octant_code[o1]);
+ s:=link(s); left_octant(s):=right_octant(s);
+ end
+
+@ Now it remains to insert the redundant
+transition information into the |left_transition|
+and |right_transition| fields between adjacent octants, in the octant
+boundary nodes that have just been inserted between |link(p)| and~|q|.
+The turning number is easily computed from these transitions.
+
+@<Fix up the transition fields and adjust the turning number@>=
+p:=link(p);
+repeat s:=link(p);
+o1:=octant_number[right_octant(p)]; o2:=octant_number[left_octant(s)];
+if abs(o1-o2)=1 then
+ begin if o2<o1 then o2:=o1;
+ if odd(o2) then right_transition(p):=axis
+ else right_transition(p):=diagonal;
+ end
+else begin if o1=8 then incr(turning_number)@+else decr(turning_number);
+ right_transition(p):=axis;
+ end;
+left_transition(s):=right_transition(p);
+p:=s;
+until p=q
+
+@* \[22] Filling a contour.
+Given the low-level machinery for making moves and for transforming a
+cyclic path into a cycle spec, we're almost able to fill a digitized path.
+All we need is a high-level routine that walks through the cycle spec and
+controls the overall process.
+
+Our overall goal is to plot the integer points $\bigl(\round(x(t)),
+\round(y(t))\bigr)$ and to connect them by rook moves, assuming that
+$\round(x(t))$ and $\round(y(t))$ don't both jump simultaneously from
+one integer to another as $t$~varies; these rook moves will be the edge
+of the contour that will be filled. We have reduced this problem to the
+case of curves that travel in first octant directions, i.e., curves
+such that $0\L y'(t)\L x'(t)$, by transforming the original coordinates.
+
+\def\xtilde{{\tilde x}} \def\ytilde{{\tilde y}}
+Another transformation makes the problem still simpler. We shall say that
+we are working with {\sl biased coordinates\/} when $(x,y)$ has been
+replaced by $(\xtilde,\ytilde)=(x-y,y+{1\over2})$. When a curve travels
+in first octant directions, the corresponding curve with biased
+coordinates travels in first {\sl quadrant\/} directions; the latter
+condition is symmetric in $x$ and~$y$, so it has advantages for the
+design of algorithms. The |make_spec| routine gives us skewed coordinates
+$(x-y,y)$, hence we obtain biased coordinates by simply adding $1\over2$
+to the second component.
+
+The most important fact about biased coordinates is that we can determine the
+rounded unbiased path $\bigl(\round(x(t)),\round(y(t))\bigr)$ from the
+truncated biased path $\bigl(\lfloor\xtilde(t)\rfloor,\lfloor\ytilde(t)\rfloor
+\bigr)$ and information about the initial and final endpoints. If the
+unrounded and unbiased
+path begins at $(x_0,y_0)$ and ends at $(x_1,y_1)$, it's possible to
+prove (by induction on the length of the truncated biased path) that the
+rounded unbiased path is obtained by the following construction:
+
+\yskip\textindent{1)} Start at $\bigl(\round(x_0),\round(y_0)\bigr)$.
+
+\yskip\textindent{2)} If $(x_0+{1\over2})\bmod1\G(y_0+{1\over2})\bmod1$,
+move one step right.
+
+\yskip\textindent{3)} Whenever the path
+$\bigl(\lfloor\xtilde(t)\rfloor,\lfloor\ytilde(t)\rfloor\bigr)$
+takes an upward step (i.e., when
+$\lfloor\xtilde(t+\epsilon)\rfloor=\lfloor\xtilde(t)\rfloor$ and
+$\lfloor\ytilde(t+\epsilon)\rfloor=\lfloor\ytilde(t)\rfloor+1$),
+move one step up and then one step right.
+
+\yskip\textindent{4)} Whenever the path
+$\bigl(\lfloor\xtilde(t)\rfloor,\lfloor\ytilde(t)\rfloor\bigr)$
+takes a rightward step (i.e., when
+$\lfloor\xtilde(t+\epsilon)\rfloor=\lfloor\xtilde(t)\rfloor+1$ and
+$\lfloor\ytilde(t+\epsilon)\rfloor=\lfloor\ytilde(t)\rfloor$),
+move one step right.
+
+\yskip\textindent{5)} Finally, if
+$(x_1+{1\over2})\bmod1\G(y_1+{1\over2})\bmod1$, move one step left (thereby
+cancelling the previous move, which was one step right). You will now be
+at the point $\bigl(\round(x_1),\round(y_1)\bigr)$.
+
+@ In order to validate the assumption that $\round(x(t))$ and $\round(y(t))$
+don't both jump simultaneously, we shall consider that a coordinate pair
+$(x,y)$ actually represents $(x+\epsilon,y+\epsilon\delta)$, where
+$\epsilon$ and $\delta$ are extremely small positive numbers---so small
+that their precise values never matter. This convention makes rounding
+unambiguous, since there is always a unique integer point nearest to any
+given scaled numbers~$(x,y)$.
+
+When coordinates are transformed so that \MF\ needs to work only in ``first
+octant'' directions, the transformations involve negating~$x$, negating~$y$,
+and/or interchanging $x$ with~$y$. Corresponding adjustments to the
+rounding conventions must be made so that consistent values will be
+obtained. For example, suppose that we're working with coordinates that
+have been transformed so that a third-octant curve travels in first-octant
+directions. The skewed coordinates $(x,y)$ in our data structure represent
+unskewed coordinates $(-y,x+y)$, which are actually $(-y+\epsilon,
+x+y+\epsilon\delta)$. We should therefore round as if our skewed coordinates
+were $(x+\epsilon+\epsilon\delta,y-\epsilon)$ instead of $(x,y)$. The following
+table shows how the skewed coordinates should be perturbed when rounding
+decisions are made:
+$$\vcenter{\halign{#\hfil&&\quad$#$\hfil&\hskip4em#\hfil\cr
+|first_octant|&(x+\epsilon-\epsilon\delta,y+\epsilon\delta)&
+ |fifth_octant|&(x-\epsilon+\epsilon\delta,y-\epsilon\delta)\cr
+|second_octant|&(x-\epsilon+\epsilon\delta,y+\epsilon)&
+ |sixth_octant|&(x+\epsilon-\epsilon\delta,y-\epsilon)\cr
+|third_octant|&(x+\epsilon+\epsilon\delta,y-\epsilon)&
+ |seventh_octant|&(x-\epsilon-\epsilon\delta,y+\epsilon)\cr
+|fourth_octant|&(x-\epsilon-\epsilon\delta,y+\epsilon\delta)&
+ |eighth_octant|&(x+\epsilon+\epsilon\delta,y-\epsilon\delta)\cr}}$$
+
+Four small arrays are set up so that the rounding operations will be
+fairly easy in any given octant.
+
+@<Glob...@>=
+@!y_corr,@!xy_corr,@!z_corr:array[first_octant..sixth_octant] of 0..1;
+@!x_corr:array[first_octant..sixth_octant] of -1..1;
+
+@ Here |xy_corr| is 1 if and only if the $x$ component of a skewed coordinate
+is to be decreased by an infinitesimal amount; |y_corr| is similar, but for
+the $y$ components. The other tables are set up so that the condition
+$$(x+y+|half_unit|)\bmod|unity|\G(y+|half_unit|)\bmod|unity|$$
+is properly perturbed to the condition
+$$(x+y+|half_unit|-|x_corr|-|y_corr|)\bmod|unity|\G
+ (y+|half_unit|-|y_corr|)\bmod|unity|+|z_corr|.$$
+
+@<Set init...@>=
+x_corr[first_octant]:=0; y_corr[first_octant]:=0;
+xy_corr[first_octant]:=0;@/
+x_corr[second_octant]:=0; y_corr[second_octant]:=0;
+xy_corr[second_octant]:=1;@/
+x_corr[third_octant]:=-1; y_corr[third_octant]:=1;
+xy_corr[third_octant]:=0;@/
+x_corr[fourth_octant]:=1; y_corr[fourth_octant]:=0;
+xy_corr[fourth_octant]:=1;@/
+x_corr[fifth_octant]:=0; y_corr[fifth_octant]:=1;
+xy_corr[fifth_octant]:=1;@/
+x_corr[sixth_octant]:=0; y_corr[sixth_octant]:=1;
+xy_corr[sixth_octant]:=0;@/
+x_corr[seventh_octant]:=1; y_corr[seventh_octant]:=0;
+xy_corr[seventh_octant]:=1;@/
+x_corr[eighth_octant]:=-1; y_corr[eighth_octant]:=1;
+xy_corr[eighth_octant]:=0;@/
+for k:=1 to 8 do z_corr[k]:=xy_corr[k]-x_corr[k];
+
+@ Here's a procedure that handles the details of rounding at the
+endpoints: Given skewed coordinates |(x,y)|, it sets |(m1,n1)|
+to the corresponding rounded lattice points, taking the current
+|octant| into account. Global variable |d1| is also set to 1 if
+$(x+y+{1\over2})\bmod1\G(y+{1\over2})\bmod1$.
+
+@p procedure end_round(@!x,@!y:scaled);
+begin y:=y+half_unit-y_corr[octant];
+x:=x+y-x_corr[octant];
+m1:=floor_unscaled(x); n1:=floor_unscaled(y);
+if x-unity*m1>=y-unity*n1+z_corr[octant] then d1:=1@+else d1:=0;
+end;
+
+@ The outputs |(m1,n1,d1)| of |end_round| will sometimes be moved
+to |(m0,n0,d0)|.
+
+@<Glob...@>=
+@!m0,@!n0,@!m1,@!n1:integer; {lattice point coordinates}
+@!d0,@!d1:0..1; {displacement corrections}
+
+@ We're ready now to fill the pixels enclosed by a given cycle spec~|h|;
+the knot list that represents the cycle is destroyed in the process.
+The edge structure that gets all the resulting data is |cur_edges|,
+and the edges are weighted by |cur_wt|.
+
+@p procedure fill_spec(@!h:pointer);
+var @!p,@!q,@!r,@!s:pointer; {for list traversal}
+begin if internal[tracing_edges]>0 then begin_edge_tracing;
+p:=h; {we assume that |left_type(h)=endpoint|}
+repeat octant:=left_octant(p);
+@<Set variable |q| to the node at the end of the current octant@>;
+if q<>p then
+ begin @<Determine the starting and ending
+ lattice points |(m0,n0)| and |(m1,n1)|@>;
+ @<Make the moves for the current octant@>;
+ move_to_edges(m0,n0,m1,n1);
+ end;
+p:=link(q);
+until p=h;
+toss_knot_list(h);
+if internal[tracing_edges]>0 then end_edge_tracing;
+end;
+
+@ @<Set variable |q| to the node at the end of the current octant@>=
+q:=p;
+while right_type(q)<>endpoint do q:=link(q)
+
+@ @<Determine the starting and ending lattice points |(m0,n0)| and |(m1,n1)|@>=
+end_round(x_coord(p),y_coord(p)); m0:=m1; n0:=n1; d0:=d1;@/
+end_round(x_coord(q),y_coord(q))
+
+@ Finally we perform the five-step process that was explained at
+the very beginning of this part of the program.
+
+@<Make the moves for the current octant@>=
+if n1-n0>=move_size then overflow("move table size",move_size);
+@:METAFONT capacity exceeded move table size}{\quad move table size@>
+move[0]:=d0; move_ptr:=0; r:=p;
+repeat s:=link(r);@/
+make_moves(x_coord(r),right_x(r),left_x(s),x_coord(s),@|
+ y_coord(r)+half_unit,right_y(r)+half_unit,left_y(s)+half_unit,
+ y_coord(s)+half_unit,@| xy_corr[octant],y_corr[octant]);
+r:=s;
+until r=q;
+move[move_ptr]:=move[move_ptr]-d1;
+if internal[smoothing]>0 then smooth_moves(0,move_ptr)
+
+@* \[23] Polygonal pens.
+The next few parts of the program deal with the additional complications
+associated with ``envelopes,'' leading up to an algorithm that fills a
+contour with respect to a pen whose boundary is a convex polygon. The
+mathematics underlying this algorithm is based on simple aspects of the
+theory of tracings developed by Leo Guibas, Lyle Ramshaw, and Jorge
+Stolfi [``A kinetic framework for computational geometry,''
+{\sl Proc.\ IEEE Symp.\ Foundations of Computer Science\/ \bf24} (1983),
+100--111].
+@^Guibas, Leonidas Ioannis@>
+@^Ramshaw, Lyle Harold@>
+@^Stolfi, Jorge@>
+
+If the vertices of the polygon are $w_0$, $w_1$, \dots, $w_{n-1}$, $w_n=w_0$,
+in counterclockwise order, the convexity condition requires that ``left
+turns'' are made at each vertex when a person proceeds from $w_0$ to
+$w_1$ to $\cdots$ to~$w_n$. The envelope is obtained if we offset a given
+curve $z(t)$ by $w_k$ when that curve is traveling in a direction
+$z'(t)$ lying between the directions $w_k-w_{k-1}$ and $w\k-w_k$.
+At times~$t$ when the curve direction $z'(t)$ increases past
+$w\k-w_k$, we temporarily stop plotting the offset curve and we insert
+a straight line from $z(t)+w_k$ to $z(t)+w\k$; notice that this straight
+line is tangent to the offset curve. Similarly, when the curve direction
+decreases past $w_k-w_{k-1}$, we stop plotting and insert a straight
+line from $z(t)+w_k$ to $z(t)+w_{k-1}$; the latter line is actually a
+``retrograde'' step, which won't be part of the final envelope under
+\MF's assumptions. The result of this construction is a continuous path
+that consists of alternating curves and straight line segments. The
+segments are usually so short, in practice, that they blend with the
+curves; after all, it's possible to represent any digitized path as
+a sequence of digitized straight lines.
+
+The nicest feature of this approach to envelopes is that it blends
+perfectly with the octant subdivision process we have already developed.
+The envelope travels in the same direction as the curve itself, as we
+plot it, and we need merely be careful what offset is being added.
+Retrograde motion presents a problem, but we will see that there is
+a decent way to handle it.
+
+@ We shall represent pens by maintaining eight lists of offsets,
+one for each octant direction. The offsets at the boundary points
+where a curve turns into a new octant will appear in the lists for
+both octants. This means that we can restrict consideration to
+segments of the original polygon whose directions aim in the first
+octant, as we have done in the simpler case when envelopes were not
+required.
+
+An example should help to clarify this situation: Consider the
+quadrilateral whose vertices are $w_0=(0,-1)$, $w_1=(3,-1)$,
+$w_2=(6,1)$, and $w_3=(1,2)$. A curve that travels in the first octant
+will be offset by $w_1$ or $w_2$, unless its slope drops to zero
+en route to the eighth octant; in the latter case we should switch to $w_0$ as
+we cross the octant boundary. Our list for the first octant will
+contain the three offsets $w_0$, $w_1$,~$w_2$. By convention we will
+duplicate a boundary offset if the angle between octants doesn't
+explicitly appear; in this case there is no explicit line of slope~1
+at the end of the list, so the full list is
+$$w_0\;w_1\;w_2\;w_2\;=\;(0,-1)\;(3,-1)\;(6,1)\;(6,1).$$
+With skewed coordinates $(u-v,v)$ instead of $(u,v)$ we obtain the list
+$$w_0\;w_1\;w_2\;w_2\;\mapsto\;(1,-1)\;(4,-1)\;(5,1)\;(5,1),$$
+which is what actually appears in the data structure. In the second
+octant there's only one offset; we list it twice (with coordinates
+interchanged, so as to make the second octant look like the first),
+and skew those coordinates, obtaining
+$$\tabskip\centering
+\halign to\hsize{$\hfil#\;\mapsto\;{}$\tabskip=0pt&
+ $#\hfil$&\quad in the #\hfil\tabskip\centering\cr
+w_2\;w_2&(-5,6)\;(-5,6)\cr
+\noalign{\vskip\belowdisplayskip
+\vbox{\noindent\strut as the list of transformed and skewed offsets to use
+when curves travel in the second octant. Similarly, we will have\strut}
+\vskip\abovedisplayskip}
+w_2\;w_2&(7,-6)\;(7,-6)&third;\cr
+w_2\;w_2\;w_3\;w_3&(-7,1)\;(-7,1)\;(-3,2)\;(-3,2)&fourth;\cr
+w_3\;w_3&(1,-2)\;(1,-2)&fifth;\cr
+w_3\;w_3\;w_0\;w_0&(-1,1)\;(-1,1)\;(1,0)\;(1,0)&sixth;\cr
+w_0\;w_0&(1,0)\;(1,0)&seventh;\cr
+w_0\;w_0&(-1,1)\;(-1,1)&eighth.\cr}$$
+Notice that $w_1$ is considered here to be internal to the first octant;
+it's not part of the eighth. We could equally well have taken $w_0$ out
+of the first octant list and put it into the eighth; then the first octant
+list would have been
+$$w_1\;w_1\;w_2\;w_2\;\mapsto\;(4,-1)\;(4,-1)\;(5,1)\;(5,1)$$
+and the eighth octant list would have been
+$$w_0\;w_0\;w_1\;\mapsto\;(-1,1)\;(-1,1)\;(2,1).$$
+
+Actually, there's one more complication: The order of offsets is reversed
+in even-numbered octants, because the transformation of coordinates has
+reversed counterclockwise and clockwise orientations in those octants.
+The offsets in the fourth octant, for example, are really $w_3$, $w_3$,
+$w_2$,~$w_2$, not $w_2$, $w_2$, $w_3$,~$w_3$.
+
+@ In general, the list of offsets for an octant will have the form
+$$w_0\;\;w_1\;\;\ldots\;\;w_n\;\;w_{n+1}$$
+(if we renumber the subscripts in each list), where $w_0$ and $w_{n+1}$
+are offsets common to the neighboring lists. We'll often have $w_0=w_1$
+and/or $w_n=w_{n+1}$, but the other $w$'s will be distinct. Curves
+that travel between slope~0 and direction $w_2-w_1$ will use offset~$w_1$;
+curves that travel between directions $w_k-w_{k-1}$ and $w\k-w_k$ will
+use offset~$w_k$, for $1<k<n$; curves between direction $w_n-w_{n-1}$
+and slope~1 (actually slope~$\infty$ after skewing) will use offset~$w_n$.
+In even-numbered octants, the directions are actually $w_k-w\k$ instead
+of $w\k-w_k$, because the offsets have been listed in reverse order.
+
+Each offset $w_k$ is represented by skewed coordinates $(u_k-v_k,v_k)$,
+where $(u_k,v_k)$ is the representation of $w_k$ after it has been rotated
+into a first-octant disguise.
+
+@ The top-level data structure of a pen polygon is a 10-word node containing
+a reference count followed by pointers to the eight offset lists, followed
+by an indication of the pen's range of values.
+@^reference counts@>
+
+If |p|~points to such a node, and if the
+offset list for, say, the fourth octant has entries $w_0$, $w_1$, \dots,
+$w_n$,~$w_{n+1}$, then |info(p+fourth_octant)| will equal~$n$, and
+|link(p+fourth_octant)| will point to the offset node containing~$w_0$.
+Memory location |p+fourth_octant| is said to be the {\sl header\/} of
+the pen-offset list for the fourth octant. Since this is an even-numbered
+octant, $w_0$ is the offset that goes with the fifth octant, and
+$w_{n+1}$ goes with the third.
+
+The elements of the offset list themselves are doubly linked 3-word nodes,
+containing coordinates in their |x_coord| and |y_coord| fields.
+The two link fields are called |link| and |knil|; if |w|~points to
+the node for~$w_k$, then |link(w)| and |knil(w)| point respectively
+to the nodes for $w\k$ and~$w_{k-1}$. If |h| is the list header,
+|link(h)| points to the node for~$w_0$ and |knil(link(h))| to the
+node for~$w_{n+1}$.
+
+The tenth word of a pen header node contains the maximum absolute value of
+an $x$ or $y$ coordinate among all of the unskewed pen offsets.
+
+The |link| field of a pen header node should be |null| if and only if
+the pen is a single point.
+
+@d pen_node_size=10
+@d coord_node_size=3
+@d max_offset(#)==mem[#+9].sc
+
+@ The |print_pen| subroutine illustrates these conventions by
+reconstructing the vertices of a polygon from \MF's complicated
+internal offset representation.
+
+@<Declare subroutines for printing expressions@>=
+procedure print_pen(@!p:pointer;@!s:str_number;@!nuline:boolean);
+var @!nothing_printed:boolean; {has there been any action yet?}
+@!k:1..8; {octant number}
+@!h:pointer; {offset list head}
+@!m,@!n:integer; {offset indices}
+@!w,@!ww:pointer; {pointers that traverse the offset list}
+begin print_diagnostic("Pen polygon",s,nuline);
+nothing_printed:=true; print_ln;
+for k:=1 to 8 do
+ begin octant:=octant_code[k]; h:=p+octant; n:=info(h); w:=link(h);
+ if not odd(k) then w:=knil(w); {in even octants, start at $w_{n+1}$}
+ for m:=1 to n+1 do
+ begin if odd(k) then ww:=link(w)@+else ww:=knil(w);
+ if (x_coord(ww)<>x_coord(w))or(y_coord(ww)<>y_coord(w)) then
+ @<Print the unskewed and unrotated coordinates of node |ww|@>;
+ w:=ww;
+ end;
+ end;
+if nothing_printed then
+ begin w:=link(p+first_octant); print_two(x_coord(w)+y_coord(w),y_coord(w));
+ end;
+print_nl(" .. cycle"); end_diagnostic(true);
+end;
+
+@ @<Print the unskewed and unrotated coordinates of node |ww|@>=
+begin if nothing_printed then nothing_printed:=false
+else print_nl(" .. ");
+print_two_true(x_coord(ww),y_coord(ww));
+end
+
+@ A null pen polygon, which has just one vertex $(0,0)$, is
+predeclared for error recovery. It doesn't need a proper
+reference count, because the |toss_pen| procedure below
+will never delete it from memory.
+@^reference counts@>
+
+@<Initialize table entries...@>=
+ref_count(null_pen):=null; link(null_pen):=null;@/
+info(null_pen+1):=1; link(null_pen+1):=null_coords;
+for k:=null_pen+2 to null_pen+8 do mem[k]:=mem[null_pen+1];
+max_offset(null_pen):=0;@/
+link(null_coords):=null_coords;
+knil(null_coords):=null_coords;@/
+x_coord(null_coords):=0;
+y_coord(null_coords):=0;
+
+@ Here's a trivial subroutine that inserts a copy of an offset
+on the |link| side of its clone in the doubly linked list.
+
+@p procedure dup_offset(@!w:pointer);
+var @!r:pointer; {the new node}
+begin r:=get_node(coord_node_size);
+x_coord(r):=x_coord(w);
+y_coord(r):=y_coord(w);
+link(r):=link(w); knil(link(w)):=r;
+knil(r):=w; link(w):=r;
+end;
+
+@ The following algorithm is somewhat more interesting: It converts a
+knot list for a cyclic path into a pen polygon, ignoring everything
+but the |x_coord|, |y_coord|, and |link| fields. If the given path
+vertices do not define a convex polygon, an error message is issued
+and the null pen is returned.
+
+@p function make_pen(@!h:pointer):pointer;
+label done,done1,not_found,found;
+var @!o,@!oo,@!k:small_number; {octant numbers---old, new, and current}
+@!p:pointer; {top-level node for the new pen}
+@!q,@!r,@!s,@!w,@!hh:pointer; {for list manipulation}
+@!n:integer; {offset counter}
+@!dx,@!dy:scaled; {polygon direction}
+@!mc:scaled; {the largest coordinate}
+begin @<Stamp all nodes with an octant code, compute the maximum offset,
+ and set |hh| to the node that begins the first octant;
+ |goto not_found| if there's a problem@>;
+if mc>=fraction_one-half_unit then goto not_found;
+p:=get_node(pen_node_size); q:=hh; max_offset(p):=mc; ref_count(p):=null;
+if link(q)<>q then link(p):=null+1;
+for k:=1 to 8 do @<Construct the offset list for the |k|th octant@>;
+goto found;
+not_found:p:=null_pen; @<Complain about a bad pen path@>;
+found: if internal[tracing_pens]>0 then print_pen(p," (newly created)",true);
+make_pen:=p;
+end;
+
+@ @<Complain about a bad pen path@>=
+if mc>=fraction_one-half_unit then
+ begin print_err("Pen too large");
+@.Pen too large@>
+ help2("The cycle you specified has a coordinate of 4095.5 or more.")@/
+ ("So I've replaced it by the trivial path `(0,0)..cycle'.");@/
+ end
+else begin print_err("Pen cycle must be convex");
+@.Pen cycle must be convex@>
+ help3("The cycle you specified either has consecutive equal points")@/
+ ("or turns right or turns through more than 360 degrees.")@/
+ ("So I've replaced it by the trivial path `(0,0)..cycle'.");@/
+ end;
+put_get_error
+
+@ There should be exactly one node whose octant number is less than its
+predecessor in the cycle; that is node~|hh|.
+
+The loop here will terminate in all cases, but the proof is somewhat tricky:
+If there are at least two distinct $y$~coordinates in the cycle, we will have
+|o>4| and |o<=4| at different points of the cycle. Otherwise there are
+at least two distinct $x$~coordinates, and we will have |o>2| somewhere,
+|o<=2| somewhere.
+
+@<Stamp all nodes...@>=
+q:=h; r:=link(q); mc:=abs(x_coord(h));
+if q=r then
+ begin hh:=h; right_type(h):=0; {this trick is explained below}
+ if mc<abs(y_coord(h)) then mc:=abs(y_coord(h));
+ end
+else begin o:=0; hh:=null;
+ loop@+ begin s:=link(r);
+ if mc<abs(x_coord(r)) then mc:=abs(x_coord(r));
+ if mc<abs(y_coord(r)) then mc:=abs(y_coord(r));
+ dx:=x_coord(r)-x_coord(q); dy:=y_coord(r)-y_coord(q);
+ if dx=0 then if dy=0 then goto not_found; {double point}
+ if ab_vs_cd(dx,y_coord(s)-y_coord(r),dy,x_coord(s)-x_coord(r))<0 then
+ goto not_found; {right turn}
+ @<Determine the octant code for direction |(dx,dy)|@>;
+ right_type(q):=octant; oo:=octant_number[octant];
+ if o>oo then
+ begin if hh<>null then goto not_found; {$>360^\circ$}
+ hh:=q;
+ end;
+ o:=oo;
+ if (q=h)and(hh<>null) then goto done;
+ q:=r; r:=s;
+ end;
+ done:end
+
+
+@ We want the octant for |(-dx,-dy)| to be
+exactly opposite the octant for |(dx,dy)|.
+
+@<Determine the octant code for direction |(dx,dy)|@>=
+if dx>0 then octant:=first_octant
+else if dx=0 then
+ if dy>0 then octant:=first_octant@+else octant:=first_octant+negate_x
+else begin negate(dx); octant:=first_octant+negate_x;
+ end;
+if dy<0 then
+ begin negate(dy); octant:=octant+negate_y;
+ end
+else if dy=0 then
+ if octant>first_octant then octant:=first_octant+negate_x+negate_y;
+if dx<dy then octant:=octant+switch_x_and_y
+
+@ Now |q| points to the node that the present octant shares with the previous
+octant, and |right_type(q)| is the octant code during which |q|~should advance.
+We have set |right_type(q)=0| in the special case that |q| should never advance
+(because the pen is degenerate).
+
+The number of offsets |n| must be smaller than |max_quarterword|, because
+the |fill_envelope| routine stores |n+1| in the |right_type| field
+of a knot node.
+
+@<Construct the offset list...@>=
+begin octant:=octant_code[k]; n:=0; h:=p+octant;
+loop@+ begin r:=get_node(coord_node_size);
+ skew(x_coord(q),y_coord(q),octant); x_coord(r):=cur_x; y_coord(r):=cur_y;
+ if n=0 then link(h):=r
+ else @<Link node |r| to the previous node@>;
+ w:=r;
+ if right_type(q)<>octant then goto done1;
+ q:=link(q); incr(n);
+ end;
+done1: @<Finish linking the offset nodes, and duplicate the
+ borderline offset nodes if necessary@>;
+if n>=max_quarterword then overflow("pen polygon size",max_quarterword);
+@:METAFONT capacity exceeded pen polygon size}{\quad pen polygon size@>
+info(h):=n;
+end
+
+@ Now |w| points to the node that was inserted most recently, and
+|k| is the current octant number.
+
+@<Link node |r| to the previous node@>=
+if odd(k) then
+ begin link(w):=r; knil(r):=w;
+ end
+else begin knil(w):=r; link(r):=w;
+ end
+
+@ We have inserted |n+1| nodes; it remains to duplicate the nodes at the
+ends, if slopes 0 and~$\infty$ aren't already represented. At the end of
+this section the total number of offset nodes should be |n+2|
+(since we call them $w_0$, $w_1$, \dots,~$w_{n+1}$).
+
+@<Finish linking the offset nodes, and duplicate...@>=
+r:=link(h);
+if odd(k) then
+ begin link(w):=r; knil(r):=w;
+ end
+else begin knil(w):=r; link(r):=w; link(h):=w; r:=w;
+ end;
+if (y_coord(r)<>y_coord(link(r)))or(n=0) then
+ begin dup_offset(r); incr(n);
+ end;
+r:=knil(r);
+if x_coord(r)<>x_coord(knil(r)) then dup_offset(r)
+else decr(n)
+
+@ Conversely, |make_path| goes back from a pen to a cyclic path that
+might have generated it. The structure of this subroutine is essentially
+the same as |print_pen|.
+
+@p @t\4@>@<Declare the function called |trivial_knot|@>@;
+function make_path(@!pen_head:pointer):pointer;
+var @!p:pointer; {the most recently copied knot}
+@!k:1..8; {octant number}
+@!h:pointer; {offset list head}
+@!m,@!n:integer; {offset indices}
+@!w,@!ww:pointer; {pointers that traverse the offset list}
+begin p:=temp_head;
+for k:=1 to 8 do
+ begin octant:=octant_code[k]; h:=pen_head+octant; n:=info(h); w:=link(h);
+ if not odd(k) then w:=knil(w); {in even octants, start at $w_{n+1}$}
+ for m:=1 to n+1 do
+ begin if odd(k) then ww:=link(w)@+else ww:=knil(w);
+ if (x_coord(ww)<>x_coord(w))or(y_coord(ww)<>y_coord(w)) then
+ @<Copy the unskewed and unrotated coordinates of node |ww|@>;
+ w:=ww;
+ end;
+ end;
+if p=temp_head then
+ begin w:=link(pen_head+first_octant);
+ p:=trivial_knot(x_coord(w)+y_coord(w),y_coord(w)); link(temp_head):=p;
+ end;
+link(p):=link(temp_head); make_path:=link(temp_head);
+end;
+
+@ @<Copy the unskewed and unrotated coordinates of node |ww|@>=
+begin unskew(x_coord(ww),y_coord(ww),octant);
+link(p):=trivial_knot(cur_x,cur_y); p:=link(p);
+end
+
+@ @<Declare the function called |trivial_knot|@>=
+function trivial_knot(@!x,@!y:scaled):pointer;
+var @!p:pointer; {a new knot for explicit coordinates |x| and |y|}
+begin p:=get_node(knot_node_size);
+left_type(p):=explicit; right_type(p):=explicit;@/
+x_coord(p):=x; left_x(p):=x; right_x(p):=x;@/
+y_coord(p):=y; left_y(p):=y; right_y(p):=y;@/
+trivial_knot:=p;
+end;
+
+@ That which can be created can be destroyed.
+
+@d add_pen_ref(#)==incr(ref_count(#))
+@d delete_pen_ref(#)==if ref_count(#)=null then toss_pen(#)
+ else decr(ref_count(#))
+
+@<Declare the recycling subroutines@>=
+procedure toss_pen(@!p:pointer);
+var @!k:1..8; {relative header locations}
+@!w,@!ww:pointer; {pointers to offset nodes}
+begin if p<>null_pen then
+ begin for k:=1 to 8 do
+ begin w:=link(p+k);
+ repeat ww:=link(w); free_node(w,coord_node_size); w:=ww;
+ until w=link(p+k);
+ end;
+ free_node(p,pen_node_size);
+ end;
+end;
+
+@ The |find_offset| procedure sets |(cur_x,cur_y)| to the offset associated
+with a given direction~|(x,y)| and a given pen~|p|. If |x=y=0|, the
+result is |(0,0)|. If two different offsets apply, one of them is
+chosen arbitrarily.
+
+@p procedure find_offset(@!x,@!y:scaled; @!p:pointer);
+label done,exit;
+var @!octant:first_octant..sixth_octant; {octant code for |(x,y)|}
+@!s:-1..+1; {sign of the octant}
+@!n:integer; {number of offsets remaining}
+@!h,@!w,@!ww:pointer; {list traversal registers}
+begin @<Compute the octant code; skew and rotate the coordinates |(x,y)|@>;
+if odd(octant_number[octant]) then s:=-1@+else s:=+1;
+h:=p+octant; w:=link(link(h)); ww:=link(w); n:=info(h);
+while n>1 do
+ begin if ab_vs_cd(x,y_coord(ww)-y_coord(w),@|
+ y,x_coord(ww)-x_coord(w))<>s then goto done;
+ w:=ww; ww:=link(w); decr(n);
+ end;
+done:unskew(x_coord(w),y_coord(w),octant);
+exit:end;
+
+@ @<Compute the octant code; skew and rotate the coordinates |(x,y)|@>=
+if x>0 then octant:=first_octant
+else if x=0 then
+ if y<=0 then
+ if y=0 then
+ begin cur_x:=0; cur_y:=0; return;
+ end
+ else octant:=first_octant+negate_x
+ else octant:=first_octant
+else begin x:=-x;
+ if y=0 then octant:=first_octant+negate_x+negate_y
+ else octant:=first_octant+negate_x;
+ end;
+if y<0 then
+ begin octant:=octant+negate_y; y:=-y;
+ end;
+if x>=y then x:=x-y
+else begin octant:=octant+switch_x_and_y; x:=y-x; y:=y-x;
+ end
+
+@* \[24] Filling an envelope.
+We are about to reach the culmination of \MF's digital plotting routines:
+Almost all of the previous algorithms will be brought to bear on \MF's
+most difficult task, which is to fill the envelope of a given cyclic path
+with respect to a given pen polygon.
+
+But we still must complete some of the preparatory work before taking such
+a big plunge.
+
+@ Given a pointer |c| to a nonempty list of cubics,
+and a pointer~|h| to the header information of a pen polygon segment,
+the |offset_prep| routine changes the list into cubics that are
+associated with particular pen offsets. Namely, the cubic between |p|
+and~|q| should be associated with the |k|th offset when |right_type(p)=k|.
+
+List |c| is actually part of a cycle spec, so it terminates at the
+first node whose |right_type| is |endpoint|. The cubics all have
+monotone-nondecreasing $x(t)$ and $y(t)$.
+
+@p @t\4@>@<Declare subroutines needed by |offset_prep|@>@;
+procedure offset_prep(@!c,@!h:pointer);
+label done,not_found;
+var @!n:halfword; {the number of pen offsets}
+@!p,@!q,@!r,@!lh,@!ww:pointer; {for list manipulation}
+@!k:halfword; {the current offset index}
+@!w:pointer; {a pointer to offset $w_k$}
+@<Other local variables for |offset_prep|@>@;
+begin p:=c; n:=info(h); lh:=link(h); {now |lh| points to $w_0$}
+while right_type(p)<>endpoint do
+ begin q:=link(p);
+ @<Split the cubic between |p| and |q|, if necessary, into cubics
+ associated with single offsets, after which |q| should
+ point to the end of the final such cubic@>;
+ @<Advance |p| to node |q|, removing any ``dead'' cubics that
+ might have been introduced by the splitting process@>;
+ end;
+end;
+
+@ @<Advance |p| to node |q|, removing any ``dead'' cubics...@>=
+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
+ begin remove_cubic(p);
+ if r=q then q:=p;
+ r:=p;
+ end;
+p:=r;
+until p=q
+
+@ The splitting process uses a subroutine like |split_cubic|, but
+(for ``bulletproof'' operation) we check to make sure that the
+resulting (skewed) coordinates satisfy $\Delta x\G0$ and $\Delta y\G0$
+after splitting; |make_spec| has made sure that these relations hold
+before splitting. (This precaution is surely unnecessary, now that
+|make_spec| is so much more careful than it used to be. But who
+wants to take a chance? Maybe the hardware will fail or something.)
+
+@<Declare subroutines needed by |offset_prep|@>=
+procedure split_for_offset(@!p:pointer;@!t:fraction);
+var @!q:pointer; {the successor of |p|}
+@!r:pointer; {the new node}
+begin q:=link(p); split_cubic(p,t,x_coord(q),y_coord(q)); r:=link(p);
+if y_coord(r)<y_coord(p) then y_coord(r):=y_coord(p)
+else if y_coord(r)>y_coord(q) then y_coord(r):=y_coord(q);
+if x_coord(r)<x_coord(p) then x_coord(r):=x_coord(p)
+else if x_coord(r)>x_coord(q) then x_coord(r):=x_coord(q);
+end;
+
+@ If the pen polygon has |n| offsets, and if $w_k=(u_k,v_k)$ is the $k$th
+of these, the $k$th pen slope is defined by the formula
+$$s_k={v\k-v_k\over u\k-u_k},\qquad\hbox{for $0<k<n$}.$$
+In odd-numbered octants, the numerator and denominator of this fraction
+will be nonnegative; in even-numbered octants they will both be nonpositive.
+Furthermore we always have $0=s_0\le s_1\le\cdots\le s_n=\infty$. The goal of
+|offset_prep| is to find an offset index~|k| to associate with
+each cubic, such that the slope $s(t)$ of the cubic satisfies
+$$s_{k-1}\le s(t)\le s_k\qquad\hbox{for $0\le t\le 1$.}\eqno(*)$$
+We may have to split a cubic into as many as $2n-1$ pieces before each
+piece corresponds to a unique offset.
+
+@<Split the cubic between |p| and |q|, if necessary, into cubics...@>=
+if n<=1 then right_type(p):=1 {this case is easy}
+else begin @<Prepare for derivative computations;
+ |goto not_found| if the current cubic is dead@>;
+ @<Find the initial slope, |dy/dx|@>;
+ if dx=0 then @<Handle the special case of infinite slope@>
+ else begin @<Find the index |k| such that $s_{k-1}\L\\{dy}/\\{dx}<s_k$@>;
+ @<Complete the offset splitting process@>;
+ end;
+not_found: end
+
+@ The slope 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 slopes 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 $Y_2=2^l(y_3-y_2)$.
+To test whether the slope of the cubic is $\ge s$ or $\le s$, we will test
+the sign of the quadratic ${1\over3}2^l\bigl(y'(t)-sx'(t)\bigr)$ if $s\le1$,
+or ${1\over3}2^l\bigl(y'(t)/s-x'(t)\bigr)$ if $s>1$.
+
+@<Other local variables for |offset_prep|@>=
+@!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 slopes of the pen and 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; {slope or reciprocal slope}
+
+@ @<Prepare for derivative computations...@>=
+x0:=right_x(p)-x_coord(p); {should be |>=0|}
+x2:=x_coord(q)-left_x(q); {likewise}
+x1:=left_x(q)-right_x(p); {but this might be negative}
+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); {we take |abs| just to make sure}
+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<fraction_half do
+ begin double(max_coef);
+ double(x0); double(x1); double(x2);
+ double(y0); double(y1); double(y2);
+ end
+
+@ Let us first solve a special case of the problem: Suppose we
+know an index~$k$ such that either (i)~$s(t)\G s_{k-1}$ for all~$t$
+and $s(0)<s_k$, or (ii)~$s(t)\L s_k$ for all~$t$ and $s(0)>s_{k-1}$.
+Then, in a sense, we're halfway done, since one of the two inequalities
+in $(*)$ is satisfied, and the other couldn't be satisfied for
+any other value of~|k|.
+
+The |fin_offset_prep| subroutine solves the stated subproblem.
+It has a boolean parameter called |rising| that is |true| in
+case~(i), |false| in case~(ii). When |rising=false|, parameters
+|x0| through |y2| represent the negative of the derivative of
+the cubic following |p|; otherwise they represent the actual derivative.
+The |w| parameter should point to offset~$w_k$.
+
+@<Declare subroutines needed by |offset_prep|@>=
+procedure fin_offset_prep(@!p:pointer;@!k:halfword;@!w:pointer;
+ @!x0,@!x1,@!x2,@!y0,@!y1,@!y2:integer;@!rising:boolean;@!n: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|}
+begin loop
+ begin right_type(p):=k;
+ if rising then
+ if k=n then return
+ else ww:=link(w) {a pointer to $w\k$}
+ else if k=1 then return
+ else ww:=knil(w); {a pointer to $w_{k-1}$}
+ @<Compute test coefficients |(t0,t1,t2)|
+ for $s(t)$ versus $s_k$ or $s_{k-1}$@>;
+ t:=crossing_point(t0,t1,t2);
+ if t>=fraction_one then return;
+ @<Split the cubic at $t$,
+ and split off another cubic if the derivative crosses back@>;
+ if rising then incr(k)@+else decr(k);
+ w:=ww;
+ end;
+exit:end;
+
+@ @<Compute test coefficients |(t0,t1,t2)| for $s(t)$ versus...@>=
+du:=x_coord(ww)-x_coord(w); dv:=y_coord(ww)-y_coord(w);
+if abs(du)>=abs(dv) then {$s_{k-1}\le1$ or $s_k\le1$}
+ begin s:=make_fraction(dv,du);
+ t0:=take_fraction(x0,s)-y0;
+ t1:=take_fraction(x1,s)-y1;
+ t2:=take_fraction(x2,s)-y2;
+ 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);
+ end
+
+@ The curve has crossed $s_k$ or $s_{k-1}$; its initial segment satisfies
+$(*)$, and it might cross again and return towards $s_{k-1}$ or $s_k$,
+respectively, yielding another solution of $(*)$.
+
+@<Split the cubic at $t$, and split off another...@>=
+begin split_for_offset(p,t); right_type(p):=k; p:=link(p);@/
+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);@/
+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
+ begin split_for_offset(p,t); right_type(link(p)):=k;@/
+ 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
+
+@ Now we must consider the general problem of |offset_prep|, when
+nothing is known about a given cubic. We start by finding its
+slope $s(0)$ in the vicinity of |t=0|.
+
+If $z'(t)=0$, the given cubic is numerically unstable, since the
+slope direction is probably being influenced primarily by rounding
+errors. A user who specifies such cuspy curves should expect to generate
+rather wild results. The present code tries its best to believe the
+existing data, as if no rounding errors were present.
+
+@ @<Find the initial slope, |dy/dx|@>=
+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
+
+@ The next step is to bracket the initial slope between consecutive
+slopes of the pen polygon. The most important invariant relation in the
+following loop is that |dy/dx>=@t$s_{k-1}$@>|.
+
+@<Find the index |k| such that $s_{k-1}\L\\{dy}/\\{dx}<s_k$@>=
+k:=1; w:=link(lh);
+loop@+ begin if k=n then goto done;
+ ww:=link(w);
+ if ab_vs_cd(dy,abs(x_coord(ww)-x_coord(w)),@|
+ dx,abs(y_coord(ww)-y_coord(w)))>=0 then
+ begin incr(k); w:=ww;
+ end
+ else goto done;
+ end;
+done:
+
+@ Finally we want to reduce the general problem to situations that
+|fin_offset_prep| can handle. If |k=1|, we already are in the desired
+situation. Otherwise we can split the cubic into at most three parts
+with respect to $s_{k-1}$, and apply |fin_offset_prep| to each part.
+
+@<Complete the offset splitting process@>=
+if k=1 then t:=fraction_one+1
+else begin ww:=knil(w); @<Compute test coeff...@>;
+ t:=crossing_point(-t0,-t1,-t2);
+ end;
+if t>=fraction_one then fin_offset_prep(p,k,w,x0,x1,x2,y0,y1,y2,true,n)
+else begin split_for_offset(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,k,w,x0,x1a,x2a,y0,y1a,y2a,true,n); x0:=x2a; y0:=y2a;
+ t1:=t_of_the_way(t1)(t2);
+ if t1<0 then t1:=0;
+ t:=crossing_point(0,t1,t2);
+ if t<fraction_one then
+ @<Split off another |rising| cubic for |fin_offset_prep|@>;
+ fin_offset_prep(r,k-1,ww,-x0,-x1,-x2,-y0,-y1,-y2,false,n);
+ end
+
+@ @<Split off another |rising| cubic for |fin_offset_prep|@>=
+begin split_for_offset(r,t);@/
+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),k,w,x0a,x1a,x2,y0a,y1a,y2,true,n);
+x2:=x0a; y2:=y0a;
+end
+
+@ @<Handle the special case of infinite slope@>=
+fin_offset_prep(p,n,knil(knil(lh)),-x0,-x1,-x2,-y0,-y1,-y2,false,n)
+
+@ OK, it's time now for the biggie. The |fill_envelope| routine generalizes
+|fill_spec| to polygonal envelopes. Its outer structure is essentially the
+same as before, except that octants with no cubics do contribute to
+the envelope.
+
+@p @t\4@>@<Declare the procedure called |skew_line_edges|@>@;
+@t\4@>@<Declare the procedure called |dual_moves|@>@;
+procedure fill_envelope(@!spec_head:pointer);
+label done, done1;
+var @!p,@!q,@!r,@!s:pointer; {for list traversal}
+@!h:pointer; {head of pen offset list for current octant}
+@!www:pointer; {a pen offset of temporary interest}
+@<Other local variables for |fill_envelope|@>@;
+begin if internal[tracing_edges]>0 then begin_edge_tracing;
+p:=spec_head; {we assume that |left_type(spec_head)=endpoint|}
+repeat octant:=left_octant(p); h:=cur_pen+octant;
+@<Set variable |q| to the node at the end of the current octant@>;
+@<Determine the envelope's starting and ending
+ lattice points |(m0,n0)| and |(m1,n1)|@>;
+offset_prep(p,h); {this may clobber node~|q|, if it becomes ``dead''}
+@<Set variable |q| to the node at the end of the current octant@>;
+@<Make the envelope moves for the current octant and insert them
+ in the pixel data@>;
+p:=link(q);
+until p=spec_head;
+if internal[tracing_edges]>0 then end_edge_tracing;
+toss_knot_list(spec_head);
+end;
+
+@ In even-numbered octants we have reflected the coordinates an odd number
+of times, hence clockwise and counterclockwise are reversed; this means that
+the envelope is being formed in a ``dual'' manner. For the time being, let's
+concentrate on odd-numbered octants, since they're easier to understand.
+After we have coded the program for odd-numbered octants, the changes needed
+to dualize it will not be so mysterious.
+
+It is convenient to assume that we enter an odd-numbered octant with
+an |axis| transition (where the skewed slope is zero) and leave at a
+|diagonal| one (where the skewed slope is infinite). Then all of the
+offset points $z(t)+w(t)$ will lie in a rectangle whose lower left and
+upper right corners are the initial and final offset points. If this
+assumption doesn't hold we can implicitly change the curve so that it does.
+For example, if the entering transition is diagonal, we can draw a
+straight line from $z_0+w_{n+1}$ to $z_0+w_0$ and continue as if the
+curve were moving rightward. The effect of this on the envelope is simply
+to ``doubly color'' the region enveloped by a section of the pen that
+goes from $w_0$ to $w_1$ to $\cdots$ to $w_{n+1}$ to~$w_0$. The additional
+straight line at the beginning (and a similar one at the end, where it
+may be necessary to go from $z_1+w_{n+1}$ to $z_1+w_0$) can be drawn by
+the |line_edges| routine; we are thereby saved from the embarrassment that
+these lines travel backwards from the current octant direction.
+
+Once we have established the assumption that the curve goes from
+$z_0+w_0$ to $z_1+w_{n+1}$, any further retrograde moves that might
+occur within the octant can be essentially ignored; we merely need to
+keep track of the rightmost edge in each row, in order to compute
+the envelope.
+
+Envelope moves consist of offset cubics intermixed with straight line
+segments. We record them in a separate |env_move| array, which is
+something like |move| but it keeps track of the rightmost position of the
+envelope in each row.
+
+@<Glob...@>=
+@!env_move:array[0..move_size] of integer;
+
+@ @<Determine the envelope's starting and ending...@>=
+w:=link(h);@+if left_transition(p)=diagonal then w:=knil(w);
+@!stat if internal[tracing_edges]>unity then
+ @<Print a line of diagnostic info to introduce this octant@>;
+tats@;@/
+ww:=link(h); www:=ww; {starting and ending offsets}
+if odd(octant_number[octant]) then www:=knil(www)@+else ww:=knil(ww);
+if w<>ww then skew_line_edges(p,w,ww);
+end_round(x_coord(p)+x_coord(ww),y_coord(p)+y_coord(ww));
+m0:=m1; n0:=n1; d0:=d1;@/
+end_round(x_coord(q)+x_coord(www),y_coord(q)+y_coord(www));
+if n1-n0>=move_size then overflow("move table size",move_size)
+@:METAFONT capacity exceeded move table size}{\quad move table size@>
+
+@ @<Print a line of diagnostic info to introduce this octant@>=
+begin print_nl("@@ Octant "); print(octant_dir[octant]);
+@:]]]\AT!_Octant}{\.{\AT! Octant...}@>
+print(" ("); print_int(info(h)); print(" offset");
+if info(h)<>1 then print_char("s");
+print("), from ");
+print_two_true(x_coord(p)+x_coord(w),y_coord(p)+y_coord(w));@/
+ww:=link(h);@+if right_transition(q)=diagonal then ww:=knil(ww);
+print(" to ");
+print_two_true(x_coord(q)+x_coord(ww),y_coord(q)+y_coord(ww));
+end
+
+@ A slight variation of the |line_edges| procedure comes in handy
+when we must draw the retrograde lines for nonstandard entry and exit
+conditions.
+
+@<Declare the procedure called |skew_line_edges|@>=
+procedure skew_line_edges(@!p,@!w,@!ww:pointer);
+var @!x0,@!y0,@!x1,@!y1:scaled; {from and to}
+begin if (x_coord(w)<>x_coord(ww))or(y_coord(w)<>y_coord(ww)) then
+ begin x0:=x_coord(p)+x_coord(w); y0:=y_coord(p)+y_coord(w);@/
+ x1:=x_coord(p)+x_coord(ww); y1:=y_coord(p)+y_coord(ww);@/
+ unskew(x0,y0,octant); {unskew and unrotate the coordinates}
+ x0:=cur_x; y0:=cur_y;@/
+ unskew(x1,y1,octant);@/
+ @!stat if internal[tracing_edges]>unity then
+ begin print_nl("@@ retrograde line from ");
+@:]]]\AT!_retro_}{\.{\AT! retrograde line...}@>
+ @.retrograde line...@>
+ print_two(x0,y0); print(" to "); print_two(cur_x,cur_y); print_nl("");
+ end;@+tats@;@/
+ line_edges(x0,y0,cur_x,cur_y); {then draw a straight line}
+ end;
+end;
+
+@ The envelope calculations require more local variables than we needed
+in the simpler case of |fill_spec|. At critical points in the computation,
+|w| will point to offset $w_k$; |m| and |n| will record the current
+lattice positions. The values of |move_ptr| after the initial and before
+the final offset adjustments are stored in |smooth_bot| and |smooth_top|,
+respectively.
+
+@<Other local variables for |fill_envelope|@>=
+@!m,@!n:integer; {current lattice position}
+@!mm0,@!mm1:integer; {skewed equivalents of |m0| and |m1|}
+@!k:integer; {current offset number}
+@!w,@!ww:pointer; {pointers to the current offset and its neighbor}
+@!smooth_bot,@!smooth_top:0..move_size; {boundaries of smoothing}
+@!xx,@!yy,@!xp,@!yp,@!delx,@!dely,@!tx,@!ty:scaled;
+ {registers for coordinate calculations}
+
+@ @<Make the envelope moves for the current octant...@>=
+if odd(octant_number[octant]) then
+ begin @<Initialize for ordinary envelope moves@>;
+ r:=p; right_type(q):=info(h)+1;
+ loop@+ begin if r=q then smooth_top:=move_ptr;
+ while right_type(r)<>k do
+ @<Insert a line segment to approach the correct offset@>;
+ if r=p then smooth_bot:=move_ptr;
+ if r=q then goto done;
+ move[move_ptr]:=1; n:=move_ptr; s:=link(r);@/
+ make_moves(x_coord(r)+x_coord(w),right_x(r)+x_coord(w),
+ left_x(s)+x_coord(w),x_coord(s)+x_coord(w),@|
+ y_coord(r)+y_coord(w)+half_unit,right_y(r)+y_coord(w)+half_unit,
+ left_y(s)+y_coord(w)+half_unit,y_coord(s)+y_coord(w)+half_unit,@|
+ xy_corr[octant],y_corr[octant]);@/
+ @<Transfer moves from the |move| array to |env_move|@>;
+ r:=s;
+ end;
+done: @<Insert the new envelope moves in the pixel data@>;
+ end
+else dual_moves(h,p,q);
+right_type(q):=endpoint
+
+@ @<Initialize for ordinary envelope moves@>=
+k:=0; w:=link(h); ww:=knil(w);
+mm0:=floor_unscaled(x_coord(p)+x_coord(w)-xy_corr[octant]);
+mm1:=floor_unscaled(x_coord(q)+x_coord(ww)-xy_corr[octant]);
+for n:=0 to n1-n0-1 do env_move[n]:=mm0;
+env_move[n1-n0]:=mm1; move_ptr:=0; m:=mm0
+
+@ At this point |n| holds the value of |move_ptr| that was current
+when |make_moves| began to record its moves.
+
+@<Transfer moves from the |move| array to |env_move|@>=
+repeat m:=m+move[n]-1;
+if m>env_move[n] then env_move[n]:=m;
+incr(n);
+until n>move_ptr
+
+@ Retrograde lines (when |k| decreases) do not need to be recorded in
+|env_move| because their edges are not the furthest right in any row.
+
+@<Insert a line segment to approach the correct offset@>=
+begin xx:=x_coord(r)+x_coord(w); yy:=y_coord(r)+y_coord(w)+half_unit;
+@!stat if internal[tracing_edges]>unity then
+ begin print_nl("@@ transition line "); print_int(k); print(", from ");
+@:]]]\AT!_trans_}{\.{\AT! transition line...}@>
+@.transition line...@>
+ print_two_true(xx,yy-half_unit);
+ end;@+tats@;@/
+if right_type(r)>k then
+ begin incr(k); w:=link(w);
+ xp:=x_coord(r)+x_coord(w); yp:=y_coord(r)+y_coord(w)+half_unit;
+ if yp<>yy then
+ @<Record a line segment from |(xx,yy)| to |(xp,yp)| in |env_move|@>;
+ end
+else begin decr(k); w:=knil(w);
+ xp:=x_coord(r)+x_coord(w); yp:=y_coord(r)+y_coord(w)+half_unit;
+ end;
+stat if internal[tracing_edges]>unity then
+ begin print(" to ");
+ print_two_true(xp,yp-half_unit);
+ print_nl("");
+ end;@+tats@;@/
+m:=floor_unscaled(xp-xy_corr[octant]);
+move_ptr:=floor_unscaled(yp-y_corr[octant])-n0;
+if m>env_move[move_ptr] then env_move[move_ptr]:=m;
+end
+
+@ In this step we have |xp>=xx| and |yp>=yy|.
+
+@<Record a line segment from |(xx,yy)| to |(xp,yp)| in |env_move|@>=
+begin ty:=floor_scaled(yy-y_corr[octant]); dely:=yp-yy; yy:=yy-ty;
+ty:=yp-y_corr[octant]-ty;
+if ty>=unity then
+ begin delx:=xp-xx; yy:=unity-yy;
+ loop@+ begin tx:=take_fraction(delx,make_fraction(yy,dely));
+ if ab_vs_cd(tx,dely,delx,yy)+xy_corr[octant]>0 then decr(tx);
+ m:=floor_unscaled(xx+tx);
+ if m>env_move[move_ptr] then env_move[move_ptr]:=m;
+ ty:=ty-unity;
+ if ty<unity then goto done1;
+ yy:=yy+unity; incr(move_ptr);
+ end;
+ done1:end;
+end
+
+@ @<Insert the new envelope moves in the pixel data@>=
+debug if (m<>mm1)or(move_ptr<>n1-n0) then confusion("1");@+gubed@;@/
+@:this can't happen /}{\quad 1@>
+move[0]:=d0+env_move[0]-mm0;
+for n:=1 to move_ptr do
+ move[n]:=env_move[n]-env_move[n-1]+1;
+move[move_ptr]:=move[move_ptr]-d1;
+if internal[smoothing]>0 then smooth_moves(smooth_bot,smooth_top);
+move_to_edges(m0,n0,m1,n1);
+if right_transition(q)=axis then
+ begin w:=link(h); skew_line_edges(q,knil(w),w);
+ end
+
+@ We've done it all in the odd-octant case; the only thing remaining
+is to repeat the same ideas, upside down and/or backwards.
+
+The following code has been split off as a subprocedure of |fill_envelope|,
+because some \PASCAL\ compilers cannot handle procedures as large as
+|fill_envelope| would otherwise be.
+
+@<Declare the procedure called |dual_moves|@>=
+procedure dual_moves(@!h,@!p,@!q:pointer);
+label done,done1;
+var @!r,@!s:pointer; {for list traversal}
+@<Other local variables for |fill_envelope|@>@;
+begin @<Initialize for dual envelope moves@>;
+r:=p; {recall that |right_type(q)=endpoint=0| now}
+loop@+ begin if r=q then smooth_top:=move_ptr;
+ while right_type(r)<>k do
+ @<Insert a line segment dually to approach the correct offset@>;
+ if r=p then smooth_bot:=move_ptr;
+ if r=q then goto done;
+ move[move_ptr]:=1; n:=move_ptr; s:=link(r);@/
+ make_moves(x_coord(r)+x_coord(w),right_x(r)+x_coord(w),
+ left_x(s)+x_coord(w),x_coord(s)+x_coord(w),@|
+ y_coord(r)+y_coord(w)+half_unit,right_y(r)+y_coord(w)+half_unit,
+ left_y(s)+y_coord(w)+half_unit,y_coord(s)+y_coord(w)+half_unit,@|
+ xy_corr[octant],y_corr[octant]);
+ @<Transfer moves dually from the |move| array to |env_move|@>;
+ r:=s;
+ end;
+done:@<Insert the new envelope moves dually in the pixel data@>;
+end;
+
+@ In the dual case the normal situation is to arrive with a |diagonal|
+transition and to leave at the |axis|. The leftmost edge in each row
+is relevant instead of the rightmost one.
+
+@<Initialize for dual envelope moves@>=
+k:=info(h)+1; ww:=link(h); w:=knil(ww);@/
+mm0:=floor_unscaled(x_coord(p)+x_coord(w)-xy_corr[octant]);
+mm1:=floor_unscaled(x_coord(q)+x_coord(ww)-xy_corr[octant]);
+for n:=1 to n1-n0+1 do env_move[n]:=mm1;
+env_move[0]:=mm0; move_ptr:=0; m:=mm0
+
+@ @<Transfer moves dually from the |move| array to |env_move|@>=
+repeat if m<env_move[n] then env_move[n]:=m;
+m:=m+move[n]-1;
+incr(n);
+until n>move_ptr
+
+@ Dual retrograde lines occur when |k| increases; the edges of such lines
+are not the furthest left in any row.
+
+@<Insert a line segment dually to approach the correct offset@>=
+begin xx:=x_coord(r)+x_coord(w); yy:=y_coord(r)+y_coord(w)+half_unit;
+@!stat if internal[tracing_edges]>unity then
+ begin print_nl("@@ transition line "); print_int(k); print(", from ");
+@:]]]\AT!_trans_}{\.{\AT! transition line...}@>
+@.transition line...@>
+ print_two_true(xx,yy-half_unit);
+ end;@+tats@;@/
+if right_type(r)<k then
+ begin decr(k); w:=knil(w);
+ xp:=x_coord(r)+x_coord(w); yp:=y_coord(r)+y_coord(w)+half_unit;
+ if yp<>yy then
+ @<Record a line segment from |(xx,yy)| to |(xp,yp)| dually in |env_move|@>;
+ end
+else begin incr(k); w:=link(w);
+ xp:=x_coord(r)+x_coord(w); yp:=y_coord(r)+y_coord(w)+half_unit;
+ end;
+stat if internal[tracing_edges]>unity then
+ begin print(" to ");
+ print_two_true(xp,yp-half_unit);
+ print_nl("");
+ end;@+tats@;@/
+m:=floor_unscaled(xp-xy_corr[octant]);
+move_ptr:=floor_unscaled(yp-y_corr[octant])-n0;
+if m<env_move[move_ptr] then env_move[move_ptr]:=m;
+end
+
+@ Again, |xp>=xx| and |yp>=yy|; but this time we are interested in the {\sl
+smallest\/} |m| that belongs to a given |move_ptr| position, instead of
+the largest~|m|.
+
+@<Record a line segment from |(xx,yy)| to |(xp,yp)| dually in |env_move|@>=
+begin ty:=floor_scaled(yy-y_corr[octant]); dely:=yp-yy; yy:=yy-ty;
+ty:=yp-y_corr[octant]-ty;
+if ty>=unity then
+ begin delx:=xp-xx; yy:=unity-yy;
+ loop@+ begin if m<env_move[move_ptr] then env_move[move_ptr]:=m;
+ tx:=take_fraction(delx,make_fraction(yy,dely));
+ if ab_vs_cd(tx,dely,delx,yy)+xy_corr[octant]>0 then decr(tx);
+ m:=floor_unscaled(xx+tx);
+ ty:=ty-unity; incr(move_ptr);
+ if ty<unity then goto done1;
+ yy:=yy+unity;
+ end;
+done1: if m<env_move[move_ptr] then env_move[move_ptr]:=m;
+ end;
+end
+
+@ Since |env_move| contains minimum values instead of maximum values, the
+finishing-up process is slightly different in the dual case.
+
+@<Insert the new envelope moves dually in the pixel data@>=
+debug if (m<>mm1)or(move_ptr<>n1-n0) then confusion("2");@+gubed@;@/
+@:this can't happen /}{\quad 2@>
+move[0]:=d0+env_move[1]-mm0;
+for n:=1 to move_ptr do
+ move[n]:=env_move[n+1]-env_move[n]+1;
+move[move_ptr]:=move[move_ptr]-d1;
+if internal[smoothing]>0 then smooth_moves(smooth_bot,smooth_top);
+move_to_edges(m0,n0,m1,n1);
+if right_transition(q)=diagonal then
+ begin w:=link(h); skew_line_edges(q,w,knil(w));
+ end
+
+@* \[25] Elliptical pens.
+To get the envelope of a cyclic path with respect to an ellipse, \MF\
+calculates the envelope with respect to a polygonal approximation to
+the ellipse, using an approach due to John Hobby (Ph.D. thesis,
+Stanford University, 1985).
+@^Hobby, John Douglas@>
+This has two important advantages over trying to obtain the ``exact''
+envelope:
+
+\yskip\textindent{1)}It gives better results, because the polygon has been
+designed to counteract problems that arise from digitization; the
+polygon includes sub-pixel corrections to an exact ellipse that make
+the results essentially independent of where the path falls on the raster.
+For example, the exact envelope with respect to a pen of diameter~1
+blackens a pixel if and only if the path intersects a circle of diameter~1
+inscribed in that pixel; the resulting pattern has ``blots'' when the path
+is traveling diagonally in unfortunate raster positions. A much better
+result is obtained when pixels are blackened only when the path intersects
+an inscribed {\sl diamond\/} of diameter~1. Such a diamond is precisely
+the polygon that \MF\ uses in the special case of a circle whose diameter is~1.
+
+\yskip\textindent{2)}Polygonal envelopes of cubic splines are cubic
+splines, hence it isn't necessary to introduce completely different
+routines. By contrast, exact envelopes of cubic splines with respect
+to circles are complicated curves, more difficult to plot than cubics.
+
+@ Hobby's construction involves some interesting number theory.
+If $u$ and~$v$ are relatively prime integers, we divide the
+set of integer points $(m,n)$ into equivalence classes by saying
+that $(m,n)$ belongs to class $um+vn$. Then any two integer points
+that lie on a line of slope $-u/v$ belong to the same class, because
+such points have the form $(m+tv,n-tu)$. Neighboring lines of slope $-u/v$
+that go through integer points are separated by distance $1/\psqrt{u^2+v^2}$
+from each other, and these lines are perpendicular to lines of slope~$v/u$.
+If we start at the origin and travel a distance $k/\psqrt{u^2+v^2}$ in
+direction $(u,v)$, we reach the line of slope~$-u/v$ whose points
+belong to class~$k$.
+
+For example, let $u=2$ and $v=3$. Then the points $(0,0)$, $(3,-2)$,
+$\ldots$ belong to class~0; the points $(-1,1)$, $(2,-1)$, $\ldots$ belong
+to class~1; and the distance between these two lines is $1/\sqrt{13}$.
+The point $(2,3)$ itself belongs to class~13, hence its distance from
+the origin is $13/\sqrt{13}=\sqrt{13}$ (which we already knew).
+
+Suppose we wish to plot envelopes with respect to polygons with
+integer vertices. Then the best polygon for curves that travel in
+direction $(v,-u)$ will contain the points of class~$k$ such that
+$k/\psqrt{u^2+v^2}$ is as close as possible to~$d$, where $d$ is the
+maximum distance of the given ellipse from the line $ux+vy=0$.
+
+The |fillin| correction assumes that a diagonal line has an
+apparent thickness $$2f\cdot\min(\vert u\vert,\vert v\vert)/\psqrt{u^2+v^2}$$
+greater than would be obtained with truly square pixels. (If a
+white pixel at an exterior corner is assumed to have apparent
+darkness $f_1$ and a black pixel at an interior corner is assumed
+to have apparent darkness $1-f_2$, then $f=f_1-f_2$ is the |fillin|
+parameter.) Under this assumption we want to choose $k$ so that
+$\bigl(k+2f\cdot\min(\vert u\vert,\vert v\vert)\bigr)\big/\psqrt{u^2+v^2}$
+is as close as possible to $d$.
+
+Integer coordinates for the vertices work nicely because the thickness of
+the envelope at any given slope is independent of the position of the
+path with respect to the raster. It turns out, in fact, that the same
+property holds for polygons whose vertices have coordinates that are
+integer multiples of~$1\over2$, because ellipses are symmetric about
+the origin. It's convenient to double all dimensions and require the
+resulting polygon to have vertices with integer coordinates. For example,
+to get a circle of {\sl diameter}~$r$, we shall compute integer
+coordinates for a circle of {\sl radius}~$r$. The circle of radius~$r$
+will want to be represented by a polygon that contains the boundary
+points $(0,\pm r)$ and~$(\pm r,0)$; later we will divide everything
+by~2 and get a polygon with $(0,\pm{1\over2}r)$ and $(\pm{1\over2}r,0)$
+on its boundary.
+
+@ In practice the important slopes are those having small values of
+$u$ and~$v$; these make regular patterns in which our eyes quickly
+spot irregularities. For example, horizontal and vertical lines
+(when $u=0$ and $\vert v\vert=1$, or $\vert u\vert=1$ and $v=0$)
+are the most important; diagonal lines (when $\vert u\vert=\vert v\vert=1$)
+are next; and then come lines with slope $\pm2$ or $\pm1/2$.
+
+The nicest way to generate all rational directions having small
+numerators and denominators is to generalize the Stern--Brocot tree
+[cf.~{\sl Concrete Mathematics}, section 4.5]
+@^Brocot, Achille@>
+@^Stern, Moritz Abraham@>
+to a ``Stern--Brocot wreath'' as follows: Begin with four nodes
+arranged in a circle, containing the respective directions
+$(u,v)=(1,0)$, $(0,1)$, $(-1,0)$, and~$(0,-1)$. Then between pairs of
+consecutive terms $(u,v)$ and $(u',v')$ of the wreath, insert the
+direction $(u+u',v+v')$; continue doing this until some stopping
+criterion is fulfilled.
+
+It is not difficult to verify that, regardless of the stopping
+criterion, consecutive directions $(u,v)$ and $(u',v')$ of this
+wreath will always satisfy the relation $uv'-u'v=1$. Such pairs
+of directions have a nice property with respect to the equivalence
+classes described above. Let $l$ be a line of equivalent integer points
+$(m+tv,n-tu)$ with respect to~$(u,v)$, and let $l'$ be a line of
+equivalent integer points $(m'+tv',n'-tu')$ with respect to~$(u',v')$.
+Then $l$ and~$l'$ intersect in an integer point $(m'',n'')$, because
+the determinant of the linear equations for intersection is $uv'-u'v=1$.
+Notice that the class number of $(m'',n'')$ with respect to $(u+u',v+v')$
+is the sum of its class numbers with respect to $(u,v)$ and~$(u',v')$.
+Moreover, consecutive points on~$l$ and~$l'$ belong to classes that
+differ by exactly~1 with respect to $(u+u',v+v')$.
+
+This leads to a nice algorithm in which we construct a polygon having
+``correct'' class numbers for as many small-integer directions $(u,v)$
+as possible: Assuming that lines $l$ and~$l'$ contain points of the
+correct class for $(u,v)$ and~$(u',v')$, respectively, we determine
+the intersection $(m'',n'')$ and compute its class with respect to
+$(u+u',v+v')$. If the class is too large to be the best approximation,
+we move back the proper number of steps from $(m'',n'')$ toward smaller
+class numbers on both $l$ and~$l'$, unless this requires moving to points
+that are no longer in the polygon; in this way we arrive at two points that
+determine a line~$l''$ having the appropriate class. The process continues
+recursively, until it cannot proceed without removing the last remaining
+point from the class for $(u,v)$ or the class for $(u',v')$.
+
+@ The |make_ellipse| subroutine produces a pointer to a cyclic path
+whose vertices define a polygon suitable for envelopes. The control
+points on this path will be ignored; in fact, the fields in knot nodes
+that are usually reserved for control points are occupied by other
+data that helps |make_ellipse| compute the desired polygon.
+
+Parameters |major_axis| and |minor_axis| define the axes of the ellipse;
+and parameter |theta| is an angle by which the ellipse is rotated
+counterclockwise. If |theta=0|, the ellipse has the equation
+$(x/a)^2+(y/b)^2=1$, where |a=major_axis/2| and |b=minor_axis/2|.
+In general, the points of the ellipse are generated in the complex plane
+by the formula $e^{i\theta}(a\cos t+ib\sin t)$, as $t$~ranges over all
+angles. Notice that if |major_axis=minor_axis=d|, we obtain a circle
+of diameter~|d|, regardless of the value of |theta|.
+
+The method sketched above is used to produce the elliptical polygon,
+except that the main work is done only in the halfplane obtained from
+the three starting directions $(0,-1)$, $(1,0)$,~$(0,1)$. Since the ellipse
+has circular symmetry, we use the fact that the last half of the polygon
+is simply the negative of the first half. Furthermore, we need to compute only
+one quarter of the polygon if the ellipse has axis symmetry.
+
+@p function make_ellipse(@!major_axis,@!minor_axis:scaled;
+ @!theta:angle):pointer;
+label done,done1,found;
+var @!p,@!q,@!r,@!s:pointer; {for list manipulation}
+@!h:pointer; {head of the constructed knot list}
+@!alpha,@!beta,@!gamma,@!delta:integer; {special points}
+@!c,@!d:integer; {class numbers}
+@!u,@!v:integer; {directions}
+@!symmetric:boolean; {should the result be symmetric about the axes?}
+begin @<Initialize the ellipse data structure by beginning with
+ directions $(0,-1)$, $(1,0)$, $(0,1)$@>;
+@<Interpolate new vertices in the ellipse data structure until
+ improvement is impossible@>;
+if symmetric then
+ @<Complete the half ellipse by reflecting the quarter already computed@>;
+@<Complete the ellipse by copying the negative of the half already computed@>;
+make_ellipse:=h;
+end;
+
+@ A special data structure is used only with |make_ellipse|: The
+|right_x|, |left_x|, |right_y|, and |left_y| fields of knot nodes
+are renamed |right_u|, |left_v|, |right_class|, and |left_length|,
+in order to store information that simplifies the necessary computations.
+
+If |p| and |q| are consecutive knots in this data structure, the
+|x_coord| and |y_coord| fields of |p| and~|q| contain current vertices
+of the polygon; their values are integer multiples
+of |half_unit|. Both of these vertices belong to equivalence class
+|right_class(p)| with respect to the direction
+$\bigl($|right_u(p),left_v(q)|$\bigr)$. The number of points of this class
+on the line from vertex~|p| to vertex~|q| is |1+left_length(q)|.
+In particular, |left_length(q)=0| means that |x_coord(p)=x_coord(q)|
+and |y_coord(p)=y_coord(q)|; such duplicate vertices will be
+discarded during the course of the algorithm.
+
+The contents of |right_u(p)| and |left_v(q)| are integer multiples
+of |half_unit|, just like the coordinate fields. Hence, for example,
+the point $\bigl($|x_coord(p)-left_v(q),y_coord(p)+right_u(p)|$\bigr)$
+also belongs to class number |right_class(p)|. This point is one
+step closer to the vertex in node~|q|; it equals that vertex
+if and only if |left_length(q)=1|.
+
+The |left_type| and |right_type| fields are not used, but |link|
+has its normal meaning.
+
+To start the process, we create four nodes for the three directions
+$(0,-1)$, $(1,0)$, and $(0,1)$. The corresponding vertices are
+$(-\alpha,-\beta)$, $(\gamma,-\beta)$, $(\gamma,\beta)$, and
+$(\alpha,\beta)$, where $(\alpha,\beta)$ is a half-integer approximation
+to where the ellipse rises highest above the $x$-axis, and where
+$\gamma$ is a half-integer approximation to the maximum $x$~coordinate
+of the ellipse. The fourth of these nodes is not actually calculated
+if the ellipse has axis symmetry.
+
+@d right_u==right_x {|u| value for a pen edge}
+@d left_v==left_x {|v| value for a pen edge}
+@d right_class==right_y {equivalence class number of a pen edge}
+@d left_length==left_y {length of a pen edge}
+
+@<Initialize the ellipse data structure...@>=
+@<Calculate integers $\alpha$, $\beta$, $\gamma$ for the vertex
+ coordinates@>;
+p:=get_node(knot_node_size); q:=get_node(knot_node_size);
+r:=get_node(knot_node_size);
+if symmetric then s:=null@+else s:=get_node(knot_node_size);
+h:=p; link(p):=q; link(q):=r; link(r):=s; {|s=null| or |link(s)=null|}
+@<Revise the values of $\alpha$, $\beta$, $\gamma$, if necessary,
+ so that degenerate lines of length zero will not be obtained@>;
+x_coord(p):=-alpha*half_unit;
+y_coord(p):=-beta*half_unit;
+x_coord(q):=gamma*half_unit;@/
+y_coord(q):=y_coord(p); x_coord(r):=x_coord(q);@/
+right_u(p):=0; left_v(q):=-half_unit;@/
+right_u(q):=half_unit; left_v(r):=0;@/
+right_u(r):=0;
+right_class(p):=beta; right_class(q):=gamma; right_class(r):=beta;@/
+left_length(q):=gamma+alpha;
+if symmetric then
+ begin y_coord(r):=0; left_length(r):=beta;
+ end
+else begin y_coord(r):=-y_coord(p); left_length(r):=beta+beta;@/
+ x_coord(s):=-x_coord(p); y_coord(s):=y_coord(r);@/
+ left_v(s):=half_unit; left_length(s):=gamma-alpha;
+ end
+
+@ One of the important invariants of the pen data structure is that
+the points are distinct. We may need to correct the pen specification
+in order to avoid this. (The result of \&{pencircle} will always be at
+least one pixel wide and one pixel tall, although \&{makepen} is
+capable of producing smaller pens.)
+
+@<Revise the values of $\alpha$, $\beta$, $\gamma$, if necessary...@>=
+if beta=0 then beta:=1;
+if gamma=0 then gamma:=1;
+if gamma<=abs(alpha) then
+ if alpha>0 then alpha:=gamma-1
+ else alpha:=1-gamma
+
+@ If $a$ and $b$ are the semi-major and semi-minor axes,
+the given ellipse rises highest above the $x$-axis at the point
+$\bigl((a^2-b^2)\sin\theta\cos\theta/\rho\bigr)+i\rho$, where
+$\rho=\sqrt{(a\sin\theta)^2+(b\cos\theta)^2}$. It reaches
+furthest to the right of~the $y$-axis at the point
+$\sigma+i(a^2-b^2)\sin\theta\cos\theta/\sigma$, where
+$\sigma=\sqrt{(a\cos\theta)^2+(b\sin\theta)^2}$.
+
+@<Calculate integers $\alpha$, $\beta$, $\gamma$...@>=
+if (major_axis=minor_axis)or(theta mod ninety_deg=0) then
+ begin symmetric:=true; alpha:=0;
+ if odd(theta div ninety_deg) then
+ begin beta:=major_axis; gamma:=minor_axis;
+ n_sin:=fraction_one; n_cos:=0; {|n_sin| and |n_cos| are used later}
+ end
+ else begin beta:=minor_axis; gamma:=major_axis; theta:=0;
+ end; {|n_sin| and |n_cos| aren't needed in this case}
+ end
+else begin symmetric:=false;
+ n_sin_cos(theta); {set up $|n_sin|=\sin\theta$ and $|n_cos|=\cos\theta$}
+ gamma:=take_fraction(major_axis,n_sin);
+ delta:=take_fraction(minor_axis,n_cos);
+ beta:=pyth_add(gamma,delta);
+ alpha:=take_fraction(take_fraction(major_axis,
+ make_fraction(gamma,beta)),n_cos)@|
+ -take_fraction(take_fraction(minor_axis,
+ make_fraction(delta,beta)),n_sin);
+ alpha:=(alpha+half_unit) div unity;
+ gamma:=pyth_add(take_fraction(major_axis,n_cos),
+ take_fraction(minor_axis,n_sin));
+ end;
+beta:=(beta+half_unit) div unity;
+gamma:=(gamma+half_unit) div unity
+
+@ Now |p|, |q|, and |r| march through the list, always representing
+three consecutive vertices and two consecutive slope directions.
+When a new slope is interpolated, we back up slightly, until
+further refinement is impossible; then we march forward again.
+The somewhat magical operations performed in this part of the
+algorithm are justified by the theory sketched earlier.
+Complications arise only from the need to keep zero-length lines
+out of the final data structure.
+
+@<Interpolate new vertices in the ellipse data structure...@>=
+loop@+ begin u:=right_u(p)+right_u(q); v:=left_v(q)+left_v(r);
+ c:=right_class(p)+right_class(q);@/
+ @<Compute the distance |d| from class~0 to the edge of the ellipse
+ in direction |(u,v)|, times $\psqrt{u^2+v^2}$,
+ rounded to the nearest integer@>;
+ delta:=c-d; {we want to move |delta| steps back
+ from the intersection vertex~|q|}
+ if delta>0 then
+ begin if delta>left_length(r) then delta:=left_length(r);
+ if delta>=left_length(q) then
+ @<Remove the line from |p| to |q|,
+ and adjust vertex~|q| to introduce a new line@>
+ else @<Insert a new line for direction |(u,v)| between |p| and~|q|@>;
+ end
+ else p:=q;
+ @<Move to the next remaining triple |(p,q,r)|, removing and skipping past
+ zero-length lines that might be present; |goto done| if all
+ triples have been processed@>;
+ end;
+done:
+
+@ The appearance of a zero-length line means that we should advance |p|
+past it. We must not try to straddle a missing direction, because the
+algorithm works only on consecutive pairs of directions.
+
+@<Move to the next remaining triple |(p,q,r)|...@>=
+loop@+ begin q:=link(p);
+ if q=null then goto done;
+ if left_length(q)=0 then
+ begin link(p):=link(q); right_class(p):=right_class(q);
+ right_u(p):=right_u(q); free_node(q,knot_node_size);
+ end
+ else begin r:=link(q);
+ if r=null then goto done;
+ if left_length(r)=0 then
+ begin link(p):=r; free_node(q,knot_node_size); p:=r;
+ end
+ else goto found;
+ end;
+ end;
+found:
+
+@ The `\&{div} 8' near the end of this step comes from
+the fact that |delta| is scaled by~$2^{15}$ and $d$~by~$2^{16}$,
+while |take_fraction| removes a scale factor of~$2^{28}$.
+We also make sure that $d\G\max(\vert u\vert,\vert v\vert)$, so that
+the pen will always include a circular pen of diameter~1 as a subset;
+then it won't be possible to get disconnected path envelopes.
+
+@<Compute the distance |d| from class~0 to the edge of the ellipse...@>=
+delta:=pyth_add(u,v);
+if major_axis=minor_axis then d:=major_axis {circles are easy}
+else begin if theta=0 then
+ begin alpha:=u; beta:=v;
+ end
+ else begin alpha:=take_fraction(u,n_cos)+take_fraction(v,n_sin);
+ beta:=take_fraction(v,n_cos)-take_fraction(u,n_sin);
+ end;
+ alpha:=make_fraction(alpha,delta);
+ beta:=make_fraction(beta,delta);
+ d:=pyth_add(take_fraction(major_axis,alpha),
+ take_fraction(minor_axis,beta));
+ end;
+alpha:=abs(u); beta:=abs(v);
+if alpha<beta then
+ begin alpha:=abs(v); beta:=abs(u);
+ end; {now $\alpha=\max(\vert u\vert,\vert v\vert)$,
+ $\beta=\min(\vert u\vert,\vert v\vert)$}
+if internal[fillin]<>0 then
+ d:=d-take_fraction(internal[fillin],make_fraction(beta+beta,delta));
+d:=take_fraction((d+4) div 8,delta); alpha:=alpha div half_unit;
+if d<alpha then d:=alpha
+
+@ At this point there's a line of length |<=delta| from vertex~|p|
+to vertex~|q|, orthogonal to direction $\bigl($|right_u(p),left_v(q)|$\bigr)$;
+and there's a line of length |>=delta| from vertex~|q|
+to vertex~|r|, orthogonal to direction $\bigl($|right_u(q),left_v(r)|$\bigr)$.
+The best line to direction $(u,v)$ should replace the line from
+|p| to~|q|; this new line will have the same length as the old.
+
+@<Remove the line from |p| to |q|...@>=
+begin delta:=left_length(q);@/
+right_class(p):=c-delta; right_u(p):=u; left_v(q):=v;@/
+x_coord(q):=x_coord(q)-delta*left_v(r);
+y_coord(q):=y_coord(q)+delta*right_u(q);@/
+left_length(r):=left_length(r)-delta;
+end
+
+@ Here is the main case, now that we have dealt with the exception:
+We insert a new line of length |delta| for direction |(u,v)|, decreasing
+each of the adjacent lines by |delta| steps.
+
+@<Insert a new line for direction |(u,v)| between |p| and~|q|@>=
+begin s:=get_node(knot_node_size); link(p):=s; link(s):=q;@/
+x_coord(s):=x_coord(q)+delta*left_v(q);
+y_coord(s):=y_coord(q)-delta*right_u(p);@/
+x_coord(q):=x_coord(q)-delta*left_v(r);
+y_coord(q):=y_coord(q)+delta*right_u(q);@/
+left_v(s):=left_v(q); right_u(s):=u; left_v(q):=v;@/
+right_class(s):=c-delta;@/
+left_length(s):=left_length(q)-delta; left_length(q):=delta;
+left_length(r):=left_length(r)-delta;
+end
+
+@ Only the coordinates need to be copied, not the class numbers and other stuff.
+At this point either |link(p)| or |link(link(p))| is |null|.
+
+@<Complete the half ellipse...@>=
+begin s:=null; q:=h;
+loop@+ begin r:=get_node(knot_node_size); link(r):=s; s:=r;@/
+ x_coord(s):=x_coord(q); y_coord(s):=-y_coord(q);
+ if q=p then goto done1;
+ q:=link(q);
+ if y_coord(q)=0 then goto done1;
+ end;
+done1: if (link(p)<>null) then free_node(link(p),knot_node_size);
+link(p):=s; beta:=-y_coord(h);
+while y_coord(p)<>beta do p:=link(p);
+q:=link(p);
+end
+
+@ Now we use a somewhat tricky fact: The pointer |q| will be null if and
+only if the line for the final direction $(0,1)$ has been removed. If
+that line still survives, it should be combined with a possibly
+surviving line in the initial direction $(0,-1)$.
+
+@<Complete the ellipse by copying...@>=
+if q<>null then
+ begin if right_u(h)=0 then
+ begin p:=h; h:=link(h); free_node(p,knot_node_size);@/
+ x_coord(q):=-x_coord(h);
+ end;
+ p:=q;
+ end
+else q:=p;
+r:=link(h); {now |p=q|, |x_coord(p)=-x_coord(h)|, |y_coord(p)=-y_coord(h)|}
+repeat s:=get_node(knot_node_size); link(p):=s; p:=s;@/
+x_coord(p):=-x_coord(r); y_coord(p):=-y_coord(r); r:=link(r);
+until r=q;
+link(p):=h
+
+@* \[26] 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}
+@<Other local variables for |find_direction_time|@>@;
+begin @<Normalize the given direction for better accuracy;
+ but |return| with zero result if it's zero@>;
+n:=0; p:=h;
+loop@+ begin if right_type(p)=endpoint then goto not_found;
+ q:=link(p);
+ @<Rotate the cubic between |p| and |q|; then
+ |goto found| if the rotated cubic travels due east at some time |tt|;
+ but |goto not_found| if an entire cyclic path has been traversed@>;
+ p:=q; n:=n+unity;
+ end;
+not_found: find_direction_time:=-unity; return;
+found: find_direction_time:=n+tt;
+exit:end;
+
+@ @<Normalize the given direction for better accuracy...@>=
+if abs(x)<abs(y) then
+ begin x:=make_fraction(x,abs(y));
+ if y>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 $${1\over3}B'(x_0,x_1,x_2,x_3;t)=
+B(x_1-x_0,x_2-x_1,x_3-x_2;t)$$ 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|.)
+
+@<Rotate the cubic between |p| and |q|; then...@>=
+tt:=0;
+@<Set local variables |x1,x2,x3| and |y1,y2,y3| to multiples of the control
+ points of the rotated derivatives@>;
+if y1=0 then if x1>=0 then goto found;
+if n>0 then
+ begin @<Exit to |found| if an eastward direction occurs at knot |p|@>;
+ if p=h then goto not_found;
+ end;
+if (x3<>0)or(y3<>0) then phi:=n_arg(x3,y3);
+@<Exit to |found| if the curve whose derivatives are specified by
+ |x1,x2,x3,y1,y2,y3| travels eastward at some time~|tt|@>
+
+@ @<Other local variables for |find_direction_time|@>=
+@!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}
+
+@ @<Set local variables |x1,x2,x3| and |y1,y2,y3| to multiples...@>=
+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<fraction_half do
+ begin double(max); double(x1); double(x2); double(x3);
+ double(y1); double(y2); double(y3);
+ end;
+t:=x1; x1:=take_fraction(x1,x)+take_fraction(y1,y);
+y1:=take_fraction(y1,x)-take_fraction(t,y);@/
+t:=x2; x2:=take_fraction(x2,x)+take_fraction(y2,y);
+y2:=take_fraction(y2,x)-take_fraction(t,y);@/
+t:=x3; x3:=take_fraction(x3,x)+take_fraction(y3,y);
+y3:=take_fraction(y3,x)-take_fraction(t,y)
+
+@ @<Exit to |found| if an eastward direction occurs at knot |p|@>=
+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.
+
+@ @<Exit to |found| if the curve whose derivatives are specified by...@>=
+if x1<0 then if x2<0 then if x3<0 then goto done;
+if ab_vs_cd(y1,y3,y2,y2)=0 then
+ @<Handle the test for eastward directions when $y_1y_3=y_2^2$;
+ either |goto found| or |goto done|@>;
+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;
+@<Check the places where $B(y_1,y_2,y_3;t)=0$ to see if
+ $B(x_1,x_2,x_3;t)\ge0$@>;
+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 $y_1y_3<y_2^2$. The rotation process is itself
+subject to rounding errors. Yet this code optimistically tries to
+do the right thing.
+
+@d we_found_it==begin tt:=(t+@'4000) div @'10000; goto found;
+ end
+
+@<Check the places where $B(y_1,y_2,y_3;t)=0$...@>=
+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
+
+@ @<Handle the test for eastward directions when $y_1y_3=y_2^2$;
+ either |goto found| or |goto done|@>=
+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
+ @<Exit to |found| if the derivative $B(x_1,x_2,x_3;t)$ becomes |>=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.
+
+@<Exit to |found| if the derivative $B(x_1,x_2,x_3;t)$ becomes |>=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 |make_moves|.\
+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 $2^lu\submin=2^lu_0+U\submin$, 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 $t_1<t_1'$ iff
+$t_2<t_2'$; but in general, lexicographic order can be quite different,
+and the bisection algorithm would be substantially less efficient if it were
+constrained by lexicographic order.
+
+For example, suppose that an overlap has been found for $l=3$ and
+$(t_1,t_2)= (.101,.011)$ in binary, but that no overlap is produced by
+either of the alternatives $(.1010,.0110)$, $(.1010,.0111)$ at level~4.
+Then there is probably an intersection in one of the subintervals
+$(.1011,.011x)$; but lexicographic order would require us to explore
+$(.1010,.1xxx)$ and $(.1011,.00xx)$ and $(.1011,.010x)$ first. We wouldn't
+want to store all of the subdivision data for the second path, so the
+subdivisions would have to be regenerated many times. Such inefficiencies
+would be associated with every `1' in the binary representation of~$t_1$.
+
+@ The subdivision process introduces rounding errors, hence we need to
+make a more liberal test for overlap. It is not hard to show that the
+computed values of $U_i$ differ from the truth by at most~$l$, on
+level~$l$, hence $U\submin$ and $U\submax$ will be at most $3l$ in error.
+If $\beta$ is an upper bound on the absolute error in the computed
+components of $\Delta=(|delx|,|dely|)$ on level~$l$, we will replace
+the test `$X\submin-U\submax\L|delx|$' by the more liberal test
+`$X\submin-U\submax\L|delx|+|tol|$', where $|tol|=6l+\beta$.
+
+More accuracy is obtained if we try the algorithm first with |tol=0|;
+the more liberal tolerance is used only if an exact approach fails.
+It is convenient to do this double-take by letting `3' in the preceding
+paragraph be a parameter, which is first 0, then 3.
+
+@<Glob...@>=
+@!tol_step:0..6; {either 0 or 3, usually}
+
+@ We shall use an explicit stack to implement the recursive bisection
+method described above. In fact, the |bisect_stack| array is available for
+this purpose. It 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}
+
+@<Check the ``constant''...@>=
+if int_packets+17*int_increment>bistack_size then bad:=32;
+
+@ 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(#)<stack_1(#) then stack_max(#):=stack_1(#);
+ stack_min(#):=stack_1(#)+stack_2(#);
+ if stack_min(#)>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
+
+@<Glob...@>=
+@!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;
+@<Initialize for intersections at level zero@>;
+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:=half(cur_t+1); cur_tt:=half(cur_tt+1); return;
+ end;
+ double(max_t); appr_t:=cur_t; appr_tt:=cur_tt;
+ end;
+ @<Subdivide for a new level of intersection@>;
+ goto continue;
+ end;
+ if time_to_go>0 then decr(time_to_go)
+ else begin while appr_t<unity do
+ begin double(appr_t); double(appr_tt);
+ end;
+ cur_t:=appr_t; cur_tt:=appr_tt; return;
+ end;
+ @<Advance to the next pair |(cur_t,cur_tt)|@>;
+ 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.
+
+@<Glob...@>=
+@!delx,@!dely:integer; {the components of $\Delta=2^l(w_0-z_0)$}
+@!tol:integer; {bound on the uncertainty 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.
+@^overflow in arithmetic@>
+
+@<Initialize for intersections at level zero@>=
+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
+
+@ @<Subdivide for a new level of intersection@>=
+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
+
+@ @<Advance to the next pair |(cur_t,cur_tt)|@>=
+not_found: if odd(cur_tt) then
+ if odd(cur_t) then @<Descend to the previous level and |goto not_found|@>
+ 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_packets| to |r_packets|}
+ decr(cur_tt); xy:=xy-int_packets; {switch from |r_packets| to |l_packets|}
+ 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_packets| to |r_packets|}
+ end
+
+@ @<Descend to the previous level...@>=
+begin cur_t:=half(cur_t); cur_tt:=half(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 @<Change one-point paths into dead cycles@>;
+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;
+
+@ @<Change one-point paths...@>=
+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;
+
+@* \[27] Online graphic output.
+\MF\ displays images on the user's screen by means of a few primitive
+operations that are defined below. These operations have deliberately been
+kept simple so that they can be implemented without great difficulty on a
+wide variety of machines. Since \PASCAL\ has no traditional standards for
+graphic output, some system-dependent code needs to be written in order to
+support this aspect of \MF; but the necessary routines are usually quite
+easy to write.
+@^system dependencies@>
+
+In fact, there are exactly four such routines:
+
+\yskip\hang
+|init_screen| does whatever initialization is necessary to
+support the other operations; it is a boolean function that returns
+|false| if graphic output cannot be supported (e.g., if the other three
+routines have not been written, or if the user doesn't have the
+right kind of terminal).
+
+\yskip\hang
+|blank_rectangle| updates a buffer area in memory so that
+all pixels in a specified rectangle will be set to the background color.
+
+\yskip\hang
+|paint_row| assigns values to specified pixels in a row of
+the buffer just mentioned, based on ``transition'' indices explained below.
+
+\yskip\hang
+|update_screen| displays the current screen buffer; the
+effects of |blank_rectangle| and |paint_row| commands may or may not
+become visible until the next |update_screen| operation is performed.
+(Thus, |update_screen| is analogous to |update_terminal|.)
+
+\yskip\noindent
+The \PASCAL\ code here is a minimum version of |init_screen| and
+|update_screen|, usable on \MF\ installations that don't
+support screen output. If |init_screen| is changed to return |true|
+instead of |false|, the other routines will simply log the fact
+that they have been called; they won't really display anything.
+The standard test routines for \MF\ use this log information to check
+that \MF\ is working properly, but the |wlog| instructions should be
+removed from production versions of \MF.
+
+@p function init_screen:boolean;
+begin init_screen:=false;
+end;
+@#
+procedure update_screen; {will be called only if |init_screen| returns |true|}
+begin @!init wlog_ln('Calling UPDATESCREEN');@+tini {for testing only}
+end;
+
+@ The user's screen is assumed to be a rectangular area, |screen_width|
+pixels wide and |screen_depth| pixels deep. The pixel in the upper left
+corner is said to be in column~0 of row~0; the pixel in the lower right
+corner is said to be in column |screen_width-1| of row |screen_depth-1|.
+Notice that row numbers increase from top to bottom, contrary to \MF's
+other coordinates.
+
+Each pixel is assumed to have two states, referred to in this documentation
+as |black| and |white|. The background color is called |white| and the
+other color is called |black|; but any two distinct pixel values
+can actually be used. For example, the author developed \MF\ on a
+system for which |white| was black and |black| was bright green.
+
+@d white=0 {background pixels}
+@d black=1 {visible pixels}
+
+@<Types...@>=
+@!screen_row=0..screen_depth; {a row number on the screen}
+@!screen_col=0..screen_width; {a column number on the screen}
+@!trans_spec=array[screen_col] of screen_col; {a transition spec, see below}
+@!pixel_color=white..black; {specifies one of the two pixel values}
+
+@ We'll illustrate the |blank_rectangle| and |paint_row| operations by
+pretending to declare a screen buffer called |screen_pixel|. This code
+is actually commented out, but it does specify the intended effects.
+
+@<Glob...@>=
+@{@+@!screen_pixel:array[screen_row,screen_col] of pixel_color@t; @>@}
+
+@ The |blank_rectangle| routine simply whitens all pixels that lie in
+columns |left_col| through |right_col-1|, inclusive, of rows
+|top_row| through |bot_row-1|, inclusive, given four parameters that satisfy
+the relations
+$$\hbox{|0<=left_col<=right_col<=screen_width|,\quad
+ |0<=top_row<=bot_row<=screen_depth|.}$$
+If |left_col=right_col| or |top_row=bot_row|, nothing happens.
+
+The commented-out code in the following procedure is for illustrative
+purposes only.
+@^system dependencies@>
+
+@p procedure blank_rectangle(@!left_col,@!right_col:screen_col;
+ @!top_row,@!bot_row:screen_row);
+var @!r:screen_row;
+@!c:screen_col;
+begin @{@+for r:=top_row to bot_row-1 do
+ for c:=left_col to right_col-1 do
+ screen_pixel[r,c]:=white;@+@}@/
+@!init wlog_cr; {this will be done only after |init_screen=true|}
+wlog_ln('Calling BLANKRECTANGLE(',left_col:1,',',
+ right_col:1,',',top_row:1,',',bot_row:1,')');@+tini
+end;
+
+@ The real work of screen display is done by |paint_row|. But it's not
+hard work, because the operation affects only
+one of the screen rows, and it affects only a contiguous set of columns
+in that row. There are four parameters: |r|~(the row),
+|b|~(the initial color),
+|a|~(the array of transition specifications),
+and |n|~(the number of transitions). The elements of~|a| will satisfy
+$$0\L a[0]<a[1]<\cdots<a[n]\L |screen_width|;$$
+the value of |r| will satisfy |0<=r<screen_depth|; and |n| will be positive.
+
+The general idea is to paint blocks of pixels in alternate colors;
+the precise details are best conveyed by means of a \PASCAL\
+program (see the commented-out code below).
+@^system dependencies@>
+
+@p procedure paint_row(@!r:screen_row;@!b:pixel_color;var @!a:trans_spec;
+ @!n:screen_col);
+var @!k:screen_col; {an index into |a|}
+@!c:screen_col; {an index into |screen_pixel|}
+begin @{@+k:=0; c:=a[0];
+repeat incr(k);
+ repeat screen_pixel[r,c]:=b; incr(c);
+ until c=a[k];
+ b:=black-b; {$|black|\swap|white|$}
+ until k=n;@+@}@/
+@!init wlog('Calling PAINTROW(',r:1,',',b:1,';');
+ {this is done only after |init_screen=true|}
+for k:=0 to n do
+ begin wlog(a[k]:1); if k<>n then wlog(',');
+ end;
+wlog_ln(')');@+tini
+end;
+
+@ The remainder of \MF's screen routines are system-independent calls
+on the four primitives just defined.
+
+First we have a global boolean variable that tells if |init_screen|
+has been called, and another one that tells if |init_screen| has
+given a |true| response.
+
+@<Glob...@>=
+@!screen_started:boolean; {have the screen primitives been initialized?}
+@!screen_OK:boolean; {is it legitimate to call |blank_rectangle|,
+ |paint_row|, and |update_screen|?}
+
+@ @d start_screen==begin if not screen_started then
+ begin screen_OK:=init_screen; screen_started:=true;
+ end;
+ end
+
+@<Set init...@>=
+screen_started:=false; screen_OK:=false;
+
+@ \MF\ provides the user with 16 ``window'' areas on the screen, in each
+of which it is possible to produce independent displays.
+
+It should be noted that \MF's windows aren't really independent
+``clickable'' entities in the sense of multi-window graphic workstations;
+\MF\ simply maps them into subsets of a single screen image that is
+controlled by |init_screen|, |blank_rectangle|, |paint_row|, and
+|update_screen| as described above. Implementations of \MF\ on a
+multi-window workstation probably therefore make use of only two
+windows in the other sense: one for the terminal output and another
+for the screen with \MF's 16 areas. Henceforth we shall
+use the term window only in \MF's sense.
+
+@<Types...@>=
+@!window_number=0..15;
+
+@ A user doesn't have to use any of the 16 windows. But when a window is
+``opened,'' it is allocated to a specific rectangular portion of the screen
+and to a specific rectangle with respect to \MF's coordinates. The relevant
+data is stored in global arrays |window_open|, |left_col|, |right_col|,
+|top_row|, |bot_row|, |m_window|, and |n_window|.
+
+The |window_open| array is boolean, and its significance is obvious. The
+|left_col|, \dots, |bot_row| arrays contain screen coordinates that
+can be used to blank the entire window with |blank_rectangle|. And the
+other two arrays just mentioned handle the conversion between
+actual coordinates and screen coordinates: \MF's pixel in column~$m$
+of row~$n$ will appear in screen column |m_window+m| and in screen row
+|n_window-n|, provided that these lie inside the boundaries of the window.
+
+Another array |window_time| holds the number of times this window has
+been updated.
+
+@<Glob...@>=
+@!window_open:array[window_number] of boolean;
+ {has this window been opened?}
+@!left_col:array[window_number] of screen_col;
+ {leftmost column position on screen}
+@!right_col:array[window_number] of screen_col;
+ {rightmost column position, plus~1}
+@!top_row:array[window_number] of screen_row;
+ {topmost row position on screen}
+@!bot_row:array[window_number] of screen_row;
+ {bottommost row position, plus~1}
+@!m_window:array[window_number] of integer;
+ {offset between user and screen columns}
+@!n_window:array[window_number] of integer;
+ {offset between user and screen rows}
+@!window_time:array[window_number] of integer;
+ {it has been updated this often}
+
+@ @<Set init...@>=
+for k:=0 to 15 do
+ begin window_open[k]:=false; window_time[k]:=0;
+ end;
+
+@ Opening a window isn't like opening a file, because you can open it
+as often as you like, and you never have to close it again. The idea is
+simply to define special points on the current screen display.
+
+Overlapping window specifications may cause complex effects that can
+be understood only by scrutinizing \MF's display algorithms; thus it
+has been left undefined in the \MF\ user manual, although the behavior
+@:METAFONTbook}{\sl The {\logos METAFONT\/}book@>
+is in fact predictable.
+
+Here is a subroutine that implements the command `\&{openwindow}~|k|
+\&{from}~$(\\{r0},\\{c0})$ \&{to}~$(\\{r1},\\{c1})$ \&{at}~$(x,y)$'.
+
+@p procedure open_a_window(@!k:window_number;@!r0,@!c0,@!r1,@!c1:scaled;
+ @!x,@!y:scaled);
+var @!m,@!n:integer; {pixel coordinates}
+begin @<Adjust the coordinates |(r0,c0)| and |(r1,c1)| so that
+ they lie in the proper range@>;
+window_open[k]:=true; incr(window_time[k]);@/
+left_col[k]:=c0; right_col[k]:=c1; top_row[k]:=r0; bot_row[k]:=r1;@/
+@<Compute the offsets between screen coordinates and actual coordinates@>;
+start_screen;
+if screen_OK then
+ begin blank_rectangle(c0,c1,r0,r1); update_screen;
+ end;
+end;
+
+@ A window whose coordinates don't fit the existing screen size will be
+truncated until they do.
+
+@<Adjust the coordinates |(r0,c0)| and |(r1,c1)|...@>=
+if r0<0 then r0:=0@+else r0:=round_unscaled(r0);
+r1:=round_unscaled(r1);
+if r1>screen_depth then r1:=screen_depth;
+if r1<r0 then
+ if r0>screen_depth then r0:=r1@+else r1:=r0;
+if c0<0 then c0:=0@+else c0:=round_unscaled(c0);
+c1:=round_unscaled(c1);
+if c1>screen_width then c1:=screen_width;
+if c1<c0 then
+ if c0>screen_width then c0:=c1@+else c1:=c0
+
+@ Three sets of coordinates are rampant, and they must be kept straight!
+(i)~\MF's main coordinates refer to the edges between pixels. (ii)~\MF's
+pixel coordinates (within edge structures) say that the pixel bounded by
+$(m,n)$, $(m,n+1)$, $(m+1,n)$, and~$(m+1,n+1)$ is in pixel row number~$n$
+and pixel column number~$m$. (iii)~Screen coordinates, on the other hand,
+have rows numbered in increasing order from top to bottom, as mentioned
+above.
+@^coordinates, explained@>
+
+The program here first computes integers $m$ and $n$ such that
+pixel column~$m$ of pixel row~$n$ will be at the upper left corner
+of the window. Hence pixel column |m-c0| of pixel row |n+r0|
+will be at the upper left corner of the screen.
+
+@<Compute the offsets between screen coordinates and actual coordinates@>=
+m:=round_unscaled(x); n:=round_unscaled(y)-1;@/
+m_window[k]:=c0-m; n_window[k]:=r0+n
+
+@ Now here comes \MF's most complicated operation related to window
+display: Given the number~|k| of an open window, the pixels of positive
+weight in |cur_edges| will be shown as |black| in the window; all other
+pixels will be shown as |white|.
+
+@p procedure disp_edges(@!k:window_number);
+label done,found;
+var @!p,@!q:pointer; {for list manipulation}
+@!already_there:boolean; {is a previous incarnation in the window?}
+@!r:integer; {row number}
+@<Other local variables for |disp_edges|@>@;
+begin if screen_OK then
+ if left_col[k]<right_col[k] then if top_row[k]<bot_row[k] then
+ begin already_there:=false;
+ if last_window(cur_edges)=k then
+ if last_window_time(cur_edges)=window_time[k] then
+ already_there:=true;
+ if not already_there then
+ blank_rectangle(left_col[k],right_col[k],top_row[k],bot_row[k]);
+ @<Initialize for the display computations@>;
+ p:=link(cur_edges); r:=n_window[k]-(n_min(cur_edges)-zero_field);
+ while (p<>cur_edges)and(r>=top_row[k]) do
+ begin if r<bot_row[k] then
+ @<Display the pixels of edge row |p| in screen row |r|@>;
+ p:=link(p); decr(r);
+ end;
+ update_screen;
+ incr(window_time[k]);
+ last_window(cur_edges):=k; last_window_time(cur_edges):=window_time[k];
+ end;
+end;
+
+@ Since it takes some work to display a row, we try to avoid recomputation
+whenever we can.
+
+@<Display the pixels of edge row |p| in screen row |r|@>=
+begin if unsorted(p)>void then sort_edges(p)
+else if unsorted(p)=void then if already_there then goto done;
+unsorted(p):=void; {this time we'll paint, but maybe not next time}
+@<Set up the parameters needed for |paint_row|;
+ but |goto done| if no painting is needed after all@>;
+paint_row(r,b,row_transition,n);
+done: end
+
+@ The transition-specification parameter to |paint_row| is always the same
+array.
+
+@<Glob...@>=
+@!row_transition:trans_spec; {an array of |black|/|white| transitions}
+
+@ The job remaining is to go through the list |sorted(p)|, unpacking the
+|info| fields into |m| and weight, then making |black| the pixels whose
+accumulated weight~|w| is positive.
+
+@<Other local variables for |disp_edges|@>=
+@!n:screen_col; {the highest active index in |row_transition|}
+@!w,@!ww:integer; {old and new accumulated weights}
+@!b:pixel_color; {status of first pixel in the row transitions}
+@!m,@!mm:integer; {old and new screen column positions}
+@!d:integer; {edge-and-weight without |min_halfword| compensation}
+@!m_adjustment:integer; {conversion between edge and screen coordinates}
+@!right_edge:integer; {largest edge-and-weight that could affect the window}
+@!min_col:screen_col; {the smallest screen column number in the window}
+
+@ Some precomputed constants make the display calculations faster.
+
+@<Initialize for the display computations@>=
+m_adjustment:=m_window[k]-m_offset(cur_edges);@/
+right_edge:=8*(right_col[k]-m_adjustment);@/
+min_col:=left_col[k]
+
+@ @<Set up the parameters needed for |paint_row|...@>=
+n:=0; ww:=0; m:=-1; w:=0;
+q:=sorted(p); row_transition[0]:=min_col;
+loop@+ begin if q=sentinel then d:=right_edge
+ else d:=ho(info(q));
+ mm:=(d div 8)+m_adjustment;
+ if mm<>m then
+ begin @<Record a possible transition in column |m|@>;
+ m:=mm; w:=ww;
+ end;
+ if d>=right_edge then goto found;
+ ww:=ww+(d mod 8)-zero_w;
+ q:=link(q);
+ end;
+found:@<Wind up the |paint_row| parameter calculation by inserting the
+ final transition; |goto done| if no painting is needed@>;
+
+@ Now |m| is a screen column |<right_col[k]|.
+
+@<Record a possible transition in column |m|@>=
+if w<=0 then
+ begin if ww>0 then if m>min_col then
+ begin if n=0 then
+ if already_there then
+ begin b:=white; incr(n);
+ end
+ else b:=black
+ else incr(n);
+ row_transition[n]:=m;
+ end;
+ end
+else if ww<=0 then if m>min_col then
+ begin if n=0 then b:=black;
+ incr(n); row_transition[n]:=m;
+ end
+
+@ If the entire row is |white| in the window area, we can omit painting it
+when |already_there| is false, since it has already been blanked out in
+that case.
+
+When the following code is invoked, |row_transition[n]| will be
+strictly less than |right_col[k]|.
+
+@<Wind up the |paint_row|...@>=
+if already_there or(ww>0) then
+ begin if n=0 then
+ if ww>0 then b:=black
+ else b:=white;
+ incr(n); row_transition[n]:=right_col[k];
+ end
+else if n=0 then goto done
+
+@* \[28] Dynamic linear equations.
+\MF\ 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 if serial_no>el_gordo-s_scale then
+ overflow("independent variables",serial_no div s_scale);
+@:METAFONT capacity exceeded independent variables}{\quad independent variables@>
+ type(#):=independent; serial_no:=serial_no+s_scale;
+ value(#):=serial_no;
+ end
+
+@<Glob...@>=
+@!serial_no:integer; {the most recent serial number, times |s_scale|}
+
+@ @<Make variable |q+s| newly independent@>=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 $v_1$; 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. \MF\ 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}
+
+@<Initialize table entries...@>= 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.
+
+@<Declare subroutines for printing expressions@>=
+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;
+ @<Print the coefficient, unless it's $\pm1.0$@>;
+ 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;
+
+@ @<Print the coefficient, unless it's $\pm1.0$@>=
+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 \MF\ 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
+
+@<Glob...@>=
+@!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}
+
+@ @<Set init...@>=
+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|}
+
+@<Declare basic dependency-list subroutines@>=
+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 @<Contribute a term from |p|, plus |f| times the
+ corresponding term from |q|@>
+ else if value(pp)<value(qq) then
+ @<Contribute a term from |q|, multiplied by~|f|@>
+ 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;
+
+@ @<Contribute a term from |p|, plus |f|...@>=
+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)<threshold then free_node(s,dep_node_size)
+else begin 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
+
+@ @<Contribute a term from |q|, multiplied by~|f|@>=
+begin if tt=dependent then v:=take_fraction(f,value(q))
+else v:=take_scaled(f,value(q));
+if abs(v)>half(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 @<Contribute a term from |p|, plus the
+ corresponding term from |q|@>
+ else if value(pp)<value(qq) then
+ begin s:=get_node(dep_node_size); info(s):=qq; value(s):=value(q);
+ q:=link(q); qq:=info(q); link(r):=s; r:=s;
+ end
+ else begin link(r):=p; r:=p; p:=link(p); pp:=info(p);
+ end;
+done: value(p):=slow_add(value(p),value(q));
+link(r):=p; dep_final:=p; p_plus_q:=link(temp_head);
+end;
+
+@ @<Contribute a term from |p|, plus the...@>=
+begin v:=value(p)+value(q);
+value(p):=v; s:=p; p:=link(p); pp:=info(p);
+if abs(v)<threshold then free_node(s,dep_node_size)
+else begin 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.
+
+@<Declare basic dependency-list subroutines@>=
+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.
+
+@<Declare basic dependency-list subroutines@>=
+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).
+
+@<Declare basic dependency-list subroutines@>=
+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.
+
+@<Declare basic dependency-list subroutines@>=
+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;
+ @<Run through the dependency list for variable |t|, fixing
+ all nodes, and ending with final link~|q|@>;
+ 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|}
+
+@<Run through the dependency list for variable |t|...@>=
+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)<independent_being_fixed then
+ begin p:=get_avail; link(p):=s; s:=p;
+ info(s):=x; type(x):=independent_being_fixed;
+ end;
+ value(q):=value(q) div 4;
+ if value(q)=0 then
+ begin link(r):=link(q); free_node(q,dep_node_size); q:=r;
+ end;
+ end;
+ r:=q;
+ end;
+done:
+
+@ The |new_dep| routine installs a dependency list~|p| into the value node~|q|,
+linking it into the list of all known dependencies. We assume that
+|dep_final| points to the final node of list~|p|.
+
+@p procedure new_dep(@!q,@!p:pointer);
+var @!r:pointer; {what used to be the first dependency}
+begin dep_list(q):=p; prev_dep(q):=dep_head;
+r:=link(dep_head); link(dep_final):=r; prev_dep(r):=dep_final;
+link(dep_head):=q;
+end;
+
+@ Here is one of the ways a dependency list gets started.
+The |const_dependency| routine produces a list that has nothing but
+a constant term.
+
+@p function const_dependency(@!v:scaled):pointer;
+begin dep_final:=get_node(dep_node_size);
+value(dep_final):=v; info(dep_final):=null;
+const_dependency:=dep_final;
+end;
+
+@ And here's a more interesting way to start a dependency list from scratch:
+The parameter to |single_dependency| is the location of an
+independent variable~|x|, and the result is the simple dependency list
+`|x+0|'.
+
+In the unlikely event that the given independent variable has been doubled so
+often that we can't refer to it with a nonzero coefficient,
+|single_dependency| returns the simple list `0'. This case can be
+recognized by testing that the returned list pointer is equal to
+|dep_final|.
+
+@p function single_dependency(@!p:pointer):pointer;
+var @!q:pointer; {the new dependency list}
+@!m:integer; {the number of doublings}
+begin m:=value(p) mod s_scale;
+if m>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 @<Find a node |q| in list |p| whose coefficient |v| is largest@>;
+x:=info(q); n:=value(x) mod s_scale;@/
+@<Divide list |p| by |-v|, removing node |q|@>;
+if internal[tracing_equations]>0 then @<Display the new dependency@>;
+@<Simplify all existing dependencies by substituting for |x|@>;
+@<Change variable |x| from |independent| to |dependent| or |known|@>;
+if fix_needed then fix_dependencies;
+end;
+
+@ @<Find a node |q| in list |p| whose coefficient |v| is largest@>=
+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|.
+
+@<Divide list |p| by |-v|, removing node |q|@>=
+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)
+
+@ @<Display the new dependency@>=
+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
+
+@ @<Simplify all existing dependencies by substituting for |x|@>=
+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
+
+@ @<Change variable |x| from |independent| to |dependent| or |known|@>=
+if n>0 then @<Divide list |p| by $2^n$@>;
+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
+
+@ @<Divide list |p| by $2^n$@>=
+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 \MF\ is being
+debugged, makes sure that the current dependency lists are well formed.
+
+@<Check the list of linear dependencies@>=
+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); r:=inf_val;
+ repeat if value(info(p))>=value(r) then
+ begin print_nl("Out of order at "); print_int(p);
+@.Out of order...@>
+ end;
+ r:=info(p); q:=p; p:=link(q);
+ until r=null;
+ end
+
+@* \[29] 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 \MF\ 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.
+
+@<Declare the recycling subroutines@>=
+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: begin value(q):=v; add_pen_ref(v);
+ end;
+path_type: value(q):=copy_path(v);
+picture_type: value(q):=copy_edges(v);
+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 @<Exclaim about a redundant equation@>;
+ return;
+ end;
+ r:=value(r);
+ end;
+r:=value(p); value(p):=value(q); value(q):=r;
+exit:end;
+
+@ @<Exclaim about a redundant equation@>=
+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
+
+@* \[30] Introduction to the syntactic routines.
+Let's pause a moment now and try to look at the Big Picture.
+The \MF\ 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 \MF\ that
+actually controls the activities.
+
+Our current goal is to come to grips with the |get_next| procedure,
+which is the keystone of \MF'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,
+\MF\ 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.
+
+@<Glob...@>=
+@!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 \MF\ primitive into |hash| and |eqtb|. Therefore almost
+all of this procedure appears elsewhere in the program, together with the
+corresponding |primitive| calls.
+
+@<Declare the procedure called |print_cmd_mod|@>=
+procedure print_cmd_mod(@!c,@!m:integer);
+begin case c of
+@t\4@>@<Cases of |print_cmd_mod| for symbolic printing of primitives@>@/
+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;
+
+@* \[31] Input stacks and states.
+The state of \MF'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:
+
+@<Types...@>=
+@!in_state_record = record
+ @!index_field: quarterword;
+ @!start_field,@!loc_field, @!limit_field, @!name_field: halfword;
+ end;
+
+@ @<Glob...@>=
+@!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 \MF\ 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. \MF\ 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 a text file. It is 0 if we
+are reading from the terminal for normal input, or 1 if we are executing a
+\&{readstring} command, or 2 if we are reading a string that was
+moved into the buffer by \&{scantokens}.
+
+@ 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 font}', we will have |index=1| while reading
+the file \.{font.mf}. 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.
+
+The global variable |in_open| is equal to the |index|
+value of the highest non-token-list level. 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=0|,
+and |cur_file| as an abbreviation for |input_file[index]|.
+
+The global variable |line| contains the line number in the topmost
+open file, for use in error messages. If we are not reading from
+the terminal, |line_stack[index]| holds the line number for the
+enclosing level, so that |line| can be restored when the current
+file has been 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|, maintained for enclosing levels in
+`\ignorespaces|@!page_stack:array[1..max_in_open] of integer|\unskip'
+by analogy with |line_stack|.
+@^system dependencies@>
+
+@d terminal_input==(name=0) {are we reading from the terminal?}
+@d cur_file==input_file[index] {the current |alpha_file| variable}
+
+@<Glob...@>=
+@!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 : integer; {current line number in the current source file}
+@!line_stack : array[1..max_in_open] of integer;
+
+@ 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 macro
+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.
+
+@<Glob...@>=
+@!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|}
+
+@ 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 \MF'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.
+
+@<Glob...@>=
+@!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}
+@<Local variables for formatting calculations@>@/
+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}
+ @<Display the current context@>;
+ if file_state then
+ if (name>2) or (file_ptr=0) then goto done;
+ decr(file_ptr);
+ end;
+done: cur_input:=input_stack[input_ptr]; {restore original state}
+end;
+
+@ @<Display the current context@>=
+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 @<Print location of current line@>;
+ @<Pseudoprint the line@>;
+ end
+ else begin @<Print type of token list@>;
+ @<Pseudoprint the token list@>;
+ end;
+ selector:=old_setting; {stop pseudoprinting}
+ @<Print two lines using the tricky pseudoprinted information@>;
+ 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@>
+
+@<Print location of current line@>=
+if name<=1 then
+ if terminal_input and(file_ptr=0) then print_nl("<*>")
+ else print_nl("<insert>")
+else if name=2 then print_nl("<scantokens>")
+else begin print_nl("l."); print_int(line);
+ end;
+print_char(" ")
+
+@ @<Print type of token list@>=
+case token_type of
+forever_text: print_nl("<forever> ");
+loop_text: @<Print the current loop value@>;
+parameter: print_nl("<argument> ");
+backed_up: if loc=null then print_nl("<recently read> ")
+ else print_nl("<to be read again> ");
+inserted: print_nl("<inserted text> ");
+macro: begin print_ln;
+ if name<>null then slow_print(text(name))
+ else @<Print the name of a \&{vardef}'d macro@>;
+ 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.
+
+@<Print the current loop value@>=
+begin print_nl("<for("); p:=param_stack[param_start];
+if p<>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.
+
+@<Print the name of a \&{vardef}'d macro@>=
+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 \MF'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<trick_count|, otherwise character |k| is dropped. Initially we set
+|tally:=0| and |trick_count:=1000000|; then when we reach the
+point where transition from line 1 to line 2 should occur, we
+set |first_count:=tally| and |trick_count:=@tmax@>(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.
+
+@<Local variables for formatting...@>=
+@!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<error_line then trick_count:=error_line;
+ end
+
+@ And the following code uses the information after it has been gathered.
+
+@<Print two lines using the tricky pseudoprinted information@>=
+if trick_count=1000000 then set_trick_count;
+ {|set_trick_count| must be performed}
+if tally<trick_count then m:=tally-first_count
+else m:=trick_count-first_count; {context on line 2}
+if l+first_count<=half_error_line then
+ begin p:=0; n:=l+first_count;
+ end
+else begin print("..."); p:=l+first_count-half_error_line+3;
+ n:=half_error_line;
+ end;
+for q:=p to first_count-1 do print_char(trick_buf[q mod error_line]);
+print_ln;
+for q:=1 to n do print_char(" "); {print |n| spaces to begin line~2}
+if m+n<=error_line then p:=first_count+m else p:=first_count+(error_line-n-3);
+for q:=first_count to p-1 do print_char(trick_buf[q mod error_line]);
+if m+n>error_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.
+
+@<Pseudoprint the line@>=
+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
+
+@ @<Pseudoprint the token list@>=
+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:
+
+@<Do magic computation@>=set_trick_count
+
+@* \[32] 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);
+@:METAFONT 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@>@<Declare the procedure called |make_exp_copy|@>@;@/
+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 \MF\ 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);
+@:METAFONT capacity exceeded text input levels}{\quad text input levels@>
+if first=buf_size then overflow("buffer size",buf_size);
+@:METAFONT capacity exceeded buffer size}{\quad buffer size@>
+incr(in_open); push_input; index:=in_open;
+line_stack[index]:=line; start:=first;
+name:=0; {|terminal_input| is now |true|}
+end;
+
+@ Conversely, the variables must be downdated when such a level of input
+is finished:
+
+@p procedure end_file_reading;
+begin first:=start; line:=line_stack[index];
+if index<>in_open then confusion("endinput");
+@:this can't happen endinput}{\quad endinput@>
+if name>2 then a_close(cur_file); {forget it}
+pop_input; decr(in_open);
+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 \MF's whole input mechanism going, we perform the following
+actions.
+
+@<Initialize the input routines@>=
+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:=0;
+force_eof:=false;
+if not init_terminal then goto final_end;
+limit:=last; first:=last+1; {|init_terminal| has set |loc| and |last|}
+end;
+
+@* \[33] Getting the next token.
+The heart of \MF'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 \MF's
+eyes and mouth, reading the source files and gobbling them up. And it also
+helps \MF\ 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
+\MF\ 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}
+
+@<Glob...@>=
+@!scanner_status:normal..loop_defining; {are we scanning at high speed?}
+@!warning_info:integer; {if so, what else do we need to know,
+ in case an error occurs?}
+
+@ @<Initialize the input routines@>=
+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 begin deletions_allowed:=false;
+ @<Back up an outer symbolic token so that it can be reread@>;
+ if scanner_status>skipping then
+ @<Tell the user what has run away and try to recover@>
+ 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;
+
+@ @<Back up an outer symbolic token so that it can be reread@>=
+if cur_sym<>0 then
+ begin p:=get_avail; info(p):=cur_sym;
+ back_list(p); {prepare to read the symbolic token again}
+ end
+
+@ @<Tell the user what has run away...@>=
+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@>@<Complete the error message,
+ and set |cur_sym| to a token that might help recover from the error@>@;
+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.
+
+@<Complete the error message,...@>=
+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 slow_print(text(warning_info))
+ else print_variable_name(warning_info);
+ cur_sym:=frozen_end_def;
+ end;
+loop_defining: begin print("the text of a "); slow_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 \MF\ began its special |scanner_status|, if that text has been saved.
+
+@<Declare the procedure called |runaway|@>=
+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.
+
+@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}
+ 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
+@<Input from external file; |goto restart| if no input found,
+ or |return| if a non-symbolic token is found@>
+else @<Input from token list; |goto restart| if end of list or
+ if a parameter needs to be expanded,
+ or |return| if a non-symbolic token is found@>;
+@<Finish getting the symbolic token in |cur_sym|;
+ |goto restart| if it is illegal@>;
+exit:end;
+
+@ When a symbolic token is declared to be `\&{outer}', its command code
+is increased by |outer_tag|.
+@^inner loop@>
+
+@<Finish getting the symbolic token in |cur_sym|...@>=
+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@>
+
+@<Input from external file;...@>=
+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<period_class then {|class=digit_class|}
+ begin n:=0; goto start_decimal_token;
+ end;
+@:. }{\..\ token@>
+ end;
+space_class: goto switch;
+percent_class: begin @<Move to next line of file,
+ or |goto restart| if there is no next line@>;
+ check_interrupt;
+ goto switch;
+ end;
+string_class: @<Get a string token and |return|@>;
+isolated_classes: begin k:=loc-1; goto found;
+ end;
+invalid_class: @<Decry the invalid character and |goto restart|@>;
+othercases do_nothing {letters, etc.}
+endcases;@/
+k:=loc-1;
+while char_class[buffer[loc]]=class do incr(loc);
+goto found;
+start_numeric_token:@<Get the integer part |n| of a numeric token;
+ set |f:=0| and |goto fin_numeric_token| if there is no decimal point@>;
+start_decimal_token:@<Get the fraction part |f| of a numeric token@>;
+fin_numeric_token:@<Pack the numeric and fraction parts of a numeric token
+ and |return|@>;
+found: cur_sym:=id_lookup(k,loc-k);
+end
+
+@ We go to |restart| instead of to |switch|, because we might enter
+|token_state| after the error has been dealt with
+(cf.\ |clear_for_error_prompt|).
+
+@<Decry the invalid...@>=
+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
+
+@ @<Get a string token and |return|@>=
+begin if buffer[loc]="""" then cur_mod:=""
+else begin k:=loc; buffer[limit+1]:="""";
+ repeat incr(loc);
+ until buffer[loc]="""";
+ if loc>limit then @<Decry the missing string delimiter and |goto restart|@>;
+ if (loc=k+1) and (length(buffer[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.
+
+@<Decry the missing string delimiter and |goto restart|@>=
+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
+
+@ @<Get the integer part |n| of a numeric token...@>=
+n:=c-"0";
+while char_class[buffer[loc]]=digit_class do
+ begin if n<4096 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)
+
+@ @<Get the fraction part |f| of a numeric token@>=
+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
+
+@ @<Pack the numeric and fraction parts of a numeric token and |return|@>=
+if n<4096 then cur_mod:=n*unity+f
+else begin print_err("Enormous number has been reduced");
+@.Enormous number...@>
+ help2("I can't handle numbers bigger than about 4095.99998;")@/
+ ("so I've changed your constant to that maximum amount.");@/
+ deletions_allowed:=false; error; deletions_allowed:=true;
+ cur_mod:=@'1777777777;
+ end;
+cur_cmd:=numeric_token; return
+
+@ Let's consider now what happens when |get_next| is looking at a token list.
+@^inner loop@>
+
+@<Input from token list;...@>=
+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
+ @<Insert a suffix or text parameter and |goto restart|@>
+ 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
+ @<Get a stored numeric or string or capsule token and |return|@>
+else begin {we are done with this token list}
+ end_token_list; goto restart; {resume previous level}
+ end
+
+@ @<Insert a suffix or text parameter...@>=
+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
+
+@ @<Get a stored numeric or string or capsule token...@>=
+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.
+
+@<Move to next line of file, or |goto restart|...@>=
+if name>2 then @<Read next line of file into |buffer|, or
+ |goto restart| if the file has ended@>
+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 selector<log_only then open_log_file;
+ if interaction>nonstop_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.
+
+@<Glob...@>=
+@!force_eof:boolean; {should the next \&{input} be aborted early?}
+
+@ @<Read next line of file into |buffer|, or
+ |goto restart| if the file has ended@>=
+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 print_char(")"); decr(open_parens);
+ update_terminal; {show user that file has been read}
+ force_eof:=false;
+ end_file_reading; {resume previous level}
+ if check_outer_validity then goto restart@+else goto restart;
+ end;
+buffer[limit]:="%"; first:=limit+1; loc:=start; {ready to read}
+end
+
+@ 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 `\.{=>}'.
+\MF\ 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<limit then for k:=start to limit-1 do print(buffer[k]);
+ first:=limit; prompt_input("=>"); {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;
+
+@* \[34] Scanning macro definitions.
+\MF\ 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}}
+
+@<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd...@>=
+macro_def:if m<=var_def then
+ if m=start_def then print("def")
+ else if m<start_def then print("enddef")
+ else print("vardef")
+ else if m=secondary_primary_macro then print("primarydef")
+ else if m=tertiary_secondary_macro then print("secondarydef")
+ else print("tertiarydef");
+iteration: if m<=start_forever then
+ if m=start_forever then print("forever")@+else print("endfor")
+ else if m=expr_base then print("for")@+else print("forsuffixes");
+
+@ Different macro-absorbing operations have different syntaxes, but they
+also have a lot in common. There is a list of special symbols that are to
+be replaced by parameter tokens; there is a special command code that
+ends the definition; the quotation conventions are identical. Therefore
+it makes sense to have most of the work done by a single subroutine. That
+subroutine is called |scan_toks|.
+
+The first parameter to |scan_toks| is the command code that will
+terminate scanning (either |macro_def| or |iteration|).
+
+The second parameter, |subst_list|, points to a (possibly empty) list
+of two-word nodes whose |info| and |value| fields specify symbol tokens
+before and after replacement. The list will be returned to free storage
+by |scan_toks|.
+
+The third parameter is simply appended to the token list that is built.
+And the final parameter tells how many of the special operations
+\.{\#\AT!}, \.{\AT!}, and \.{\AT!\#} are to be replaced by suffix parameters.
+When such parameters are present, they are called \.{(SUFFIX0)},
+\.{(SUFFIX1)}, and \.{(SUFFIX2)}.
+
+@p function scan_toks(@!terminator:command_code;
+ @!subst_list,@!tail_end:pointer;@!suffix_count:small_number):pointer;
+label done,found;
+var @!p:pointer; {tail of the token list being built}
+@!q:pointer; {temporary for link management}
+@!balance:integer; {left delimiters minus right delimiters}
+begin p:=hold_head; balance:=1; link(hold_head):=null;
+loop@+ begin get_next;
+ if cur_sym>0 then
+ begin @<Substitute for |cur_sym|, if it's on the |subst_list|@>;
+ if cur_cmd=terminator then
+ @<Adjust the balance; |goto done| if it's zero@>
+ else if cur_cmd=macro_special then
+ @<Handle quoted symbols, \.{\#\AT!}, \.{\AT!}, or \.{\AT!\#}@>;
+ 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;
+
+@ @<Substitute for |cur_sym|...@>=
+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
+
+@ @<Adjust the balance; |goto done| if it's zero@>=
+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!\#}}
+
+@<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd...@>=
+macro_special: case m of
+ macro_prefix: print("#@@");
+ macro_at: print_char("@@");
+ macro_suffix: print("@@#");
+ othercases print("quote")
+ endcases;
+
+@ @<Handle quoted...@>=
+begin if cur_mod=quote then get_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 \MF's tables won't get fouled up.
+
+@p procedure get_symbol; {sets |cur_sym| to a safe symbol}
+label restart;
+begin restart: get_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_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}.
+
+@<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd...@>=
+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<secondary_macro then print("primary")
+ else if m=secondary_macro then print("secondary")
+ else print("tertiary");
+
+@ Let's turn next to the more complex processing associated with \&{def}
+and \&{vardef}. When the following procedure is called, |cur_mod|
+should be either |start_def| or |var_def|.
+
+@p @t\4@>@<Declare the procedure called |check_delimiter|@>@;
+@t\4@>@<Declare the function called |scan_declared_variable|@>@;
+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;@/
+@<Scan the token or variable to be defined;
+ set |n|, |scanner_status|, and |warning_info|@>;
+k:=n;
+if cur_cmd=left_delimiter then
+ @<Absorb delimited parameters, putting them into lists |q| and |r|@>;
+if cur_cmd=param_type then
+ @<Absorb undelimited parameters, putting them into list |r|@>;
+check_equals;
+p:=get_avail; info(p):=c; link(q):=p;
+@<Attach the replacement text to the tail of node |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}'.
+
+@<Attach the replacement text to the tail of node |p|@>=
+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))
+
+@ @<Glob...@>=
+@!bg_loc,@!eg_loc:1..hash_end;
+ {hash addresses of `\.{begingroup}' and `\.{endgroup}'}
+
+@ @<Scan the token or variable to be defined;...@>=
+if m=start_def then
+ begin get_clear_symbol; warning_info:=cur_sym; get_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 @<Change to `\.{a bad variable}'@>;
+ scanner_status:=var_defining; n:=2;
+ if cur_cmd=macro_special then if cur_mod=macro_suffix then {\.{\AT!\#}}
+ begin n:=3; get_next;
+ end;
+ type(warning_info):=unsuffixed_macro-2+n; value(warning_info):=q;
+ end {|suffixed_macro=unsuffixed_macro+1|}
+
+@ @<Change to `\.{a bad variable}'@>=
+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
+
+@ @<Initialize table entries...@>=
+name_type(bad_vardef):=root; link(bad_vardef):=frozen_bad_vardef;
+equiv(frozen_bad_vardef):=bad_vardef; eq_type(frozen_bad_vardef):=tag_token;
+
+@ @<Absorb delimited parameters, putting them into lists |q| and |r|@>=
+repeat l_delim:=cur_sym; r_delim:=cur_mod; get_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;
+@<Absorb parameter tokens for type |base|@>;
+check_delimiter(l_delim,r_delim);
+get_next;
+until cur_cmd<>left_delimiter
+
+@ @<Absorb parameter tokens for type |base|@>=
+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);
+@:METAFONT capacity exceeded parameter stack size}{\quad parameter stack size@>
+incr(k); link(p):=r; r:=p; get_next;
+until cur_cmd<>comma
+
+@ @<Absorb undelimited parameters, putting them into list |r|@>=
+begin p:=get_node(token_node_size);
+if cur_mod<expr_base then
+ begin c:=cur_mod; value(p):=expr_base+k;
+ end
+else begin value(p):=cur_mod+k;
+ if cur_mod=expr_base then c:=expr_macro
+ else if cur_mod=suffix_base then c:=suffix_macro
+ else c:=text_macro;
+ end;
+if k=param_size then overflow("parameter stack size",param_size);
+incr(k); get_symbol; info(p):=cur_sym; link(p):=r; r:=p; get_next;
+if c=expr_macro then if cur_cmd=of_token then
+ begin c:=of_macro; p:=get_node(token_node_size);
+ if k=param_size then overflow("parameter stack size",param_size);
+ value(p):=expr_base+k; get_symbol; info(p):=cur_sym;
+ link(p):=r; r:=p; get_next;
+ end;
+end
+
+@* \[35] Expanding the next token.
+Only a few command codes |<min_command| can possibly be returned by
+|get_next|; in increasing order, they are
+|if_test|, |fi_or_else|, |input|, |iteration|, |repeat_loop|,
+|exit_test|, |relax|, |scan_tokens|, |expand_after|, and |defined_macro|.
+
+\MF\ usually gets the next token of input by saying |get_x_next|. This is
+like |get_next| except that it keeps getting more tokens until
+finding |cur_cmd>=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@>@<Declare the procedure called |macro_call|@>@;@/
+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:@<Terminate the current conditional and skip to \&{fi}@>;
+input:@<Initiate or terminate input from a file@>;
+iteration:if cur_mod=end_for then
+ @<Scold the user for having an extra \&{endfor}@>
+ else begin_iteration; {this procedure is discussed in Part 37 below}
+repeat_loop: @<Repeat a loop@>;
+exit_test: @<Exit a loop if the proper time has come@>;
+relax: do_nothing;
+expand_after: @<Expand the token after the next token@>;
+scan_tokens: @<Put a string into the input buffer@>;
+defined_macro:macro_call(cur_mod,null,cur_sym);
+end; {there are no other cases}
+end;
+
+@ @<Scold the user...@>=
+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.
+
+@<Put each...@>=
+primitive("input",input,0);@/
+@!@:input_}{\&{input} primitive@>
+primitive("endinput",input,1);@/
+@!@:end_input_}{\&{endinput} primitive@>
+
+@ @<Cases of |print_cmd_mod|...@>=
+input: if m=0 then print("input")@+else print("endinput");
+
+@ @<Initiate or terminate input...@>=
+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.
+
+@<Repeat a loop@>=
+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
+
+@ @<Exit a loop if the proper time has come@>=
+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 @<Exit prematurely from an iteration@>
+else if cur_cmd<>semicolon then
+ begin missing_err(";");@/
+@.Missing `;'@>
+ help2("After `exitif <boolean expr>' 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|.
+
+@<Exit prematurely...@>=
+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 37 below}
+end
+
+@ @<Expand the token after the next token@>=
+begin get_next;
+p:=cur_tok; get_next;
+if cur_cmd<min_command then expand else back_input;
+back_list(p);
+end
+
+@ @<Put a string into the input buffer@>=
+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 @<Pretend we're reading a new one-line file@>;
+ end;
+end
+
+@ @<Pretend we're reading a new one-line file@>=
+begin begin_file_reading; name:=2;
+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);
+@:METAFONT capacity exceeded buffer size}{\quad buffer size@>
+ end;
+ max_buf_stack:=k+1;
+ end;
+j:=str_start[cur_exp]; limit:=k;
+while first<limit do
+ begin buffer[first]:=so(str_pool[j]); incr(j); incr(first);
+ end;
+buffer[limit]:="%"; first:=limit+1; loc:=start; flush_cur_exp(0);
+end
+
+@ Here finally is |get_x_next|.
+
+The expression scanning routines to be considered later
+communicate via the global quantities |cur_type| and |cur_exp|;
+we must be very careful to save and restore these quantities while
+macros are being expanded.
+@^inner loop@>
+
+@p procedure get_x_next;
+var @!save_exp:pointer; {a capsule to save |cur_type| and |cur_exp|}
+begin get_next;
+if cur_cmd<min_command then
+ begin save_exp:=stash_cur_exp;
+ repeat if cur_cmd=defined_macro then macro_call(cur_mod,null,cur_sym)
+ else expand;
+ get_next;
+ until 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 \MF's
+input stack, so that |get_next| will proceed to read it next.
+
+@<Declare the procedure called |macro_call|@>=
+@t\4@>@<Declare the procedure called |print_macro_name|@>@;
+@t\4@>@<Declare the procedure called |print_arg|@>@;
+@t\4@>@<Declare the procedure called |scan_text_arg|@>@;
+procedure macro_call(@!def_ref,@!arg_list,@!macro_name:pointer);
+ {invokes a user-defined sequence of commands}
+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 @<Determine the number |n| of arguments already supplied,
+ and set |tail| to the tail of |arg_list|@>;
+if internal[tracing_macros]>0 then
+ @<Show the text of the macro being expanded, and the existing arguments@>;
+@<Scan the remaining arguments, if any; set |r| to the first token
+ of the replacement text@>;
+@<Feed the arguments and replacement text to the scanner@>;
+end;
+
+@ @<Show the text of the macro...@>=
+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
+
+@ @<Declare the procedure called |print_macro_name|@>=
+procedure print_macro_name(@!a,@!n:pointer);
+var @!p,@!q:pointer; {they traverse the first part of |a|}
+begin if n<>null then slow_print(text(n))
+else begin p:=info(a);
+ if p=null then slow_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;
+
+@ @<Declare the procedure called |print_arg|@>=
+procedure print_arg(@!q:pointer;@!n:integer;@!b:pointer);
+begin if link(q)=void then print_nl("(EXPR")
+else if (b<text_base)and(b<>text_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;
+
+@ @<Determine the number |n| of arguments already supplied...@>=
+begin n:=1; tail:=arg_list;
+while link(tail)<>null do
+ begin incr(n); tail:=link(tail);
+ end;
+end
+
+@ @<Scan the remaining arguments, if any; set |r|...@>=
+cur_cmd:=comma+1; {anything |<>comma| will do}
+while info(r)>=expr_base do
+ begin @<Scan the delimited argument represented by |info(r)|@>;
+ 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 `"); slow_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 @<Scan undelimited argument(s)@>;
+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.)
+
+@<Scan the delimited argument represented by |info(r)|@>=
+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;
+@<Scan the argument represented by |info(r)|@>;
+if cur_cmd<>comma then @<Check that the proper right delimiter was present@>;
+found: @<Append the current expression to |arg_list|@>
+
+@ @<Check that the proper right delim...@>=
+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 have been scanned as
+a token list pointed to by |cur_exp|, in which case we will have
+|cur_type=token_list|.
+
+@<Append the current expression to |arg_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
+
+@ @<Scan the argument represented by |info(r)|@>=
+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.
+
+@<Declare the procedure called |scan_text_arg|@>=
+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_next;
+ if l_delim=0 then @<Adjust the balance for an undelimited argument;
+ |goto done| if done@>
+ else @<Adjust the balance for a delimited argument;
+ |goto done| if done@>;
+ link(p):=cur_tok; p:=link(p);
+ end;
+done: cur_exp:=link(hold_head); cur_type:=token_list;
+scanner_status:=normal;
+end;
+
+@ @<Adjust the balance for a delimited argument...@>=
+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
+
+@ @<Adjust the balance for an undelimited...@>=
+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
+
+@ @<Scan undelimited argument(s)@>=
+begin if info(r)<text_macro then
+ begin get_x_next;
+ 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:@<Scan an expression followed by `\&{of} $\langle$primary$\rangle$'@>;
+suffix_macro:@<Scan a suffix with optional delimiters@>;
+text_macro:scan_text_arg(0,0);
+end; {there are no other cases}
+back_input; @<Append the current expression to |arg_list|@>;
+end
+
+@ @<Scan an expression followed by `\&{of} $\langle$primary$\rangle$'@>=
+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
+
+@ @<Scan a suffix with optional delimiters@>=
+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.
+
+@<Feed the arguments and replacement text to the scanner@>=
+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);
+@:METAFONT 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);
+@:METAFONT capacity exceeded parameter stack size}{\quad parameter stack size@>
+ end;
+param_stack[param_ptr]:=p; incr(param_ptr);
+end;
+
+@* \[36] 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}}
+
+@<Glob...@>=
+@!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}
+
+@ @<Set init...@>=
+cond_ptr:=null; if_limit:=normal; cur_if:=0; if_line:=0;
+
+@ @<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd_mod|...@>=
+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 the current level of $\&{if}\ldots\&{fi}$
+nesting. After it has acted, |cur_mod| will indicate the token that
+was found.
+
+\MF'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:=line;
+loop@+ begin get_next;
+ if cur_cmd<=fi_or_else then
+ if cur_cmd<fi_or_else then incr(l)
+ else begin if l=0 then goto done;
+ if cur_mod=fi_code then decr(l);
+ end
+ else @<Decrease the string reference count,
+ if the current token is a string@>;
+ end;
+done: scanner_status:=normal;
+end;
+
+@ @<Decrease the string reference count...@>=
+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 \MF\ confused.
+
+@<Push the condition stack@>=
+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:=line; cur_if:=if_code;
+end
+
+@ @<Pop the condition stack@>=
+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, \MF\ 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 @<Push the condition stack@>;@+save_cond_ptr:=cond_ptr;
+reswitch: get_boolean; new_if_limit:=else_if_code;
+if internal[tracing_commands]>unity then
+ @<Display the boolean value of |cur_exp|@>;
+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;
+@<Skip to \&{elseif} or \&{else} or \&{fi}, then |goto done|@>;
+done: cur_if:=cur_mod; if_line:=line;
+if cur_mod=fi_code then @<Pop the condition stack@>
+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.
+
+@<Skip to \&{elseif}...@>=
+loop@+ begin pass_text;
+ if cond_ptr=save_cond_ptr then goto done
+ else if cur_mod=fi_code then @<Pop the condition stack@>;
+ end
+
+
+@ @<Display the boolean value...@>=
+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.
+
+@<Terminate the current conditional and skip to \&{fi}@>=
+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}}
+ @<Pop the condition stack@>;
+ end
+
+@* \[37] Iterations.
+To bring our treatment of |get_x_next| to a close, we need to consider what
+\MF\ 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)=p>void| means that |value(p)|,
+|step_size(p)|, and |final_value(p)| contain the data for an arithmetic
+progression.
+
+\yskip\noindent In the latter case, |p| points to a ``progression node''
+whose first word is not used. (No value could be stored there 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}
+
+@<Glob...@>=
+@!loop_ptr:pointer; {top of the loop-control-node stack}
+
+@ @<Set init...@>=
+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 \MF\ 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 author
+didn't write it until later. The reader may wish to come back to it.)
+
+@p procedure begin_iteration;
+label continue,done,found;
+var @!m:halfword; {|expr_base| (\&{for}) or |suffix_base| (\&{forsuffixes})}
+@!n:halfword; {hash address of the current symbol}
+@!p,@!q,@!s,@!pp:pointer; {link manipulation registers}
+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; goto found;
+ end;
+get_symbol; p:=get_node(token_node_size); info(p):=cur_sym; value(p):=m;@/
+get_x_next;
+if (cur_cmd<>equals)and(cur_cmd<>assignment) then
+ begin missing_err("=");@/
+@.Missing `='@>
+ help3("The next thing in this loop should have been `=' or `:='.")@/
+ ("But don't worry; I'll pretend that an equals sign")@/
+ ("was present, and I'll look for the values next.");@/
+ back_error;
+ end;
+@<Scan the values to be used in the loop@>;
+found:@<Check for the presence of a colon@>;
+@<Scan the loop text and put it on the loop control stack@>;
+resume_iteration;
+end;
+
+@ @<Check for the presence of a colon@>=
+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 \MF's scanner
+at the proper time to cause the loop to be repeated.
+
+(A user who tries some shenanigan like `\&{for} $\ldots$ \&{let} \&{endfor}'
+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.)
+
+@ @<Scan the loop text...@>=
+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
+
+@ @<Initialize table...@>=
+eq_type(frozen_repeat_loop):=repeat_loop+outer_tag;
+text(frozen_repeat_loop):=" ENDFOR";
+
+@ The loop text is inserted into \MF'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>void then {|p| points to a progression node}
+ begin cur_exp:=value(p);
+ if @<The arithmetic progression has ended@> 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<void 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 begin begin_token_list(info(loop_ptr),forever_text); return;
+ end;
+begin_token_list(info(loop_ptr),loop_text);
+stack_argument(q);
+if internal[tracing_commands]>unity then @<Trace the start of a loop@>;
+return;
+not_found:stop_iteration;
+exit:end;
+
+@ @<The arithmetic progression has ended@>=
+((step_size(p)>0)and(cur_exp>final_value(p)))or@|
+ ((step_size(p)<0)and(cur_exp<final_value(p)))
+
+@ @<Trace the start of a loop@>=
+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
+
+@ 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>void then free_node(p,progression_node_size)
+else if p<void 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;
+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|).
+
+@<Scan the values to be used in the loop@>=
+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
+ @<Prepare for step-until construction and |goto done|@>;
+ 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:
+
+@ @<Prepare for step-until construction and |goto 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_type(s):=pp; goto done;
+end
+
+@* \[38] 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@>
+
+\MF\ 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
+`\.{.mf}' unless otherwise specified; it is `\.{.log}' on the
+transcript file that records each run of \MF; it is `\.{.tfm}' on the font
+metric files that describe characters in the fonts \MF\ creates; it is
+`\.{.gf}' on the output files that specify generic font information; and it
+is `\.{.base}' on the base files written by \.{INIMF} to initialize \MF.
+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, \MF\ will look for it on a special system
+area; this special area is intended for commonly used input files.
+
+Simple uses of \MF\ 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 \MF\ 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 \MF. 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.
+
+@ \MF\ uses the same conventions that have proved to be satisfactory for
+\TeX. In order to isolate the system-dependent aspects of file names, the
+@^system dependencies@>
+system-independent parts of \MF\ 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 \MF\ 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|.
+
+@<Glob...@>=
+@!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 \.{""}}
+
+@ 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:
+
+@<Glob...@>=
+@!area_delimiter:pool_pointer; {the most recent `\.>' or `\.:', if any}
+@!ext_delimiter:pool_pointer; {the relevant `\..', if any}
+
+@ Input files that can't be found in the user's area may appear in a standard
+system area called |MF_area|.
+This system area name will, of course, vary from place to place.
+@^system dependencies@>
+
+@d MF_area=="MFinputs:"
+@.MFinputs@>
+
+@ Here now is the first of the system-dependent routines for file name scanning.
+@^system dependencies@>
+
+@p procedure begin_name;
+begin area_delimiter:=0; ext_delimiter:=0;
+end;
+
+@ And here's the second.
+@^system dependencies@>
+
+@p 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; ext_delimiter:=0;
+ end
+ else if (c=".")and(ext_delimiter=0) then ext_delimiter:=pool_ptr;
+ str_room(1); append_char(c); {contribute |c| to the current string}
+ more_name:=true;
+ end;
+end;
+
+@ The third.
+@^system dependencies@>
+
+@p procedure end_name;
+begin if str_ptr+3>max_str_ptr then
+ begin if str_ptr+3>max_strings then
+ overflow("number of strings",max_strings-init_str_ptr);
+@:METAFONT capacity exceeded number of strings}{\quad number of strings@>
+ max_str_ptr:=str_ptr+3;
+ end;
+if area_delimiter=0 then cur_area:=""
+else begin cur_area:=str_ptr; incr(str_ptr);
+ str_start[str_ptr]:=area_delimiter+1;
+ end;
+if ext_delimiter=0 then
+ begin cur_ext:=""; cur_name:=make_string;
+ end
+else begin cur_name:=str_ptr; incr(str_ptr);
+ str_start[str_ptr]:=ext_delimiter; cur_ext:=make_string;
+ end;
+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@>
+
+@<Basic printing...@>=
+procedure print_file_name(@!n,@!a,@!e:integer);
+begin slow_print(a); slow_print(n); slow_print(e);
+end;
+
+@ Another system-dependent routine is needed to convert three internal
+\MF\ 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
+
+@p 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_start[a+1]-1 do append_to_name(so(str_pool[j]));
+for j:=str_start[n] to str_start[n+1]-1 do append_to_name(so(str_pool[j]));
+for j:=str_start[e] to str_start[e+1]-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 base file names must be scanned
+before \MF's string mechanism has been initialized. We shall use the
+global variable |MF_base_default| to supply the text for default system areas
+and extensions related to base files.
+@^system dependencies@>
+
+@d base_default_length=18 {length of the |MF_base_default| string}
+@d base_area_length=8 {length of its area part}
+@d base_ext_length=5 {length of its `\.{.base}' part}
+@d base_extension=".base" {the extension, as a \.{WEB} constant}
+
+@<Glob...@>=
+@!MF_base_default:packed array[1..base_default_length] of char;
+
+@ @<Set init...@>=
+MF_base_default:='MFbases:plain.base';
+@.MFbases@>
+@.plain@>
+@^system dependencies@>
+
+@ @<Check the ``constant'' values for consistency@>=
+if base_default_length>file_name_size then bad:=41;
+
+@ Here is the messy routine that was just mentioned. It sets |name_of_file|
+from the first |n| characters of |MF_base_default|, followed by
+|buffer[a..b]|, followed by the last |base_ext_length| characters of
+|MF_base_default|.
+
+We dare not give error messages here, since \MF\ 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 |MF_base_default|}
+begin if n+b-a+1+base_ext_length>file_name_size then
+ b:=a+file_name_size-n-1-base_ext_length;
+k:=0;
+for j:=1 to n do append_to_name(xord[MF_base_default[j]]);
+for j:=a to b do append_to_name(buffer[j]);
+for j:=base_default_length-base_ext_length+1 to base_default_length do
+ append_to_name(xord[MF_base_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'' \MF\ is trying to get going, just after
+the preliminary initialization, or when the user is substituting another
+base file by typing `\.\&' after the initial `\.{**}' prompt. The buffer
+contains the first line of input in |buffer[loc..(last-1)]|, where
+|loc<last| and |buffer[loc]<>" "|.
+
+@<Declare the function called |open_base_file|@>=
+function open_base_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(base_file) then goto found;
+ pack_buffered_name(base_area_length,loc,j-1);
+ {now try the system base file area}
+ if w_open_in(base_file) then goto found;
+ wake_up_terminal;
+ wterm_ln('Sorry, I can''t find that base;',' 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(base_default_length-base_ext_length,1,0);
+if not w_open_in(base_file) then
+ begin wake_up_terminal;
+ wterm_ln('I can''t find the PLAIN base file!');
+@.I can't find PLAIN...@>
+@.plain@>
+ open_base_file:=false; return;
+ end;
+found:loc:=j; open_base_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 \MF\ 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 dare not use `|str_room|'.
+
+@p function make_name_string:str_number;
+var @!k:1..file_name_size; {index into |name_of_file|}
+begin if (pool_ptr+name_length>pool_size)or(str_ptr=max_strings) then
+ make_name_string:="?"
+else begin 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 \MF\ 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 \MF. The manual 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;
+
+@ The global variable |job_name| contains the file name that was first
+\&{input} by the user. This name is extended by `\.{.log}' and `\.{.gf}' and
+`\.{.base}' and `\.{.tfm}' in the names of \MF's output files.
+
+@<Glob...@>=
+@!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.
+
+@<Initialize the output...@>=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"|, |".gf"|,
+ |".tfm"|, or |base_extension|}
+begin cur_area:=""; cur_ext:=s;
+cur_name:=job_name; pack_cur_name;
+end;
+
+@ Actually the main output file extension is usually something like
+|".300gf"| instead of just |".gf"|; the additional number indicates the
+resolution in pixels per inch, based on the setting of |hppp| when
+the file is opened.
+
+@<Glob...@>=
+@!gf_ext:str_number; {default extension for the output file}
+
+@ If some trouble arises when \MF\ 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=".mf" then show_context;
+print_nl("Please type another "); print(s);
+@.Please type...@>
+if interaction<scroll_mode then
+ fatal_error("*** (job aborted, file error in nonstop mode)");
+@.job aborted, file error...@>
+clear_terminal; prompt_input(": "); @<Scan file name in the buffer@>;
+if cur_ext="" then cur_ext:=e;
+pack_cur_name;
+end;
+
+@ @<Scan file name in the buffer@>=
+begin begin_name; k:=first;
+while (buffer[k]=" ")and(k<last) do incr(k);
+loop@+ begin if k=last then goto done;
+ if not more_name(buffer[k]) then goto done;
+ incr(k);
+ end;
+done:end_name;
+end
+
+@ The |open_log_file| routine is used to open the transcript file and to help
+it catch up to what has previously been printed on the terminal.
+
+@p procedure open_log_file;
+var @!old_setting:0..max_selector; {previous |selector| setting}
+@!k:0..buf_size; {index into |months| and |buffer|}
+@!l:0..buf_size; {end of first input line}
+@!m:integer; {the current month}
+@!months:packed array [1..36] of char; {abbreviations of month names}
+begin old_setting:=selector;
+if job_name=0 then job_name:="mfput";
+@.mfput@>
+pack_job_name(".log");
+while not a_open_out(log_file) do @<Try to get a different log file name@>;
+log_name:=a_make_name_string(log_file);
+selector:=log_only; log_opened:=true;
+@<Print the banner line, including the date and time@>;
+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 \MF\ 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.
+
+@<Try to get a different log file name@>=
+begin selector:=term_only;
+prompt_file_name("transcript file name",".log");
+end
+
+@ @<Print the banner...@>=
+begin wlog(banner);
+slow_print(base_ident); print(" ");
+print_int(sys_day); print_char(" ");
+months:='JANFEBMARAPRMAYJUNJULAUGSEPOCTNOVDEC';
+for k:=3*sys_month-2 to 3*sys_month do wlog(months[k]);
+print_char(" "); print_int(sys_year); print_char(" ");
+print_dd(sys_time div 60); print_char(":"); print_dd(sys_time mod 60);
+end
+
+@ Here's an example of how these file-name-parsing routines work in practice.
+We shall use the macro |set_output_file_name| when it is time to
+crank up the output file.
+
+@d set_output_file_name==
+ begin if job_name=0 then open_log_file;
+ pack_job_name(gf_ext);
+ while not b_open_out(gf_file) do
+ prompt_file_name("file name for output",gf_ext);
+ output_file_name:=b_make_name_string(gf_file);
+ end
+
+@<Glob...@>=
+@!gf_file: byte_file; {the generic font output goes here}
+@!output_file_name: str_number; {full name of the output file}
+
+@ @<Initialize the output...@>=output_file_name:=0;
+
+@ Let's turn now to the procedure that is used to initiate file reading
+when an `\.{input}' command is being processed.
+Beware: For historic reasons, this code foolishly conserves a tiny bit
+of string pool space; but that can confuse the interactive `\.E' option.
+@^system dependencies@>
+
+@p procedure start_input; {\MF\ will \.{input} something}
+label done;
+begin @<Put the desired file name in |(cur_name,cur_ext,cur_area)|@>;
+if cur_ext="" then cur_ext:=".mf";
+pack_cur_name;
+loop@+ begin begin_file_reading; {set up |cur_file| and new level of input}
+ if a_open_in(cur_file) then goto done;
+ if cur_area="" then
+ begin pack_file_name(cur_name,MF_area,cur_ext);
+ if a_open_in(cur_file) then goto done;
+ end;
+ end_file_reading; {remove the level that didn't work}
+ prompt_file_name("input file name",".mf");
+ end;
+done: name:=a_make_name_string(cur_file); str_ref[cur_name]:=max_str_ref;
+if job_name=0 then
+ begin job_name:=cur_name; 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); slow_print(name); update_terminal;
+if name=str_ptr-1 then {conserve string pool space (but see note above)}
+ begin flush_string(name); name:=cur_name;
+ end;
+@<Read the first line of the new file@>;
+end;
+
+@ 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@>
+
+@<Read the first line...@>=
+begin line:=1;
+if input_ln(cur_file,false) then do_nothing;
+firm_up_the_line;
+buffer[limit]:="%"; first:=limit+1; loc:=start;
+end
+
+@ @<Put the desired file name in |(cur_name,cur_ext,cur_area)|@>=
+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
+
+@* \[39] Introduction to the parsing routines.
+We come now to the central nervous system that sparks many of \MF's activities.
+By evaluating expressions, from their primary constituents to ever larger
+subexpressions, \MF\ builds the structures that ultimately define 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.
+
+@<Glob...@>=
+@!cur_type:small_number; {the type of the expression just found}
+@!cur_exp:integer; {the value of the expression just found}
+
+@ @<Set init...@>=
+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
+a ring of equivalent booleans whose value has not yet been defined.
+
+\smallskip\hang
+|cur_type=string_type| means that |cur_exp| is a string number (i.e., an
+integer in the range |0<=cur_exp<str_ptr|). That string's reference count
+includes this particular reference.
+
+\smallskip\hang
+|cur_type=unknown_string| means that |cur_exp| points to a capsule
+node that is in
+a ring of equivalent strings whose value has not yet been defined.
+
+\smallskip\hang
+|cur_type=pen_type| means that |cur_exp| points to a pen header node. This
+node contains a reference count, which takes account of this particular
+reference.
+
+\smallskip\hang
+|cur_type=unknown_pen| means that |cur_exp| points to a capsule
+node that is in
+a ring of equivalent pens whose value has not yet been defined.
+
+\smallskip\hang
+|cur_type=future_pen| means that |cur_exp| points to a knot list that
+should eventually be made into a pen. Nobody else points to this particular
+knot list. The |future_pen| option occurs only as an output of |scan_primary|
+and |scan_secondary|, not as an output of |scan_tertiary| or |scan_expression|.
+
+\smallskip\hang
+|cur_type=path_type| means that |cur_exp| points to the first node of
+a path; nobody else points to this particular path. The control points of
+the path will have been chosen.
+
+\smallskip\hang
+|cur_type=unknown_path| means that |cur_exp| points to a capsule
+node that is in
+a ring of equivalent paths whose value has not yet been defined.
+
+\smallskip\hang
+|cur_type=picture_type| means that |cur_exp| points to an edges header node.
+Nobody else points to this particular set of edges.
+
+\smallskip\hang
+|cur_type=unknown_picture| means that |cur_exp| points to a capsule
+node that is in
+a ring of equivalent pictures whose value has not yet been defined.
+
+\smallskip\hang
+|cur_type=transform_type| means that |cur_exp| points to a |transform_type|
+capsule node. The |value| part of this capsule
+points to a transform node that contains six numeric values,
+each of which is |independent|, |dependent|, |proto_dependent|, or |known|.
+
+\smallskip\hang
+|cur_type=pair_type| means that |cur_exp| points to a capsule
+node whose type is |pair_type|. The |value| part of this capsule
+points to a pair node that contains two numeric values,
+each of which is |independent|, |dependent|, |proto_dependent|, or |known|.
+
+\smallskip\hang
+|cur_type=known| means that |cur_exp| is a |scaled| value.
+
+\smallskip\hang
+|cur_type=dependent| means that |cur_exp| points to a capsule node whose type
+is |dependent|. The |dep_list| field in this capsule points to the associated
+dependency list.
+
+\smallskip\hang
+|cur_type=proto_dependent| means that |cur_exp| points to a |proto_dependent|
+capsule node. The |dep_list| field in this capsule
+points to the associated dependency list.
+
+\smallskip\hang
+|cur_type=independent| means that |cur_exp| points to a capsule node
+whose type is |independent|. This somewhat unusual case can arise, for
+example, in the expression
+`$x+\&{begingroup}\penalty0\,\&{string}\,x; 0\,\&{endgroup}$'.
+
+\smallskip\hang
+|cur_type=token_list| means that |cur_exp| points to a linked list of
+tokens.
+
+\smallskip\noindent
+The possible settings of |cur_type| have been listed here in increasing
+numerical order. Notice that |cur_type| will never be |numeric_type| or
+|suffixed_macro| or |unsuffixed_macro|, although variables of those types
+are allowed. Conversely, \MF\ has no variables of type |vacuous| or
+|token_list|.
+
+@ Capsules are two-word nodes that have a similar meaning
+to |cur_type| and |cur_exp|. Such nodes have |name_type=capsule|,
+and their |type| field is one of the possibilities for |cur_type| listed above.
+Also |link<=void| in capsules that aren't part of a token list.
+
+The |value| field of a capsule is, in most cases, the value that
+corresponds to its |type|, as |cur_exp| corresponds to |cur_type|.
+However, when |cur_exp| would point to a capsule,
+no extra layer of indirection is present; the |value|
+field is what would have been called |value(cur_exp)| if it had not been
+encapsulated. Furthermore, if the type is |dependent| or
+|proto_dependent|, the |value| field of a capsule is replaced by
+|dep_list| and |prev_dep| fields, since dependency lists in capsules are
+always part of the general |dep_list| structure.
+
+The |get_x_next| routine is careful not to change the values of |cur_type|
+and |cur_exp| when it gets an expanded token. However, |get_x_next| might
+call a macro, which might parse an expression, which might execute lots of
+commands in a group; hence it's possible that |cur_type| might change
+from, say, |unknown_boolean| to |boolean_type|, or from |dependent| to
+|known| or |independent|, during the time |get_x_next| is called. The
+programs below are careful to stash sensitive intermediate results in
+capsules, so that \MF's generality doesn't cause trouble.
+
+Here's a procedure that illustrates these conventions. It takes
+the contents of $(|cur_type|\kern-.3pt,|cur_exp|\kern-.3pt)$
+and stashes them away in a
+capsule. It is not used when |cur_type=token_list|.
+After the operation, |cur_type=vacuous|; hence there is no need to
+copy path lists or to update reference counts, etc.
+
+The special link |void| is put on the capsule returned by
+|stash_cur_exp|, because this procedure is used to store macro parameters
+that must be easily distinguishable from token lists.
+
+@<Declare the stashing/unstashing routines@>=
+function stash_cur_exp:pointer;
+var @!p:pointer; {the capsule that will be returned}
+begin case cur_type of
+unknown_types,transform_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 \MF\ 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.
+
+@<Declare the stashing/unstashing...@>=
+procedure unstash_cur_exp(@!p:pointer);
+begin cur_type:=type(p);
+case cur_type of
+unknown_types,transform_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.
+@.linearform@>
+
+@<Declare subroutines for printing expressions@>=
+@t\4@>@<Declare the procedure called |print_dp|@>@;
+@t\4@>@<Declare the stashing/unstashing routines@>@;
+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<dependent then v:=value(p)@+else if t<independent then v:=dep_list(p);
+@<Print an abbreviated value of |v| with format depending on |t|@>;
+if restore_cur_exp then unstash_cur_exp(p);
+end;
+
+@ @<Print an abbreviated value of |v| with format depending on |t|@>=
+case t of
+vacuous:print("vacuous");
+boolean_type:if v=true_code then print("true")@+else print("false");
+unknown_types,numeric_type:@<Display a variable
+ that's been declared but not defined@>;
+string_type:begin print_char(""""); slow_print(v); print_char("""");
+ end;
+pen_type,future_pen,path_type,picture_type:@<Display a complex type@>;
+transform_type,pair_type:if v=null then print_type(t)
+ else @<Display a big node@>;
+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
+
+@ @<Display a big node@>=
+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}.
+
+@<Display a complex type@>=
+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);
+ future_pen:print_path(v," (future pen)",false);
+ path_type:print_path(v,"",false);
+ picture_type:begin cur_edges:=v; print_edges("",false,0,0);
+ end;
+ end; {there are no other cases}
+ end
+
+@ @<Declare the procedure called |print_dp|@>=
+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");
+@.linearform@>
+end;
+
+@ The displayed name of a variable in a ring will not be a capsule unless
+the ring consists entirely of capsules.
+
+@<Display a variable that's been declared but not defined@>=
+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}
+
+@<Declare subroutines for printing expressions@>=
+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.
+
+@<Declare the procedure called |flush_cur_exp|@>=
+procedure flush_cur_exp(@!v:scaled);
+begin case cur_type of
+unknown_types,transform_type,pair_type,@|dependent,proto_dependent,independent:
+ begin recycle_value(cur_exp); free_node(cur_exp,value_node_size);
+ end;
+pen_type: delete_pen_ref(cur_exp);
+string_type:delete_str_ref(cur_exp);
+future_pen,path_type: toss_knot_list(cur_exp);
+picture_type:toss_edges(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.
+
+@<Declare the recycling subroutines@>=
+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 then v:=value(p);
+case t of
+undefined,vacuous,boolean_type,known,numeric_type:do_nothing;
+unknown_types:ring_delete(p);
+string_type:delete_str_ref(v);
+pen_type:delete_pen_ref(v);
+path_type,future_pen:toss_knot_list(v);
+picture_type:toss_edges(v);
+pair_type,transform_type:@<Recycle a big node@>;
+dependent,proto_dependent:@<Recycle a dependency list@>;
+independent:@<Recycle an independent variable@>;
+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;
+
+@ @<Recycle a big node@>=
+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
+
+@ @<Recycle a dependency list@>=
+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 Andr\'es@>
+
+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.
+
+@<Recycle an independent variable@>=
+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
+ @<Record a new maximum coefficient of type |t|@>
+ 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
+ @<Choose a dependent variable to take the place of the disappearing
+ independent variable, and change all remaining dependencies
+ accordingly@>;
+end
+
+@ The code for independency removal makes use of three two-word arrays.
+
+@<Glob...@>=
+@!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|}
+
+@ @<Record a new maximum coefficient...@>=
+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
+
+@ @<Choose a dependent...@>=
+begin if (max_c[dependent] div @'10000 >=
+ max_c[proto_dependent]) then
+ t:=dependent
+else t:=proto_dependent;
+@<Determine the dependency list |s| to substitute for the independent
+ variable~|p|@>;
+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 @<Substitute new dependencies in place of |p|@>
+else @<Substitute new proto-dependencies in place of |p|@>;
+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.
+
+@<Determine the dep...@>=
+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 @<Show the transformed dependency@>
+
+@ Now $(-v)$ times the formerly independent variable~|p| is being replaced
+by the dependency list~|s|.
+
+@<Show the transformed...@>=
+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.
+
+@<Substitute new dependencies...@>=
+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
+
+@ @<Substitute new proto...@>=
+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.)
+
+@<Declare the procedure called |flush_cur_exp|@>=
+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 called |var_flag| is set to a special command code
+just before \MF\ 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.
+
+
+@<Glob...@>=
+@!var_flag:0..max_command_code; {command that wants a variable}
+
+@ @<Set init...@>=
+var_flag:=0;
+
+@* \[40] 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 signalled.
+
+@<Declare the basic parsing subroutines@>=
+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 |var_flag|}
+@!l_delim,@!r_delim:pointer; {hash addresses of a delimiter pair}
+@<Other local variables for |scan_primary|@>@;
+begin my_var_flag:=var_flag; var_flag:=0;
+restart:check_arith;
+@<Supply diagnostic information, if requested@>;
+case cur_cmd of
+left_delimiter:@<Scan a delimited primary@>;
+begin_group:@<Scan a grouped primary@>;
+string_token:@<Scan a string constant@>;
+numeric_token:@<Scan a primary that starts with a numeric token@>;
+nullary:@<Scan a nullary operation@>;
+unary,type_name,cycle,plus_or_minus:@<Scan a unary operation@>;
+primary_binary:@<Scan a binary operation with `\&{of}' between its operands@>;
+str_op:@<Convert a suffix to a string@>;
+internal_quantity:@<Scan an internal numeric quantity@>;
+capsule_token:make_exp_copy(cur_mod);
+tag_token:@<Scan a variable primary;
+ |goto restart| if it turns out to be a macro@>;
+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 @<Scan a mediation construction@>;
+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;
+
+@ @<Supply diagnostic information, if requested@>=
+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
+
+@ @<Scan a delimited primary@>=
+begin l_delim:=cur_sym; r_delim:=cur_mod; get_x_next; scan_expression;
+if (cur_cmd=comma) and (cur_type>=known) then
+ @<Scan the second of a pair of numerics@>
+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
+ @<Stash an independent |cur_exp| into a big node@>
+ 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.
+
+@ @<Stash an independent |cur_exp|...@>=
+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
+
+@ @<Scan the second of a pair of numerics@>=
+begin p:=get_node(value_node_size); type(p):=pair_type; name_type(p):=capsule;
+init_big_node(p); q:=value(p); stash_in(x_part_loc(q));@/
+get_x_next; scan_expression;
+if cur_type<known then
+ begin exp_err("Nonnumeric ypart has been replaced by 0");
+@.Nonnumeric...replaced by 0@>
+ help4("I thought you were giving me a pair `(x,y)'; but")@/
+ ("after finding a nice xpart `x' I found a ypart `y'")@/
+ ("that isn't of numeric type. So I've changed y to zero.")@/
+ ("(The y that I didn't like appears above the error message.)");
+ put_get_flush_error(0);
+ end;
+stash_in(y_part_loc(q));
+check_delimiter(l_delim,r_delim);
+cur_type:=pair_type; cur_exp:=p;
+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.
+
+@<Other local variables for |scan_primary|@>=
+@!group_line:integer; {where a group began}
+
+@ @<Scan a grouped primary@>=
+begin group_line:=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
+
+@ @<Scan a string constant@>=
+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.
+
+@<Scan a nullary operation@>=do_nullary(cur_mod)
+
+@ @<Scan a unary operation@>=
+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.
+
+@<Other local variables for |scan_primary|@>=
+@!num,@!denom:scaled; {for primaries that are fractions, like `1/2'}
+
+@ @<Scan a primary that starts with a numeric token@>=
+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 @<Protest division by zero@>
+ else cur_exp:=make_scaled(num,denom);
+ check_arith; get_x_next;
+ end;
+if cur_cmd>=min_primary_command then
+ if cur_cmd<numeric_token then {in particular, |cur_cmd<>plus_or_minus|}
+ begin p:=stash_cur_exp; scan_primary;
+ if (abs(num)>=abs(denom))or(cur_type<pair_type) then do_binary(p,times)
+ else begin frac_mult(num,denom);
+ free_node(p,value_node_size);
+ end;
+ end;
+goto done;
+end
+
+@ @<Protest division...@>=
+begin print_err("Division by zero");
+@.Division by zero@>
+help1("I'll pretend that you meant to divide by 1."); error;
+end
+
+@ @<Scan a binary operation with `\&{of}' between its operands@>=
+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
+
+@ @<Convert a suffix to a string@>=
+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.)
+
+@<Scan an internal...@>=
+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<tt<unsuffixed_macro|, the precise value of |tt| isn't critical.
+
+@ @<Other local variables for |scan_primary|@>=
+@!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}
+
+@ @<Scan a variable primary...@>=
+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 @<Find the approximate type |tt| and corresponding~|q|@>;
+ if tt>=unsuffixed_macro then
+ @<Either begin an unsuffixed macro call or
+ prepare for a suffixed one@>;
+ end;
+ get_x_next; tail:=t;
+ if cur_cmd=left_bracket then
+ @<Scan for a subscript; replace |cur_cmd| by |numeric_token| if found@>;
+ if cur_cmd>max_suffix_token then goto done1;
+ if cur_cmd<min_suffix_token then goto done1;
+ end; {now |cur_cmd| is |internal_quantity|, |tag_token|, or |numeric_token|}
+done1:@<Handle unusual cases that masquerade as variables, and |goto restart|
+ or |goto done| if appropriate;
+ otherwise make a copy of the variable and |goto done|@>;
+end
+
+@ @<Either begin an unsuffixed macro call or...@>=
+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 @<Set up unsuffixed macro call and |goto restart|@>;
+end
+
+@ @<Scan for a subscript; replace |cur_cmd| by |numeric_token| if found@>=
+begin get_x_next; scan_expression;
+if cur_cmd<>right_bracket then
+ @<Put the left bracket and the expression back to be rescanned@>
+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.
+
+@<Put the left bracket and the expression back to be rescanned@>=
+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.
+
+@<Find the approximate type |tt| and corresponding~|q|@>=
+@^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.
+
+@<Handle unusual cases that masquerade as variables...@>=
+if post_head<>null then @<Set up suffixed macro call and |goto restart|@>;
+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.
+
+@<Set up unsuffixed macro call and |goto restart|@>=
+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.
+
+@<Set up suffixed macro call and |goto restart|@>=
+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.
+
+@<Declare the procedure called |make_exp_copy|@>=
+@t\4@>@<Declare subroutines needed by |make_exp_copy|@>@;
+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;
+pen_type:begin cur_exp:=value(p); add_pen_ref(cur_exp);
+ end;
+picture_type:cur_exp:=copy_edges(value(p));
+path_type,future_pen:cur_exp:=copy_path(value(p));
+transform_type,pair_type:@<Copy the big node |p|@>;
+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,dep_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|.
+
+@<Declare subroutines needed by |make_exp_copy|@>=
+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} or \&{transform} variable; we must copy several fields,
+each of which can be |independent|, |dependent|, |proto_dependent|,
+or |known|.
+
+@<Copy the big node |p|@>=
+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.
+
+@<Declare subroutines needed by |make_exp_copy|@>=
+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,dep_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.
+
+@<Scan a mediation...@>=
+begin p:=stash_cur_exp; get_x_next; scan_expression;
+if cur_cmd<>comma then
+ begin @<Put the left bracket and the expression back...@>;
+ 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.
+
+@<Declare the basic parsing subroutines@>=
+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
+ @<Scan a bracketed subscript and set |cur_cmd:=numeric_token|@>;
+ 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;
+
+@ @<Scan a bracketed subscript and set |cur_cmd:=numeric_token|@>=
+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
+
+@* \[41] 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.
+
+@<Declare the basic parsing subroutines@>=
+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_cmd<min_primary_command)or@|
+ (cur_cmd>max_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.
+
+@<Declare the basic parsing subroutines@>=
+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_cmd<min_primary_command)or@|
+ (cur_cmd>max_primary_command) then
+ bad_exp("A tertiary");
+@.A tertiary expression...@>
+scan_secondary;
+if cur_type=future_pen then materialize_pen;
+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;
+
+@ A |future_pen| becomes a full-fledged pen here.
+
+@p procedure materialize_pen;
+label common_ending;
+var @!a_minus_b,@!a_plus_b,@!major_axis,@!minor_axis:scaled; {ellipse variables}
+@!theta:angle; {amount by which the ellipse has been rotated}
+@!p:pointer; {path traverser}
+@!q:pointer; {the knot list to be made into a pen}
+begin q:=cur_exp;
+if left_type(q)=endpoint then
+ begin print_err("Pen path must be a cycle");
+@.Pen path must be a cycle@>
+ help2("I can't make a pen from the given path.")@/
+ ("So I've replaced it by the trivial path `(0,0)..cycle'.");
+ put_get_error; cur_exp:=null_pen; goto common_ending;
+ end
+else if left_type(q)=open then
+ @<Change node |q| to a path for an elliptical pen@>;
+cur_exp:=make_pen(q);
+common_ending: toss_knot_list(q); cur_type:=pen_type;
+end;
+
+@ We placed the three points $(0,0)$, $(1,0)$, $(0,1)$ into a \&{pencircle},
+and they have now been transformed to $(u,v)$, $(A+u,B+v)$, $(C+u,D+v)$;
+this gives us enough information to deduce the transformation
+$(x,y)\mapsto(Ax+Cy+u,Bx+Dy+v)$.
+
+Given ($A,B,C,D)$ we can always find $(a,b,\theta,\phi)$ such that
+$$\eqalign{A&=a\cos\phi\cos\theta-b\sin\phi\sin\theta;\cr
+B&=a\cos\phi\sin\theta+b\sin\phi\cos\theta;\cr
+C&=-a\sin\phi\cos\theta-b\cos\phi\sin\theta;\cr
+D&=-a\sin\phi\sin\theta+b\cos\phi\cos\theta.\cr}$$
+In this notation, the unit circle $(\cos t,\sin t)$ is transformed into
+$$\bigl(a\cos(\phi+t)\cos\theta-b\sin(\phi+t)\sin\theta,\;
+a\cos(\phi+t)\sin\theta+b\sin(\phi+t)\cos\theta\bigr)\;+\;(u,v),$$
+which is an ellipse with semi-axes~$(a,b)$, rotated by~$\theta$ and
+shifted by~$(u,v)$. To solve the stated equations, we note that it is
+necessary and sufficient to solve
+$$\eqalign{A-D&=(a-b)\cos(\theta-\phi),\cr
+B+C&=(a-b)\sin(\theta-\phi),\cr}
+\qquad
+\eqalign{A+D&=(a+b)\cos(\theta+\phi),\cr
+B-C&=(a+b)\sin(\theta+\phi);\cr}$$
+and it is easy to find $a-b$, $a+b$, $\theta-\phi$, and $\theta+\phi$
+from these formulas.
+
+The code below uses |(txx,tyx,txy,tyy,tx,ty)| to stand for
+$(A,B,C,D,u,v)$.
+
+@<Change node |q|...@>=
+begin tx:=x_coord(q); ty:=y_coord(q);
+txx:=left_x(q)-tx; tyx:=left_y(q)-ty;
+txy:=right_x(q)-tx; tyy:=right_y(q)-ty;
+a_minus_b:=pyth_add(txx-tyy,tyx+txy); a_plus_b:=pyth_add(txx+tyy,tyx-txy);
+major_axis:=half(a_minus_b+a_plus_b); minor_axis:=half(abs(a_plus_b-a_minus_b));
+if major_axis=minor_axis then theta:=0 {circle}
+else theta:=half(n_arg(txx-tyy,tyx+txy)+n_arg(txx+tyy,tyx-txy));
+free_node(q,knot_node_size);
+q:=make_ellipse(major_axis,minor_axis,theta);
+if (tx<>0)or(ty<>0) then @<Shift the coordinates of path |q|@>;
+end
+
+@ @<Shift the coordinates of path |q|@>=
+begin p:=q;
+repeat x_coord(p):=x_coord(p)+tx; y_coord(p):=y_coord(p)+ty; p:=link(p);
+until p=q;
+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}
+
+@<Declare the basic parsing subroutines@>=
+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_cmd<min_primary_command)or@|
+ (cur_cmd>max_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<ampersand)or((d=ampersand)and@|
+ ((type(p)=pair_type)or(type(p)=path_type))) then
+ @<Scan a path construction operation;
+ but |return| if |p| has the wrong type@>
+ 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.
+
+@<Scan a path construction operation...@>=
+begin cycle_hit:=false;
+@<Convert the left operand, |p|, into a partial path ending at~|q|;
+ but |return| if |p| doesn't have a suitable type@>;
+continue_path: @<Determine the path join parameters;
+ but |goto finish_path| if there's only a direction specifier@>;
+if cur_cmd=cycle then @<Get ready to close a cycle@>
+else begin scan_tertiary;
+ @<Convert the right operand, |cur_exp|,
+ into a partial path from |pp| to~|qq|@>;
+ end;
+@<Join the partial paths and reset |p| and |q| to the head and tail
+ of the result@>;
+if cur_cmd>=min_expression_command then
+ if cur_cmd<=ampersand then if not cycle_hit then goto continue_path;
+finish_path:
+@<Choose control points for the path and put the result into |cur_exp|@>;
+end
+
+@ @<Convert the left operand, |p|, into a partial path ending at~|q|...@>=
+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 \MF\ discovers that the pair is part of a path.
+
+@p@t\4@>@<Declare the procedure called |known_pair|@>@;
+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.
+
+@<Declare the procedure called |known_pair|@>=
+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);
+ @<Make sure that both |x| and |y| parts of |p| are known;
+ copy them into |cur_x| and |cur_y|@>;
+ flush_cur_exp(0);
+ end;
+end;
+
+@ @<Make sure that both |x| and |y| parts of |p| are known...@>=
+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|.
+
+@<Determine the path join parameters...@>=
+if cur_cmd=left_brace then
+ @<Put the pre-join direction information into node |q|@>;
+d:=cur_cmd;
+if d=path_join then @<Determine the tension and/or control points@>
+else if d<>ampersand then goto finish_path;
+get_x_next;
+if cur_cmd=left_brace then
+ @<Put the post-join direction information into |x| and |t|@>
+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 @<Scan a curl specification@>
+else @<Scan a given direction@>;
+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;
+
+@ @<Scan a curl specification@>=
+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
+
+@ @<Scan a given direction@>=
+begin scan_expression;
+if cur_type>pair_type then @<Get given directions separated by commas@>
+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
+
+@ @<Get given directions separated by commas@>=
+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 operation. We must maintain
+the value of |right_type(q)| in cases such as
+`\.{..\{curl2\}z\{0,0\}..}'.
+
+@<Put the pre-join...@>=
+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.
+
+@<Put the post-join...@>=
+begin t:=scan_direction;
+if right_type(q)<>explicit then x:=cur_exp
+else t:=explicit; {the direction information is superfluous}
+end
+
+@ @<Determine the tension and/or...@>=
+begin get_x_next;
+if cur_cmd=tension then @<Set explicit tensions@>
+else if cur_cmd=controls then @<Set explicit control points@>
+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
+
+@ @<Set explicit tensions@>=
+begin get_x_next; y:=cur_cmd;
+if cur_cmd=at_least then get_x_next;
+scan_primary;
+@<Make sure that the current expression is a valid tension setting@>;
+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;
+ @<Make sure that the current expression is a valid tension setting@>;
+ if y=at_least then negate(cur_exp);
+ end;
+y:=cur_exp;
+end
+
+@ @d min_tension==three_quarter_unit
+
+@<Make sure that the current expression is a valid tension setting@>=
+if (cur_type<>known)or(cur_exp<min_tension) then
+ begin exp_err("Improper tension has been set to 1");
+@.Improper tension@>
+ help1("The expression above should have been a number >=3/4.");
+ put_get_flush_error(unity);
+ end
+
+@ @<Set explicit control points@>=
+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
+
+@ @<Convert the right operand, |cur_exp|, into a partial path...@>=
+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.
+
+@<Get ready to close a cycle@>=
+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
+
+@ @<Join the partial paths and reset |p| and |q|...@>=
+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;
+@<Plug an opening in |right_type(pp)|, if possible@>;
+if d=ampersand then @<Splice independent paths together@>
+else begin @<Plug an opening in |right_type(q)|, if possible@>;
+ 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
+
+@ @<Plug an opening in |right_type(q)|...@>=
+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
+
+@ @<Plug an opening in |right_type(pp)|...@>=
+if right_type(pp)=open then
+ if (t=curl)or(t=given) then
+ begin right_type(pp):=t; right_given(pp):=x;
+ end
+
+@ @<Splice independent paths together@>=
+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
+
+@ @<Choose control points for the path...@>=
+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|.
+
+@<Declare the basic parsing subroutines@>=
+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;
+
+@* \[42] 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 \MF, 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.
+
+@<Put each...@>=
+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("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("totalweight",unary,total_weight_op);@/
+@!@:total_weight_}{\&{totalweight} 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("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("angle",unary,angle_op);@/
+@!@:angle_}{\&{angle} primitive@>
+primitive("cycle",cycle,cycle_op);@/
+@!@:cycle_}{\&{cycle} 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("and",and_command,and_op);@/
+@!@:and_}{\&{and} primitive@>
+primitive("or",tertiary_binary,or_op);@/
+@!@:or_}{\&{or} 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("&",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("intersectiontimes",tertiary_binary,intersect);@/
+@!@:intersection_times_}{\&{intersectiontimes} primitive@>
+
+@ @<Cases of |print_cmd...@>=
+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 procedure do_nullary(@!c:quarterword);
+var @!k:integer; {all-purpose loop index}
+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:=null_pen;
+ end;
+normal_deviate:begin cur_type:=known; cur_exp:=norm_rand;
+ end;
+pen_circle:@<Make a special knot node for \&{pencircle}@>;
+job_name_op: begin if job_name=0 then open_log_file;
+ cur_type:=string_type; cur_exp:=job_name;
+ end;
+read_string_op:@<Read a string from the terminal@>;
+end; {there are no other cases}
+check_arith;
+end;
+
+@ @<Make a special knot node for \&{pencircle}@>=
+begin cur_type:=future_pen; cur_exp:=get_node(knot_node_size);
+left_type(cur_exp):=open; right_type(cur_exp):=open;
+link(cur_exp):=cur_exp;@/
+x_coord(cur_exp):=0; y_coord(cur_exp):=0;@/
+left_x(cur_exp):=unity; left_y(cur_exp):=0;@/
+right_x(cur_exp):=0; right_y(cur_exp):=unity;@/
+end
+
+@ @<Read a string...@>=
+begin if interaction<=nonstop_mode then
+ fatal_error("*** (cannot readstring in nonstop modes)");
+begin_file_reading; name:=1; prompt_input("");
+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@>@<Declare unary action procedures@>@;
+procedure do_unary(@!c:quarterword);
+var @!p,@!q:pointer; {for list manipulation}
+@!x:integer; {a temporary register}
+begin check_arith;
+if internal[tracing_commands]>two then
+ @<Trace the current unary operation@>;
+case c of
+plus:if cur_type<pair_type then
+ if cur_type<>picture_type then bad_unary(plus);
+minus:@<Negate the current expression@>;
+@t\4@>@<Additional cases of unary operators@>@;
+end; {there are no other cases}
+check_arith;
+end;
+
+@ The |nice_pair| function returns |true| if both components of a pair
+are known.
+
+@<Declare unary action procedures@>=
+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;
+
+@ @<Declare unary action...@>=
+procedure print_known_or_unknown_type(@!t:small_number;@!v:integer);
+begin print_char("(");
+if t<dependent then
+ if t<>pair_type then print_type(t)
+ else if nice_pair(v,pair_type) then print("pair")
+ else print("unknown pair")
+else print("unknown numeric");
+print_char(")");
+end;
+
+@ @<Declare unary action...@>=
+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;
+
+@ @<Trace the current unary operation@>=
+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).
+
+@<Negate the current expression@>=
+case cur_type of
+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
+ begin p:=value(cur_exp);
+ if type(x_part_loc(p))=known then negate(value(x_part_loc(p)))
+ else negate_dep_list(dep_list(x_part_loc(p)));
+ if type(y_part_loc(p))=known then negate(value(y_part_loc(p)))
+ else negate_dep_list(dep_list(y_part_loc(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);
+picture_type:negate_edges(cur_exp);
+othercases bad_unary(minus)
+endcases
+
+@ @<Declare unary action...@>=
+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;
+
+@ @<Additional cases of unary operators@>=
+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
+
+@<Additional cases of unary operators@>=
+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:@<Determine if a character has been shipped out@>;
+ end; {there are no other cases}
+
+@ @<Additional cases of unary operators@>=
+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|.
+
+@<Declare unary action...@>=
+procedure pair_to_path;
+begin cur_exp:=new_knot; cur_type:=path_type;
+end;
+
+@ @<Additional cases of unary operators@>=
+x_part,y_part:if (cur_type<=pair_type)and(cur_type>=transform_type) then
+ take_part(c)
+ else bad_unary(c);
+xx_part,xy_part,yx_part,yy_part: if cur_type=transform_type then take_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.
+
+@<Declare unary action...@>=
+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+2*(c-x_part));
+recycle_value(temp_val);
+end;
+
+@ @<Initialize table entries...@>=
+name_type(temp_val):=capsule;
+
+@ @<Additional cases of unary...@>=
+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;
+ if length(cur_exp)<>1 then
+ begin str_room(1); append_char(cur_exp); cur_exp:=make_string;
+ end;
+ 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);
+
+@ @<Declare unary action...@>=
+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_start[cur_exp+1]-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;
+ @<Give error messages if |bad_char| or |n>=4096|@>;
+ end;
+flush_cur_exp(n*unity);
+end;
+
+@ @<Give error messages if |bad_char|...@>=
+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
+ begin print_err("Number too large ("); print_int(n); print_char(")");
+@.Number too large@>
+ help1("I have trouble with numbers greater than 4095; watch out.");
+ put_get_error;
+ end
+
+@ The length operation is somewhat unusual in that it applies to a variety
+of different types of operands.
+
+@<Additional cases of unary...@>=
+length_op: if cur_type=string_type then flush_cur_exp(length(cur_exp)*unity)
+ else if cur_type=path_type then flush_cur_exp(path_length)
+ else if cur_type=known then cur_exp:=abs(cur_exp)
+ else 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);
+
+@ @<Declare unary action...@>=
+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;
+
+@ The turning number is computed only with respect to null pens. A different
+pen might affect the turning number, in degenerate cases, because autorounding
+will produce a slightly different path, or because excessively large coordinates
+might be truncated.
+
+@<Additional cases of unary...@>=
+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_pen:=null_pen; cur_path_type:=contour_code;
+ cur_exp:=make_spec(cur_exp,
+ fraction_one-half_unit-1-el_gordo,0);
+ flush_cur_exp(turning_number*unity); {convert to |scaled|}
+ 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
+
+@<Additional cases of unary operators@>=
+boolean_type: type_range(boolean_type)(unknown_boolean);
+string_type: type_range(string_type)(unknown_string);
+pen_type: type_range(pen_type)(future_pen);
+path_type: type_range(path_type)(unknown_path);
+picture_type: type_range(picture_type)(unknown_picture);
+transform_type,pair_type: type_test(c);
+numeric_type: type_range(known)(independent);
+known_op,unknown_op: test_known(c);
+
+@ @<Declare unary action procedures@>=
+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,future_pen,path_type,picture_type,
+ known: b:=true_code;
+transform_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;
+
+@ @<Additional cases of unary operators@>=
+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;
+
+@ @<Additional cases of unary operators@>=
+make_pen_op: begin if cur_type=pair_type then pair_to_path;
+ if cur_type=path_type then cur_type:=future_pen
+ else bad_unary(make_pen_op);
+ end;
+make_path_op: begin if cur_type=future_pen then materialize_pen;
+ if cur_type<>pen_type then bad_unary(make_path_op)
+ else begin flush_cur_exp(make_path(cur_exp)); cur_type:=path_type;
+ end;
+ end;
+total_weight_op: if cur_type<>picture_type then bad_unary(total_weight_op)
+ else flush_cur_exp(total_weight(cur_exp));
+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);
+
+@ Finally, we have the operations that combine a capsule~|p|
+with the current expression.
+
+@p @t\4@>@<Declare binary action procedures@>@;
+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
+ @<Trace the current binary operation@>;
+@<Sidestep |independent| cases in capsule |p|@>;
+@<Sidestep |independent| cases in the current expression@>;
+case c of
+plus,minus:@<Add or subtract the current expression from |p|@>;
+@t\4@>@<Additional cases of binary operators@>@;
+end; {there are no other cases}
+recycle_value(p); free_node(p,value_node_size); {|return| to avoid this}
+exit:check_arith; @<Recycle any sidestepped |independent| capsules@>;
+end;
+
+@ @<Declare binary action...@>=
+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;
+
+@ @<Trace the current binary operation@>=
+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.
+
+@<Recycle any sidestepped |independent| capsules@>=
+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.
+
+@<Sidestep |independent| cases in capsule |p|@>=
+case type(p) of
+transform_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;
+
+@ @<Sidestep |independent| cases in the current expression@>=
+case cur_type of
+transform_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
+
+@ @<Declare binary action...@>=
+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;
+
+@ @<Add or subtract the current expression from |p|@>=
+if (cur_type<pair_type)or(type(p)<pair_type) then
+ if (cur_type=picture_type)and(type(p)=picture_type) then
+ begin if c=minus then negate_edges(cur_exp);
+ cur_edges:=cur_exp; merge_edges(value(p));
+ end
+ else bad_binary(p,c)
+else if cur_type=pair_type then
+ if type(p)<>pair_type then bad_binary(p,c)
+ else begin q:=value(p); r:=value(cur_exp);
+ add_or_subtract(x_part_loc(q),x_part_loc(r),c);
+ add_or_subtract(y_part_loc(q),y_part_loc(r),c);
+ end
+ else if type(p)=pair_type then bad_binary(p,c)
+ else add_or_subtract(p,null,c)
+
+@ The first argument to |add_or_subtract| is the location of a value node
+in a capsule or pair node that will soon be recycled. The second argument
+is either a location within a pair or transform node of |cur_exp|,
+or it is null (which means that |cur_exp| itself should be the second
+argument). The third argument is either |plus| or |minus|.
+
+The sum or difference of the numeric quantities will replace the second
+operand. Arithmetic overflow may go undetected; users aren't supposed to
+be monkeying around with really big values.
+@^overflow in arithmetic@>
+
+@<Declare binary action...@>=
+@t\4@>@<Declare the procedure called |dep_finish|@>@;
+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<dependent then v:=cur_exp@+else v:=dep_list(cur_exp);
+ end
+else begin t:=type(q);
+ if t<dependent then v:=value(q)@+else v:=dep_list(q);
+ end;
+if t=known then
+ begin if c=minus then negate(v);
+ if type(p)=known then
+ begin v:=slow_add(value(p),v);
+ if q=null then cur_exp:=v@+else value(q):=v;
+ return;
+ end;
+ @<Add a known value to the constant term of |dep_list(p)|@>;
+ end
+else begin if c=minus then negate_dep_list(v);
+ @<Add operand |p| to the dependency list |v|@>;
+ end;
+exit:end;
+
+@ @<Add a known value to the constant term of |dep_list(p)|@>=
+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.
+
+@<Add operand |p| to the dependency list |v|@>=
+if type(p)=known then
+ @<Add the known |value(p)| to the constant term of |v|@>
+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)<coef_bound then
+ begin v:=p_plus_q(v,r,dependent); goto done;
+ end; {|fix_needed| will necessarily be false}
+ t:=proto_dependent; v:=p_over_v(v,unity,dependent,proto_dependent);
+ end;
+ if s=proto_dependent then v:=p_plus_q(v,r,proto_dependent)
+ else v:=p_plus_fq(v,unity,r,proto_dependent,dependent);
+ done: @<Output the answer, |v| (which might have become |known|)@>;
+ end
+
+@ @<Add the known |value(p)| to the constant term of |v|@>=
+begin while info(v)<>null do v:=link(v);
+value(v):=slow_add(value(p),value(v));
+end
+
+@ @<Output the answer, |v| (which might have become |known|)@>=
+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|.
+
+@<Declare the procedure called |dep_finish|@>=
+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.
+
+@<Additional cases of binary operators@>=
+less_than,less_or_equal,greater_than,greater_or_equal,equal_to,unequal_to:
+ begin@t@>@;
+ 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
+ @<Check if unknowns have been equated@>
+ else if (cur_type=pair_type)or(cur_type=transform_type) then
+ @<Reduce comparison of big nodes to comparison of scalars@>
+ else if cur_type=boolean_type then flush_cur_exp(cur_exp-value(p))
+ else begin bad_binary(p,c); goto done;
+ end;
+ @<Compare the current expression with zero@>;
+done: end;
+
+@ @<Compare the current expression with zero@>=
+if cur_type<>known then
+ begin if cur_type<known then
+ begin disp_err(p,"");
+ help1("The quantities shown above have not been equated.")@/
+ end
+ else help2("Oh dear. I can't decide if the expression above is positive,")@/
+ ("negative, or zero. So this comparison test won't be `true'.");
+ exp_err("Unknown relation will be considered false");
+@.Unknown relation...@>
+ 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.
+
+@<Check if unknowns have been equated@>=
+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
+
+@ @<Reduce comparison of big nodes to comparison of scalars@>=
+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(x_part+half(r-value(cur_exp)));
+end
+
+@ Here we use the sneaky fact that |and_op-false_code=or_op-true_code|.
+
+@<Additional cases of binary operators@>=
+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);
+
+@ @<Additional cases of binary operators@>=
+times: if (cur_type<pair_type)or(type(p)<pair_type) then bad_binary(p,times)
+ else if (cur_type=known)or(type(p)=known) then
+ @<Multiply when at least one operand is known@>
+ else if (nice_pair(p,type(p))and(cur_type>pair_type))
+ or(nice_pair(cur_exp,cur_type)and(type(p)>pair_type)) then
+ begin hard_times(p); return;
+ end
+ else bad_binary(p,times);
+
+@ @<Multiply when at least one operand is known@>=
+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 then
+ begin p:=value(cur_exp);
+ dep_mult(x_part_loc(p),v,true);
+ dep_mult(y_part_loc(p),v,true);
+ end
+else dep_mult(null,v,true);
+return;
+end
+
+@ @<Declare binary action...@>=
+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>=pair_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
+ @<Trace the fraction multiplication@>;
+case cur_type of
+transform_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);
+ dep_mult(x_part_loc(p),v,false);
+ dep_mult(y_part_loc(p),v,false);
+ 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;
+
+@ @<Trace the fraction multiplication@>=
+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 pair by a dependency list.
+
+@<Declare binary action procedures@>=
+procedure hard_times(@!p:pointer);
+var @!q:pointer; {a copy of the dependent variable |p|}
+@!r:pointer; {the big node for the nice pair}
+@!u,@!v:scaled; {the known values of the nice pair}
+begin if type(p)=pair_type then
+ begin q:=stash_cur_exp; unstash_cur_exp(p); p:=q;
+ end; {now |cur_type=pair_type|}
+r:=value(cur_exp); u:=value(x_part_loc(r)); v:=value(y_part_loc(r));
+@<Move the dependent variable |p| into both parts of the pair node |r|@>;
+dep_mult(x_part_loc(r),u,true); dep_mult(y_part_loc(r),v,true);
+end;
+
+@ @<Move the dependent variable |p|...@>=
+type(y_part_loc(r)):=type(p);
+new_dep(y_part_loc(r),copy_dep_list(dep_list(p)));@/
+type(x_part_loc(r)):=type(p);
+mem[value_loc(x_part_loc(r))]:=mem[value_loc(p)];
+link(prev_dep(p)):=x_part_loc(r);
+free_node(p,value_node_size)
+
+@ @<Additional cases of binary operators@>=
+over: if (cur_type<>known)or(type(p)<pair_type) then bad_binary(p,over)
+ else begin v:=cur_exp; unstash_cur_exp(p);
+ if v=0 then @<Squeal about division by zero@>
+ 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);
+ dep_div(x_part_loc(p),v);
+ dep_div(y_part_loc(p),v);
+ end
+ else dep_div(null,v);
+ end;
+ return;
+ end;
+
+@ @<Declare binary action...@>=
+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;
+
+@ @<Squeal about division by zero@>=
+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
+
+@ @<Additional cases of binary operators@>=
+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.
+
+@<Additional cases of binary operators@>=
+rotated_by,slanted_by,scaled_by,shifted_by,transformed_by,
+ x_scaled,y_scaled,z_scaled: @t@>@;@/
+ if (type(p)=path_type)or(type(p)=future_pen)or(type(p)=pen_type) then
+ begin path_trans(p,c); 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 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.
+
+@<Declare binary action...@>=
+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
+ @<Put the current transform into |cur_exp|@>;
+@<If the current transform is entirely known, stash it in global variables;
+ otherwise |return|@>;
+exit:end;
+
+@ @<Glob...@>=
+@!txx,@!txy,@!tyx,@!tyy,@!tx,@!ty:scaled; {current transform coefficients}
+
+@ @<Put the current transform...@>=
+begin p:=stash_cur_exp; cur_exp:=id_transform; cur_type:=transform_type;
+q:=value(cur_exp);
+case c of
+@<For each of the eight cases, change the relevant fields of |cur_exp|
+ and |goto done|;
+ but do nothing if capsule |p| doesn't have the appropriate type@>@;
+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
+
+@ @<If the current transform is entirely known, ...@>=
+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)
+
+@ @<For each of the eight cases...@>=
+rotated_by:if type(p)=known then
+ @<Install sines and cosines, then |goto done|@>;
+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
+ @<Install a complex multiplier, then |goto done|@>;
+transformed_by:do_nothing;
+
+@ @<Install sines and cosines, then |goto done|@>=
+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
+
+@ @<Install a complex multiplier, then |goto done|@>=
+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.
+
+@<Declare binary action...@>=
+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|.
+
+@<Declare binary action...@>=
+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 |null_pen| remains unchanged if it isn't
+being shifted.
+
+@<Declare binary action...@>=
+procedure path_trans(@!p:pointer;@!c:quarterword);
+label exit;
+var @!q:pointer; {list traverser}
+begin set_up_known_trans(c); unstash_cur_exp(p);
+if cur_type=pen_type then
+ begin if max_offset(cur_exp)=0 then if tx=0 then if ty=0 then return;
+ flush_cur_exp(make_path(cur_exp)); cur_type:=future_pen;
+ end;
+q:=cur_exp;
+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|}
+q:=link(q);
+until q=cur_exp;
+exit:end;
+
+@ The next simplest transformation procedure applies to edges.
+It is simple primarily because \MF\ doesn't allow very general
+transformations to be made, and because the tricky subroutines
+for edge transformation have already been written.
+
+@<Declare binary action...@>=
+procedure edges_trans(@!p:pointer;@!c:quarterword);
+label exit;
+begin set_up_known_trans(c); unstash_cur_exp(p); cur_edges:=cur_exp;
+if empty_edges(cur_edges) then return; {the empty set is easy to transform}
+if txx=0 then if tyy=0 then
+ if txy mod unity=0 then if tyx mod unity=0 then
+ begin xy_swap_edges; txx:=txy; tyy:=tyx; txy:=0; tyx:=0;
+ if empty_edges(cur_edges) then return;
+ end;
+if txy=0 then if tyx=0 then
+ if txx mod unity=0 then if tyy mod unity=0 then
+ @<Scale the edges, shift them, and |return|@>;
+print_err("That transformation is too hard");
+@.That transformation...@>
+help3("I can apply complicated transformations to paths,")@/
+ ("but I can only do integer operations on pictures.")@/
+ ("Proceed, and I'll omit the transformation.");
+put_get_error;
+exit:end;
+
+@ @<Scale the edges, shift them, and |return|@>=
+begin if (txx=0)or(tyy=0) then
+ begin toss_edges(cur_edges);
+ cur_exp:=get_node(edge_header_size); init_edges(cur_exp);
+ end
+else begin if txx<0 then
+ begin x_reflect_edges; txx:=-txx;
+ end;
+ if tyy<0 then
+ begin y_reflect_edges; tyy:=-tyy;
+ end;
+ if txx<>unity then x_scale_edges(txx div unity);
+ if tyy<>unity then y_scale_edges(tyy div unity);
+ @<Shift the edges by |(tx,ty)|, rounded@>;
+ end;
+return;
+end
+
+@ @<Shift the edges...@>=
+tx:=round_unscaled(tx); ty:=round_unscaled(ty);
+if (m_min(cur_edges)+tx<=0)or(m_max(cur_edges)+tx>=8192)or@|
+ (n_min(cur_edges)+ty<=0)or(n_max(cur_edges)+ty>=8191)or@|
+ (abs(tx)>=4096)or(abs(ty)>=4096) then
+ begin print_err("Too far to shift");
+@.Too far to shift@>
+ help3("I can't shift the picture as requested---it would")@/
+ ("make some coordinates too large or too small.")@/
+ ("Proceed, and I'll omit the transformation.");
+ put_get_error;
+ end
+else begin if tx<>0 then
+ begin if not valid_range(m_offset(cur_edges)-tx) then fix_offset;
+ m_min(cur_edges):=m_min(cur_edges)+tx;
+ m_max(cur_edges):=m_max(cur_edges)+tx;
+ m_offset(cur_edges):=m_offset(cur_edges)-tx;
+ last_window_time(cur_edges):=0;
+ end;
+ if ty<>0 then
+ begin n_min(cur_edges):=n_min(cur_edges)+ty;
+ n_max(cur_edges):=n_max(cur_edges)+ty;
+ n_pos(cur_edges):=n_pos(cur_edges)+ty;
+ last_window_time(cur_edges):=0;
+ end;
+ end
+
+@ The hard cases of transformation occur when big nodes are involved,
+and when some of their components are unknown.
+
+@<Declare binary action...@>=
+@t\4@>@<Declare subroutines needed by |big_trans|@>@;
+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 @<Transform an unknown big node and |return|@>;
+until r=q;
+@<Transform a known big node@>;
+exit:end; {node |p| will now be recycled by |do_binary|}
+
+@ @<Transform an unknown big node and |return|@>=
+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 $p\cdot t+q\cdot u+\delta$.
+
+@<Declare subroutines needed by |big_trans|@>=
+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 @<Ensure that |type(p)=proto_dependent|@>;
+ 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;
+
+@ @<Ensure that |type(p)=proto_dependent|@>=
+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
+
+@ @<Transform a known big node@>=
+set_up_trans(c);
+if cur_type=known then @<Transform known by known@>
+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|.
+
+@<Declare subroutines needed by |big_trans|@>=
+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 $p\cdot t+v\cdot u+q$.
+
+@<Declare subroutines needed by |big_trans|@>=
+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;
+
+@ @<Transform known by known@>=
+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|.
+
+@<Declare subroutines needed by |big_trans|@>=
+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;
+
+@ @<Additional cases of binary operators@>=
+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;
+
+@ @<Declare binary action...@>=
+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_start[a+1]-1 do append_char(so(str_pool[k]));
+for k:=str_start[b] to str_start[b+1]-1 do append_char(so(str_pool[k]));
+cur_exp:=make_string; delete_str_ref(b);
+end;
+
+@ @<Declare binary action...@>=
+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;
+
+@ @<Declare binary action...@>=
+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;
+@<Dispense with the cases |a<0| and/or |b>l|@>;
+q:=cur_exp;
+while a>=unity do
+ begin q:=link(q); a:=a-unity; b:=b-unity;
+ end;
+if b=a then @<Construct a path from |pp| to |qq| of length zero@>
+else @<Construct a path from |pp| to |qq| of length $\lceil b\rceil$@>;
+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;
+
+@ @<Dispense with the cases |a<0| and/or |b>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
+
+@ @<Construct a path from |pp| to |qq| of length $\lceil b\rceil$@>=
+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,x_coord(pp),y_coord(pp)); 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,x_coord(qq),y_coord(qq));
+ free_node(qq,knot_node_size);
+ qq:=link(rr);
+ end;
+end
+
+@ @<Construct a path from |pp| to |qq| of length zero@>=
+begin if a>0 then
+ begin qq:=link(q);
+ split_cubic(q,a*@'10000,x_coord(qq),y_coord(qq)); q:=link(q);
+ end;
+pp:=copy_knot(q); qq:=pp;
+end
+
+@ The |pair_value| routine changes the current expression to a
+given ordered pair of values.
+
+@<Declare binary action...@>=
+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;
+
+@ @<Additional cases of binary operators@>=
+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: begin if cur_type=future_pen then materialize_pen;
+ if (cur_type=pen_type)and nice_pair(p,type(p)) then
+ set_up_offset(value(p))
+ else bad_binary(p,pen_offset_of);
+ end;
+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;
+
+@ @<Declare binary action...@>=
+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;
+
+@ @<Declare binary action...@>=
+procedure find_point(@!v:scaled;@!c:quarterword);
+var @!p:pointer; {the path}
+@!n:scaled; {its length}
+@!q:pointer; {successor of |p|}
+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 @<Insert a fractional node by splitting the cubic@>;
+@<Set the current expression to the desired path coordinates@>;
+end;
+
+@ @<Insert a fractional node...@>=
+begin q:=link(p); split_cubic(p,v*@'10000,x_coord(q),y_coord(q)); p:=link(p);
+end
+
+@ @<Set the current expression to the desired path coordinates...@>=
+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}
+
+@ @<Additional cases of bin...@>=
+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;
+
+@* \[43] Statements and commands.
+The chief executive of \MF\ is the |do_statement| routine, which
+contains the master switch that causes all the various pieces of \MF\
+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 @t\4@>@<Declare generic font output procedures@>@;
+@t\4@>@<Declare action procedures for use by |do_statement|@>@;
+procedure do_statement; {governs \MF's activities}
+begin cur_type:=vacuous; get_x_next;
+if cur_cmd>max_primary_command then @<Worry about bad statement@>
+else if cur_cmd>max_statement_command then
+ @<Do an equation, assignment, title, or
+ `$\langle\,$expression$\,\rangle\,$\&{endgroup}'@>
+else @<Do a statement that doesn't begin with an expression@>;
+if cur_cmd<semicolon then
+ @<Flush unparsable junk that was found after the statement@>;
+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.
+
+@<Worry about bad statement@>=
+begin if cur_cmd<semicolon then
+ begin print_err("A statement can't begin with `");
+@.A statement can't begin with x@>
+ 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.
+
+@<Flush unparsable junk that was found after the 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_next;
+@<Decrease the string reference count...@>;
+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.
+
+@<Do a statement that doesn't...@>=
+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@>@<Cases of |do_statement| that invoke particular commands@>@;
+end; {there are no other cases}
+cur_type:=vacuous;
+end
+
+@ The most important statements begin with expressions.
+
+@<Do an equation, assignment, title, or...@>=
+begin var_flag:=assignment; scan_expression;
+if cur_cmd<end_group then
+ begin if cur_cmd=equals then do_equation
+ else if cur_cmd=assignment then do_assignment
+ else if cur_type=string_type then @<Do a title@>
+ 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
+
+@ @<Do a title@>=
+begin if internal[tracing_titles]>0 then
+ begin print_nl(""); slow_print(cur_exp); update_terminal;
+ end;
+if internal[proofing]>0 then
+ @<Send the current expression as a title to the output file@>;
+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).
+
+@<Declare action procedures for use by |do_statement|@>=
+@t\4@>@<Declare the procedure called |try_eq|@>@;
+@t\4@>@<Declare the procedure called |make_eq|@>@;
+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 @<Trace the current equation@>;
+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_equation|:
+
+@<Declare action procedures for use by |do_statement|@>=
+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 @<Trace the current assignment@>;
+ if info(lhs)>hash_end then
+ @<Assign the current expression to an internal variable@>
+ else @<Assign the current expression to the variable |lhs|@>;
+ flush_node_list(lhs);
+ end;
+end;
+
+@ @<Trace the current equation@>=
+begin begin_diagnostic; print_nl("{("); print_exp(lhs,0);
+print(")=("); print_exp(null,0); print(")}"); end_diagnostic(false);
+end
+
+@ @<Trace the current assignment@>=
+begin begin_diagnostic; print_nl("{");
+if info(lhs)>hash_end then slow_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
+
+@ @<Assign the current expression to an internal variable@>=
+if cur_type=known then internal[info(lhs)-(hash_end)]:=cur_exp
+else begin exp_err("Internal quantity `");
+@.Internal quantity...@>
+ slow_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
+
+@ @<Assign the current expression to the variable |lhs|@>=
+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.
+
+@<Declare the procedure called |make_eq|@>=
+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@>@<For each type |t|, make an equation and |goto done| unless |cur_type|
+ is incompatible with~|t|@>@;
+end; {all cases have been listed}
+@<Announce that the equation cannot be performed@>;
+done:check_arith; recycle_value(lhs); free_node(lhs,value_node_size);
+end;
+
+@ @<Announce that the equation cannot be performed@>=
+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
+
+@ @<For each type |t|, make an equation and |goto done| unless...@>=
+boolean_type,string_type,pen_type,path_type,picture_type:
+ if cur_type=t+unknown_tag then
+ begin nonlinear_eq(v,cur_exp,false); unstash_cur_exp(cur_exp); goto done;
+ end
+ else if cur_type=t then
+ @<Report redundant or inconsistent equation and |goto done|@>;
+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,pair_type:if cur_type=t then
+ @<Do multiple equations and |goto done|@>;
+known,dependent,proto_dependent,independent:if cur_type>=known then
+ begin try_eq(lhs,null); goto done;
+ end;
+vacuous:do_nothing;
+
+@ @<Report redundant or inconsistent equation and |goto done|@>=
+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;
+ @<Exclaim about a redundant equation@>; 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
+
+@ @<Do multiple equations and |goto done|@>=
+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.
+
+@<Declare the procedure called |try_eq|@>=
+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 @<Remove the left operand from its container, negate it, and
+ put it into dependency list~|p| with constant term~|q|@>;
+@<Add the right operand to list |p|@>;
+if info(p)=null then @<Deal with redundant or inconsistent equation@>
+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;
+
+@ @<Remove the left operand from its container, negate it, and...@>=
+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
+
+@ @<Deal with redundant or inconsistent equation@>=
+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 @<Exclaim about a redundant equation@>;
+free_node(p,dep_node_size);
+end
+
+@ @<Add the right operand to list |p|@>=
+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;
+@<Add dependency list |pp| of type |tt| to dependency list~|p| of type~|t|@>;
+if copied then flush_node_list(pp);
+done1:
+
+@ @<Add dependency list |pp| of type |tt| to dependency list~|p| of type~|t|@>=
+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|.
+
+@<Declare the function called |scan_declared_variable|@>=
+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 @<Descend past a collective subscript@>
+ else goto done;
+ link(t):=get_avail; t:=link(t); info(t):=cur_sym;
+ end;
+done: if eq_type(x) mod outer_tag<>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.
+
+@<Descend past a collective subscript@>=
+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.
+
+@<Put each...@>=
+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("pair",type_name,pair_type);@/
+@!@:pair_}{\&{pair} primitive@>
+
+@ @<Cases of |print_cmd...@>=
+type_name: print_type(m);
+
+@ Now we are ready to handle type declarations, assuming that a
+|type_name| has just been scanned.
+
+@<Declare action procedures for use by |do_statement|@>=
+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<comma then @<Flush spurious symbols after the declared variable@>;
+until end_of_statement;
+end;
+
+@ @<Flush spurious symbols after the declared variable@>=
+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_next;
+@<Decrease the string reference count...@>;
+until cur_cmd>=comma; {either |end_of_statement| or |cur_cmd=comma|}
+scanner_status:=normal;
+end
+
+@ \MF'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;
+
+@ @<Put each...@>=
+primitive("end",stop,0);@/
+@!@:end_}{\&{end} primitive@>
+primitive("dump",stop,1);@/
+@!@:dump_}{\&{dump} primitive@>
+
+@ @<Cases of |print_cmd...@>=
+stop:if m=0 then print("end")@+else print("dump");
+
+@* \[44] 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:
+
+@<Cases of |do_statement|...@>=
+random_seed: do_random_seed;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_random_seed;
+begin get_x_next;
+if cur_cmd<>assignment then
+ begin missing_err(":=");
+@.Missing `:='@>
+ help1("Always say `randomseed:=<numeric expression>'.");
+ 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 @<Initialize the random seed to |cur_exp|@>;
+end;
+
+@ @<Initialize the random seed to |cur_exp|@>=
+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):
+
+@<Cases of |do_statement|...@>=
+mode_command: begin print_ln; interaction:=cur_mod;
+ @<Initialize the print |selector| based on |interaction|@>;
+ if log_opened then selector:=selector+2;
+ get_x_next;
+ end;
+
+@ @<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd_mod|...@>=
+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.
+
+@<Cases of |do_statement|...@>=
+protection_command: do_protection;
+
+@ @<Put each...@>=
+primitive("inner",protection_command,0);@/
+@!@:inner_}{\&{inner} primitive@>
+primitive("outer",protection_command,1);@/
+@!@:outer_}{\&{outer} primitive@>
+
+@ @<Cases of |print_cmd...@>=
+protection_command: if m=0 then print("inner")@+else print("outer");
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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 t<outer_tag then eq_type(cur_sym):=t+outer_tag;
+ get_x_next;
+until cur_cmd<>comma;
+end;
+
+@ \MF\ never defines the tokens `\.(' and `\.)' to be primitives, but
+plain \MF\ 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.
+
+@<Cases of |do_statement|...@>=
+delimiters: def_delims;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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 \MF\ has reached a point
+where some right delimiter is mandatory.
+
+@<Declare the procedure called |check_delimiter|@>=
+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 `"); slow_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.
+
+@<Cases of |do_statement|...@>=
+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;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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 slow_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.
+
+@<Declare action procedures for use by |do_statement|@>=
+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;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_new_internal;
+begin repeat if int_ptr=max_internal then
+ overflow("number of internals",max_internal);
+@:METAFONT 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}
+
+@<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd...@>=
+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;
+
+@ @<Cases of |do_statement|...@>=
+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.
+
+@<Declare action procedures for use by |do_statement|@>=
+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;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure disp_token;
+begin print_nl("> ");
+@.>\relax@>
+if cur_sym=0 then @<Show a numeric or string or capsule token@>
+else begin slow_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;
+
+@ @<Show a numeric or string or capsule token@>=
+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(""""); slow_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 necessarily correspond to
+primitive tokens.
+
+@<Cases of |print_cmd_...@>=
+left_delimiter,right_delimiter: begin if c=left_delimiter then print("lef")
+ else print("righ");
+ print("t delimiter that matches "); slow_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:slow_print(int_name[m]);
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_show_token;
+begin repeat get_next; disp_token;
+get_x_next;
+until cur_cmd<>comma;
+end;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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 ");
+print_int(str_ptr-init_str_ptr); print_char("&");
+print_int(pool_ptr-init_pool_ptr);
+print(" (");
+print_int(max_strings-max_str_ptr); print_char("&");
+print_int(pool_size-max_pool_ptr); print(" still 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|.
+
+@<Declare action procedures for use by |do_statement|@>=
+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 @<Descend the structure@>
+else if type(p)>=unsuffixed_macro then @<Display a variable macro@>
+else if type(p)<>undefined then
+ begin print_nl(""); print_variable_name(p); print_char("=");
+ print_exp(p,0);
+ end;
+end;
+
+@ @<Descend the structure@>=
+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
+
+@ @<Display a variable macro@>=
+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
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_show_var;
+label done;
+begin repeat get_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;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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.
+
+@<Declare action procedures for use by |do_statement|@>=
+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<error_stop_mode then
+ begin help0; decr(error_count);
+ end
+ else help1("This isn't an error message; I'm just showing something.");
+ if cur_cmd=semicolon then error@+else put_get_error;
+ end;
+end;
+
+@ The `\&{addto}' command needs the following additional primitives:
+
+@d drop_code=0 {command modifier for `\&{dropping}'}
+@d keep_code=1 {command modifier for `\&{keeping}'}
+
+@<Put each...@>=
+primitive("contour",thing_to_add,contour_code);@/
+@!@:contour_}{\&{contour} primitive@>
+primitive("doublepath",thing_to_add,double_path_code);@/
+@!@:double_path_}{\&{doublepath} primitive@>
+primitive("also",thing_to_add,also_code);@/
+@!@:also_}{\&{also} primitive@>
+primitive("withpen",with_option,pen_type);@/
+@!@:with_pen_}{\&{withpen} primitive@>
+primitive("withweight",with_option,known);@/
+@!@:with_weight_}{\&{withweight} primitive@>
+primitive("dropping",cull_op,drop_code);@/
+@!@:dropping_}{\&{dropping} primitive@>
+primitive("keeping",cull_op,keep_code);@/
+@!@:keeping_}{\&{keeping} primitive@>
+
+@ @<Cases of |print_cmd...@>=
+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 print("withweight");
+cull_op:if m=drop_code then print("dropping")
+ else print("keeping");
+
+@ @<Declare action procedures for use by |do_statement|@>=
+function scan_with:boolean;
+var @!t:small_number; {|known| or |pen_type|}
+@!result:boolean; {the value to return}
+begin t:=cur_mod; cur_type:=vacuous; get_x_next; scan_expression;
+result:=false;
+if cur_type<>t then @<Complain about improper type@>
+else if cur_type=pen_type then result:=true
+else @<Check the tentative weight@>;
+scan_with:=result;
+end;
+
+@ @<Complain about improper type@>=
+begin exp_err("Improper type");
+@.Improper type@>
+help2("Next time say `withweight <known numeric expression>';")@/
+ ("I'll ignore the bad `with' clause and look for another.");
+if t=pen_type then
+ help_line[1]:="Next time say `withpen <known pen expression>';";
+put_get_flush_error(0);
+end
+
+@ @<Check the tentative weight@>=
+begin cur_exp:=round_unscaled(cur_exp);
+if (abs(cur_exp)<4)and(cur_exp<>0) then result:=true
+else begin print_err("Weight must be -3, -2, -1, +1, +2, or +3");
+@.Weight must be...@>
+ help1("I'll ignore the bad `with' clause and look for another.");
+ put_get_flush_error(0);
+ end;
+end
+
+@ One of the things we need to do when we've parsed an \&{addto} or
+similar command is set |cur_edges| to the header of a supposed \&{picture}
+variable, given a token list for that variable.
+
+@<Declare action procedures for use by |do_statement|@>=
+procedure find_edges_var(@!t:pointer);
+var @!p:pointer;
+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 cur_edges:=value(p);
+flush_node_list(t);
+end;
+
+@ @<Cases of |do_statement|...@>=
+add_to_command: do_add_to;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_add_to;
+label done, not_found;
+var @!lhs,@!rhs:pointer; {variable on left, path on right}
+@!w:integer; {tentative weight}
+@!p:pointer; {list manipulation register}
+@!q:pointer; {beginning of second half of doubled path}
+@!add_to_type:double_path_code..also_code; {modifier of \&{addto}}
+begin get_x_next; var_flag:=thing_to_add; scan_primary;
+if cur_type<>token_list then
+ @<Abandon edges command because there's no variable@>
+else begin lhs:=cur_exp; add_to_type:=cur_mod;@/
+ cur_type:=vacuous; get_x_next; scan_expression;
+ if add_to_type=also_code then @<Augment some edges by others@>
+ else @<Get ready to fill a contour, and fill it@>;
+ end;
+end;
+
+@ @<Abandon edges command because there's no variable@>=
+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
+
+@ @<Augment some edges by others@>=
+begin find_edges_var(lhs);
+if cur_edges=null then flush_cur_exp(0)
+else 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 merge_edges(cur_exp); flush_cur_exp(0);
+ end;
+end
+
+@ @<Get ready to fill a contour...@>=
+begin 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 been a known path.")@/
+ ("So I'll not change anything just now.");
+ put_get_flush_error(0); flush_token_list(lhs);
+ end
+else begin rhs:=cur_exp; w:=1; cur_pen:=null_pen;
+ while cur_cmd=with_option do
+ if scan_with then
+ if cur_type=known then w:=cur_exp
+ else @<Change the tentative pen@>;
+ @<Complete the contour filling operation@>;
+ delete_pen_ref(cur_pen);
+ end;
+end
+
+@ We could say `|add_pen_ref(cur_pen)|; |flush_cur_exp(0)|' after changing
+|cur_pen| here. But that would have no effect, because the current expression
+will not be flushed. Thus we save a bit of code (at the risk of being too
+tricky).
+
+@<Change the tentative pen@>=
+begin delete_pen_ref(cur_pen); cur_pen:=cur_exp;
+end
+
+@ @<Complete the contour filling...@>=
+find_edges_var(lhs);
+if cur_edges=null then toss_knot_list(rhs)
+else begin lhs:=null; cur_path_type:=add_to_type;
+ if left_type(rhs)=endpoint then
+ if cur_path_type=double_path_code then @<Double the path@>
+ else @<Complain about non-cycle and |goto not_found|@>
+ else if cur_path_type=double_path_code then lhs:=htap_ypoc(rhs);
+ cur_wt:=w; rhs:=make_spec(rhs,max_offset(cur_pen),internal[tracing_specs]);
+ @<Check the turning number@>;
+ if max_offset(cur_pen)=0 then fill_spec(rhs)
+ else fill_envelope(rhs);
+ if lhs<>null then
+ begin rev_turns:=true;
+ lhs:=make_spec(lhs,max_offset(cur_pen),internal[tracing_specs]);
+ rev_turns:=false;
+ if max_offset(cur_pen)=0 then fill_spec(lhs)
+ else fill_envelope(lhs);
+ end;
+not_found: end
+
+@ @<Double the path@>=
+if link(rhs)=rhs then @<Make a trivial one-point path cycle@>
+else begin p:=htap_ypoc(rhs); q:=link(p);@/
+ right_x(path_tail):=right_x(q); right_y(path_tail):=right_y(q);
+ right_type(path_tail):=right_type(q);
+ link(path_tail):=link(q); free_node(q,knot_node_size);@/
+ right_x(p):=right_x(rhs); right_y(p):=right_y(rhs);
+ right_type(p):=right_type(rhs);
+ link(p):=link(rhs); free_node(rhs,knot_node_size);@/
+ rhs:=p;
+ end
+
+@ @<Make a trivial one-point path cycle@>=
+begin right_x(rhs):=x_coord(rhs); right_y(rhs):=y_coord(rhs);
+left_x(rhs):=x_coord(rhs); left_y(rhs):=y_coord(rhs);
+left_type(rhs):=explicit; right_type(rhs):=explicit;
+end
+
+@ @<Complain about non-cycle...@>=
+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;
+toss_knot_list(rhs); goto not_found;
+end
+
+@ @<Check the turning number@>=
+if turning_number<=0 then
+ if cur_path_type<>double_path_code then if internal[turning_check]>0 then
+ if (turning_number<0)and(link(cur_pen)=null) then negate(cur_wt)
+ else begin if turning_number=0 then
+ if (internal[turning_check]<=unity)and(link(cur_pen)=null) then goto done
+ else print_strange("Strange path (turning number is zero)")
+@.Strange path...@>
+ else print_strange("Backwards path (turning number is negative)");
+@.Backwards path...@>
+ help3("The path doesn't have a counterclockwise orientation,")@/
+ ("so I'll probably have trouble drawing it.")@/
+ ("(See Chapter 27 of The METAFONTbook for more help.)");
+@:METAFONTbook}{\sl The {\logos METAFONT\/}book@>
+ put_get_error;
+ end;
+done:
+
+@ @<Cases of |do_statement|...@>=
+ship_out_command: do_ship_out;
+display_command: do_display;
+open_window: do_open_window;
+cull_command: do_cull;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+@t\4@>@<Declare the function called |tfm_check|@>@;
+procedure do_ship_out;
+label exit;
+var @!c:integer; {the character code}
+begin get_x_next; var_flag:=semicolon; scan_expression;
+if cur_type<>token_list then
+ if cur_type=picture_type then cur_edges:=cur_exp
+ else begin @<Abandon edges command because there's no variable@>;
+ return;
+ end
+else begin find_edges_var(cur_exp); cur_type:=vacuous;
+ end;
+if cur_edges<>null then
+ begin c:=round_unscaled(internal[char_code]) mod 256;
+ if c<0 then c:=c+256;
+ @<Store the width information for character code~|c|@>;
+ if internal[proofing]>=0 then ship_out(c);
+ end;
+flush_cur_exp(0);
+exit:end;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_display;
+label not_found,common_ending,exit;
+var @!e:pointer; {token list for a picture variable}
+begin get_x_next; var_flag:=in_window; scan_primary;
+if cur_type<>token_list then
+ @<Abandon edges command because there's no variable@>
+else begin e:=cur_exp; cur_type:=vacuous;
+ get_x_next; scan_expression;
+ if cur_type<>known then goto common_ending;
+ cur_exp:=round_unscaled(cur_exp);
+ if cur_exp<0 then goto not_found;
+ if cur_exp>15 then goto not_found;
+ if not window_open[cur_exp] then goto not_found;
+ find_edges_var(e);
+ if cur_edges<>null then disp_edges(cur_exp);
+ return;
+ not_found: cur_exp:=cur_exp*unity;
+ common_ending: exp_err("Bad window number");
+@.Bad window number@>
+ help1("It should be the number of an open window.");
+ put_get_flush_error(0); flush_token_list(e);
+ end;
+exit:end;
+
+@ The only thing difficult about `\&{openwindow}' is that the syntax
+allows the user to go astray in many ways. The following subroutine
+helps keep the necessary program reasonably short and sweet.
+
+@<Declare action procedures for use by |do_statement|@>=
+function get_pair(@!c:command_code):boolean;
+var @!p:pointer; {a pair of values that are known (we hope)}
+@!b:boolean; {did we find such a pair?}
+begin if cur_cmd<>c then get_pair:=false
+else begin get_x_next; scan_expression;
+ if nice_pair(cur_exp,cur_type) then
+ begin p:=value(cur_exp);
+ cur_x:=value(x_part_loc(p)); cur_y:=value(y_part_loc(p));
+ b:=true;
+ end
+ else b:=false;
+ flush_cur_exp(0); get_pair:=b;
+ end;
+end;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_open_window;
+label not_found,exit;
+var @!k:integer; {the window number in question}
+@!r0,@!c0,@!r1,@!c1:scaled; {window coordinates}
+begin get_x_next; scan_expression;
+if cur_type<>known then goto not_found;
+k:=round_unscaled(cur_exp);
+if k<0 then goto not_found;
+if k>15 then goto not_found;
+if not get_pair(from_token) then goto not_found;
+r0:=cur_x; c0:=cur_y;
+if not get_pair(to_token) then goto not_found;
+r1:=cur_x; c1:=cur_y;
+if not get_pair(at_token) then goto not_found;
+open_a_window(k,r0,c0,r1,c1,cur_x,cur_y); return;
+not_found:print_err("Improper `openwindow'");
+@.Improper `openwindow'@>
+help2("Say `openwindow k from (r0,c0) to (r1,c1) at (x,y)',")@/
+ ("where all quantities are known and k is between 0 and 15.");
+put_get_error;
+exit:end;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_cull;
+label not_found,exit;
+var @!e:pointer; {token list for a picture variable}
+@!keeping:drop_code..keep_code; {modifier of |cull_op|}
+@!w,@!w_in,@!w_out:integer; {culling weights}
+begin w:=1;
+get_x_next; var_flag:=cull_op; scan_primary;
+if cur_type<>token_list then
+ @<Abandon edges command because there's no variable@>
+else begin e:=cur_exp; cur_type:=vacuous; keeping:=cur_mod;
+ if not get_pair(cull_op) then goto not_found;
+ while (cur_cmd=with_option)and(cur_mod=known) do
+ if scan_with then w:=cur_exp;
+ @<Set up the culling weights,
+ or |goto not_found| if the thresholds are bad@>;
+ find_edges_var(e);
+ if cur_edges<>null then
+ cull_edges(floor_unscaled(cur_x+unity-1),floor_unscaled(cur_y),w_out,w_in);
+ return;
+ not_found: print_err("Bad culling amounts");
+@.Bad culling amounts@>
+ help1("Always cull by known amounts that exclude 0.");
+ put_get_error; flush_token_list(e);
+ end;
+exit:end;
+
+@ @<Set up the culling weights, or |goto not_found| if the thresholds are bad@>=
+if cur_x>cur_y then goto not_found;
+if keeping=drop_code then
+ begin if (cur_x>0)or(cur_y<0) then goto not_found;
+ w_out:=w; w_in:=0;
+ end
+else begin if (cur_x<=0)and(cur_y>=0) then goto not_found;
+ w_out:=0; w_in:=w;
+ end
+
+@ The \&{everyjob} command simply assigns a nonzero value to the global variable
+|start_sym|.
+
+@<Cases of |do_statement|...@>=
+every_job_command: begin get_symbol; start_sym:=cur_sym; get_x_next;
+ end;
+
+@ @<Glob...@>=
+@!start_sym:halfword; {a symbolic token to insert at beginning of job}
+
+@ @<Set init...@>=
+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
+
+@<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd...@>=
+message_command: if m<err_message_code then print("message")
+ else if m=err_message_code then print("errmessage")
+ else print("errhelp");
+
+@ @<Cases of |do_statement|...@>=
+message_command: do_message;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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
+ begin exp_err("Not a string");
+@.Not a string@>
+ help1("A message should be a known string expression.");
+ put_get_error;
+ end
+else case m of
+ message_code:begin print_nl(""); slow_print(cur_exp);
+ end;
+ err_message_code:@<Print string |cur_exp| as an error message@>;
+ err_help_code:@<Save string |cur_exp| as the |err_help|@>;
+ end; {there are no other cases}
+flush_cur_exp(0);
+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.
+
+@<Save string |cur_exp| as the |err_help|@>=
+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.
+
+@<Glob...@>=
+@!long_help_seen:boolean; {has the long \&{errmessage} help been used?}
+
+@ @<Set init...@>=long_help_seen:=false;
+
+@ @<Print string |cur_exp| as an error message@>=
+begin print_err(""); slow_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<error_stop_mode then long_help_seen:=true;
+ help4("This error message was generated by an `errmessage'")@/
+ ("command, so I can't give any explicit help.")@/
+ ("Pretend that you're Miss Marple: Examine all clues,")@/
+@^Marple, Jane@>
+ ("and deduce the truth by inspired guesses.");
+ end;
+put_get_error; use_err_help:=false;
+end
+
+@* \[45] 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 \MF'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 \MF\ 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.
+
+@<Glob...@>=
+@!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 \MF\ 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 \MF\ 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.
+@^design size@>
+
+@ 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.
+
+\def\hangg#1 {\hang\hbox{#1 }}
+\yskip\hangg|tag=0| (|no_tag|) means that |remainder| is unused.\par
+\hangg|tag=1| (|lig_tag|) means that this character has a ligature/kerning
+program starting at location |remainder| in the |lig_kern| array.\par
+\hangg|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
+\hangg|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 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 boundary character at the
+left, 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
+$$\hbox{|256*op_byte+remainder<nl|.}$$
+If such an instruction is encountered during
+normal program execution, it denotes an unconditional halt; no ligature
+or kerning command is performed.
+
+@d stop_flag=128+min_quarterword
+ {value indicating `\.{STOP}' in a lig/kern program}
+@d kern_flag=128+min_quarterword {op code for a kern step}
+@d skip_byte(#)==lig_kern[#].b0
+@d next_char(#)==lig_kern[#].b1
+@d op_byte(#)==lig_kern[#].b2
+@d rem_byte(#)==lig_kern[#].b3
+
+@ Extensible characters are specified by an |@!extensible_recipe|, which
+consists of four bytes called |@!top|, |@!mid|, |@!bot|, and |@!rep| (in this
+order). These bytes are the character codes of individual pieces used to
+build up a large symbol. If |top|, |mid|, or |bot| are zero, they are not
+present in the built-up result. For example, an extensible vertical line is
+like an extensible bracket, except that the top and bottom pieces are missing.
+
+Let $T$, $M$, $B$, and $R$ denote the respective pieces, or an empty box
+if the piece isn't present. Then the extensible characters have the form
+$TR^kMR^kB$ from top to bottom, for some |k>=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.
+@^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 \MF'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}
+
+@<Glob...@>=
+@!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; {\&{fontdimen} parameters}
+@!np:0..max_font_dimen; {the largest \&{fontdimen} 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|}
+
+@ @<Set init...@>=
+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;
+
+@ @<Declare the function called |tfm_check|@>=
+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;
+
+@ @<Store the width information for character code~|c|@>=
+if c<bc then bc:=c;
+if c>ec then ec:=c;
+char_exists[c]:=true;
+gf_dx[c]:=internal[char_dx]; gf_dy[c]:=internal[char_dy];
+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 \MF's special \.{TFM}-oriented commands.
+
+@<Cases of |do_statement|...@>=
+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
+
+@<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd...@>=
+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;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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;
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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 @<Complain about a character tag conflict@>;
+end;
+
+@ @<Complain about a character tag conflict@>=
+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
+
+@ @<Declare action procedures for use by |do_statement|@>=
+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: @<Store a list of ligature/kern steps@>;
+extensible_code: @<Define an extensible recipe@>;
+header_byte_code, font_dimen_code: begin c:=cur_mod; get_x_next;
+ scan_expression;
+ if (cur_type<>known)or(cur_exp<half_unit) then
+ begin exp_err("Improper location");
+@.Improper location@>
+ 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 fontdimen location.");
+ back_error;
+ end;
+ if c=header_byte_code then @<Store a list of header bytes@>
+ else @<Store a list of font dimensions@>;
+ end;
+ end;
+end; {there are no other cases}
+end;
+
+@ @<Store a list of ligature/kern steps@>=
+begin lk_started:=false;
+continue: get_x_next;
+if(cur_cmd=skip_to)and lk_started then
+ @<Process a |skip_to| command and |goto done|@>;
+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
+ @<Record a label in a lig/kern subprogram and |goto continue|@>;
+if cur_cmd=lig_kern_token then @<Compile a ligature/kern command@>
+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);
+@:METAFONT capacity exceeded ligtable size}{\quad ligtable size@>
+incr(nl);
+if cur_cmd=comma then goto continue;
+if skip_byte(nl-1)<stop_flag then skip_byte(nl-1):=stop_flag;
+done:end
+
+@ @<Put each...@>=
+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@>
+
+@ @<Cases of |print_cmd...@>=
+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
+
+@<Process a |skip_to| command and |goto done|@>=
+begin c:=get_code;
+if nl-skip_table[c]>128 then
+ begin skip_error(skip_table[c]); skip_table[c]:=undefined_label;
+ end;
+if skip_table[c]=undefined_label then skip_byte(nl-1):=qi(0)
+else skip_byte(nl-1):=qi(nl-skip_table[c]-1);
+skip_table[c]:=nl-1; goto done;
+end
+
+@ @<Record a label in a lig/kern subprogram and |goto continue|@>=
+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]<undefined_label then
+ begin ll:=skip_table[c]; skip_table[c]:=undefined_label;
+ repeat lll:=qo(skip_byte(ll));
+ if nl-ll>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
+
+@ @<Compile a ligature/kern...@>=
+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);
+@:METAFONT 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
+
+@<Define an extensible recipe@>=
+begin if ne=256 then overflow("extensible",256);
+@:METAFONT 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
+
+@ @<Store a list of header bytes@>=
+repeat if j>header_size then overflow("headerbyte",header_size);
+@:METAFONT capacity exceeded headerbyte}{\quad headerbyte@>
+header_byte[j]:=get_code; incr(j);
+until cur_cmd<>comma
+
+@ @<Store a list of font dimensions@>=
+repeat if j>max_font_dimen then overflow("fontdimen",max_font_dimen);
+@:METAFONT 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.
+
+\MF\ 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.
+
+@<Initialize table entries...@>=
+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 \MF'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 v<value(q) then
+ begin r:=get_node(value_node_size); value(r):=v; link(r):=q; link(p):=r;
+ end;
+sort_in:=link(p);
+end;
+
+@ Now we come to the interesting part, where we reduce the list if necessary
+until it has the required size. The |min_cover| routine is basic to this
+process; it computes the minimum number~|m| such that the values of the
+current sorted list can be covered by |m|~intervals of width~|d|. It
+also sets the global value |perturbation| to the smallest value $d'>d$
+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 then perturbation:=value(p)-l;
+ end;
+min_cover:=m;
+end;
+
+@ @<Glob...@>=
+@!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
+ @<Replace an interval of values by its midpoint@>;
+ q:=p; p:=link(p);
+ end;
+skimp:=m;
+end;
+
+@ @<Replace an interval...@>=
+begin repeat p:=link(p); info(p):=m;
+decr(excess);@+if excess=0 then d:=0;
+until value(link(p))>l+d;
+v:=l+half(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 \MF\ 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.
+
+@<Massage the \.{TFM} widths@>=
+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)
+
+@ @<Glob...@>=
+@!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.
+
+@<Massage the \.{TFM} heights, depths, and italic corrections@>=
+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)
+
+@ @<Initialize table entries...@>=
+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.
+@^design size@>
+
+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<unity)or(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]-1-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:=max_tfm_dimen@+else x:=-max_tfm_dimen;
+ end;
+x:=make_scaled(x*16,internal[design_size]);
+dimen_out:=x;
+end;
+
+@ @<Glob...@>=
+@!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 @<Compute a check sum in |(b1,b2,b3,b4)|@>;
+ 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;
+
+@ @<Compute a check sum in |(b1,b2,b3,b4)|@>=
+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;
+
+@ @<Finish the \.{TFM} file@>=
+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);
+@<Output the subfile sizes and header bytes@>;
+@<Output the character information bytes, then
+ output the dimensions themselves@>;
+@<Output the ligature/kern program@>;
+@<Output the extensible character recipes and the font metric parameters@>;
+@!stat if internal[tracing_stats]>0 then
+ @<Log the subfile sizes of the \.{TFM} file@>;@;@+tats@/
+print_nl("Font metrics written on "); slow_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.
+
+@<Output the subfile sizes and header bytes@>=
+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|}
+@<Compute the ligature/kern program offset and implant the
+ left boundary label@>;
+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
+
+@ @<Output the character information bytes...@>=
+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 $-1=|label_loc|[0]<|label_loc|[1]\le\cdots
+\le|label_loc|[|label_ptr]|$.
+
+@<Compute the ligature/kern program offset...@>=
+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;
+@<Find the minimum |lk_offset| and adjust all remainders@>;
+if bch_label<undefined_label then
+ begin skip_byte(nl):=qi(255); next_char(nl):=qi(0);
+ op_byte(nl):=qi(((bch_label+lk_offset)div 256));
+ rem_byte(nl):=qi(((bch_label+lk_offset)mod 256));
+ incr(nl); {possibly |nl=lig_table_size+1|}
+ end
+
+@ @<Find the minimum |lk_offset|...@>=
+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
+
+@ @<Output the ligature/kern program@>=
+for k:=0 to 255 do if skip_table[k]<undefined_label then
+ begin print_nl("(local label "); print_int(k); print(":: was missing)");
+@.local label l:: was missing@>
+ 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]<ll;
+ end;
+for k:=0 to nl-1 do tfm_qqqq(lig_kern[k]);
+for k:=0 to nk-1 do tfm_four(dimen_out(kern[k]))
+
+@ @<Output the extensible character recipes...@>=
+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])<fraction_half then tfm_four(param[1]*16)
+ else begin incr(tfm_changed);
+ if 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
+
+@ @<Log the subfile sizes of the \.{TFM} file@>=
+begin wlog_ln(' ');
+if bch_label<undefined_label then decr(nl);
+wlog_ln('(You used ',nw:1,'w,',@| nh:1,'h,',@| nd:1,'d,',@| ni:1,'i,',@|
+ nl:1,'l,',@| nk:1,'k,',@| ne:1,'e,',@|
+ np:1,'p metric file positions');
+wlog_ln(' out of ',@| '256w,16h,16d,64i,',@|
+ lig_table_size:1,'l,',max_kerns:1,'k,256e,',@|
+ max_font_dimen:1,'p)');
+end
+
+@* \[46] Generic font file format.
+The most important output produced by a typical run of \MF\ is the
+``generic font'' (\.{GF}) file that specifies the bit patterns of the
+characters that have been drawn. The term {\sl generic\/} indicates that
+this file format doesn't match the conventions of any name-brand manufacturer;
+but it is easy to convert \.{GF} files to the special format required by
+almost all digital phototypesetting equipment. There's a strong analogy
+between the \.{DVI} files written by \TeX\ and the \.{GF} files written
+by \MF; and, in fact, the file formats have a lot in common.
+
+A \.{GF} file is a stream of 8-bit bytes that may be
+regarded as a series of commands in a machine-like language. The first
+byte of each command is the operation code, and this code is followed by
+zero or more bytes that provide parameters to the command. The parameters
+themselves may consist of several consecutive bytes; for example, the
+`|boc|' (beginning of character) command has six parameters, each of
+which is four bytes long. Parameters are usually regarded as nonnegative
+integers; but four-byte-long parameters can be either positive or
+negative, hence they range in value from $-2^{31}$ to $2^{31}-1$.
+As in \.{TFM} files, numbers that occupy
+more than one byte position appear in BigEndian order,
+and negative numbers appear in two's complement notation.
+
+A \.{GF} file consists of a ``preamble,'' followed by a sequence of one or
+more ``characters,'' followed by a ``postamble.'' The preamble is simply a
+|pre| command, with its parameters that introduce the file; this must come
+first. Each ``character'' consists of a |boc| command, followed by any
+number of other commands that specify ``black'' pixels,
+followed by an |eoc| command. The characters appear in the order that \MF\
+generated them. If we ignore no-op commands (which are allowed between any
+two commands in the file), each |eoc| command is immediately followed by a
+|boc| command, or by a |post| command; in the latter case, there are no
+more characters in the file, and the remaining bytes form the postamble.
+Further details about the postamble will be explained later.
+
+Some parameters in \.{GF} commands are ``pointers.'' These are four-byte
+quantities that give the location number of some other byte in the file;
+the first file byte is number~0, then comes number~1, and so on.
+
+@ The \.{GF} format is intended to be both compact and easily interpreted
+by a machine. Compactness is achieved by making most of the information
+relative instead of absolute. When a \.{GF}-reading program reads the
+commands for a character, it keeps track of two quantities: (a)~the current
+column number,~|m|; and (b)~the current row number,~|n|. These are 32-bit
+signed integers, although most actual font formats produced from \.{GF}
+files will need to curtail this vast range because of practical
+limitations. (\MF\ output will never allow $\vert m\vert$ or $\vert
+n\vert$ to get extremely large, but the \.{GF} format tries to be more general.)
+
+How do \.{GF}'s row and column numbers correspond to the conventions
+of \TeX\ and \MF? Well, the ``reference point'' of a character, in \TeX's
+view, is considered to be at the lower left corner of the pixel in row~0
+and column~0. This point is the intersection of the baseline with the left
+edge of the type; it corresponds to location $(0,0)$ in \MF\ programs.
+Thus the pixel in \.{GF} row~0 and column~0 is \MF's unit square, comprising the
+region of the plane whose coordinates both lie between 0 and~1. The
+pixel in \.{GF} row~|n| and column~|m| consists of the points whose \MF\
+coordinates |(x,y)| satisfy |m<=x<=m+1| and |n<=y<=n+1|. Negative values of
+|m| and~|x| correspond to columns of pixels {\sl left\/} of the reference
+point; negative values of |n| and~|y| correspond to rows of pixels {\sl
+below\/} the baseline.
+
+Besides |m| and |n|, there's also a third aspect of the current
+state, namely the @!|paint_switch|, which is always either |black| or
+|white|. Each \\{paint} command advances |m| by a specified amount~|d|,
+and blackens the intervening pixels if |paint_switch=black|; then
+the |paint_switch| changes to the opposite state. \.{GF}'s commands are
+designed so that |m| will never decrease within a row, and |n| will never
+increase within a character; hence there is no way to whiten a pixel that
+has been blackened.
+
+@ Here is a list of all the commands that may appear in a \.{GF} file. Each
+command is specified by its symbolic name (e.g., |boc|), its opcode byte
+(e.g., 67), and its parameters (if any). The parameters are followed
+by a bracketed number telling how many bytes they occupy; for example,
+`|d[2]|' means that parameter |d| is two bytes long.
+
+\yskip\hang|paint_0| 0. This is a \\{paint} command with |d=0|; it does
+nothing but change the |paint_switch| from \\{black} to \\{white} or vice~versa.
+
+\yskip\hang\\{paint\_1} through \\{paint\_63} (opcodes 1 to 63).
+These are \\{paint} commands with |d=1| to~63, defined as follows: If
+|paint_switch=black|, blacken |d|~pixels of the current row~|n|,
+in columns |m| through |m+d-1| inclusive. Then, in any case,
+complement the |paint_switch| and advance |m| by~|d|.
+
+\yskip\hang|paint1| 64 |d[1]|. This is a \\{paint} command with a specified
+value of~|d|; \MF\ uses it to paint when |64<=d<256|.
+
+\yskip\hang|@!paint2| 65 |d[2]|. Same as |paint1|, but |d|~can be as high
+as~65535.
+
+\yskip\hang|@!paint3| 66 |d[3]|. Same as |paint1|, but |d|~can be as high
+as $2^{24}-1$. \MF\ never needs this command, and it is hard to imagine
+anybody making practical use of it; surely a more compact encoding will be
+desirable when characters can be this large. But the command is there,
+anyway, just in case.
+
+\yskip\hang|boc| 67 |c[4]| |p[4]| |min_m[4]| |max_m[4]| |min_n[4]|
+|max_n[4]|. Beginning of a character: Here |c| is the character code, and
+|p| points to the previous character beginning (if any) for characters having
+this code number modulo 256. (The pointer |p| is |-1| if there was no
+prior character with an equivalent code.) The values of registers |m| and |n|
+defined by the instructions that follow for this character must
+satisfy |min_m<=m<=max_m| and |min_n<=n<=max_n|. (The values of |max_m| and
+|min_n| need not be the tightest bounds possible.) When a \.{GF}-reading
+program sees a |boc|, it can use |min_m|, |max_m|, |min_n|, and |max_n| to
+initialize the bounds of an array. Then it sets |m:=min_m|, |n:=max_n|, and
+|paint_switch:=white|.
+
+\yskip\hang|boc1| 68 |c[1]| |@!del_m[1]| |max_m[1]| |@!del_n[1]| |max_n[1]|.
+Same as |boc|, but |p| is assumed to be~$-1$; also |del_m=max_m-min_m|
+and |del_n=max_n-min_n| are given instead of |min_m| and |min_n|.
+The one-byte parameters must be between 0 and 255, inclusive.
+\ (This abbreviated |boc| saves 19~bytes per character, in common cases.)
+
+\yskip\hang|eoc| 69. End of character: All pixels blackened so far
+constitute the pattern for this character. In particular, a completely
+blank character might have |eoc| immediately following |boc|.
+
+\yskip\hang|skip0| 70. Decrease |n| by 1 and set |m:=min_m|,
+|paint_switch:=white|. \ (This finishes one row and begins another,
+ready to whiten the leftmost pixel in the new row.)
+
+\yskip\hang|skip1| 71 |d[1]|. Decrease |n| by |d+1|, set |m:=min_m|, and set
+|paint_switch:=white|. This is a way to produce |d| all-white rows.
+
+\yskip\hang|@!skip2| 72 |d[2]|. Same as |skip1|, but |d| can be as large
+as 65535.
+
+\yskip\hang|@!skip3| 73 |d[3]|. Same as |skip1|, but |d| can be as large
+as $2^{24}-1$. \MF\ obviously never needs this command.
+
+\yskip\hang|new_row_0| 74. Decrease |n| by 1 and set |m:=min_m|,
+|paint_switch:=black|. \ (This finishes one row and begins another,
+ready to {\sl blacken\/} the leftmost pixel in the new row.)
+
+\yskip\hang|@!new_row_1| through |@!new_row_164| (opcodes 75 to 238). Same as
+|new_row_0|, but with |m:=min_m+1| through |min_m+164|, respectively.
+
+\yskip\hang|xxx1| 239 |k[1]| |x[k]|. This command is undefined in
+general; it functions as a $(k+2)$-byte |no_op| unless special \.{GF}-reading
+programs are being used. \MF\ generates \\{xxx} commands when encountering
+a \&{special} string; this occurs in the \.{GF} file only between
+characters, after the preamble, and before the postamble. However,
+\\{xxx} commands might appear within characters,
+in \.{GF} files generated by other
+processors. It is recommended that |x| be a string having the form of a
+keyword followed by possible parameters relevant to that keyword.
+
+\yskip\hang|@!xxx2| 240 |k[2]| |x[k]|. Like |xxx1|, but |0<=k<65536|.
+
+\yskip\hang|xxx3| 241 |k[3]| |x[k]|. Like |xxx1|, but |0<=k<@t$2^{24}$@>|.
+\MF\ uses this when sending a \&{special} string whose length exceeds~255.
+
+\yskip\hang|@!xxx4| 242 |k[4]| |x[k]|. Like |xxx1|, but |k| can be
+ridiculously large; |k| mustn't be negative.
+
+\yskip\hang|yyy| 243 |y[4]|. This command is undefined in general;
+it functions as a 5-byte |no_op| unless special \.{GF}-reading programs
+are being used. \MF\ puts |scaled| numbers into |yyy|'s, as a
+result of \&{numspecial} commands; the intent is to provide numeric
+parameters to \\{xxx} commands that immediately precede.
+
+\yskip\hang|@!no_op| 244. No operation, do nothing. Any number of |no_op|'s
+may occur between \.{GF} commands, but a |no_op| cannot be inserted between
+a command and its parameters or between two parameters.
+
+\yskip\hang|char_loc| 245 |c[1]| |dx[4]| |dy[4]| |w[4]| |p[4]|.
+This command will appear only in the postamble, which will be explained shortly.
+
+\yskip\hang|@!char_loc0| 246 |c[1]| |@!dm[1]| |w[4]| |p[4]|.
+Same as |char_loc|, except that |dy| is assumed to be zero, and the value
+of~|dx| is taken to be |65536*dm|, where |0<=dm<256|.
+
+\yskip\hang|pre| 247 |i[1]| |k[1]| |x[k]|.
+Beginning of the preamble; this must come at the very beginning of the
+file. Parameter |i| is an identifying number for \.{GF} format, currently
+131. The other information is merely commentary; it is not given
+special interpretation like \\{xxx} commands are. (Note that \\{xxx}
+commands may immediately follow the preamble, before the first |boc|.)
+
+\yskip\hang|post| 248. Beginning of the postamble, see below.
+
+\yskip\hang|post_post| 249. Ending of the postamble, see below.
+
+\yskip\noindent Commands 250--255 are undefined at the present time.
+
+@d gf_id_byte=131 {identifies the kind of \.{GF} files described here}
+
+@ \MF\ refers to the following opcodes explicitly.
+
+@d paint_0=0 {beginning of the \\{paint} commands}
+@d paint1=64 {move right a given number of columns, then
+ black${}\swap{}$white}
+@d boc=67 {beginning of a character}
+@d boc1=68 {short form of |boc|}
+@d eoc=69 {end of a character}
+@d skip0=70 {skip no blank rows}
+@d skip1=71 {skip over blank rows}
+@d new_row_0=74 {move down one row and then right}
+@d max_new_row=164 {the largest \\{new\_row} command is |new_row_164|}
+@d xxx1=239 {for \&{special} strings}
+@d xxx3=241 {for long \&{special} strings}
+@d yyy=243 {for \&{numspecial} numbers}
+@d char_loc=245 {character locators in the postamble}
+@d pre=247 {preamble}
+@d post=248 {postamble beginning}
+@d post_post=249 {postamble ending}
+
+@ The last character in a \.{GF} file is followed by `|post|'; this command
+introduces the postamble, which summarizes important facts that \MF\ has
+accumulated. The postamble has the form
+$$\vbox{\halign{\hbox{#\hfil}\cr
+ |post| |p[4]| |@!ds[4]| |@!cs[4]| |@!hppp[4]| |@!vppp[4]|
+ |@!min_m[4]| |@!max_m[4]| |@!min_n[4]| |@!max_n[4]|\cr
+ $\langle\,$character locators$\,\rangle$\cr
+ |post_post| |q[4]| |i[1]| 223's$[{\G}4]$\cr}}$$
+Here |p| is a pointer to the byte following the final |eoc| in the file
+(or to the byte following the preamble, if there are no characters);
+it can be used to locate the beginning of \\{xxx} commands
+that might have preceded the postamble. The |ds| and |cs| parameters
+@^design size@> @^check sum@>
+give the design size and check sum, respectively, which are exactly the
+values put into the header of the \.{TFM} file that \MF\ produces (or
+would produce) on this run. Parameters |hppp| and |vppp| are the ratios of
+pixels per point, horizontally and vertically, expressed as |scaled| integers
+(i.e., multiplied by $2^{16}$); they can be used to correlate the font
+with specific device resolutions, magnifications, and ``at sizes.'' Then
+come |min_m|, |max_m|, |min_n|, and |max_n|, which bound the values that
+registers |m| and~|n| assume in all characters in this \.{GF} file.
+(These bounds need not be the best possible; |max_m| and |min_n| may, on the
+other hand, be tighter than the similar bounds in |boc| commands. For
+example, some character may have |min_n=-100| in its |boc|, but it might
+turn out that |n| never gets lower than |-50| in any character; then
+|min_n| can have any value |<=-50|. If there are no characters in the file,
+it's possible to have |min_m>max_m| and/or |min_n>max_n|.)
+
+@ Character locators are introduced by |char_loc| commands,
+which specify a character residue~|c|, character escapements (|dx,dy|),
+a character width~|w|, and a pointer~|p|
+to the beginning of that character. (If two or more characters have the
+same code~|c| modulo 256, only the last will be indicated; the others can be
+located by following backpointers. Characters whose codes differ by a
+multiple of 256 are assumed to share the same font metric information,
+hence the \.{TFM} file contains only residues of character codes modulo~256.
+This convention is intended for oriental languages, when there are many
+character shapes but few distinct widths.)
+@^oriental characters@>@^Chinese characters@>@^Japanese characters@>
+
+The character escapements (|dx,dy|) are the values of \MF's \&{chardx}
+and \&{chardy} parameters; they are in units of |scaled| pixels;
+i.e., |dx| is in horizontal pixel units times $2^{16}$, and |dy| is in
+vertical pixel units times $2^{16}$. This is the intended amount of
+displacement after typesetting the character; for \.{DVI} files, |dy|
+should be zero, but other document file formats allow nonzero vertical
+escapement.
+
+The character width~|w| duplicates the information in the \.{TFM} file; it
+is a |fix_word| value relative to the design size, and it should be
+independent of magnification.
+
+The backpointer |p| points to the character's |boc|, or to the first of
+a sequence of consecutive \\{xxx} or |yyy| or |no_op| commands that
+immediately precede the |boc|, if such commands exist; such ``special''
+commands essentially belong to the characters, while the special commands
+after the final character belong to the postamble (i.e., to the font
+as a whole). This convention about |p| applies also to the backpointers
+in |boc| commands, even though it wasn't explained in the description
+of~|boc|. @^backpointers@>
+
+Pointer |p| might be |-1| if the character exists in the \.{TFM} file
+but not in the \.{GF} file. This unusual situation can arise in \MF\ output
+if the user had |proofing<0| when the character was being shipped out,
+but then made |proofing>=0| in order to get a \.{GF} file.
+
+@ The last part of the postamble, following the |post_post| byte that
+signifies the end of the character locators, contains |q|, a pointer to the
+|post| command that started the postamble. An identification byte, |i|,
+comes next; this currently equals~131, as in the preamble.
+
+The |i| byte is followed by four or more bytes that are all equal to
+the decimal number 223 (i.e., @'337 in octal). \MF\ puts out four to seven of
+these trailing bytes, until the total length of the file is a multiple of
+four bytes, since this works out best on machines that pack four bytes per
+word; but any number of 223's is allowed, as long as there are at least four
+of them. In effect, 223 is a sort of signature that is added at the very end.
+@^Fuchs, David Raymond@>
+
+This curious way to finish off a \.{GF} file makes it feasible for
+\.{GF}-reading programs to find the postamble first, on most computers,
+even though \MF\ wants to write the postamble last. Most operating
+systems permit random access to individual words or bytes of a file, so
+the \.{GF} reader can start at the end and skip backwards over the 223's
+until finding the identification byte. Then it can back up four bytes, read
+|q|, and move to byte |q| of the file. This byte should, of course,
+contain the value 248 (|post|); now the postamble can be read, so the
+\.{GF} reader can discover all the information needed for individual characters.
+
+Unfortunately, however, standard \PASCAL\ does not include the ability to
+@^system dependencies@>
+access a random position in a file, or even to determine the length of a file.
+Almost all systems nowadays provide the necessary capabilities, so \.{GF}
+format has been designed to work most efficiently with modern operating systems.
+But if \.{GF} files have to be processed under the restrictions of standard
+\PASCAL, one can simply read them from front to back. This will
+be adequate for most applications. However, the postamble-first approach
+would facilitate a program that merges two \.{GF} files, replacing data
+from one that is overridden by corresponding data in the other.
+
+@* \[47] Shipping characters out.
+The |ship_out| procedure, to be described below, is given a pointer to
+an edge structure. Its mission is to describe the positive pixels
+in \.{GF} form, outputting a ``character'' to |gf_file|.
+
+Several global variables hold information about the font file as a whole:\
+|gf_min_m|, |gf_max_m|, |gf_min_n|, and |gf_max_n| are the minimum and
+maximum \.{GF} coordinates output so far; |gf_prev_ptr| is the byte number
+following the preamble or the last |eoc| command in the output;
+|total_chars| is the total number of characters (i.e., |boc..eoc| segments)
+shipped out. There's also an array, |char_ptr|, containing the starting
+positions of each character in the file, as required for the postamble. If
+character code~|c| has not yet been output, |char_ptr[c]=-1|.
+
+@<Glob...@>=
+@!gf_min_m,@!gf_max_m,@!gf_min_n,@!gf_max_n:integer; {bounding rectangle}
+@!gf_prev_ptr:integer; {where the present/next character started/starts}
+@!total_chars:integer; {the number of characters output so far}
+@!char_ptr:array[eight_bits] of integer; {where individual characters started}
+@!gf_dx,@!gf_dy:array[eight_bits] of integer; {device escapements}
+
+@ @<Set init...@>=
+gf_prev_ptr:=0; total_chars:=0;
+
+@ The \.{GF} bytes are output to a buffer instead of being sent
+byte-by-byte to |gf_file|, because this tends to save a lot of
+subroutine-call overhead. \MF\ uses the same conventions for |gf_file|
+as \TeX\ uses for its \\{dvi\_file}; hence if system-dependent
+changes are needed, they should probably be the same for both programs.
+
+The output buffer is divided into two parts of equal size; the bytes found
+in |gf_buf[0..half_buf-1]| constitute the first half, and those in
+|gf_buf[half_buf..gf_buf_size-1]| constitute the second. The global
+variable |gf_ptr| points to the position that will receive the next
+output byte. When |gf_ptr| reaches |gf_limit|, which is always equal
+to one of the two values |half_buf| or |gf_buf_size|, the half buffer that
+is about to be invaded next is sent to the output and |gf_limit| is
+changed to its other value. Thus, there is always at least a half buffer's
+worth of information present, except at the very beginning of the job.
+
+Bytes of the \.{GF} file are numbered sequentially starting with 0;
+the next byte to be generated will be number |gf_offset+gf_ptr|.
+
+@<Types...@>=
+@!gf_index=0..gf_buf_size; {an index into the output buffer}
+
+@ Some systems may find it more efficient to make |gf_buf| a |packed|
+array, since output of four bytes at once may be facilitated.
+@^system dependencies@>
+
+@<Glob...@>=
+@!gf_buf:array[gf_index] of eight_bits; {buffer for \.{GF} output}
+@!half_buf:gf_index; {half of |gf_buf_size|}
+@!gf_limit:gf_index; {end of the current half buffer}
+@!gf_ptr:gf_index; {the next available buffer address}
+@!gf_offset:integer; {|gf_buf_size| times the number of times the
+ output buffer has been fully emptied}
+
+@ Initially the buffer is all in one piece; we will output half of it only
+after it first fills up.
+
+@<Set init...@>=
+half_buf:=gf_buf_size div 2; gf_limit:=gf_buf_size; gf_ptr:=0;
+gf_offset:=0;
+
+@ The actual output of |gf_buf[a..b]| to |gf_file| is performed by calling
+|write_gf(a,b)|. It is safe to assume that |a| and |b+1| will both be
+multiples of 4 when |write_gf(a,b)| is called; therefore it is possible on
+many machines to use efficient methods to pack four bytes per word and to
+output an array of words with one system call.
+@^system dependencies@>
+
+@<Declare generic font output procedures@>=
+procedure write_gf(@!a,@!b:gf_index);
+var k:gf_index;
+begin for k:=a to b do write(gf_file,gf_buf[k]);
+end;
+
+@ To put a byte in the buffer without paying the cost of invoking a procedure
+each time, we use the macro |gf_out|.
+
+@d gf_out(#)==@+begin gf_buf[gf_ptr]:=#; incr(gf_ptr);
+ if gf_ptr=gf_limit then gf_swap;
+ end
+
+@<Declare generic font output procedures@>=
+procedure gf_swap; {outputs half of the buffer}
+begin if gf_limit=gf_buf_size then
+ begin write_gf(0,half_buf-1); gf_limit:=half_buf;
+ gf_offset:=gf_offset+gf_buf_size; gf_ptr:=0;
+ end
+else begin write_gf(half_buf,gf_buf_size-1); gf_limit:=gf_buf_size;
+ end;
+end;
+
+@ Here is how we clean out the buffer when \MF\ is all through; |gf_ptr|
+will be a multiple of~4.
+
+@<Empty the last bytes out of |gf_buf|@>=
+if gf_limit=half_buf then write_gf(half_buf,gf_buf_size-1);
+if gf_ptr>0 then write_gf(0,gf_ptr-1)
+
+@ The |gf_four| procedure outputs four bytes in two's complement notation,
+without risking arithmetic overflow.
+
+@<Declare generic font output procedures@>=
+procedure gf_four(@!x:integer);
+begin if x>=0 then gf_out(x div three_bytes)
+else begin x:=x+@'10000000000;
+ x:=x+@'10000000000;
+ gf_out((x div three_bytes) + 128);
+ end;
+x:=x mod three_bytes; gf_out(x div unity);
+x:=x mod unity; gf_out(x div @'400);
+gf_out(x mod @'400);
+end;
+
+@ Of course, it's even easier to output just two or three bytes.
+
+@<Declare generic font output procedures@>=
+procedure gf_two(@!x:integer);
+begin gf_out(x div @'400); gf_out(x mod @'400);
+end;
+@#
+procedure gf_three(@!x:integer);
+begin gf_out(x div unity); gf_out((x mod unity) div @'400);
+gf_out(x mod @'400);
+end;
+
+@ We need a simple routine to generate a \\{paint}
+command of the appropriate type.
+
+@<Declare generic font output procedures@>=
+procedure gf_paint(@!d:integer); {here |0<=d<65536|}
+begin if d<64 then gf_out(paint_0+d)
+else if d<256 then
+ begin gf_out(paint1); gf_out(d);
+ end
+else begin gf_out(paint1+1); gf_two(d);
+ end;
+end;
+
+@ And |gf_string| outputs one or two strings. If the first string number
+is nonzero, an \\{xxx} command is generated.
+
+@<Declare generic font output procedures@>=
+procedure gf_string(@!s,@!t:str_number);
+var @!k:pool_pointer;
+@!l:integer; {length of the strings to output}
+begin if s<>0 then
+ begin l:=length(s);
+ if t<>0 then l:=l+length(t);
+ if l<=255 then
+ begin gf_out(xxx1); gf_out(l);
+ end
+ else begin gf_out(xxx3); gf_three(l);
+ end;
+ for k:=str_start[s] to str_start[s+1]-1 do gf_out(so(str_pool[k]));
+ end;
+if t<>0 then for k:=str_start[t] to str_start[t+1]-1 do gf_out(so(str_pool[k]));
+end;
+
+@ The choice between |boc| commands is handled by |gf_boc|.
+
+@d one_byte(#)== #>=0 then if #<256
+
+@<Declare generic font output procedures@>=
+procedure gf_boc(@!min_m,@!max_m,@!min_n,@!max_n:integer);
+label exit;
+begin if min_m<gf_min_m then gf_min_m:=min_m;
+if max_n>gf_max_n then gf_max_n:=max_n;
+if boc_p=-1 then if one_byte(boc_c) then
+ if one_byte(max_m-min_m) then if one_byte(max_m) then
+ if one_byte(max_n-min_n) then if one_byte(max_n) then
+ begin gf_out(boc1); gf_out(boc_c);@/
+ gf_out(max_m-min_m); gf_out(max_m);
+ gf_out(max_n-min_n); gf_out(max_n); return;
+ end;
+gf_out(boc); gf_four(boc_c); gf_four(boc_p);@/
+gf_four(min_m); gf_four(max_m); gf_four(min_n); gf_four(max_n);
+exit: end;
+
+@ Two of the parameters to |gf_boc| are global.
+
+@<Glob...@>=
+@!boc_c,@!boc_p:integer; {parameters of the next |boc| command}
+
+@ Here is a routine that gets a \.{GF} file off to a good start.
+
+@d check_gf==@t@>@+if output_file_name=0 then init_gf
+
+@<Declare generic font output procedures@>=
+procedure init_gf;
+var @!k:eight_bits; {runs through all possible character codes}
+@!t:integer; {the time of this run}
+begin gf_min_m:=4096; gf_max_m:=-4096; gf_min_n:=4096; gf_max_n:=-4096;
+for k:=0 to 255 do char_ptr[k]:=-1;
+@<Determine the file extension, |gf_ext|@>;
+set_output_file_name;
+gf_out(pre); gf_out(gf_id_byte); {begin to output the preamble}
+old_setting:=selector; selector:=new_string; print(" METAFONT output ");
+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);@/
+selector:=old_setting; gf_out(cur_length);
+gf_string(0,make_string); decr(str_ptr);
+pool_ptr:=str_start[str_ptr]; {flush that string from memory}
+gf_prev_ptr:=gf_offset+gf_ptr;
+end;
+
+@ @<Determine the file extension...@>=
+if internal[hppp]<=0 then gf_ext:=".gf"
+else begin old_setting:=selector; selector:=new_string; print_char(".");
+ print_int(make_scaled(internal[hppp],59429463));
+ {$2^{32}/72.27\approx59429463.07$}
+ print("gf"); gf_ext:=make_string; selector:=old_setting;
+ end
+
+@ With those preliminaries out of the way, |ship_out| is not especially
+difficult.
+
+@<Declare generic font output procedures@>=
+procedure ship_out(@!c:eight_bits);
+label done;
+var @!f:integer; {current character extension}
+@!prev_m,@!m,@!mm:integer; {previous and current pixel column numbers}
+@!prev_n,@!n:integer; {previous and current pixel row numbers}
+@!p,@!q:pointer; {for list traversal}
+@!prev_w,@!w,@!ww:integer; {old and new weights}
+@!d:integer; {data from edge-weight node}
+@!delta:integer; {number of rows to skip}
+@!cur_min_m:integer; {starting column, relative to the current offset}
+@!x_off,@!y_off:integer; {offsets, rounded to integers}
+begin check_gf; f:=round_unscaled(internal[char_ext]);@/
+x_off:=round_unscaled(internal[x_offset]);
+y_off:=round_unscaled(internal[y_offset]);
+if term_offset>max_print_line-9 then print_ln
+else if (term_offset>0)or(file_offset>0) then print_char(" ");
+print_char("["); print_int(c);
+if f<>0 then
+ begin print_char("."); print_int(f);
+ end;
+update_terminal;
+boc_c:=256*f+c; boc_p:=char_ptr[c]; char_ptr[c]:=gf_prev_ptr;@/
+if internal[proofing]>0 then @<Send nonzero offsets to the output file@>;
+@<Output the character represented in |cur_edges|@>;
+gf_out(eoc); gf_prev_ptr:=gf_offset+gf_ptr; incr(total_chars);
+print_char("]"); update_terminal; {progress report}
+if internal[tracing_output]>0 then
+ print_edges(" (just shipped out)",true,x_off,y_off);
+end;
+
+@ @<Send nonzero offsets to the output file@>=
+begin if x_off<>0 then
+ begin gf_string("xoffset",0); gf_out(yyy); gf_four(x_off*unity);
+ end;
+if y_off<>0 then
+ begin gf_string("yoffset",0); gf_out(yyy); gf_four(y_off*unity);
+ end;
+end
+
+@ @<Output the character represented in |cur_edges|@>=
+prev_n:=4096; p:=knil(cur_edges); n:=n_max(cur_edges)-zero_field;
+while p<>cur_edges do
+ begin @<Output the pixels of edge row |p| to font row |n|@>;
+ p:=knil(p); decr(n);
+ end;
+if prev_n=4096 then @<Finish off an entirely blank character@>
+else if prev_n+y_off<gf_min_n then
+ gf_min_n:=prev_n+y_off
+
+@ @<Finish off an entirely blank...@>=
+begin gf_boc(0,0,0,0);
+if gf_max_m<0 then gf_max_m:=0;
+if gf_min_n>0 then gf_min_n:=0;
+end
+
+@ In this loop, |prev_w| represents the weight at column |prev_m|, which is
+the most recent column reflected in the output so far; |w| represents the
+weight at column~|m|, which is the most recent column in the edge data.
+Several edges might cancel at the same column position, so we need to
+look ahead to column~|mm| before actually outputting anything.
+
+@<Output the pixels of edge row |p| to font row |n|@>=
+if unsorted(p)>void then sort_edges(p);
+q:=sorted(p); w:=0; prev_m:=-fraction_one; {$|fraction_one|\approx\infty$}
+ww:=0; prev_w:=0; m:=prev_m;
+repeat if q=sentinel then mm:=fraction_one
+else begin d:=ho(info(q)); mm:=d div 8; ww:=ww+(d mod 8)-zero_w;
+ end;
+if mm<>m then
+ begin if prev_w<=0 then
+ begin if w>0 then @<Start black at $(m,n)$@>;
+ end
+ else if w<=0 then @<Stop black at $(m,n)$@>;
+ m:=mm;
+ end;
+w:=ww; q:=link(q);
+until mm=fraction_one;
+if w<>0 then {this should be impossible}
+ print_nl("(There's unbounded black in character shipped out!)");
+@.There's unbounded black...@>
+if prev_m-m_offset(cur_edges)+x_off>gf_max_m then
+ gf_max_m:=prev_m-m_offset(cur_edges)+x_off
+
+
+@ @<Start black at $(m,n)$@>=
+begin if prev_m=-fraction_one then @<Start a new row at $(m,n)$@>
+else gf_paint(m-prev_m);
+prev_m:=m; prev_w:=w;
+end
+
+@ @<Stop black at $(m,n)$@>=
+begin gf_paint(m-prev_m); prev_m:=m; prev_w:=w;
+end
+
+@ @<Start a new row at $(m,n)$@>=
+begin if prev_n=4096 then
+ begin gf_boc(m_min(cur_edges)+x_off-zero_field,
+ m_max(cur_edges)+x_off-zero_field,@|
+ n_min(cur_edges)+y_off-zero_field,n+y_off);
+ cur_min_m:=m_min(cur_edges)-zero_field+m_offset(cur_edges);
+ end
+else if prev_n>n+1 then @<Skip down |prev_n-n| rows@>
+else @<Skip to column $m$ in the next row and |goto done|, or skip zero rows@>;
+gf_paint(m-cur_min_m); {skip to column $m$, painting white}
+done:prev_n:=n;
+end
+
+@ @<Skip to column $m$ in the next row...@>=
+begin delta:=m-cur_min_m;
+if delta>max_new_row then gf_out(skip0)
+else begin gf_out(new_row_0+delta); goto done;
+ end;
+end
+
+@ @<Skip down...@>=
+begin delta:=prev_n-n-1;
+if delta<@'400 then
+ begin gf_out(skip1); gf_out(delta);
+ end
+else begin gf_out(skip1+1); gf_two(delta);
+ end;
+end
+
+@ 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.
+
+@<Cases of |do_statement|...@>=
+special_command: do_special;
+
+@ @<Put each...@>=
+primitive("special",special_command,string_type);@/
+@!@:special_}{\&{special} primitive@>
+primitive("numspecial",special_command,known);@/
+@!@:num_special_}{\&{numspecial} primitive@>
+
+@ @<Declare action procedures for use by |do_statement|@>=
+procedure do_special;
+var @!m:small_number; {either |string_type| or |known|}
+begin m:=cur_mod; get_x_next; scan_expression;
+if internal[proofing]>=0 then
+ if cur_type<>m then @<Complain about improper special operation@>
+ else begin check_gf;
+ if m=string_type then gf_string(cur_exp,0)
+ else begin gf_out(yyy); gf_four(cur_exp);
+ end;
+ end;
+flush_cur_exp(0);
+end;
+
+@ @<Complain about improper special operation@>=
+begin exp_err("Unsuitable expression");
+@.Unsuitable expression@>
+help1("The expression shown above has the wrong type to be output.");
+put_get_error;
+end
+
+@ @<Send the current expression as a title to the output file@>=
+begin check_gf; gf_string("title ",cur_exp);
+@.title@>
+end
+
+@ @<Cases of |print_cmd...@>=
+special_command:if m=known then print("numspecial")
+ else print("special");
+
+@ @<Determine if a character has been shipped out@>=
+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
+
+@ At the end of the program we must finish things off by writing the postamble.
+The \.{TFM} information should have been computed first.
+
+An integer variable |k| and a |scaled| variable |x| will be declared for
+use by this routine.
+
+@<Finish the \.{GF} file@>=
+begin gf_out(post); {beginning of the postamble}
+gf_four(gf_prev_ptr); gf_prev_ptr:=gf_offset+gf_ptr-5; {|post| location}
+gf_four(internal[design_size]*16);
+for k:=1 to 4 do gf_out(header_byte[k]); {the check sum}
+gf_four(internal[hppp]);
+gf_four(internal[vppp]);@/
+gf_four(gf_min_m); gf_four(gf_max_m);
+gf_four(gf_min_n); gf_four(gf_max_n);
+for k:=0 to 255 do if char_exists[k] then
+ begin x:=gf_dx[k] div unity;
+ if (gf_dy[k]=0)and(x>=0)and(x<256)and(gf_dx[k]=x*unity) then
+ begin gf_out(char_loc+1); gf_out(k); gf_out(x);
+ end
+ else begin gf_out(char_loc); gf_out(k);
+ gf_four(gf_dx[k]); gf_four(gf_dy[k]);
+ end;
+ x:=value(tfm_width[k]);
+ if abs(x)>max_tfm_dimen then
+ if x>0 then x:=three_bytes-1@+else x:=1-three_bytes
+ else x:=make_scaled(x*16,internal[design_size]);
+ gf_four(x); gf_four(char_ptr[k]);
+ end;
+gf_out(post_post); gf_four(gf_prev_ptr); gf_out(gf_id_byte);@/
+k:=4+((gf_buf_size-gf_ptr) mod 4); {the number of 223's}
+while k>0 do
+ begin gf_out(223); decr(k);
+ end;
+@<Empty the last bytes out of |gf_buf|@>;
+print_nl("Output written on "); slow_print(output_file_name);
+@.Output written...@>
+print(" ("); print_int(total_chars); print(" character");
+if total_chars<>1 then print_char("s");
+print(", "); print_int(gf_offset+gf_ptr); print(" bytes).");
+b_close(gf_file);
+end
+
+@* \[48] Dumping and undumping the tables.
+After \.{INIMF} has seen a collection of macros, it
+can write all the necessary information on an auxiliary file so
+that production versions of \MF\ 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.
+@.INIMF@>
+
+The global variable |base_ident| is a string that is printed right
+after the |banner| line when \MF\ is ready to start. For \.{INIMF} this
+string says simply `\.{(INIMF)}'; for other versions of \MF\ it says,
+for example, `\.{(preloaded base=plain 1984.2.29)}', showing the year,
+month, and day that the base file was created. We have |base_ident=0|
+before \MF's tables are loaded.
+
+@<Glob...@>=
+@!base_ident:str_number;
+
+@ @<Set init...@>=
+base_ident:=0;
+
+@ @<Initialize table entries...@>=
+base_ident:=" (INIMF)";
+
+@ @<Declare act...@>=
+@!init procedure store_base_file;
+var @!k:integer; {all-purpose index}
+@!p,@!q: pointer; {all-purpose pointers}
+@!x: integer; {something to dump}
+@!w: four_quarters; {four ASCII codes}
+begin @<Create the |base_ident|, open the base file,
+ and inform the user that dumping has begun@>;
+@<Dump constants for consistency check@>;
+@<Dump the string pool@>;
+@<Dump the dynamic memory@>;
+@<Dump the table of equivalents and the hash table@>;
+@<Dump a few more things and the closing check word@>;
+@<Close the base file@>;
+end;
+tini
+
+@ Corresponding to the procedure that dumps a base file, we also have a function
+that reads~one~in. The function returns |false| if the dumped base is
+incompatible with the present \MF\ table sizes, etc.
+
+@d off_base=6666 {go here if the base 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@>@<Declare the function called |open_base_file|@>@;
+function load_base_file:boolean;
+label off_base,exit;
+var @!k:integer; {all-purpose index}
+@!p,@!q: pointer; {all-purpose pointers}
+@!x: integer; {something undumped}
+@!w: four_quarters; {four ASCII codes}
+begin @<Undump constants for consistency check@>;
+@<Undump the string pool@>;
+@<Undump the dynamic memory@>;
+@<Undump the table of equivalents and the hash table@>;
+@<Undump a few more things and the closing check word@>;
+load_base_file:=true; return; {it worked!}
+off_base: wake_up_terminal;
+ wterm_ln('(Fatal base file error; I''m stymied)');
+@.Fatal base file error@>
+load_base_file:=false;
+exit:end;
+
+@ Base files consist of |memory_word| items, and we use the following
+macros to dump words of different types:
+
+@d dump_wd(#)==begin base_file^:=#; put(base_file);@+end
+@d dump_int(#)==begin base_file^.int:=#; put(base_file);@+end
+@d dump_hh(#)==begin base_file^.hh:=#; put(base_file);@+end
+@d dump_qqqq(#)==begin base_file^.qqqq:=#; put(base_file);@+end
+
+@<Glob...@>=
+@!base_file:word_file; {for input or output of base 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|.
+System error messages should be suppressed when undumping.
+@^system dependencies@>
+
+@d undump_wd(#)==begin get(base_file); #:=base_file^;@+end
+@d undump_int(#)==begin get(base_file); #:=base_file^.int;@+end
+@d undump_hh(#)==begin get(base_file); #:=base_file^.hh;@+end
+@d undump_qqqq(#)==begin get(base_file); #:=base_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 constants for consistency check@>=
+dump_int(@$);@/
+dump_int(mem_min);@/
+dump_int(mem_top);@/
+dump_int(hash_size);@/
+dump_int(hash_prime);@/
+dump_int(max_in_open)
+
+@ Sections of a \.{WEB} program that are ``commented out'' still contribute
+strings to the string pool; therefore \.{INIMF} and \MF\ will have
+the same strings. (And it is, of course, a good thing that they do.)
+@.WEB@>
+@^string pool@>
+
+@<Undump constants for consistency check@>=
+x:=base_file^.int;
+if x<>@$ 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
+
+@ @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)
+
+@<Dump the string pool@>=
+dump_int(pool_ptr);
+dump_int(str_ptr);
+for k:=0 to str_ptr do dump_int(str_start[k]);
+k:=0;
+while k+4<pool_ptr do
+ begin dump_four_ASCII; k:=k+4;
+ end;
+k:=pool_ptr-4; dump_four_ASCII;
+print_ln; print_int(str_ptr); print(" strings of total length ");
+print_int(pool_ptr)
+
+@ @d undump_four_ASCII==
+ undump_qqqq(w);
+ str_pool[k]:=si(qo(w.b0)); str_pool[k+1]:=si(qo(w.b1));
+ str_pool[k+2]:=si(qo(w.b2)); str_pool[k+3]:=si(qo(w.b3))
+
+@<Undump the string pool@>=
+undump_size(0)(pool_size)('string pool size')(pool_ptr);
+undump_size(0)(max_strings)('max strings')(str_ptr);
+for k:=0 to str_ptr do
+ begin undump(0)(pool_ptr)(str_start[k]); str_ref[k]:=max_str_ref;
+ end;
+k:=0;
+while k+4<pool_ptr do
+ begin undump_four_ASCII; k:=k+4;
+ end;
+k:=pool_ptr-4; undump_four_ASCII;
+init_str_ptr:=str_ptr; init_pool_ptr:=pool_ptr;
+max_str_ptr:=str_ptr; max_pool_ptr:=pool_ptr
+
+@ By sorting the list of available spaces in the variable-size portion of
+|mem|, we are usually able to get by without having to dump very much
+of the dynamic memory.
+
+We recompute |var_used| and |dyn_used|, so that \.{INIMF} dumps valid
+information even when it has not been gathering statistics.
+
+@<Dump the dynamic memory@>=
+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 the dynamic memory@>=
+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 the table of equivalents and the hash table@>=
+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 the table of equivalents and the hash table@>=
+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 from appearing again.
+
+@<Dump a few more things and the closing check word@>=
+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(base_ident);
+dump_int(bg_loc); dump_int(eg_loc); dump_int(serial_no); dump_int(69069);
+internal[tracing_stats]:=0
+
+@ @<Undump a few more things and the closing check word@>=
+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)(base_ident);
+undump(1)(hash_end)(bg_loc);
+undump(1)(hash_end)(eg_loc);
+undump_int(serial_no);@/
+undump_int(x);@+if (x<>69069)or eof(base_file) then goto off_base
+
+@ @<Create the |base_ident|...@>=
+selector:=new_string;
+print(" (preloaded base="); print(job_name); print_char(" ");
+print_int(round_unscaled(internal[year])); 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); base_ident:=make_string; str_ref[base_ident]:=max_str_ref;@/
+pack_job_name(base_extension);
+while not w_open_out(base_file) do
+ prompt_file_name("base file name",base_extension);
+print_nl("Beginning to dump on file ");
+@.Beginning to dump...@>
+slow_print(w_make_name_string(base_file)); flush_string(str_ptr-1);
+print_nl(""); slow_print(base_ident)
+
+@ @<Close the base file@>=
+w_close(base_file)
+
+@* \[49] The main program.
+This is it: the part of \MF\ 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 @<Declare the basic parsing subroutines@>@;
+@<Declare miscellaneous procedures that were declared |forward|@>@;
+@<Last-minute procedures@>
+
+@ We've noted that there are two versions of \MF84. One, called \.{INIMF},
+@.INIMF@>
+has to be run first; it initializes everything from scratch, without
+reading a base file, and it has the capability of dumping a base file.
+The other one is called `\.{VIRMF}'; it is a ``virgin'' program that needs
+@.VIRMF@>
+to input a base file in order to get started. \.{VIRMF} typically has
+a bit more memory capacity than \.{INIMF}, because it does not need the
+space consumed by the dumping/undumping routines and the numerous calls on
+|primitive|, etc.
+
+The \.{VIRMF} program cannot read a base file instantaneously, of course;
+the best implementations therefore allow for production versions of \MF\ that
+not only avoid the loading routine for \PASCAL\ object code, they also have
+a base 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 \.{VIRMF} is first
+loaded.\quad(2)~After we have read in a base file and initialized
+everything, we set |ready_already:=314159|.\quad(3)~Soon \.{VIRMF}
+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@>
+
+On systems that allow such preloading, the standard program called \.{MF}
+should be the one that has \.{plain} base preloaded, since that agrees
+with {\sl The {\logos METAFONT\/}book}. Other versions, e.g., \.{CMMF},
+should also be provided for commonly used bases such as \.{cmbase}.
+@:METAFONTbook}{\sl The {\logos METAFONT\/}book@>
+@.cmbase@>
+@.plain@>
+
+@<Glob...@>=
+@!ready_already:integer; {a sacrifice of purity for economy}
+
+@ Now this is really it: \MF\ 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_MF;
+@<Check the ``constant'' values...@>@;
+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_ptr:=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_MF: @<Initialize the output routines@>;
+@<Get the first line of input and prepare to start@>;
+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_MF: close_files_and_terminate;
+final_end: ready_already:=0;
+end.
+
+@ Here we do whatever is needed to complete \MF'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@>
+
+If |final_cleanup| is bypassed, this program doesn't bother to close
+the input files that may still be open.
+
+@<Last-minute...@>=
+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}
+@!x:scaled; {a |tfm_width| value being output to the \.{GF} file}
+begin
+@!stat if internal[tracing_stats]>0 then
+ @<Output statistics about this job@>;@;@+tats@/
+wake_up_terminal; @<Finish the \.{TFM} and \.{GF} files@>;
+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...@>
+ slow_print(log_name); print_char(".");
+ end;
+ end;
+end;
+
+@ We want to finish the \.{GF} file if and only if it has already been started;
+this will be true if and only if |gf_prev_ptr| is positive.
+We want to produce a \.{TFM} file if and only if |fontmaking| is positive.
+The \.{TFM} widths must be computed if there's a \.{GF} file, even if
+there's going to be no \.{TFM}~file.
+
+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.
+
+@<Finish the \.{TFM} and \.{GF} files@>=
+if (gf_prev_ptr>0)or(internal[fontmaking]>0) then
+ begin @<Make the dynamic memory into one big available node@>;
+ @<Massage the \.{TFM} widths@>;
+ fix_design_size; fix_check_sum;
+ if internal[fontmaking]>0 then
+ begin @<Massage the \.{TFM} heights, depths, and italic corrections@>;
+ internal[fontmaking]:=0; {avoid loop in case of fatal error}
+ @<Finish the \.{TFM} file@>;
+ end;
+ if gf_prev_ptr>0 then @<Finish the \.{GF} file@>;
+ end
+
+@ @<Make the dynamic memory into one big available node@>=
+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 \MF\ is being used.
+
+@<Output statistics...@>=
+if log_opened then
+ begin wlog_ln(' ');
+ wlog_ln('Here is how much of METAFONT''s memory',' you used:');
+@.Here is how much...@>
+ wlog(' ',max_str_ptr-init_str_ptr:1,' string');
+ if max_str_ptr<>init_str_ptr+1 then wlog('s');
+ wlog_ln(' out of ', max_strings-init_str_ptr:1);@/
+ wlog_ln(' ',max_pool_ptr-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_rounding_ptr:1,'r,',@|
+ max_param_stack:1,'p,',@|
+ max_buf_stack+1:1,'b stack positions out of ',@|
+ stack_size:1,'i,',
+ max_internal:1,'n,',
+ max_wiggle:1,'r,',
+ param_size:1,'p,',
+ buf_size:1,'b');
+ end
+
+@ We get to the |final_cleanup| routine when \&{end} or \&{dump} has
+been scanned.
+
+@<Last-minute...@>=
+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); loop_ptr:=cond_ptr;
+ cond_ptr:=link(cond_ptr); free_node(loop_ptr,if_node_size);
+ end;
+if history<>spotless then
+ if ((history=warning_issued)or(interaction<error_stop_mode)) then
+ if selector=term_and_log then
+ begin selector:=term_only;
+ print_nl("(see the transcript file for additional information)");
+@.see the transcript file...@>
+ selector:=term_and_log;
+ end;
+if c=1 then
+ begin @!init store_base_file; return;@+tini@/
+ print_nl("(dump is performed only by INIMF)"); return;
+@.dump...only by INIMF@>
+ end;
+exit:end;
+
+@ @<Last-minute...@>=
+@!init procedure init_prim; {initialize all the primitives}
+begin
+@<Put each...@>;
+end;
+@#
+procedure init_tab; {initialize other tables}
+var @!k:integer; {all-purpose index}
+begin @<Initialize table entries (done by \.{INIMF} only)@>@;
+end;
+tini
+
+@ When we begin the following code, \MF'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, \MF\ is ready to call on the
+|main_control| routine to do its work.
+
+@<Get the first line...@>=
+begin @<Initialize the input routines@>;
+if (base_ident=0)or(buffer[loc]="&") then
+ begin if base_ident<>0 then initialize; {erase preloaded base}
+ if not open_base_file then goto final_end;
+ if not load_base_file then
+ begin w_close(base_file); goto final_end;
+ end;
+ w_close(base_file);
+ while (loc<limit)and(buffer[loc]=" ") do incr(loc);
+ end;
+buffer[limit]:="%";@/
+fix_date_and_time; init_randoms(sys_time+sys_day*unity);@/
+@<Initialize the print |selector|...@>;
+if loc<limit then if buffer[loc]<>"\" then start_input; {\&{input} assumed}
+end
+
+@* \[50] Debugging.
+Once \MF\ 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 \MF\ 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 \MF\ 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}
+
+@<Last-minute...@>=
+@!debug procedure debug_help; {routine to display various things}
+label breakpoint,exit;
+var @!k,@!l,@!m,@!n:integer;
+begin clear_terminal;
+ 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 declared label at least once}
+ breakpoint: m:=0; @{'BREAKPOINT'@}@/
+ end
+ else begin read(term_in,n);
+ case m of
+ @t\4@>@<Numbered cases for |debug_help|@>@;
+ othercases print("?")
+ endcases;
+ end;
+ end;
+exit:end;
+gubed
+
+@ @<Numbered cases...@>=
+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: slow_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;
+
+@* \[51] System-dependent changes.
+This section should be replaced, if necessary, by any special
+modifications of the program
+that are necessary to make \MF\ 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@>
+
+@* \[52] 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 \MF\ 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 \MF's running time, exclusive of input and output.