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-rw-r--r--Master/texmf-dist/source/generic/mfpic/grafbase.dtx1211
1 files changed, 1020 insertions, 191 deletions
diff --git a/Master/texmf-dist/source/generic/mfpic/grafbase.dtx b/Master/texmf-dist/source/generic/mfpic/grafbase.dtx
index a309f242f6e..8b7346720dd 100644
--- a/Master/texmf-dist/source/generic/mfpic/grafbase.dtx
+++ b/Master/texmf-dist/source/generic/mfpic/grafbase.dtx
@@ -1,17 +1,17 @@
% \iffalse
% File: grafbase.dtx
-% A part of mfpic 1.06 2011/03/08
+% A part of mfpic 1.10 2012/12/03
%
% -------------------------------------------------------------------
%
-% Copyright 2002--2011, Daniel H. Luecking
+% Copyright 2002--2012, Daniel H. Luecking
%
% Mfpic may be distributed and/or modified under the conditions of the
% LaTeX Project Public License, either version 1.3b of this license or (at
% your option) any later version. The latest version of this license is in
% <http://www.latex-project.org/lppl.txt>
-% and version 1.3b or later is part of all distributions of LaTeX version
-% 2003/12/01 or later.
+% and version 1.3c or later is part of all distributions of LaTeX version
+% 2008/12/01 or later.
%
% Mfpic has maintenance status "author-maintained". The Current Maintainer
% is Daniel H. Luecking. There are several Base Interpreters: plain TeX, LaTeX,
@@ -19,7 +19,7 @@
%
%<*driver>
\ProvidesFile{grafbase.dtx}
- [2011/03/08 v1.06. Metafont/post macros to interface with mfpic.]%
+ [2012/12/03 v1.10. Metafont/post macros to interface with mfpic.]%
\documentclass{ltxdoc}
\usepackage{docmfp}
@@ -51,6 +51,7 @@
\renewcommand\|{${}\mathrel{|}{}$}
\makeatletter
+\let\HD@SetMacroIndent\@gobble
\newcommand\bsl{{\mytt\@backslashchar}}
% Stupid lists!
\def\@listi{\leftmargin\leftmargini
@@ -113,7 +114,7 @@
%</driver>
%\fi
%
-% \CheckSum{1473}
+% \CheckSum{1631}
% \CharacterTable
% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
@@ -146,14 +147,17 @@
% This file documents the \grafbase{} source code. The user manual for
% \mfpic{} is distributed as \file{mfpic-doc.pdf}, produced from
% \file{mfpic-doc.tex}. An introductory guide to \mfpic{} is available
-% in \file{mfpguide.pdf}, produced from \file{mfpguide.pdf}
+% in \file{mfpguide.pdf}, produced from \file{mfpguide.tex}
% \end{abstract}
%
% \StopEventually{\PrintIndex}
% \tableofcontents
%
+%
+%
% \section{Introduction}\label{intro}
%
+%
% \subsection{Identification and checks}\label{checks}
%
% \DescribeVariable{grafbaseversion} We use \mfc{grafbaseversion} to
@@ -187,13 +191,14 @@ fi
boolean grafbase; grafbase := true;
string fileversion, filedate;
-fileversion := "1.06"; filedate := "2011/02/25";
+fileversion := "1.10"; filedate := "2012/12/03";
message " Loading grafbase macros, version " & fileversion & ", " &
filedate & ".";
message " ";
-def GBmsg expr s = message "Grafbase (" & jobname & "): " & s; enddef;
+def GBmsg expr s = message "Grafbase (" & jobname & "): " & s;
+enddef;
def GBwarn expr s = GBmsg "Warning, " & s; enddef;
def GBerrmsg (expr s) expr t = errhelp t;
errmessage "Grafbase (" & jobname & "): " & s; errhelp "";
@@ -236,7 +241,7 @@ def checkversions (expr g)=
fi
enddef;
-checkversions (106);
+checkversions (110);
% \end{macrocode}
%
@@ -298,29 +303,30 @@ enddef;
% In \MF{} `\mfc{color X}' is an identifier (presumably unknown) with the
% base name \mfc{color} and suffix \mfc{X}.
% \begin{macrocode}
-boolean METAPOST;
-METAPOST := known color Geamparalele din Babadag;
+boolean METAFONT, METAPOST;
+METAPOST := known color Carl Philipp Emanuel Bach;
+if METAPOST: METAFONT := false; else: METAFONT := true; fi
%<*MF>
if METAPOST:
GBerrmsg ("wrong compiler.")
- "This file is for Metafont. For Metapost use grafbase.mp.";
+ "This file is for Metafont. For Metapost, use grafbase.mp.";
fi
%</MF>
%<*MP>
-if not METAPOST:
+if METAFONT:
GBerrmsg ("wrong compiler.")
- "This file is for Metapost. For Metafont use grafbase.mf.";
+ "This file is for Metapost. For Metafont, use grafbase.mf.";
fi
% \end{macrocode}
%
% \MP{} now exists in a couple of slightly incompatible versions.
-% Versions 1.000 and later (beta versions 0.900 also) have native support for
-% \texttt{CMYK} colors with a \mfc{cmykcolor} data type. It also
-% supports grayscale colors (i.e., \mfc{withcolor} will accept a numeric
-% expression), and has the alias \mfc{rgbcolor} for \mfc{color}.
+% Versions 1.000 and later (beta versions 0.9xx also) have native support for
+% \texttt{CMYK} colors with a \mfc{cmykcolor} data type. They also
+% support grayscale colors (i.e., \mfc{withcolor} will accept a numeric
+% expression), and have the alias \mfc{rgbcolor} for \mfc{color}.
% It also has a means to set the name of the output file: the
% \mfc{filenametemplate} command. In versions 1.2 and later, this is
% deprecated in favor of setting the internal string variable
@@ -351,6 +357,7 @@ fi
%</MP>
% \end{macrocode}
%
+%
% \subsection{Setting up the font, \MF{} only}\label{font}
%
% Font-related housekeeping is only for \MF{}. \MF{} only produces
@@ -360,9 +367,10 @@ fi
% \DescribeVariable{GBgeneric}
% We intercept the \mfc{mode} variable before \mfc{mode_setup} can set
% \mfc{proof} mode. We used to set \mfc{mode := cx} (and later
-% \mfc{ljfour}) if it was unknown. For a while we just issued an error
-% message. In this version we define a 600dpi mode called \mfc{GBgeneric}
-% as a fallback (neither \gbc{mode} nor \gbc{localfont} known).
+% \mfc{ljfour}) if \mfc{mode} was unknown. For a while we just issued an
+% error message. In this version we define a 600dpi mode called
+% \mfc{GBgeneric} as a fallback (neither \gbc{mode} nor \gbc{localfont}
+% known).
%
% The font identifier and coding scheme are just for information and end
% up as comments in the \file{.tfm} file (in all capitals). The design
@@ -430,6 +438,7 @@ interim warningcheck := 0;
% \end{macrocode}
%
+%
% \subsection{Initializations}\label{init}
%
% \VariableIndex{unitlen}
@@ -460,15 +469,17 @@ yneg := 0; ypos := 10;
% We support both degrees and radians for angles. In \MF, one degree is
% the unit of angle.
% \DescribeVariable{radian}
-% One radian is $180/\pi$ degrees. We also define \gbc{pi} so a user can say
-% \gbc{90} or \gbc{90deg} or \gbc{pi/2*radian}
+% One radian is $180/\pi$ degrees. We also define \gbc{pi} so a user can
+% say \gbc{pi/2*radian} for almost the same effect as \gbc{90deg}.
% \DescribeVariable{pi}
-% for the same effect. Actually, not quite: because of \MF{}'s precision
-% limits, the latter is about 90.00025 degrees. \MF{}'s precision is 16
-% binary places, or slightly under 5 decimals. The accuracy of \gbc{pi}
-% and \gbc{radian} is the maximum possible. If we \emph{define}
-% \gbc{radian} to be \gbc{90/(pi/2)} or \gbc{180/pi} the value of
-% \gbc{pi/2*radian} is even less accurate.
+% But not quite: because of \MF{}'s precision limits, the former is about
+% 90.00025 degrees. \MF{}'s precision is 16 binary places, or slightly
+% under 5 decimals. The accuracy of \gbc{pi} and \gbc{radian} below is the
+% maximum possible. If we \emph{define} \gbc{radian} by its mathematical
+% definition \gbc{radian:=180/pi}, then \gbc{radian} and formulas
+% containing it are even less accurate. (Coincidentally, defining
+% \gbc{radian} as below, and then \gbc{pi := 180/radian} produces exactly
+% the same value for \gbc{pi} as below.)
% \begin{macrocode}
newinternal deg, pi, radian;
deg := 1; pi := 3.14159;
@@ -500,7 +511,7 @@ numeric degree; degree := deg;
%
% \VariableIndex{hatchwd}
% The default \gbc{hatchwd} used to be larger, but it seemed ugly to me.
-% (Backward compatibility? What's that?).
+% (Backward compatibility---what's that?).
% \begin{macrocode}
newinternal penwd; penwd := 0.5pt;
pen drawpen;
@@ -572,6 +583,7 @@ boolean showbbox; showbbox := false;
% \end{macrocode}
%
+%
% \subsubsection{Colors}\label{colors}
%
% Of course colors are only recognized by \MP. The colors \mfc{black},
@@ -595,7 +607,7 @@ let cmykcolor = numeric;
black := 0; white := 1;
def withcolor text t = enddef;
%</MF>
-%<MP>if not has_cmyk: let rgbcolor = color; let cmykcolor = color; fi
+%<MP>if not has_cmyk: let rgbcolor = color; let cmykcolor = color; fi
def _wc_ = withcolor enddef;
% \end{macrocode}
@@ -653,32 +665,61 @@ enddef;
% two. In \MF{} \gbc{white} is a numeric and \gbc{cmyk} returns a
% numeric, so these can be used with \MF{}, and both produce the same
% result.
-%
-% \DescribeRoutine{grayscalegray}
-% The grayscale version should return a numeric in recent \MP{}, so
-% it needs a different definition for early \MP{}. Thus, it occurs
-% in the conditional code. Oddly, its definition is the same for \MF{}
-% and recent \MP{}.
% \begin{macrocode}
vardef rgbgray (expr g) = (snapto g) * white enddef;
vardef cmykgray (expr g) = cmyk(0,0,0,1 - snapto g) enddef;
% \end{macrocode}
%
+%
+% \DescribeRoutine{colorchoice}
+% The \gbc{colorchoice} function (like \cs{mathchoice} in \TeX{}, after
+% which it was named) returns one of four bits of code: \gbc{D} (default)
+% if the first argument is unknown or not one of the recognized color
+% models, \gbc{N} if it is numeric, \gbc{R} if it is \mfc{rgbcolor}, and
+% \gbc{C} if it is \mfc{cmykcolor}. These arguments have to be \mfc{text}:
+% if they were `\mfc{expr}' \MP{} would try to evaluate them, with
+% possible errors since some of them apply functions that are not relevant
+% to the other types.
+%
+% Since this is mostly used to return values inside vardef's, it is
+% important this not be followed by a semicolon. If it is used in
+% another context, semicolons would normally be part of the arguments.
+% \begin{macrocode}
+%<*MP>
+def colorchoice (expr clr) (text D)(text N)(text R)(text C) =
+ if unknown clr: D
+ elseif numeric clr: N
+ elseif rgbcolor clr: R
+ elseif cmykcolor clr: C
+ else: D
+ fi
+enddef;
+
+% \end{macrocode}
% In recent \MP{}, all the color functions are essentially no-ops. In
% early \MP{}, they all return an \opt{rgb} color expression. In \MF{}
% they all return a numeric. It is easiest if we simply separate the three
-% cases and write the code for each, rather than load all the functions
-% with three-way booleans (often containing nested booleans).
+% cases (MF, old MP and recent MP) and write the code for each, rather
+% than load all the functions with three-way booleans (often containing
+% nested booleans).
%
+% \DescribeRoutine{gray}
+% \DescribeRoutine{cmyk}
% For all three engines we require a definition of the color functions
% \gbc{gray(g)}, \gbc{rgb(r,g,b)}, and \gbc{cmyk(c,m,y,k)}, as well as
-% conversion functions \gbc{makegray(x)}, \gbc{makergb(x)}, and
-% \gbc{makecmyk(x)}, and the boolean \gbc{iscolor clr}. The first three
-% have to return numerics for \MF{}, colors for early \MP{}, and the
-% associated color type for recent \MP{}.
+% conversion functions (see below), and the boolean function
+% \gbc{iscolor}. The first three have to return numerics for \MF{},
+% colors for early \MP{}, and the associated color type for recent \MP{}.
+% We delay the definition of \gbc{rgb} because it only requires
+% distinguishing \MF{} from \MP.
+%
+% \DescribeRoutine{grayscalegray}
+% The grayscale version should return a numeric in recent \MP{}, so
+% it needs a different definition for early \MP{}. Thus, it occurs
+% in the conditional code. Oddly, its definition is the same for \MF{}
+% and recent \MP{}.
% \begin{macrocode}
-%<*MP>
if has_cmyk :
vardef grayscalegray (expr g) = snapto g enddef;
vardef gray (expr g) = grayscalegray (g) enddef;
@@ -688,34 +729,13 @@ if has_cmyk :
% \end{macrocode}
%
-% \DescribeRoutine{colorchoice}
-% The \gbc{colorchoice} function (like \cs{mathchoice} in \TeX{}, after
-% which it was named) returns one of four bits of code: \gbc{D} (default)
-% if the first argument is unknown or not one of the recognized color
-% models, \gbc{N} if it is numeric, \gbc{R} if it is \mfc{rgbcolor}, and
-% \gbc{C} if it is \mfc{cmykcolor}. These arguments have to be \mfc{text}:
-% if they were `\mfc{expr}' \MP{} would try to evaluate them, with
-% possible errors since some of them apply functions that are not relevant
-% to the other types.
-% \begin{macrocode}
- def colorchoice (expr clr) (text D)(text N)(text R)(text C) =
- if unknown clr: D
- elseif numeric clr: N
- elseif rgbcolor clr: R
- elseif cmykcolor clr: C
- else: D
- fi
- enddef;
-
-% \end{macrocode}
-%
% \DescribeRoutine{makecmyk}
% \DescribeRoutine{makergb}
% \DescribeRoutine{makegray}
% In \gbc{makecmyk} and all the other `\gbc{make}' conversions, the
% default is to return black in the appropriate model, numerics produce
-% gray, and cmyk or rgb is either retained unchanged or converted to the
-% appropriate model.
+% gray in the appropriate model, and cmyk or rgb is either retained
+% unchanged or converted to the appropriate model.
%
% \DescribeRoutine{iscolor}
% A variable is taken to be a color if it can validly follow
@@ -729,14 +749,14 @@ if has_cmyk :
enddef;
vardef makergb primary clr =
colorchoice (clr)(rgbblack)(rgbgray(clr))(clr)
- (cmyktorgb(cyanpart clr, magentapart clr, yellowpart clr,
- blackpart clr))
+ (cmyktorgb(cyanpart clr, magentapart clr,
+ yellowpart clr, blackpart clr))
enddef;
vardef makegray primary clr =
colorchoice (clr)(grayscaleblack)(grayscalegray(clr))
(rgbtogray (redpart clr, greenpart clr, bluepart clr))
- (cmyktogray(cyanpart clr, magentapart clr, yellowpart clr,
- blackpart clr))
+ (cmyktogray(cyanpart clr, magentapart clr,
+ yellowpart clr, blackpart clr))
enddef;
vardef iscolor expr clr =
(rgbcolor clr) or (cmykcolor clr) or (numeric clr) or (boolean clr)
@@ -744,10 +764,13 @@ if has_cmyk :
else:
% \end{macrocode}
%
-% \DescribeRoutine{colorchoice}
-% In early \MP{} \gbc{colorchoice} is a three-way choice, since
-% \mfc{cmykcolor} is not an available data type, but numeric can still be
-% interpreted as a gray.
+% In early \MP{} \gbc{colorchoice} is only a three-way choice, since
+% \mfc{cmykcolor} is not a data type, but numeric can still be
+% interpreted as a gray. For a numeric or an actual rgbcolor, the first or
+% second branch would be taken. If \gbc{clr} is neither of those then
+% \mfc{cmykcolor}, being equal to \mfc{rgbcolor}, would also be false.
+% Therefore, in the context of early \MP{}, it is irrelevant what goes in
+% the last argument, so we leave it empty.
%
% \DescribeRoutine{makecmyk}
% \DescribeRoutine{makergb}
@@ -762,19 +785,12 @@ else:
vardef gray (expr g) = rgbgray(g) enddef;
vardef grayscalegray (expr g) = rgbgray(g) enddef;
vardef cmyk (expr c, m, y, k) = rgb (1-c-k, 1-m-k, 1-y-k) enddef;
- def colorchoice (expr clr) (text D)(text N)(text R) =
- if unknown clr: D
- elseif numeric clr: N
- elseif rgbcolor clr: R
- else: D
- fi
- enddef;
vardef makergb primary clr =
- colorchoice (clr)(rgbblack)(rgbgray(clr))(clr)
+ colorchoice (clr)(rgbblack)(rgbgray(clr))(clr)()
enddef;
vardef makegray primary clr =
colorchoice (clr)(rgbblack)(rgbgray(clr))
- (rgbtogray (redpart clr, greenpart clr, bluepart clr))
+ (rgbtogray (redpart clr, greenpart clr, bluepart clr))()
enddef;
def makecmyk = makergb enddef;
vardef iscolor expr clr = (color clr) enddef;
@@ -783,7 +799,7 @@ fi
%</MP>
% \end{macrocode}
%
-% \DescribeMacro{knowncolor}
+% \DescribeRoutine{knowncolor}
% Once we have \gbc{iscolor} all we need to do is add a test for
% \mfc{known} to get this boolean test.
% \begin{macrocode}
@@ -808,7 +824,7 @@ vardef iscolor expr clr = (color clr) enddef;
%</MF>
% \end{macrocode}
%
-% \DescribeRoutine{forcecolor}
+% \DescribeRoutine{forceclr}
% \DescribeRoutine{named}
% This is only used in the \gbc{named} function to force a color. In
% \MF{} the tests are all `\mfc{if numeric}'. In early \MP{} `\mfc{if
@@ -870,6 +886,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Arrays}\label{arrays}
%
% \gbc{ClipPath} is a typical example of an array. Arrays are based on the
@@ -999,6 +1016,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Utilities}\label{utilities}
%
% \DescribeRoutine{chpair}
@@ -1025,13 +1043,14 @@ vardef chpair (text proc) (expr p) =
(proc (xpart p), proc (ypart p))
enddef;
-vardef floorpair (expr p) = (floor (xpart p), floor (ypart p)) enddef;
-vardef ceilingpair (expr p) =
- (ceiling (xpart p), ceiling (ypart p))
+vardef floorpair (expr p) = (floor (xpart p), floor (ypart p))
+enddef;
+vardef ceilingpair (expr p) = (ceiling (xpart p), ceiling (ypart p))
enddef;
%<*MF>
-def hroundpair (expr p) = (hround (xpart p), hround (ypart p)) enddef;
+def hroundpair (expr p) = (hround (xpart p), hround (ypart p))
+enddef;
vardef goodpair (expr p) = hroundpair(p.t_) enddef;
%</MF>
@@ -1090,7 +1109,8 @@ enddef;
% things, twice the area of the triangle with two sides $z\sb1$ and
% $z\sb2$. It is used only in \gbc{mkconvex}.
% \begin{macrocode}
-primarydef Z xprod W = (xpart Z * ypart W - xpart W * ypart Z) enddef;
+primarydef Z xprod W = (xpart Z * ypart W - xpart W * ypart Z)
+enddef;
% \end{macrocode}
%
@@ -1262,7 +1282,7 @@ def endimage =
enddef;
def makeimage (suffix name) (expr refpt) =
- setpair (_image_reference_point) zconv(refpt);
+ setpair (_image_reference_point) zconv (refpt);
setpicture (name) beginimage
enddef;
def concludeimage =
@@ -1481,6 +1501,8 @@ enddef;
% \end{macrocode}
%
+%
+%
% \section{The \grafbase{} Coordinate System}\label{coordinate}
%
% We need to make a distinction between graph units, sharped units, and
@@ -1526,6 +1548,7 @@ enddef;
% want the lower left corner of the graph space to have device coordinates
% $(0,0)$.
%
+%
% \subsection{The main transforms}\label{ztr}
%
% \DescribeVariable{vtr}
@@ -1621,6 +1644,7 @@ vardef invvconv (expr v) = v transformed (inverse vtr) enddef;
% \end{macrocode}
%
+%
% \subsection{The \gbc{mfpic} environment}\label{mfpic}
%
% \DescribeRoutine{active_plane}
@@ -1643,7 +1667,7 @@ def active_plane = currentpicture enddef;
%
% If \gbc{underlaylabels} is true, we try to make them part of the
% background, adding them to the picture variable \gbc{background_labels}.
-% Just before shipout, the picture is placed on top.
+% Just before shipout, the picture is placed on top of these labels.
%
% If \gbc{overlaylabels} is \gbc{true}, we try to make labels in \MP{}
% behave the same as labels in \TeX{} (for \mfpic) by adding the labels
@@ -1695,8 +1719,8 @@ enddef;
% \mfpic.
%
% \DescribeRoutine{bounds}
-% This used to be for compatibility also, but I decided it was a
-% convenient abbreviation, so \mfpic{} uses it now.
+% This also used to be unused, for compatibility only, but I decided it was a
+% convenient abbreviation and \mfpic{} uses it again.
% \begin{macrocode}
def mfpicenv = enddef;
def endmfpicenv = enddef;
@@ -1844,6 +1868,7 @@ enddef;
% \end{macrocode}
%
%
+%
% \section{Text}\label{text}
%
% In the \MP{} version, \gbc{label_adjust}, \gbc{label_sep} and
@@ -1879,6 +1904,7 @@ label_sep := 0; labelpath_sep := 0;
% \end{macrocode}
%
+%
% \subsection{Placement of text, \MP{} only}\label{placement}
%
% \DescribeRoutine{newgblabel}
@@ -1996,6 +2022,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Decorating the text, \MF{} or \MP{}}\label{decorating}
%
% The three macros \gbc{textrect}, \gbc{textoval} and \gbc{textellipse}
@@ -2111,7 +2138,8 @@ vardef textrectx (expr a, b, c, rot, xy, lbl, rad, loc) =
fi
readjustdims (ll, ur) (label_sep - labelpath_sep);
- invvconv (thegblabel (ref_shift(a, b, c, ll, ur), rot, f)) shifted loc
+ invvconv (thegblabel (ref_shift(a, b, c, ll, ur), rot, f))
+ shifted loc
enddef;
% \end{macrocode}
@@ -2160,7 +2188,8 @@ vardef xellipse (expr aspect, a, b, c, r, xy, lbl, mult, loc) =
f := ellipse (cc, aa, bb, 0);
fi
readjustdims (ll, ur) (label_sep - labelpath_sep);
- invvconv (thegblabel (ref_shift(a, b, c, ll, ur), r, f)) shifted loc
+ invvconv (thegblabel (ref_shift(a, b, c, ll, ur), r, f))
+ shifted loc
fi
enddef;
@@ -2197,6 +2226,7 @@ enddef;
% \end{macrocode}
%
%
+%
% \section{Additional Functions}\label{functions}
%
% Complex variable functions are provided, which interpret a pair $(x, y)$
@@ -2214,8 +2244,12 @@ enddef;
% in \MF.)
%
% The value \gbc{eps/2 + epsilon} is the smallest value with
-% reciprocal less than \mfc{infinity}. I set \gbc{nottoosmall} a speck
-% bigger to ensure that the same is true of \gbc{2*(nottoosmall/2)}.
+% reciprocal less than \mfc{infinity}. I set \gbc{nottoosmall} to
+% \gbc{eps/2 + 2epsilon} to ensure that the same is true of
+% \gbc{2*(nottoosmall/2)}. This is probably not necessary as
+% \mfc{epsilon/2} should round up to \mfc{epsilon} and not be lost. But
+% it also ensures that \gbc{nottoosmall} equals \gbc{2*(nottoosmall/2)},
+% which could be useful.
%
% We set \gbc{secd x = 1/(cosd x)} unless \gbc{cosd x} is less than
% \gbc{reallysmall}, then we set it equal to \gbc{1/reallysmall}. We do a
@@ -2330,9 +2364,9 @@ vardef radians (expr t) = t/radian enddef;
% \RoutineIndex{invcos}\gbc{invcos} and \RoutineIndex{invtan}\gbc{invtan})
% that return angles in radians.
% \begin{macrocode}
-vardef invcos primary X = (acos X)/radian enddef;
-vardef invsin primary X = (asin X)/radian enddef;
-vardef invtan primary X = (atan X)/radian enddef;
+vardef invcos primary X = radians (acos X) enddef;
+vardef invsin primary X = radians (asin X) enddef;
+vardef invtan primary X = radians (atan X) enddef;
% \end{macrocode}
%
@@ -2346,10 +2380,10 @@ vardef invtan primary X = (atan X)/radian enddef;
% \begin{macrocode}
vardef exp primary X = mexp (256 * X) enddef;
vardef ln primary X = (mlog X) / 256 enddef;
-def log = ln enddef;
+vardef log primary X = ln (X) enddef;
vardef logbase (expr B) primary X = (mlog X)/(mlog B) enddef;
-def logtwo = logbase( 2) enddef;
-def logten = logbase(10) enddef;
+vardef logtwo primary X = logbase( 2) (X) enddef;
+vardef logten primary X = logbase(10) (X) enddef;
% \end{macrocode}
%
@@ -2437,28 +2471,53 @@ enddef;
% These give some of the more basic functions of standard complex
% analysis: \RoutineIndex{Arg}\gbc{Arg}, \RoutineIndex{Log}\gbc{Log},
% \RoutineIndex{cis}\gbc{cis}, \RoutineIndex{zexp}\gbc{zexp},
-% \RoutineIndex{sgn}\gbc{sgn}, and \RoutineIndex{conj}\gbc{conj}.
+% \RoutineIndex{sgn}\gbc{sgn}, \RoutineIndex{zsqrt}\gbc{zsqrt} and
+% \RoutineIndex{conj}\gbc{conj}.
% \begin{macrocode}
vardef Arg primary Z = (angle Z)/radian enddef;
vardef Log primary Z = (ln (abs Z), Arg Z) enddef;
vardef cis primary T = dir (T*radian) enddef;
vardef zexp primary Z = (exp (xpart Z)) * cis (ypart Z) enddef;
-vardef sgn primary Z = if not (Z = origin): unitvector fi Z enddef;
+vardef sgn primary Z = if not (Z = origin): unitvector fi Z
+enddef;
+vardef zsqrt primary Z =
+ if Z = origin: origin else: sqrt(abs(Z)) * dir ((angle Z)/2) fi
+enddef;
vardef conj primary Z = (xpart Z, -ypart Z) enddef;
% \end{macrocode}
%
+% DescribeRoutine{zmul}
+% Unfortunately, while \MF{} will happily add and subtract pairs, it
+% will not multiply or divide them without help. We provide alternatives
+% \DescribeRoutine{zdiv} here.
+% \begin{macrocode}
+primarydef Z zmul W = Z zscaled W enddef;
+primarydef Z zdiv W =
+ Z zmul ( unitvector (conj W) / (abs W) )
+enddef;
+
+% \end{macrocode}
+%
% \DescribeRoutine{Moebius}
% A less basic operation: the Moebius shift which takes the disk $|z| <
% 1$ onto itself. It is a hyperbolic geometry analog of shifting points
% in Euclidean geometry. Its mathematical definition (all variables are
% complex numbers):
% \[
-% M_a(z) = \frac{z + a}{1 - \bar az}
+% M_a(z) = \frac{z + a}{1 + \bar az}
% \]
+% Its inverse is $M_{-a}$.
+%
% \DescribeRoutine{pshdist}
% Related to \gbc{Moebius} is the pseudohyperbolic metric. The distance
-% between $z$ and $w$ in this metric is $|z-w|/|1 - \bar wz|$.
+% between $z$ and $w$ in this metric is $|z-w|/|1 - \bar wz|$. There is
+% \DescribeRoutine{pshdist_hp}
+% also a version of this for the upper half-plane: $|z-w|/|z-\bar w|$.
+%
+% Closely related to all this is Kelvin transform. In complex notation
+% it is simply $1/\bar z = z/|z|^2$. The term ``Kelvin transform'' is
+% normally only used in real variables (of any dimension greater than 1).
% \begin{macrocode}
vardef Moebius (expr A) primary Z =
save _D; pair _D;
@@ -2466,6 +2525,17 @@ vardef Moebius (expr A) primary Z =
(Z + A)/(abs _D) rotated (- angle _D)
enddef;
vardef pshdist (expr Z,W) = abs(Moebius(-W)(Z)) enddef;
+vardef pshdist_hp (expr Z,W) = abs(Z-W)/abs(Z-conj(W)) enddef;
+vardef kelvin (expr Z) =
+ save tmp_; tmp_ = abs(Z);
+ if tmp_ = 0:
+ (infinity, infinity)
+ elseif tmp_ < reallysmall:
+ infinity*unitvector Z
+ else:
+ (1/tmp_)*unitvector Z
+ fi
+enddef;
% \end{macrocode}
%
@@ -2485,6 +2555,13 @@ def id (expr x) = x enddef;
% integer (that is, satisfy \mfc{x=floor x}). Here we redefine
% \prog{plain}'s \mfc{**}, intercepting the case of a positive integer
% power of an integer.
+%
+% There are some negative powers, and some integer powers of nonintegers
+% that can also be calculated exactly within \MF{}'s limited precision,
+% but it is difficult to determine those cases programmatically. Computing
+% every integer power by repeated multiplication or division might
+% actually reduce accuracy in the nonexact cases, so we limit ourselves to
+% this one special case.
% \begin{macrocode}
primarydef x**y =
if y=2: x*x
@@ -2497,6 +2574,7 @@ let ^ = **;
% \end{macrocode}
%
+%
% \section{Coordinate Systems and Transformations}\label{systems}
%
% \DescribeVariable{T_stack}
@@ -2532,6 +2610,7 @@ def ecoords = hide ( T_pop (ztr); vtr := vectorpart ztr ) enddef;
% \end{macrocode}
%
+%
% \subsection{Coordinate changes}\label{changes}
%
% \DescribeRoutine{apply_t}
@@ -2545,7 +2624,8 @@ def ecoords = hide ( T_pop (ztr); vtr := vectorpart ztr ) enddef;
% phrase which, were it to follow a path, would produce a transformed
% path. Knuth calls such a phrase a \emph{transformer}.
% \begin{macrocode}
-vardef vectorpart primary T = T shifted -(origin transformed T) enddef;
+vardef vectorpart primary T = T shifted -(origin transformed T)
+enddef;
def apply_t (text Transformer) =
ztr := identity Transformer transformed ztr;
@@ -2589,6 +2669,7 @@ def boost primary X = zslant (cosh X, sinh X) enddef;
% \end{macrocode}
%
+%
% \subsection{Path transformation}\label{transformation}
%
% These are functions that accept and return a path in graph coordinates.
@@ -2797,8 +2878,10 @@ enddef;
% \end{macrocode}
%
%
+%
% \section{Picture-level Operations}\label{picture}
%
+%
% \subsection{Bitwise logical operations}\label{logical}
%
% None of these operations are available in \MP. Mostly these are used by
@@ -2905,6 +2988,7 @@ enddef;
%</MF>
% \end{macrocode}
%
+%
% \subsection{Producing and modifying pictures}\label{pictures}
%
% Here we define some slightly higher level commands that make use (in \MF)
@@ -2935,12 +3019,10 @@ enddef;
% \gbc{active_plane}, we have eliminated that parameter from
% \gbc{coloraddon}.
%
-% \DescribeRoutine{_orto}
-% This version of \gbc{orto} saves memory by passing \emph{both}
-% parameters by name. This also allows the application of \gbc{mono} to
-% both parameters. In addition to \gbc{coloraddon}, it is used in
-% \gbc{shade} and \gbc{tess}.
-% \DescribeRoutine{_subto}
+% The command \gbc{_orto} is like \gbc{orto}, but saves memory by passing
+% \emph{both} parameters by name. This also allows the application of
+% \gbc{mono} to both parameters. In addition to \gbc{coloraddon}, it is
+% used in \gbc{shade} and \gbc{tess}.
% We also have \gbc{_subto}, an analogous version of \gbc{subto}.
% \begin{macrocode}
def coloraddto (expr clr) (suffix u) (expr v) =
@@ -3020,6 +3102,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Clipping}\label{basicclipping}
%
% \DescribeRoutine{clipto}
@@ -3312,9 +3395,9 @@ enddef;
% calling routine must make sure that picture is initialized (it need
% not be \mfc{nullpicture}).
%
-% One might do this with two nested loops, but it turns out to be much
-% faster (surprisingly much!) to do two separate loops: the second one
-% stacking copies of the row built by the first loop.
+% One might do this with one loop nested in another, but it turns out to
+% be much faster (surprisingly much!) to do two separate loops: the second
+% one stacking copies of the row built by the first loop.
%
% We try to do any rounding that might have been forgotten. This code
% takes a mode's aspect ratio into account so that (most) calling routines
@@ -3351,6 +3434,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Hatching}\label{basichatching}
%
% \DescribeRoutine{thatchf}
@@ -3374,7 +3458,7 @@ enddef;
% gets one to \gbc{ypart b}. We make the starting value an integer
% multiple of \gbc{_sp} to make sure adjacent regions don't have jarringly
% misaligned hatch lines. (I guess that's the reason; this algorithm
-% predates me.)
+% predates my involvement with \mfpic{}.)
% \begin{macrocode}
def thatchf (suffix v) (expr CT, sp, a, b) =
begingroup
@@ -3389,6 +3473,178 @@ enddef;
% \end{macrocode}
%
+%
+% \subsection{Gradient fills}\label{basicgradient}
+%
+% \CMP{} cannot do true gradients without some external help. Level-3
+% PostScript permits it, so recent \MP{} could do it by inserting
+% appropriate PS prologues and/or \MP{} specials. Doing that runs the
+% risk of introducing code not recognized by post-processors that expect
+% only what \MP\ natively offers. Therefore, we implement gradients by
+% filling a lot of thin regions with a range of different
+% colors.
+%
+% We have to drop down to pretty low-level operations since, before now,
+% we didn't need a command that added a colored region to a named
+% picture.
+%
+% For maximum flexibility, all our gradients pass variation in colors as a
+% function \mfc{clr} which must be previously \gbc{vardef}-ed and must
+% produce a color for each parameter value between $0$ and $1$.
+%
+% \DescribeRoutine{axialgradientf}
+% A linear gradient has colored rectangular strips that vary along a
+% single axis. The function parameter of \gbc{axialgradientf} takes one
+% variable and produces the color of each strip. \gbc{v} is a known
+% picture variable to which the resulting picture will be assigned,
+% \gbc{theta} is an angle, \gbc{a} and \gbc{b} are the opposite corners of
+% a rectangle. What is returned in \gbc{v} is a rectangular picture
+% rotated by \gbc{theta}.
+%
+% Normally, this is called by the \gbc{axialgradient} command which
+% declares the picture variable \gbc{v}, passes its angle parameter
+% \gbc{theta}, and computes the bounding box of a cyclic path for \gbc{a}
+% and \gbc{b}. The calling command will clip the result to the appropriate
+% path.
+%
+% The calculations with \gbc{signof} is for the same reason as in
+% \gbc{thatchf}. The other messy calculations try to cover the rectangle
+% exactly with an integer number of strips, with the first and last having
+% exactly the colors \gbc{clr(0)} and \gbc{clr(1)}.
+%
+% If the thickness of the strip is too small, memory problems might
+% result and appearence might suffer. Nevertheless we make no attempt to
+% enforce a minimum value.
+% \begin{macrocode}
+def axialgradientf (suffix clr, v) (expr theta, sp, a, b) =
+ begingroup
+ save _hh, _sp, _nn, _y;
+ _hh := ypart b - ypart a;
+ _sp := signof (_hh) abs(sp);
+ _nn := emax (1, round (_hh/_sp));
+ _sp := _hh/_nn + signof (_hh) epsilon;
+ _nn := _nn-1;
+ setpath (_p) rect ((xpart a, 0),(xpart b, _sp));
+ _y := ypart a;
+ for _i = 0 upto _nn:
+%<*MF>
+ if (clr(_i/_nn)) < white :
+ addto v also shaded (clr(_i/_nn)) ( _p shifted (0,_y))
+ rotated theta;
+ fi
+%</MF>
+%<*MP>
+ addto v contour (_p shifted (0,_y)) rotated theta
+ withcolor clr(_i/_nn);
+%</MP>
+ _y := _y + _sp;
+ endfor
+%<MF> mono (v);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{areagradientf}
+% The command \gbc{areagradientf} fills the rectangle determined by
+% corners \gbc{a} and \gbc{b} with pixels of dimension \gbc{sp} by
+% \gbc{tp}. Each pixel is filled with the color determined by \gbc{clr}.
+% This suffix parameter must be the name of a function taking two
+% parameters.
+%
+% The resulting rectangle is built on the picture variable whose name is
+% passed as the second parameter \gbc{v}. The calling routine is
+% \gbc{areagradient}, which determine the rectangle and initializes
+% the picture variable. It passes its other parameters unchanged.
+% \begin{macrocode}
+def areagradientf (suffix clr, v) (expr sp, tp, a, b) =
+begingroup
+ save _ww, _hh, _sp, _tp, _nn, _mm, _x, _y;
+ _ww := xpart b - xpart a;
+ _hh := ypart b - ypart a;
+ _sp := signof (_ww) abs(sp);
+ _tp := signof (_hh) abs(tp);
+ _nn := emax (1, round (_ww/_sp));
+ _mm := emax (1, round (_hh/_tp));
+ _sp := _ww/_nn + signof (_ww) epsilon;
+ _tp := _hh/_mm + signof (_hh) epsilon;
+ _mm := _mm-1; _nn := _nn-1;
+ setpath (_p) rect (origin,(_sp,_tp));
+ _x := xpart a; y_a := ypart a;
+ for _i = 0 upto _nn:
+ _y := y_a;
+ for _j = 0 upto _mm:
+%<*MF>
+ if (clr(_i/_nn,_j/_mm)) < white:
+ addto v also shaded (clr(_i/_nn,_j/_mm)) (_p shifted (_x,_y));
+ fi
+%</MF>
+%<*MP>
+ addto v contour (_p shifted (_x,_y)) withcolor
+ clr(_i/_nn,_j/_mm);
+%</MP>
+ _y := _y + _tp;
+ endfor
+ _x := _x + _sp;
+ endfor
+%<MF> mono (v);
+endgroup
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{radialgradientf}
+% The command \gbc{radialgradientf} fills the a circle determined by
+% center \gbc{ctr} and radius \gbc{rad} with concentric circular strips of
+% thickness \gbc{sp}. Each strip is filled with the color determined by
+% \gbc{clr}. This suffix parameter must be the name of a function of one
+% parameter.
+%
+% This command is called by \gbc{radialgradient}, which determines the
+% radius of a circle needed to cover a region and clips the picture
+% returned in \gbc{v} to that region.
+% \begin{macrocode}
+path unitcircle;
+unitcircle := fullcircle scaled 2;
+def radialgradientf (suffix clr, v) (expr sp, ctr, rad) =
+ begingroup
+ save _sp, _r, _nn;
+ _nn := emax (1, round (rad/sp));
+ _sp := rad/_nn + epsilon;
+ _nn := _nn - 1;
+ _r := _sp;
+ % fill the small center circle first
+%<*MF>
+ if (clr(0)) < white :
+ addto v also shaded (clr(0)) (unitcircle scaled _r shifted ctr);
+ fi
+%</MF>
+%<*MP>
+ addto v contour (unitcircle scaled _r shifted ctr)
+ withcolor clr(0);
+%</MP>
+ for _i = 1 upto _nn:
+%<*MF>
+ if (clr(_i/_nn)) < white :
+ addto v also shaded (clr(_i/_nn))
+ (unitcircle scaled (_r + _sp) -- reverse unitcircle scaled _r
+ --cycle) shifted ctr;
+ fi
+%</MF>
+%<*MP>
+ addto v contour
+ (unitcircle scaled (_r + _sp) -- reverse unitcircle scaled _r
+ --cycle) shifted ctr withcolor clr(_i/_nn);
+%</MP>
+ _r := _r + _sp;
+ endfor
+%<MF> mono (v);
+ endgroup
+enddef;
+
+% \end{macrocode}
+%
+%
% \subsection{Tiles}\label{tiles}
%
% Tesselations are a type of fill in which a rectangular pattern is
@@ -3478,6 +3734,8 @@ enddef;
% \end{macrocode}
%
+%
+%
% \section{Bounding Boxes of Paths}\label{bboxes}
%
% To fill a region with other than a solid fill, we normally fill a
@@ -3522,6 +3780,9 @@ enddef;
% This description applies only to \MF, because \MP{} has built-in
% facilities for determining the bounding box.
%
+% I have changed \gbc{ctrlsbbox} to have the same syntax as \gbc{getbbox}.
+% I don't know why I defined it differently.
+%
% \RoutineIndex{pnt}
% \RoutineIndex{pre}
% \RoutineIndex{post}
@@ -3532,6 +3793,7 @@ vardef pnt@# (expr p) = point @# of p enddef;
vardef pre@# (expr p) = precontrol @# of p enddef;
vardef post@# (expr p) = postcontrol @# of p enddef;
+numeric bbox_split; bbox_split := 4;
def getbbox (suffix ll, ur) expr g =
%<MP> ll := llcorner g; ur := urcorner g;
%<*MF>
@@ -3541,15 +3803,14 @@ def getbbox (suffix ll, ur) expr g =
ll := pairmin (ll, pnt[_j] (g)); ur := pairmax (ur, pnt[_j] (g));
endfor
for _j = 1 upto _s*(length g):
- ctrlsbbox (subpath ((_j-1)/_s, _j/_s) of g) (ll, ur);
+ ctrlsbbox (ll, ur) subpath ((_j-1)/_s, _j/_s) of g;
endfor
%</MF>
if showbbox: noclip ( safedraw rect (ll, ur) ); fi
enddef;
%<*MF>
-numeric bbox_split; bbox_split := 2;
-def ctrlsbbox (expr p) (suffix ll, ur) =
+def ctrlsbbox (suffix ll, ur) expr p =
ll := pairmin ( pairmin (ll, post0 (p)), pre 1 (p) );
ur := pairmax ( pairmax (ur, post0 (p)), pre 1 (p) );
enddef;
@@ -3557,6 +3818,31 @@ enddef;
%</MF>
% \end{macrocode}
%
+% \DescribeRoutine{getradius}
+% This is very similar to \gbc{getbbox}, but gets a ``bounding circle''
+% instead of a box. It is used to get nearly the smallest circle with a
+% given center that contains a path. The path is shifted to place the
+% center at the origin and then this function is called. Similarly,
+% \DescribeRoutine{ctrlsradius}
+% \gbc{ctrlsradius} is used like \gbc{ctrlsbbox}.
+% \begin{macrocode}
+def getradius (suffix rad) expr g =
+ setsplit (_s) bbox_split;
+ rad := abs (pnt0 (g));
+ for _j = 1 upto length g:
+ rad := emax(rad, abs(pnt[_j] (g)));
+ endfor
+ for _j = 1 upto _s*(length g):
+ ctrlsradius (rad) subpath ((_j-1)/_s, _j/_s) of g;
+ endfor
+enddef;
+
+def ctrlsradius (suffix rad) expr p =
+ rad := emax( emax (rad, abs(post0 (p))), abs(pre1 (p) ))
+enddef;
+
+% \end{macrocode}
+%
% We also have \gbc{tightbbox} and \gbc{tbbox} in \MF{} but these are no
% longer used so we'll omit them from \grafbase, but keep them in the
% documentation for now.
@@ -3568,11 +3854,9 @@ enddef;
% \mfc{.5} (accurate enough, assuming pixel units). This is only called by
% \gbc{tbbox}, which is never used.
%
-% \DescribeRoutine{_xlimit}
% \gbc{xlimit(x)} returns a value of true if the path \gbc{g} doesn't
-% cross the vertical line at \gbc{x}.
-% \DescribeRoutine{_ylimit}
-% \gbc{ylimit(y)} is the same for the horizontal line at \gbc{y}.
+% cross the vertical line at \gbc{x}. \gbc{ylimit(y)} is the same for the
+% horizontal line at \gbc{y}.
% \begin{macrocode}
%<*unused>
def tightbbox (expr g) (suffix ll, ur) =
@@ -3618,6 +3902,8 @@ enddef;
%</unused>
% \end{macrocode}
%
+%
+%
% \section{Device Coordinate Rendering Commands}\label{basicrendering}
%
% We use the word `rendering' to refer to commands that accept a path
@@ -3625,6 +3911,7 @@ enddef;
% All the commands in this section expect paths, pairs and dimensions in
% device coordinates.
%
+%
% \subsection{Drawing}\label{basicdrawing}
%
% \DescribeRoutine{safedraw}
@@ -3654,6 +3941,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Filling}\label{basicfilling}
%
% \DescribeRoutine{NoCycle}
@@ -3712,6 +4000,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Clipping}\label{clipping}
%
% \DescribeRoutine{safeclip}
@@ -3727,6 +4016,8 @@ enddef;
% \end{macrocode}
%
+%
+%
% \section{Graph Coordinate Rendering}\label{rendering}
%
% \DescribeRoutine{store}
@@ -3764,6 +4055,7 @@ vardef stored (suffix fs) expr f = store (fs) f; f enddef;
% \end{macrocode}
%
+%
% \subsection{Drawing}\label{drawing}
%
% \DescribeRoutine{drawn}
@@ -3797,18 +4089,23 @@ enddef;
%
% The reason for using a loop (at the end) that draws the \gbc{sinewave}
% path in pieces, is that all the turning can quickly exceed \MF{}'s limit
-% on the autorounding stack. I'd never heard of this stack until I ran
+% on the ``rounding table size''. I'd never heard of this until I ran
% this without a loop and received the ``capacity exceeded'' message. This
% turns out to be a problem mostly when the ratio of \gbc{len} to
% \gbc{wid} is too small and the `humps' of the sine are more like
% `bulbs'. However it is always a problem with \gbc{corkscrew} (below).
+%
+% There is no need for the loop in \MP{}, nor in \MF{} if
+% \mfc{autorounding} is set to $0$, but \mfpic's curved paths definitely
+% look better with the default \mfc{autorounding=2}.
% \begin{macrocode}
def zigzag = colorzigzag (drawcolor) enddef;
def colorzigzag (expr clr) = colorwiggle (false, clr, 0) enddef;
def sinewave = colorsinewave (drawcolor) enddef;
def colorsinewave = colorwiggle (true) enddef;
-vardef colorwiggle (expr smth, clr, tens, blen, elen, len, wid) expr f =
+vardef colorwiggle (expr smth, clr, tens, blen, elen, len, wid) expr f
+=
convertpath (g) f;
setuplengtharray (cumlen, totlen, ct) g;
save B;
@@ -3919,6 +4216,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Filling, unfilling and clipping}\label{filling}
%
% \DescribeRoutine{filled}
@@ -3943,6 +4241,7 @@ vardef Clip expr c = safeclip zconv (c); c enddef;
% \end{macrocode}
%
+%
% \subsection{Shading}\label{shading}
%
% \DescribeRoutine{shade}
@@ -4117,6 +4416,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Hatching}\label{hatching}
%
% \DescribeRoutine{thatch}
@@ -4183,6 +4483,88 @@ enddef;
% \end{macrocode}
%
+%
+% \subsection{Gradients}
+%
+% \DescribeRoutine{axialgradient}
+% We pass a \mfc{vardef}-ed function that is to provide the range of
+% colors. It can output colors of different types if desired. Two
+% natural methods are: (1)~interpolate between colors of the same type:\\
+% \indent\mfc{vardef clrgrad (expr t) = (t)[red,blue] enddef}\\
+% and (2)~extract colors from a previously built array of colors:\\
+% \indent\mfc{vardef clrgrad (expr t)= A[round(t*N)]}\\
+% where, \mfc{A0}, \mfc{A1},\dots \mfc{A[N]} are colors (necessarily of
+% the same type).
+%
+% Since we simply fill strips with a single color, \gbc{sp} is the
+% thickness of the strip (in device units) and \gbc{theta} is the angle
+% by which these strips differ from being horizontal.
+% \begin{macrocode}
+vardef axialgradient (suffix clr) (expr sp, theta) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("axialgradient") g;
+ else:
+ newpicture (_grd);
+ setbbox (ll, ur) g rotated -theta;
+ axialgradientf (clr, _grd) (theta, sp, ll, ur);
+ DoClip (_grd); clipto (_grd) (g);
+%<MF> safeunfill g;
+ _orto (active_plane, _grd);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{areagradient}
+% This fills a cyclic path with colored pixels, with the color
+% determined by the \mfc{vardef}-ed function \gbc{clr} which takes two
+% parameters. The size of the pixels is given in the last two parameters
+% \gbc{sp} and \gbc{tp} which are specified in device units.
+% \begin{macrocode}
+vardef areagradient (suffix clr) (expr sp, tp) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("areagradient") g;
+ else:
+ newpicture (_agr);
+ setbbox (ll, ur) g;
+ areagradientf (clr, _agr) (sp, tp, ll, ur);
+ DoClip (_agr); clipto (_agr) (g);
+%<MF> safeunfill g;
+ _orto (active_plane, _agr);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{radialgradient}\label{getrad}
+% This fills a cyclic path with colored circular strips, with the color
+% determined by the \mfc{vardef}-ed function \gbc{clr} which takes one
+% parameters. The thickness of the strips is given in the last parameter
+% \gbc{sp} which are specified in device units. The command
+% \gbc{getradius} finds the distance from the center to the farthest point
+% of \gbc{f}. It was added (see section~\ref{bboxes}) solely for this use.
+% \begin{macrocode}
+vardef radialgradient (suffix clr) (expr sp, ctr) expr f =
+ convertpath (g) f;
+ if not cycle g: NoCycle("radialgradient") g;
+ else:
+ setpair (_ctr) zconv (ctr);
+ newpicture (_agr);
+ save _rad;
+ getradius (_rad) g shifted - _ctr;
+ radialgradientf (clr, _agr) (sp, _ctr, _rad);
+ DoClip (_agr); clipto (_agr) (g);
+%<MF> safeunfill g;
+ _orto (active_plane, _agr);
+ fi
+ f
+enddef;
+
+% \end{macrocode}
+%
+%
% \subsection{Tesselations}\label{tess}
%
% \DescribeRoutine{tess}
@@ -4215,6 +4597,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Dots and dashes}\label{dashes}
%
% \MP{} already has commands for drawing a dashed or dotted curve,
@@ -4470,8 +4853,8 @@ vardef makelengtharray (suffix clen) suffix p =
numeric clen[];
clen := _s * length p; clen0 := 0;
for _i = 1 upto clen:
- clen[_i] := clen[_i-1] + abs (pnt[_i/_s] (p) - pnt[(_i-1)/_s] (p)) /
- _rescale_factor;
+ clen[_i] := clen[_i-1] + abs (pnt[_i/_s] (p) - pnt[(_i-1)/_s] (p))
+ / _rescale_factor;
endfor
clen[clen]
enddef;
@@ -4725,6 +5108,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Double-line drawing}\label{doubleline}
%
% \DescribeRoutine{doubledraw}
@@ -4743,6 +5127,8 @@ enddef;
% \end{macrocode}
%
+%
+%
% \section{Points Symbols and Other Pictures}\label{symbols}
%
% \DescribeRoutine{centerit}
@@ -4914,6 +5300,8 @@ enddef;
% \end{macrocode}
%
+%
+%
% \section{Axes, Tic Marks, and Grids}\label{axes}
%
% \DescribeRoutine{arrowdraw}
@@ -5482,6 +5870,8 @@ enddef;
% \end{macrocode}
%
+%
+%
% \section{Path Construction}\label{pathconstruction}
%
% This section is devoted to commands that accept a list or array of
@@ -5490,6 +5880,7 @@ enddef;
% points, lines and circles associated with a triangle. No \mfpic{}
% interface is yet available for the triangle commands.
%
+%
% \subsection{Piecewise linear paths}\label{linear}
%
% \DescribeRoutine{rect}
@@ -5658,7 +6049,9 @@ enddef;
% \DescribeRoutine{mkpoly}
% This produces the path of line segments connecting \gbc{pts1},
% \gbc{pts2}, etc., closing it up if the boolean \gbc{cyclic} is true.
-% It is also used with an array of paths instead of points.
+% It can also be used with an array of paths instead of points, connecting
+% the end of each with the beginning of the next. We do this in \mfpic{}'s
+% \cs{connect} \dots\ \cs{endconnect} construct.
% \begin{macrocode}
vardef mkpoly (expr cyclic) (suffix pts) =
for _i = 1 upto pts-1: pts[_i]-- endfor
@@ -5672,8 +6065,9 @@ enddef;
% list of pair expressions, forms an array from them and calls
% \gbc{mkpoly}.
% \DescribeRoutine{NoPoints}
-% \mfc{NoPoints} prints a warning and sets the array to a single point,
-% the origin.
+% \mfc{NoPoints} is called when an array of points is defined (using
+% setpairs) that returns $0$ for the number of pairs. It prints a warning
+% and sets the array to a single point, the origin.
% \begin{macrocode}
vardef polyline (expr cyclic) (text t) =
setpairs (_pl) (t);
@@ -5718,10 +6112,32 @@ enddef;
% I needed the following to illustrate Brownian motion. It takes a given
% starting point, a given number of steps and a scaling factor. It
% generates a sequence of random points, each one being chosen randomly
-% using a Gaussian distribution centered at the previous point. Strictly
+% using a Gaussian distribution centered at the previous point. The
+% standard deviation of the random distance is the scale factor. Strictly
% speaking this is a Gaussian random walk, not Brownian motion. A true
% Brownian motion would be a limit of these, with \gbc{num} tending to
% $\infty$ and \gbc{sc} tending to 0.
+%
+% \DescribeRoutine{randomwalk}
+% This is like \gbc{brownianpath}, but the distance from one point to
+% the next is always the same, only the direction is random. It takes
+% the same arguments as \gbc{brownianpath}
+%
+% \DescribeRoutine{browniangraph}
+% This command takes a given number of steps \gbc{num} and a scaling
+% factor/step size \gbc{scst}. It generates a sequence of points, each one
+% being chosen right of the previous one by the step size \gbc{scst} and
+% randomly up or down using a Gaussian distribution centered at the
+% previous $y-value$. The Gaussian distribution has standard deviation
+% equal to \gbc{scst}. The path starts at $(0,0)$. One needs to transform
+% the path to get a different start or a scale factor different from the
+% step size.
+%
+% In \MF{} we run into capacity problems when \gbc{num} is greater than
+% 500 or so. This is the \mfc{autorounding} problem again (see the
+% discussion at \gbc{sinewave}. We can't use the same technique we used
+% there since it is the drawing that invokes \mfc{autorounding} and these
+% macros only construct paths; they don't draw them.
% \begin{macrocode}
vardef brownianpath (expr start, num, sc) =
setnumeric (_brp) 1;
@@ -5733,9 +6149,29 @@ vardef brownianpath (expr start, num, sc) =
endfor
mkpoly (false, _brp)
enddef;
+vardef randomwalk (expr start, num, dst) =
+ setnumeric (_rdw) 1;
+ setpair (_tmp) start;
+ pair _rdw[]; _rdw1 := _tmp;
+ for _idx := 1 upto num:
+ _tmp := _tmp + dst*dir(uniformdeviate(360));
+ _rdw[incr _rdw] := _tmp;
+ endfor
+ mkpoly (false, _rdw)
+enddef;
+vardef browniangraph (expr num, scst) =
+ setnumeric (_brg) 1;
+ pair _tmp, _brg[]; _tmp := _brg1 := (0,0);
+ for _idx := 1 upto num:
+ _tmp := _tmp + scst*(1,normaldeviate);
+ _brg[incr _brg] := _tmp;
+ endfor
+ mkpoly (false, _brg)
+enddef;
% \end{macrocode}
%
+%
% \subsection{Smooth paths}\label{smooth}
%
% We added an optional parameter for the tension of smooth curves to
@@ -5839,8 +6275,8 @@ vardef mkconvex (expr tens, cyclic) (suffix pts) =
_B[_j] := sqrt(abs((pts[_j]-pts[_j-1])xprod(pts[_j+1]-pts[_j])));
endfor
if cyclic:
- _B1 := sqrt(abs((pts1 - pts[pts])xprod(pts2 - pts1)));
- _B[pts] := sqrt(abs((pts[pts]-pts[pts-1])xprod(pts1 - pts[pts])));
+ _B1 := sqrt(abs((pts1 - pts[pts])xprod(pts2 - pts1)));
+ _B[pts] := sqrt(abs((pts[pts]-pts[pts-1])xprod(pts1 - pts[pts])));
else:
_B1 := _B2;
_B[pts] := _B[pts-1];
@@ -5903,12 +6339,16 @@ enddef;
numeric default_tension; default_tension := 1;
def curve = tcurve (default_tension) enddef;
vardef tcurve (expr tens, cyclic) (text t) =
- setpairs (_tc) (t); mksmooth (tens, cyclic, _tc)
+ setpairs (_tc) (t);
+ if _tc=0: NoPoints("curve", _tc); fi
+ mksmooth (tens, cyclic, _tc)
enddef;
def ccurve = tccurve (default_tension) enddef;
vardef tccurve (expr tens, cyclic) (text t) =
- setuniquepairs (_tcc) (t); mkconvex (tens, cyclic, _tcc)
+ setuniquepairs (_tcc) (t);
+ if _tcc=0: NoPoints("ccurve", _tcc); fi
+ mkconvex (tens, cyclic, _tcc)
enddef;
% \end{macrocode}
@@ -5978,8 +6418,8 @@ vardef mkqbezier (expr cyclic) (suffix pts) =
if pts=1: {0,0}
else:
for _i = 2 step 2 until pts - 1:
- ..controls 1/3[pts[_i], pts[_i-1] ] and 1/3[pts[_i], pts[_i+1] ]..
- pts[_i+1]
+ ..controls 1/3[pts[_i], pts[_i-1]]
+ and 1/3[pts[_i], pts[_i+1]].. pts[_i+1]
endfor
if cyclic:
..controls 1/3[ pts[pts], pts[pts - 1] ]
@@ -6114,12 +6554,14 @@ def tfcncurve = functioncurve enddef;
vardef functioncurve (expr ftens) (text t) =
settension (_ftens) ftens; if _ftens < 1/3: _ftens := 1/3; fi
setuniquepairs (_fc) (t);
+ if _fc=0: NoPoints ("functioncurve", _fc); fi
if _fc > 1: _fc0 := _fc1; _fc[_fc+1] := _fc[_fc]; fi
mkfcnpath (_ftens) (_fc)
enddef;
% \end{macrocode}
%
+%
% \subsection{Splines with explicit controls}\label{splines}
%
% For these quadratic B-splines, a list of pairs representing the control
@@ -6240,6 +6682,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Splines with computed controls}\label{computedsplines}
%
% A cubic spline through a set of points is a curve obtained by joining
@@ -6352,9 +6795,9 @@ enddef;
% than pair values. Such are often used to interpolate functions. That is,
% given pairs $(x\sb j,y\sb{j})$, and assuming they lie on the graph of
% some function (generally unknown), fill in the graph with $y = f(x)$
-% where $f$ is a cubic function of $x$ in each interval $x\sb j < x < x\sb
-% {j+1}$, making sure that the resulting graph is as smooth as possible at
-% the points $x\sb j$.
+% where $f$ is a cubic function of $x$ in each interval $x\sb j \le x
+% \le x\sb {j+1}$, making sure that the resulting graph is as smooth as
+% possible at the points $(x\sb j, y\sb j)$.
%
% The requirements on our $2$-dimensional path are the following:
% \begin{enumerate}
@@ -6481,6 +6924,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Arcs, circles and ellipses}\label{arcs}
%
% We have multiple commands that generate circular arcs, differing in
@@ -6502,7 +6946,9 @@ enddef;
%
% There is really no problem with \gbc{mkarc} itself: if you can express
% both \gbc{center} and \gbc{begpt} in \MF, then the other values on the
-% arc should be no problem.
+% arc should normally be no problem. (Of course, if the radius is near
+% \mfc{infinity}, there could be points on the arc with coordinates near
+% \mfc{2infinity}, causing overflow in \MF{}. One hopes this is rare.)
%
% Care has been taken that changing the sign of various parameters
% produces reasonable results. And there should be no more problem for
@@ -6527,7 +6973,7 @@ enddef;
% is mainly to ensure that the arc begins at \gbc{begpt} and ends at
% \gbc{endpt} (exactly). A \gbc{sweep} of $0$ is actually incompatible
% with any case where \gbc{begpt<>endpt} unless \gbc{center} is
-% literally at infinity, but we allow it even though I am pretty sure
+% literally at $\infty$, but we allow it even though I am pretty sure
% the other arc commands all filter out that case.
% \begin{macrocode}
vardef mkarc (expr center, begpt, endpt, sweep) =
@@ -6546,7 +6992,7 @@ enddef;
%
% \DescribeRoutine{arc}
% The most basic: center of circle, starting point of arc, and angle
-% subtended. Another name for \gbc{arc} is \gbc{arccps}, (\gbc{cps} is
+% subtended. Another name for \gbc{arc} is \gbc{arccps}, (``\gbc{cps}'' is
% for ``center, point, sweep'').
% \begin{macrocode}
vardef arc (expr center, begpt, sweep) =
@@ -6573,7 +7019,7 @@ def arccps = arc enddef;
% used them and the several cases that they had to consider are reduced
% because the \mfc{if} in this command takes care of some of them.
%
-% The code for finding \gbc{m} uses the fact that chord and the line
+% The code for finding \gbc{m} uses the fact that the chord and the line
% from one of its endpoints to the midpoint subtend a circular arc of
% \gbc{sweep/2} and so the angle between them is half that, \gbc{sweep/4}.
% The code gets the intersection between the line in that direction and
@@ -6589,7 +7035,7 @@ def arccps = arc enddef;
% circle. This gives the radius mentioned above.
% \begin{macrocode}
vardef arcpps (expr begpt, endpt, sweep) =
- if begpt = endpt: begpt--endpt
+ if (begpt = endpt) or (sweep = 0): begpt--endpt
else:
setpair (cd) unitvector (endpt-begpt);
if abs(sweep) <= 45:
@@ -6761,7 +7207,8 @@ vardef circlepps (expr one, two, sweep) =
enddef;
vardef circlepp (expr small, one, two, rad) =
- arcpp (small, one, two, rad) & arcpp (not small, two, one, rad) & cycle
+ arcpp (small, one, two, rad) & arcpp (not small, two, one, rad)
+ & cycle
enddef;
def circleppr (expr one, two, rad, small) =
@@ -6770,11 +7217,82 @@ enddef;
% \end{macrocode}
%
+% Now we implement a different way to specify an ellipse, essentially
+% specifying it by a parallelogram in which it is to be inscribed.
+%
+% \DescribeRoutine{quarterellipse}
+% If an ellipse is inscribed in a parallelogram, tangent to all four
+% sides at the midpoints, this command produces one ``corner'' of that
+% ellipse. The arguments \mfc{A} and \mfc{C} are the midpoints of two
+% adjacent sides and \mfc{B} is the corner between those two sides. This
+% quarter-ellipse starts at \mfc{A} in the direction \mfc{B-A} and ends at
+% \mfc{C} in the direction \mfc{C-B}. As a path \mfc{p} it has two segments, where
+% \mfc{point 0 of p} is \mfc{A}, \mfc{point 2 of p} is \mfc{C}, while
+% \mfc{point 1 of p} lies on the diagonal of the parallelogram through
+% \mfc{B} and has direction there the same as \mfc{C-A}.
+%
+% This was created for the purpose of rounding off corners of a polygonal
+% path.
+% \begin{macrocode}
+vardef quarterellipse(expr A,B,C) =
+ save T_;
+ transform T_;
+ (1,0) transformed T_ = A;
+ (1,1) transformed T_ = B;
+ (0,1) transformed T_ = C;
+ quartercircle scaled 2 transformed T_
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{halfellipse}
+% While \gbc{quarterellipse} is for corners, I don't have much use for
+% \gbc{halfellipse}. Nevertheless, it seems wise (and easy) to provide a
+% definition.
+%
+% The pairs \mfc{A}, \mfc{B}, and \mfc{C} are three midpoints of a
+% parallelogram with \mfc{A} and \mfc{C} on opposite sides and \mfc{B} on
+% a third side. This determines a unique parallelogram, and
+% \gbc{halfellipse} starts at \mfc{A}, passing through \mfc{B} then
+% \mfc{C}, tangent to the respective sides. It makes a point of building
+% it out of two \gbc{quarterellipse}\,s as \mfc{halfcircle} does with
+% \mfc{quartercircle} (at least in \MF{}). We just have to compute their
+% corners.
+% \begin{macrocode}
+vardef halfellipse (expr A,B,C) =
+ save P_; pair P_;
+ P_ = (C - A)/2;
+ quarterellipse (A, B - P_, B) & quarterellipse (B, B + P_, C)
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{fullellipse}
+% For \gbc{fullellipse} we specify the center \mfc{C} of the parallelogram
+% and the midpoints \mfc{A} and \mfc{B} of two adjacent sides. We compute
+% the midpoints of the other two sides and draw two \gbc{halfellipse}\,s.
+%
+% Note that the points \gbc{A} and \gbc{B} do not correspond to the
+% usual radii of an ellipse unless the corresponding parallelogram is
+% actually a rectangle (i.e., only if $\angle ACB$ is a right angle).
+% \begin{macrocode}
+vardef fullellipse (expr C, A, B) =
+ save P_; pair P_;
+ P_ := 2[A,C];
+ halfellipse (A,B,P_) & halfellipse (P_,2[B,C],A) & cycle
+enddef;
+
+% \end{macrocode}
+%
% \DescribeRoutine{pathcenter}
% This finds the center of a circle. For other paths, the point found
% may be meaningless (but it will also obtain the center of an arc or a
-% rectangle). It takes three supposedly distinct points on the path and
-% finds the intersection of the perpendicular bisectors of two chords.
+% rectangle). It takes three or four supposedly distinct points on the
+% path and finds the intersection of the perpendicular bisectors of two
+% chords.
+%
+% This code is rather non-robust if applied to an arc that has angular
+% measure very close to either 0 or 360.
% \begin{macrocode}
vardef pathcenter expr p =
save a, cntr, n; pair cntr, a[];
@@ -6782,7 +7300,7 @@ vardef pathcenter expr p =
a1 = pnt 0 (p);
a3 = pnt [n/2] (p);
if cycle p:
- a2 = pnt [n/4] (p);
+ a2 = pnt [ n/4] (p);
a4 = pnt [3n/4] (p);
else:
a2 := a3;
@@ -6804,8 +7322,8 @@ enddef;
% This is just the circle through the three corners.
%
% \DescribeRoutine{incircle}
-% The command \gbc{incircle} produces the circle inside the triangle that
-% is tangent to all three sides. It makes use of the fact that the two
+% The command \gbc{incircle} produces the circle that is tangent to all
+% three sides of the triangle. It makes use of the fact that the two
% tangent points on the sides adjacent to corner \gbc{A} (for example) are
% equidistant from \gbc{A}. The three equations then express the fact that
% the sum of the two distances from the tangent point to the corners on
@@ -6859,12 +7377,26 @@ enddef;
% \end{macrocode}
%
% \DescribeRoutine{pshcircle}
-% Here is a couple of circles maybe only I need. They are the
+% Here are a couple of circles maybe only I need. They are the
% pseudohyperbolic circles in the unit disk and upper half-plane.
% One supplies a point that must be inside the unit circle or above
% the $x$-axis, and a radius that must be less than $1$. Some degenerate
% cases will not generate an error. We code this with a boolean that
% determines whether the disk or the half-plane is to be assumed.
+%
+% If $\alpha=(a,b)$ is the hyperbolic center (the \mfc{ctr} parameter)
+% and $\rho$ is the pseudohyperbolic radius (the \mfc{rad parameter}),
+% the formula for the (Euclidean) center $C$ and radius $R$ of the circle
+% is, for the unit disk:
+% $$
+% C = \frac{ (1 - \rho^2)a }{1 - \rho^2|a|^2},\quad
+% R = \frac{\rho(1 - |a|^2)}{1 - \rho^2|a|^2}
+% $$
+% and for the half-plane:
+% $$
+% C = a + \frac{(1 + \rho^2}{1 - \rho^2}b,\quad
+% R = \frac{2\rho b}{1 - \rho^2}
+% $$
% \begin{macrocode}
vardef pshcircle (expr disk, ctr, rad) =
if disk:
@@ -6877,49 +7409,176 @@ vardef pshcircle (expr disk, ctr, rad) =
elseif abs(ctr) >= 1 :
if abs(ctr) > 1:
GBerrmsg ("Impossible center of pseudohyperbolic circle.")
- "The center of a pseudohyperbolic circle must be in "
+ "The center of this pseudohyperbolic circle must be in "
& "the unit disk.";
fi
onepointpath (true,ctr)
else:
- % compute Euclidean center and radius (and a denominator used twice
- % in calculations).
save _r, _dnm;
_r := abs(ctr);
_dnm := 1 - _r*_r*rad*rad;
- circle ( (1 - rad*rad)/_dnm*ctr, rad*(1 - _r*_r)/_dnm)
+ circle ((1 - rad*rad)/_dnm*ctr, rad*(1 - _r*_r)/_dnm)
fi
else:
if rad >= 1 :
- GBerrmsg ("Impossible pseudohyperbolic circle.")
+ GBerrmsg ("Impossible radius of pseudohyperbolic circle.")
"The radius of a pseudohyperbolic circle must be less than 1.";
onepointpath (true,ctr)
elseif ypart ctr <= 0:
if ypart ctr < 0:
- GBerrmsg ("Impossible pseudohyperbolic circle.")
- "The center of a pseudohyperbolic circle must be in "
+ GBerrmsg ("Impossible center of pseudohyperbolic circle.")
+ "The center of this pseudohyperbolic circle must be in "
& "the upper half-plane.";
fi
onepointpath (true,ctr)
else:
- % compute Euclidean center and radius (and a denominator used twice
- % in calculations).
- % Euclidean center at xpart ctr + (1 + R^2)/(1 - R^2)*ypart ctr
- % Euclidean radius 4R/(1 - R^2)*ypart ctr
save _y, _dnm;
_y := ypart ctr;
_dnm := 1 - rad*rad;
- circle ( (xpart ctr, (1 + rad*rad)/_dnm * _y), 2rad/_dnm*_y)
+ circle ((xpart ctr, (1 + rad*rad)/_dnm * _y), 2rad/_dnm*_y)
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{UHPgeodesic}
+% Here is another arc-producing command. What it produces is the
+% hyperbolic geodesic from one point to another in the \emph{upper
+% half-plane} (UHP). While, theoretically, the points should both be in
+% the UHP, where the hyperbolic geometry is defined, the computations make
+% sense for any pair of points. This could be useful, so I do not enforce
+% this theoretical requirement.
+%
+% Unless two points have the same xpart, there is a unique circle passing
+% through them that meets the $x$-axis at a right angle. The hyperbolic
+% geodesic is an arc of that circle. The path starts at the first listed
+% point and ends at the second. Of the two possible arcs that connect
+% these points, it is the one that doesn't cross the $x$-axis (if there
+% is one). Our computations simply determine the angle of the arc and call
+% \gbc{arcpps}.
+%
+% When the points have the same xpart, the hyperbolic geodesic is the
+% line segment connecting them. When the points have yparts with opposite
+% signs, both arcs cross the $x$-axis. Our code produces the shorter one.
+% If both are $180$ degrees, the one that lies all on the same side of the
+% vertical line through $A$ is produced ($A$ being the first argument).
+%
+% Our method is based on the fact that the reflection $C$ of $A$ (to the
+% other side of the $x$-axis) lies on the circle on which the arc lies.
+% The angle between $A$ and $B$ when viewed from this point is therefore
+% half the angle of the arc. We actually reflect the point farthest from
+% the $x$-axis, as this produces better results.
+%
+% If $A$ and $B$ are on opposite sides of the $x$-axis, then $C$ might
+% coincide with one of the points. In this case $A$ and $B$ would
+% necessarily have equal xparts, a case we will already have processed.
+%
+% If both points lie on the $x$-axis, the computations produce the
+% semicircle from the first to the second in the upper half-plane.
+% \begin{macrocode}
+vardef UHPgeodesic (expr A, B) =
+ if xpart A = xpart B:
+ A--B
+ else:
+ save ang_, C_; pair C_;
+ if abs(ypart A) < abs(ypart B):
+ C_ := conj B;
+ else:
+ C_ := conj A;
+ fi
+ if ypart C_ = 0: % both on x-axis
+ ang_ := anglefromto(up, B - A);
+ else:
+ ang_ := anglefromto(A - C_, B - C_);
fi
+ arcpps(A, B, 2ang_)
+ fi
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{UDgeodesic}
+% There is a hyperbolic geometry defined for any simply connected open
+% set. The standard examples of such are the UHP and the unit disk (UD).
+% This next macro produces the geodesic in the UD. Once again it is the
+% arc of a circle and, if the two points do not lie on the same diameter,
+% that circle is the unique one through the two points that meets the
+% boundary of $UD$ at a right angle. When the two points do lie on the
+% same then the geodesic is the straight line connecting the points.
+%
+% The method we use is also based on reflection, where the `reflection' of
+% a point $A$ is given by $C = A/|A|^2$. Computing this can cause overflow
+% if $|A|$ too near $0$. Unfortunately, overflow can also occur if either
+% point lies are outside the UD. That is because, even for modest sizes of
+% $A$ and $B$, the part of the mentioned circle that lies outside the UD
+% can approach \gbc{infinity} in size, making the arc itself impossible to
+% draw. While it is feasible to compute when this will occur, we try to
+% keep it simple by using an approach that is only guaranteed to work when
+% the points lie in the unit disk. A minor modification allows it to to
+% always work when only one of the points is outside. This is because the
+% geodesic is not unique and we can easily choose one that doesn't
+% overflow.
+%
+% We isolate several special cases: if either point is the origin or if
+% the points have the same angle, a straight line is produced. If either
+% point is on the boundary, the computation is based on the fact that the
+% arc is tangent to the direction of that point. In the remaining cases,
+% we compute two angles based on reflecting both points. In the case where
+% both points lie inside or both lie outside, these angles are
+% theoretically equal, but when one point lies inside and the other
+% outside, these angles have opposite signs and their absolute values sum
+% to 360. They correspond to going opposite ways around the circle. We
+% choose the shorter arc as being more ``geodesic-like''.
+%
+% If $C$ is the point being reflected, but it is close enough to the
+% origin to make overflow a significant problem, we rescale the triangle
+% used to find the angle: we compute the angle between $|C|A$ and $|C|B$
+% as viewed from $C/|C|$.
+% \begin{macrocode}
+vardef UDgeodesic (expr A, B) =
+ save a_, b_;
+ a_ := abs(A); b_ = abs(B);
+ if (a_ = 0) or (b_ = 0):
+ A--B
+ elseif angle A = angle B:
+ A--B
+ else: % note: A, B and B-A are all nonzero from this point
+ save ang_;
+ if a_ = 1:
+ ang_ := anglefromto (if b_>1: A else: -A fi, B-A)
+ elseif b_ = 1:
+ ang_ := anglefromto (A-B, if a_>1: B else: -B fi)
+ else:
+ save C_; pair C_;
+ % reflecting A
+ if a_ < eps:
+ C_ := unitvector A;
+ ang_1 := anglefromto(a_*A - C_, a_*B - C_);
+ else:
+ C_ := (1/a_)*unitvector A;
+ ang_1 := anglefromto(A - C_, B - C_);
+ fi
+ % reflecting B
+ if b_ < eps:
+ C_ := unitvector B;
+ ang_2 := anglefromto(b_*A - C_, b_*B - C_);
+ else:
+ C_ := (1/b_)*unitvector B;
+ ang_2 := anglefromto(A - C_, B - C_);
+ fi
+ ang_ := if abs(ang_1) < abs(ang_2): ang_1 else: ang_2 fi;
+ fi
+ arcpps(A, B, 2ang_)
fi
enddef;
% \end{macrocode}
%
% \DescribeRoutine{barycenter}
-% This is the average of the three corners of the triangle, or of any
-% path. If \gbc{t} is an open path with length $n$ and the nodes are
-% $x\sb0$ through $x\sb n$, the barycenter is
+% This is the average of the three corners of the triangle, or of all the
+% nodes of any path. If \gbc{t} is an open path with length $n$ and the
+% nodes are $x\sb0$ through $x\sb n$, the barycenter is
% $$ \frac{1}{n+1}\sum\sb{j=0}\sp{n} x\sb j. $$
% If \gbc{t} is a cycle with $x\sb n = x\sb0$, then it is
% $$ \frac{1}{n}\sum\sb{j=0}\sp{n-1} x\sb j. $$
@@ -6952,6 +7611,39 @@ enddef;
% \end{macrocode}
%
+% \DescribeRoutine{mkbrace}
+% Because it doesn't really fit anywhere else, and because it is not
+% really enough to waste a whole subsection on, we put \gbc{mkbrace} here.
+% It is a command to draw a brace (i.e., a ``$\lbrace$'' shape) with its
+% ends and its cusp at given points. The start is at \gbc{S}, the end at
+% \gbc{E} and the cusp at \gbc{C}. \gbc{C} should be close to, but not
+% on, the line from \gbc{S} to \gbc{E}. It should also not be too close to
+% \gbc{S} or \gbc{E}, as we need room to draw two quarter circles on
+% either side of \gbc{C} and one at each of \gbc{S} and \gbc{E}.
+% \begin{macrocode}
+vardef mkbrace (expr S, C, E) =
+ save R_, U_, V_, Z_;
+ pair U_, V_, Z_[];
+ U_ := unitvector (E-S);
+ V_ := U_ rotated 90;
+
+ R_ := 0.5*(C-S) dotprod V_;
+ if R_ = 0:
+ S--C
+ else:
+ if R_ < 0 : V_ := -V_; R_ := -R_; fi
+ V_ := R_*V_; U_ := R_*U_;
+ Z_1 := S + V_ + U_;
+ Z_2 := C - V_ - U_;
+ Z_3 := C - V_ + U_;
+ Z_4 := E + V_ - U_;
+ S{V_}..{U_}Z_1--Z_2{U_}..{V_}C{-V_}..{U_}Z_3--Z_4{U_}..{-V_}E
+ fi
+enddef;
+
+% \end{macrocode}
+%
+%
% \subsection{Plotting of functions}\label{functionplots}
%
% In these macros, if the boolean argument \gbc{sm} is true then the
@@ -7148,7 +7840,7 @@ tolerancefactor := .02;
vardef mklevelset (expr sm, tens, X, Y, t, a, b, c, d) =
save _inside_;
vardef _inside_ (expr U, V) =
- inside_levelset (U, V) and between (a, b) (U) and between (c, d) (V)
+ inside_levelset(U, V) and between(a, b)(U) and between(c, d)(V)
enddef;
if not _inside_ (X, Y):
GBwarn "Invalid seed point for levelset.";
@@ -7273,16 +7965,16 @@ vardef tRKIV (expr sm, tens, zstart, ds, N) (text _RHS_) =
for _idx := 2 upto _trj:
_dt := ds/emax(1,abs(_RHS_(_tt,_ztr)));
_th := _tt + .5_dt;
- _dz1 := _dt*_RHS_(_tt, _ztr); % displacement based on current point
+ _dz1 := _dt*_RHS_(_tt, _ztr); % displacement for current point
_ztmp := _ztr + .5_dz1; % 1st midpoint
% use _th instead of twice calculating (_tt + .5_dt)
- _dz2 := _dt*_RHS_(_th, _ztmp); % displacement based on 1st midpoint
+ _dz2 := _dt*_RHS_(_th, _ztmp); % displacement for 1st midpoint
_ztmp := _ztr + .5_dz2; % 2nd midpoint
- _dz3 := _dt*_RHS_(_th, _ztmp); % displacement based on 2nd midpoint
+ _dz3 := _dt*_RHS_(_th, _ztmp); % displacement for 2nd midpoint
_ztmp := _ztr + _dz3; % temporary end point
- % get time for next loop now since we need it right away in next line:
+ % get time for next loop now since we need it in the next line:
_tt := _tt + _dt;
- _dz4 := _dt*_RHS_(_tt, _ztmp); % displacement based on end point
+ _dz4 := _dt*_RHS_(_tt, _ztmp); % displacement for end point
% get next point
_ztr := _ztr + (_dz1 + 2_dz2 + 2_dz3 + _dz4)/6;
_trj[_idx] := _ztr;
@@ -7308,8 +8000,11 @@ enddef;
% \end{macrocode}
%
+%
+%
% \section{Modification of Paths}\label{modification}
%
+%
% \subsection{Closing a path}\label{closing}
%
% In \MF{} one closes a path with any legal path connection between the
@@ -7477,7 +8172,8 @@ vardef makesector expr p = (pathcenter p)--p--cycle enddef;
% The \gbc{setpairs} statement makes \gbc{pp1}, \gbc{pp2} and \gbc{pp3}
% three points on the arc \gbc{p} in order. The arc we want goes from
% \gbc{pp3} to \gbc{pp1} with angle twice that of the corner angle at
-% \gbc{pp2}.
+% \gbc{pp2}. This function can be applied to an arbitrary path, and its
+% result will be an arc, but not necessarily a meaningful one.
% \begin{macrocode}
vardef arccomplement expr p =
if cycle p: onepointpath (false, pnt0(p))
@@ -7490,6 +8186,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Trimming a path}\label{trimming}
%
% \DescribeRoutine{cutoffbefore}
@@ -7545,6 +8242,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Creating arrows}\label{arrows}
%
% First, some better \mfc{direction} commands. They makes use of the fact
@@ -7555,7 +8253,6 @@ enddef;
% has a tangent at $z\sb0$ equal to the first one of $z\sb{j} -z\sb0$
% that is nonzero.
%
-% \DescribeRoutine{__dir}
% \gbc{__dir} gets the direction at point 0 for an arbitrary path.
% \gbc{postdirection}
% \DescribeRoutine{postdirection}
@@ -7815,7 +8512,8 @@ enddef;
% centered at the center of the dot and rotated 45 degrees, will encompass
% the whole square (theoretically).
% \begin{macrocode}
-path cut_path; cut_path := (.5,0)--(.5,.71)--(-.5,.71)--(-.5,0)--cycle;
+path cut_path;
+cut_path := (.5,0)--(.5,.71)--(-.5,.71)--(-.5,0)--cycle;
% \end{macrocode}
%
@@ -7875,6 +8573,7 @@ enddef;
% \end{macrocode}
%
+%
% \subsection{Randomizing a path}
%
% In order to randomly change a path, we need to randomly change its
@@ -7964,9 +8663,8 @@ enddef;
%
% \DescribeRoutine{detrivialized}
% We start with a routine that strips out trivial segments from a path.
-% This makes some loops a lot easier. We wouldn't want to differently
-% shift the two (equal) endpoints of a trivial segment, making it
-% nontrivial.
+% This makes some loops a lot easier. It would be weird to differently
+% shift the two (equal) endpoints of a trivial segment.
% \begin{macrocode}
vardef detrivialized expr f =
save g; path p, g[]; g := 0;
@@ -8032,10 +8730,133 @@ enddef;
% \end{macrocode}
%
+%
+% \subsection{Interpolating paths}
+%
+% Given two cubic B\'eziers, it is straightforward to create a path that
+% is ``half-way between'' them: just take its control points to be
+% at the midpoint between corresponding control points of the two
+% B\'eziers. Two paths made up of an equal number of B\'ezier are also
+% easily interpolated. However, two paths with different numbers of
+% B\'ezier segments need to be subdivided until they have an equal
+% number.
+%
+% \DescribeRoutine{interpolatedpath}
+% This command accepts a number \gbc{num}, a path or pair \gbc{P} and a
+% path \gbc{Q}. It returns a path which is somewhere ``between'' \gbc{P}
+% and \gbc{Q} if the number is between $0$ and $1$. The case where \gbc{P}
+% or \gbc{Q} is trivial is passed on to another command which is
+% considerably more efficient for that case. In the more general case, the
+% paths are rewritten so that they have equal length. For example, if
+% \gbc{P} has length 2 and \gbc{Q} has length 1, then \gbc{Q} is rewritten
+% as\\
+% \indent \gbc{subpath (0,1/2) of Q \& subpath (1/2,1) of Q}\\
+% which follows the same course as \gbc{Q} but has the same number of
+% B\'ezier parts as \gbc{P}.
+%
+% The splitting of \gbc{Q} shown above can, for reasons unknown to me,
+% produce adjacent subpaths that do not always share an endpoint. One
+% would think that \gbc{subpath (s,t) of Q} and
+% \gbc{subpath (t,u) of Q} would obviously end and start, respectively,
+% at \gbc{point t of Q}. Alas, they don't always. Hence, we employ
+% \gbc{force_equal_ends} to to make them equal, shifting their endpoints a
+% microscopic amount.
+%
+% If \gbc{Q} is a cycle we want the returned path to also be a cycle
+% (but not otherwise). This is possible whenever the ends of \gbc{P} are
+% equal.
+% \begin{macrocode}
+vardef interpolatedpath (expr t, P) expr Q =
+ if not path Q:
+ GBerrmsg ("Improper argument to interpolatedpath.")
+ "The last argument to interpolatedpath must be a path.";
+ if pair P: onepointpath(false, P)
+ else:
+ if path P:
+ P
+ else:
+ onepointpath (false, origin)
+ fi
+ fi
+ elseif pair P:
+ interpolated_pair_path (t, cycle Q, P, Q)
+ elseif not path P:
+ GBerrmsg ("Improper argument to interpolatedpath.")
+ "The second argument to interpolatedpath must be a pair "
+ & "or a path.";
+ Q
+ else:
+ if t=0: Q
+ elseif t=1: P
+ else:
+ save P_, Q_; path P_, Q_;
+ P_ := detrivialized P;
+ Q_ := detrivialized Q;
+ if length P_ = 0:
+ interpolated_pair_path (t, cycle Q, pnt0(P_), Q)
+ elseif length Q_ = 0:
+ interpolated_pair_path (t, cycle Q, pnt0(Q_), P)
+ else:
+ save G, H, n, m, k, r;
+ path G[], H[];
+ G := H := 0;
+ n := length P_; m := length Q_;
+ k := gcd(n, m);
+ r := m/k;
+ for I=0 upto n-1:
+ for J=0 upto r-1:
+ G[incr G] := subpath (I+J/r, I+(J+1)/r) of P_;
+ endfor
+ endfor
+ r := n/k;
+ for I=0 upto m-1:
+ for J=0 upto r-1:
+ H[incr H] := subpath (I+J/r, I+(J+1)/r) of Q_;
+ endfor
+ endfor
+ for N = 1 upto G-1:
+ force_equal_ends(G[N], G[N+1]);
+ force_equal_ends(H[N], H[N+1]);
+ endfor
+ interpolated_segment (t, G1, H1)
+ for N = 2 upto G: & interpolated_segment (t, G[N], H[N])
+ endfor if (pnt0(G1)=pnt1(G[G])) and (cycle Q): & cycle fi
+ fi
+ fi
+ fi
+enddef;
+
+% \end{macrocode}
+% \DescribeRoutine{interpolated_pair_path}
+% Since we cannot rely on the cyclicity of \gbc{Q}, we pass a boolean
+% parameter . That is because the second argument here might actually
+% have been the first argument of \gbc{interpolatedpath}.
+% \begin{macrocode}
+vardef interpolated_pair_path (expr t, cyclic, P, Q) =
+ save N; N := length Q;
+ if N=0: onepointpath (cyclic, (t)[pnt0(Q),P])
+ else:
+ (t)[pnt0(Q),P]..controls (t)[post0(Q),P] and
+ for n=1 upto N - 1:
+ (t)[pre[n](Q),P]..(t)[pnt[n](Q),P]..controls (t)[post[n](Q),P]
+ and
+ endfor
+ (t)[pre[N](Q),P].. if cyclic: cycle else: (t)[pnt[N](Q),P] fi
+ fi
+enddef;
+
+vardef interpolated_segment (expr t, S, T) =
+ (t)[ pnt0(S), pnt0(T)]..controls
+ (t)[ post0(S), post0(T)] and (t)[ pre1(S), pre1(T)]..
+ (t)[ pnt1(S), pnt1(T)]
+enddef;
+
+% \end{macrocode}
+%
% \subsection{Parallelling a path}
%
% \DescribeRoutine{parasegment}
-% This creates a path parallel to a given cubic Bezier segment \gbc{f}.
+% This creates a path parallel to a given cubic B\'ezier segment \gbc{f}.
% It should be called by a command (such as \gbc{parapath}) that makes
% sure \gbc{f} is nontrivial (meaning the directions are non-zero). It
% splits the segment into subsegments for accuracy. Its arguments are the
@@ -8088,7 +8909,7 @@ vardef parapath (expr d) expr f =
path g[], h, p[], q[];
numeric a, s, t;
pair u, v, w, w[];
- s := emax (3, emin (segment_split, ceiling (max_points/5/length f)));
+ s := emax(3, emin(segment_split, ceiling(max_points/5/length f)));
p := 0;
for i = 1 upto length f:
h := subpath (i-1, i) of f;
@@ -8156,6 +8977,7 @@ enddef;
% \end{macrocode}
%
+%
% \section{Miscellaneous}\label{misc}
%
% \subsection{Implementation of \mfpic{}'s \cs{plotdata} command}%
@@ -8337,8 +9159,8 @@ Plus.clear := (right--(1,1)--(-1,1)--(left)--cycle) scaled .65;
Cross := ((0,0)--(dir 45)--(dir -135)--(0,0)--(dir -45)--(dir 135))
scaled .65;
-Cross.clear := ((0,0)--(dir -45)--dir(45)--(dir 135)--(dir -135)--cycle)
- scaled .65;
+Cross.clear :=
+ ((0,0)--(dir -45)--dir(45)--(dir 135)--(dir -135)--cycle) scaled .65;
Asterisk := ((0,0)--up--down--(0,0)--(dir 30)--(dir -150)
--(0,0)--(dir -30)--(dir 150)) scaled .6;
@@ -8384,7 +9206,8 @@ save _A; pair _A[];
SolidStar := mkstar (5, 2, _A) scaled .84;
Star := undo_cycle SolidStar;
Star.clear := polyline (true)
- (_A9, _A10, _A1, _A2, _A3, (xpart _A3, 1), (xpart _A9, 1)) scaled .84;
+ (_A9, _A10, _A1, _A2, _A3, (xpart _A3, 1), (xpart _A9, 1))
+ scaled .84;
SolidStar.clear := Star.clear;
forsuffixes S =
@@ -8403,7 +9226,9 @@ endfor
% have it,
% \DescribeRoutine{lcm}
% \gbc{lcm} is a snap. Since \gbc{gcd} always returns a positive result,
-% \gbc{lcm} satisfies the usual rule for signs of products.
+% \gbc{lcm} satisfies the rule for signs of products. Note that these both
+% silently accept noninteger arguments, though the results may not be very
+% meaningful.
% \begin{macrocode}
vardef gcd (expr n, m) =
save a, b, r;
@@ -8648,7 +9473,8 @@ def barchart (expr firstbar, sep, r, vert)(text data) =
path chartbar[];
chartbar := 0; barwd := r*sep;
for _itm = data:
- barend[incr chartbar] := if pair _itm: ypart _itm else: _itm fi;
+ barend[incr chartbar]
+ := if pair _itm: ypart _itm else: _itm fi;
barbegin[chartbar] := if pair _itm: xpart _itm else: 0 fi;
endfor
barbegin := barend := barlength := barstart := chartbar;
@@ -8685,8 +9511,10 @@ enddef;
% \end{macrocode}
%
+%
%^^A Overlays - taken from MFbook, p 295. (Bruce Leban)
%
+%
% \subsection{Overlays}\label{overlays}
%
% This final code predates me. When I inherited \mfpic{} it contained no
@@ -8762,6 +9590,7 @@ numeric gcode; gcode := 0;
%</MF|MP>
% \end{macrocode}
%
+%
% \subsection{Dvips names for colors}\label{dvipsnam}
%
% In order to make \file{dvipsnam.mp} useful outside grafbase, we give