summaryrefslogtreecommitdiff
diff options
context:
space:
mode:
Diffstat
-rwxr-xr-xBuild/cdbuild/ctan2tds.pl2
-rw-r--r--Master/texmf-dist/doc/metapost/splines/README44
-rw-r--r--Master/texmf-dist/doc/metapost/splines/splines.pdfbin0 -> 103753 bytes
-rw-r--r--Master/texmf-dist/metapost/splines/splines.mp139
-rw-r--r--Master/texmf-dist/metapost/splines/testsplines.mp122
-rw-r--r--Master/texmf-dist/source/metapost/splines/splines.dtx393
-rw-r--r--Master/texmf-dist/source/metapost/splines/splines.ins48
-rw-r--r--Master/texmf-dist/tpm/splines.tpm30
-rw-r--r--Master/texmf/lists/splines9
-rw-r--r--Master/texmf/tpm/collection-metapost.tpm1
10 files changed, 788 insertions, 0 deletions
diff --git a/Build/cdbuild/ctan2tds.pl b/Build/cdbuild/ctan2tds.pl
index 1998f61cf55..de288004554 100755
--- a/Build/cdbuild/ctan2tds.pl
+++ b/Build/cdbuild/ctan2tds.pl
@@ -417,6 +417,7 @@ $standardfoundry='public';
'pstricks', 'generic',
'ruhyphen', 'generic',
'skaknew', 'metapost',
+ 'splines', 'metapost',
'thumbpdf', 'generic',
'trajan', 'fonts',
'velthuis', 'fonts',
@@ -525,6 +526,7 @@ $standardfmt='latex';
'plnfss', 'plain',
'pxfonts', 'fonts',
'shapepar', 'generic',
+ 'splines', 'metapost',
'texsis', 'texsis',
'thumbpdf', 'generic',
'trajan', 'fonts',
diff --git a/Master/texmf-dist/doc/metapost/splines/README b/Master/texmf-dist/doc/metapost/splines/README
new file mode 100644
index 00000000000..20998c92f1e
--- /dev/null
+++ b/Master/texmf-dist/doc/metapost/splines/README
@@ -0,0 +1,44 @@
+Splines -- version 0.2a
+
+This package consists of a few macros useful for producing cubic splines
+in MetaPost (or Metafont).
+
+Given a sequence of N+1 points, a spline is a path $z = f(t)$, $0 \le t
+\le N$, connecting them, which is a cubic function of $t$ between
+points, and has continuous first and second derivatives.
+
+Such a spline is two dimensional and can wander about anywhere in the
+picture plane. Another type of spline is one dimensional: given a
+sequence of $N$ locations $x_0 < x_1 < \dots < x_N$ on the $x$-axis and
+corresponding values $y_j$, a cubic spline interpolant is a function $y
+= f(x)$ satisfying $f(x_j) = y_j$, which is cubic between the $x_j$ and
+which has continuous first and second derivatives.
+
+With these macros one can draw two-dimensional cubic splines and the
+graphs of cubic spline interpolants.
+
+The files of this package consist of:
+ splines.dtx The documented source code.
+ splines.ins LaTeX this to produce the file splines.mp.
+ splines.pdf The documentation.
+ testsplines.mp Uses the macros to produce a few splines.
+ README The file you are reading
+
+The files of this package are Copyright 2002--2005, Daniel H. Luecking
+
+Splines.mp may be distributed and/or modified under the conditions of
+the LaTeX Project Public License, either version 1.3 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 1.3 or later is part of all distributions of LaTeX version
+2003/12/01 or later.
+
+Splines has maintenance status "author-maintained". The Current
+Maintainer is Daniel H. Luecking. The Base Interpreter is MetaPost
+(or Metafont).
+
+You may email problem reports to luecking {at} uark {dot} edu.
+
+--
+Dan Luecking
diff --git a/Master/texmf-dist/doc/metapost/splines/splines.pdf b/Master/texmf-dist/doc/metapost/splines/splines.pdf
new file mode 100644
index 00000000000..55604fa53a4
--- /dev/null
+++ b/Master/texmf-dist/doc/metapost/splines/splines.pdf
Binary files differ
diff --git a/Master/texmf-dist/metapost/splines/splines.mp b/Master/texmf-dist/metapost/splines/splines.mp
new file mode 100644
index 00000000000..c725c8b3b93
--- /dev/null
+++ b/Master/texmf-dist/metapost/splines/splines.mp
@@ -0,0 +1,139 @@
+%%
+%% This is file `splines.mp',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% splines.dtx (with options: `package')
+%%
+%% -------------------------------------------------------------------
+%%
+%% Copyright 2002--2005, Daniel H. Luecking
+%%
+%% Splines.mp may be distributed and/or modified under the conditions of the
+%% LaTeX Project Public License, either version 1.3 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 1.3 or later is part of all distributions of LaTeX version
+%% 2003/12/01 or later.
+%%
+%% Splines has maintenance status "author-maintained". The Current Maintainer
+%% is Daniel H. Luecking. The Base Interpreter is MetaPost (or Metafont).
+%%
+if known splines_fileversion: endinput fi;
+string splines_fileversion;
+splines_fileversion := "2006/09/25, v0.2a";
+message "Loading splines.mp " & splines_fileversion;
+
+def list_to_array (suffix arr) (text list) =
+ arr := 0;
+ for _itm = list :
+ arr[incr arr] := _itm;
+ endfor
+enddef;
+
+def compute_spline (expr closed) (suffix points, pr, po) =
+ % interior equations:
+ for j= 2 upto points - 1 :
+ % equate first derivatives:
+ po[j] + pr[j] = 2 points[j];
+ % and second derivatives:
+ pr[j+1] + 2 pr[j] = 2 po[j] + po[j-1];
+ endfor
+ % for a closed curve, the first and last are also interior:
+ if closed:
+ po 1 + pr 1 = 2 points 1;
+ po[points] + pr[points] = 2 points[points];
+ pr 2 + 2 pr 1 = 2 po 1 + po[points];
+ pr 1 + 2 pr[points] = 2 po[points] + po[points-1];
+ fi
+enddef;
+
+vardef mksplinepath (expr closed) (suffix points, pr, po) =
+ points1..controls po1 and
+ for j = 2 upto points if not closed: -1 fi:
+ pr[j]..points[j]..controls po[j] and
+ endfor
+ if closed: pr 1..cycle else: pr[points]..points[points] fi
+enddef;
+
+vardef mkrelaxedspline (suffix pnts) =
+ save rs_pr, rs_po;
+ pair rs_po[], rs_pr[];
+ % Equate second derivative to zero at both end points
+ rs_pr 2 + pnts 1 = 2 rs_po 1 ;
+ pnts[pnts] + rs_po[pnts-1] = 2 rs_pr[pnts];
+ compute_spline (false) (pnts, rs_pr, rs_po);
+ mksplinepath (false) (pnts, rs_pr, rs_po)
+enddef;
+
+vardef mkclosedspline (suffix pnts) =
+ save cs_pr, cs_po;
+ pair cs_pr[], cs_po[];
+ compute_spline (true) (pnts, cs_pr, cs_po);
+ mksplinepath (true) (pnts, cs_pr, cs_po)
+enddef;
+
+vardef dospline (expr closed) (text the_list) =
+ save _sp; pair _sp[];
+ list_to_array (_sp) (the_list);
+ if closed :
+ mkclosedspline (_sp)
+ else:
+ mkrelaxedspline (_sp)
+ fi
+enddef;
+
+def compute_fcnspline (suffix points, dx, sl) =
+ % Get delta_x:
+ for j = 1 upto points - 1: dx[j] := xpart (points[j+1]-points[j]);
+ endfor
+ for j=2 upto points - 1:
+ sl[j + 1] * dx[j] + 2sl[j]*(dx[j] + dx[j-1]) + sl[j-1]*dx[j-1]
+ = 3*ypart(points[j+1] - points[j-1]);
+ endfor
+enddef;
+
+vardef mkfcnsplinepath (suffix points, dx, sl) =
+ points1..controls (points1 + (1, sl1)*dx1/3) and
+ for j = 2 upto points - 1:
+ (points[j] - (1, sl[j])*dx[j-1]/3) ..points[j]..
+ controls (points[j] + (1,sl[j])*dx[j]/3) and
+ endfor
+ (points[points] - (1,sl[points])*dx[points-1]/3)..points[points]
+enddef;
+
+vardef mkperiodicfcnspline (suffix pnts) =
+ save _sl, _dx; numeric _dx[], _sl[];
+ compute_fcnspline (pnts, _dx, _sl);
+ % periodicity equations:
+ _sl 1 = _sl[pnts];
+ _sl 2 * _dx 1 + 2 _sl 1 * _dx 1 + 2 _sl[pnts] * _dx[pnts-1]
+ + _sl[pnts-1] * _dx[pnts-1]
+ = 3 * ypart(pnts[2] - pnts[pnts-1]);
+ mkfcnsplinepath (pnts, _dx, _sl)
+enddef;
+
+vardef mkrelaxedfcnspline (suffix pnts) =
+ save _sl, _dx; numeric _dx[], _sl[];
+ compute_fcnspline (pnts, _dx, _sl);
+ % relaxation equations.
+ _sl 2 * _dx 1 + 2 _sl1 * _dx 1 = 3 * ypart(pnts2 - pnts1);
+ _sl[pnts-1] * _dx[pnts-1] + 2 _sl[pnts] * _dx[pnts-1]
+ = 3 * ypart(pnts[pnts] - pnts[pnts-1]);
+ mkfcnsplinepath (pnts, _dx, _sl)
+enddef;
+
+vardef fcnspline (expr periodic) (text the_list) =
+ save _fs; pair _fs[];
+ list_to_array (_fs) (the_list);
+ if periodic:
+ mkperiodicfcnspline (_fs)
+ else:
+ mkrelaxedfcnspline (_fs)
+ fi
+enddef;
+
+endinput.
+%%
+%% End of file `splines.mp'.
diff --git a/Master/texmf-dist/metapost/splines/testsplines.mp b/Master/texmf-dist/metapost/splines/testsplines.mp
new file mode 100644
index 00000000000..a6ad01d0adb
--- /dev/null
+++ b/Master/texmf-dist/metapost/splines/testsplines.mp
@@ -0,0 +1,122 @@
+input splines;
+
+numeric u; u:=1cm;
+
+path xaxis, yaxis;
+xaxis := (-2.5u,0)--(1.5u,0);
+yaxis := (0,-.5u)--(0,1.5u);
+
+def shiftit =
+ currentpicture := currentpicture shifted -llcorner currentpicture;
+enddef;
+
+def dodots (text t) =
+ for _loc = t:
+ fill (fullcircle scaled 2pt) shifted (u*_loc);
+ endfor
+enddef;
+
+
+
+beginfig(1);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ path p;
+ p := dospline (true)
+ ((-1,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ draw p scaled u;
+label(btex From {\tt dospline (true)}. etex,(.25u,-.75u));
+dodots ((-1,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+shiftit;
+endfig;
+
+beginfig(2);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ save A; pair A[];
+ list_to_array (A)((-1,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ path p;
+ p := mkclosedspline (A);
+ draw p scaled u;
+label(btex From {\tt mkclosedspline}. etex,(.25u,-.75u));
+dodots (A1, A2, A3, A4, A5, A6, A7);
+shiftit;
+endfig;
+
+beginfig(3);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ path p;
+ p := dospline (false)
+ ((-1,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ draw p scaled u;
+label(btex From {\tt dospline (false)}. etex,(.25u,-.75u));
+dodots ((-1,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+shiftit;
+endfig;
+
+beginfig(4);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ save A; pair A[];
+ list_to_array (A)((-1,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ path p;
+ p := mkrelaxedspline (A);
+ draw p scaled u;
+label(btex From {\tt mkrelaxedspline}. etex,(.25u,-.75u));
+dodots (A1, A2, A3, A4, A5, A6, A7);
+shiftit;
+endfig;
+
+beginfig(5);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ path p;
+ p := fcnspline (true)
+ ((-2,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ draw p scaled u;
+label(btex From {\tt fcnspline (true)}. etex,(.25u,-.75u));
+dodots ((-2,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+shiftit;
+endfig;
+
+beginfig(6);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ save A; pair A[];
+ list_to_array(A) ((-2,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ path p;
+ p := mkperiodicfcnspline (A);
+ draw p scaled u;
+label(btex From {\tt mkperiodicfcnspline}. etex,(.25u,-.75u));
+dodots (A1, A2, A3, A4, A5, A6, A7);
+shiftit;
+endfig;
+
+beginfig(7);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ path p;
+ p := fcnspline (false)
+ ((-2,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ draw p scaled u;
+label(btex From {\tt fcnspline (false)}. etex,(.25u,-.75u));
+dodots((-2,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+shiftit;
+endfig;
+
+beginfig(8);
+ drawarrow xaxis;
+ drawarrow yaxis;
+ save A; pair A[];
+ list_to_array(A) ((-2,0), (-1,1), (-.5,.8), (0,1),(.5,.4), (1,0), (1.5,1));
+ path p;
+ p := mkrelaxedfcnspline (A);
+ draw p scaled u;
+label(btex From {\tt mkrelaxedfcnspline}. etex,(.25u,-.75u));
+dodots (A1, A2, A3, A4, A5, A6, A7);
+shiftit;
+endfig;
+
+end.
+
diff --git a/Master/texmf-dist/source/metapost/splines/splines.dtx b/Master/texmf-dist/source/metapost/splines/splines.dtx
new file mode 100644
index 00000000000..da89dda74d6
--- /dev/null
+++ b/Master/texmf-dist/source/metapost/splines/splines.dtx
@@ -0,0 +1,393 @@
+% \iffalse
+% -------------------------------------------------------------------
+%
+% Copyright 2002--2004, Daniel H. Luecking
+%
+% Splines.mp may be distributed and/or modified under the conditions of the
+% LaTeX Project Public License, either version 1.3 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 1.3 or later is part of all distributions of LaTeX version
+% 2003/12/01 or later.
+%
+% Splines has maintenance status "author-maintained". The Current Maintainer
+% is Daniel H. Luecking. The Base Interpreter is MetaPost (or Metafont).
+%
+%<*driver>
+\ProvidesFile{splines.dtx}
+ [2005/02/05 v0.2 Metafont/post macros to compute splines.]%
+\documentclass[draft]{ltxdoc}
+\usepackage{docmfp}
+
+\addtolength{\textwidth}{.5878pt}
+
+\def\mytt{\upshape\mdseries\ttfamily}
+\renewcommand\marg[1]{{\mytt \{#1\}}}
+\renewcommand\oarg[1]{{\mytt [#1]}}
+\renewcommand\parg[1]{{\mytt (#1)}}
+\renewcommand{\meta}[1]{{$\langle$\rmfamily\itshape#1$\rangle$}}
+\DeclareRobustCommand\cs[1]{{\mytt\char`\\#1}}
+\def\prog#1{{\mdseries\scshape #1}}
+
+
+
+
+\def\MF{\prog{meta\-font}}
+\def\MP{\prog{meta\-post}}
+
+\def\CMF{\prog{Meta\-font}}
+\def\CMP{\prog{Meta\-post}}
+\def\opt#1{{\sffamily\upshape#1}}
+\def\mfc#1{{\mytt#1}}
+\let\env\mfc
+\let\file\mfc
+\let\gbc\mfc
+\renewcommand\{{{\mytt\char`\{}}
+\renewcommand\}{{\mytt\char`\}}}
+\renewcommand\|{${}\mathrel{|}{}$}
+
+\makeatletter
+\newcommand\bsl{{\mytt\@backslashchar}}
+% Stupid lists!
+\def\@listi{\leftmargin\leftmargini
+ \parsep \z@ \@plus\p@ \@minus\z@
+ \topsep 4\p@ \@plus\p@ \@minus2\p@
+ \itemsep\parsep}
+\let\@listI\@listi \@listi
+\renewcommand\labelitemi{\normalfont\bfseries \textendash}
+\renewcommand\labelitemii{\textasteriskcentered}
+\renewcommand\labelitemiii{\textperiodcentered}
+\leftmargini\parindent
+% Stupid index!
+\def\usage#1{\textrm{#1}}
+\def\index@prologue{\section*{Index}\markboth{Index}{Index}Numbers
+refer to the page where the corresponding entry is described.}
+\def\IndexParms{%
+ \parindent \z@ \columnsep 15pt
+ \parskip 0pt plus 1pt
+ \rightskip 5pt plus2em \mathsurround \z@
+ \parfillskip=-5pt \small
+ % less hanging:
+ \def\@idxitem{\par\hangindent 20pt}%
+ \def\subitem{\@idxitem\hspace*{15pt}}%
+ \def\subsubitem{\@idxitem\hspace*{25pt}}%
+ \def\indexspace{\par\vspace{10pt plus 2pt minus 3pt}}}
+\renewcommand\routinestring{}
+\renewcommand\variablestring{\space(var.)}
+% Why does every command have to be indexed twice?
+\renewcommand\SpecialMfpIndex[3]{\@bsphack
+ \index{%
+ \string#1\actualchar
+ \string\verb\quotechar*\verbatimchar\string#1\verbatimchar
+ #2 \encapchar usage}%
+ \@esphack}
+\def\close@crossref{\SpecialEscapechar{:}}
+\makeatother
+\def\VariableIndex#1{\SpecialMfpIndex{#1}{\variablestring}{}}
+\def\RoutineIndex #1{\SpecialMfpIndex{#1}{}{}}
+
+\title{Macros to Compute Splines\thanks{This file has version number
+ \fileversion, last revised \filedate.}}
+\author{Dan Luecking}
+\date{\filedate}
+\SpecialEscapechar{:}
+\def\bslash{:}
+\DisableCrossrefs
+\CodelineIndex
+\AlsoImplementation
+
+\begin{document}
+ \DeleteShortVerb{\|}
+ \DocInput{splines.dtx}
+\end{document}
+%</driver>
+%\fi
+%
+% \CheckSum{25}
+% \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
+% Digits \0\1\2\3\4\5\6\7\8\9
+% Exclamation \! Double quote \" Hash (number) \#
+% Dollar \$ Percent \% Ampersand \&
+% Acute accent \' Left paren \( Right paren \)
+% Asterisk \* Plus \+ Comma \,
+% Minus \- Point \. Solidus \/
+% Colon \: Semicolon \; Less than \<
+% Equals \= Greater than \> Question mark \?
+% Commercial at \@ Left bracket \[ Backslash \\
+% Right bracket \] Circumflex \^ Underscore \_
+% Grave accent \` Left brace \{ Vertical bar \|
+% Right brace \} Tilde \~}
+%
+% \catcode`\_=12
+% \GetFileInfo{splines.dtx}
+% \maketitle
+%
+% \StopEventually{\PrintIndex}
+% A cubic spline through a set of points is a curve obtained by joining
+% each point to the next with a cubic parametrized curve, where adjoining
+% cubics must have matching first and second derivative at their common
+% point.
+%
+% It is possible for \MP{} to compute the necessary controls.
+% Unfortunately, the controls are not uniquely determined unless the curve
+% is required to be closed. For open curves, there is need for two
+% additional equations at the end points. A `relaxed spline' is produced
+% if we require that the second derivative is $(0,0)$ at those points.
+% For a closed curve, the equality of the first and second derivatives at
+% the common beginning/ending point gives the needed additional equations.
+%
+% Note that this equates \emph{time} derivatives, so this works best when
+% points are relatively evenly spaced and so the speed is relatively
+% uniform. If points are differently spaced then the relatively slower
+% speed between closely spaced points allows sharper turns without large
+% second derivatives. Curves produced tend to have a more natural look,
+% and relaxed splines are most suitable for smoothing data that is
+% obtained by taking observations at evenly spaced times. Still, the
+% technique is somewhat unstable when points are closely spaced, for
+% example when a small change in the position of one point can produce a
+% large change in its direction when viewed from another point.
+%
+% Start with version control information.
+% \begin{macrocode}
+%<*package>
+if known splines_fileversion: endinput fi;
+string splines_fileversion;
+splines_fileversion := "2006/09/25, v0.2a";
+message "Loading splines.mp " & splines_fileversion;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{list_to_array}
+% Now for a command that takes a variable name \gbc{(suffix arr)} and
+% copies a list of pairs to \mfc{arr1}, \mfc{arr2}, etc..
+% The suffix must be declared by the calling program so that \mfc{arr}
+% is numeric but \mfc{arr[\,]} are pairs, for example by ``\mfc{save arr; pair
+% arr[\,];}''.
+% \begin{macrocode}
+def list_to_array (suffix arr) (text list) =
+ arr := 0;
+ for _itm = list :
+ arr[incr arr] := _itm;
+ endfor
+enddef;
+
+% \end{macrocode}
+%
+% \DescribeRoutine{compute_spline}
+% In this command we generate the equations common to all cubic
+% splines: the equality of derivatives at all interior points.
+% This command accepts three suffixes: \gbc{points}, \gbc{pr}, and
+% \gbc{po} which should represent previously declared arrays of pairs.
+% \gbc{points} is the array of points to be connected, and must be known.
+% Arrays \gbc{pr} and \gbc{po} \emph{must} be unknown and will hold the
+% computed control points. See the code of \mfc{mkrelaxedspline} for
+% an example of how this was arranged for the variables \mfc{rs_pr} and
+% \mfc{rs_po}.
+%
+% Contrary to the previous (unreleased) version, \mfc{compute_spline}
+% takes a boolean argument and appends the same equations at the first (=
+% last) point.
+%
+% \DescribeRoutine{mkrelaxedspline}
+% The first of these macros appends the necessary additional equations to
+% get zero second derivatives at the ends.
+% \DescribeRoutine{mkclosedspline}
+% The second simply calls \mfc{compute_spline} with the boolean set to
+% true. Both return the computed path. In theory the knowledgeable user
+% can call \gbc{compute_spline (false)}, append a choice of equations for
+% the ends, and then call \gbc{mksplinepath (false)}.
+%
+% \DescribeRoutine{dospline}
+% This version accepts a list of pairs and produces a spline through
+% them. It simply stores the list in an array and calls the appropriate
+% version that operates on an array.
+% \begin{macrocode}
+def compute_spline (expr closed) (suffix points, pr, po) =
+ % interior equations:
+ for j= 2 upto points - 1 :
+ % equate first derivatives:
+ po[j] + pr[j] = 2 points[j];
+ % and second derivatives:
+ pr[j+1] + 2 pr[j] = 2 po[j] + po[j-1];
+ endfor
+ % for a closed curve, the first and last are also interior:
+ if closed:
+ po 1 + pr 1 = 2 points 1;
+ po[points] + pr[points] = 2 points[points];
+ pr 2 + 2 pr 1 = 2 po 1 + po[points];
+ pr 1 + 2 pr[points] = 2 po[points] + po[points-1];
+ fi
+enddef;
+
+vardef mksplinepath (expr closed) (suffix points, pr, po) =
+ points1..controls po1 and
+ for j = 2 upto points if not closed: -1 fi:
+ pr[j]..points[j]..controls po[j] and
+ endfor
+ if closed: pr 1..cycle else: pr[points]..points[points] fi
+enddef;
+
+vardef mkrelaxedspline (suffix pnts) =
+ save rs_pr, rs_po;
+ pair rs_po[], rs_pr[];
+ % Equate second derivative to zero at both end points
+ rs_pr 2 + pnts 1 = 2 rs_po 1 ;
+ pnts[pnts] + rs_po[pnts-1] = 2 rs_pr[pnts];
+ compute_spline (false) (pnts, rs_pr, rs_po);
+ mksplinepath (false) (pnts, rs_pr, rs_po)
+enddef;
+
+vardef mkclosedspline (suffix pnts) =
+ save cs_pr, cs_po;
+ pair cs_pr[], cs_po[];
+ compute_spline (true) (pnts, cs_pr, cs_po);
+ mksplinepath (true) (pnts, cs_pr, cs_po)
+enddef;
+
+vardef dospline (expr closed) (text the_list) =
+ save _sp; pair _sp[];
+ list_to_array (_sp) (the_list);
+ if closed :
+ mkclosedspline (_sp)
+ else:
+ mkrelaxedspline (_sp)
+ fi
+enddef;
+
+% \end{macrocode}
+% The above computations produce a $2$-dimensional spline. A $1$-dimensional
+% cubic spline would be a function $f(t)$ with numeric values rather
+% 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 $x\sb j$.
+%
+% The requirements on our $2$-dimensional path are the following:
+% \begin{enumerate}
+% \item The $j$th link should connect $(x\sb{j},y\sb{j})$ to $(x\sb{j+1},
+% y\sb{j+1})$.
+% \item The $x$-part of that link should increase linearly from $x\sb{j}$ to
+% $x\sb{j+1}$ as $t$ goes from $0$ to $1$.
+% \item The $y$-part should be a cubic $y = f(x)$.
+% \item The $x$-derivatives $df/dx$ and $d^2f/dx^2$ should match at the
+% connecting points.
+% \end{enumerate}
+%
+% Two necessary equations for converting between $x$ and $t$ coordinates
+% are:
+% \begin{equation}\label{first}
+% x = x\sb{j} + t \Delta x\sb{j}
+% \end{equation}
+% (where $\Delta x\sb{j} = x\sb{j+1} - x\sb{j}$) and
+% \begin{equation}\label{second}
+% \frac{df}{dt} = \frac{dx}{dt}\frac{df}{dx} = (x\sb{j+1} -
+% x\sb{j}) \frac{df}{dx}.
+% \end{equation}
+% Thus we want to choose controls so that (\ref{first}) is maintained and
+% so that $x$-derivatives match. It turns out that this requires controls
+% at
+% \begin{equation}
+% \begin{array}{c}
+% (x\sb{j}, y\sb{j}) - (\Delta x\sb{j-1}, s\sb{j} \Delta x\sb{j-1})/3\\
+% (x\sb{j}, y\sb{j}) + (\Delta x\sb{j} , s\sb{j} \Delta x\sb{j} )/3
+% \end{array}
+% \end{equation}
+% where $s\sb{j}$ is the slope (derivative) at $x\sb{j}$. These control
+% points will produce matching slopes regardless of the values
+% chosen for the $s\sb j$. To get matching second derivatives we need the
+% same conditions as in parametric splines. But those equations simplify
+% to the form:
+% \begin{displaymath}
+% s\sb{j+1} dx\sb{j} - 2s\sb{j} (dx\sb{j} + dx\sb{j-1}) +
+% s\sb{j-1}dx\sb{j-1}
+% = 3y\sb{j+1} - 3y\sb{j-1}.
+% \end{displaymath}
+% As with 2-D splines there can be almost any equations at the end points.
+% For a relaxed spline we equate the second derivatives to 0. To get a
+% periodic function, we equate the slope and second derivative at
+% beginning to those at the end. This makes it possible to put a shifted
+% copy of the graph with starting point at the end of the original and
+% have the same smoothness at that connection as at the other points.
+%
+% \DescribeRoutine{compute_fcnspline}
+% This issues the equation for the slopes (array \gbc{sl} of
+% \emph{unknown} numerics). The array \gbc{points} contains the $(x,y)$
+% values and \gbc{dx} is a temporary numeric array which will be
+% overwritten if known.
+%
+% \DescribeRoutine{mkfcnsplinepath}
+% This simply assembles the path from the information computed by the
+% above equations (and the extra equations given in the calling command).
+%
+% \DescribeRoutine{mkrelaxedfcnspline}
+% This sets up arrays for the \gbc{dx} and \gbc{sl} parameters of
+% \mfc{compute_fcnspline}, emit the necessary endpoint equations (zero
+% second derivatives) and calls the previous two routines.
+%
+% \DescribeRoutine{mkperiodicfcnspline}
+% This does the same as the previous command, but the endpoint equations
+% make the first and second derivatives at the start equal to those at
+% the end.
+%
+% \DescribeRoutine{fcnspline}
+% Finally, this command copies a list of pairs into an array and calls the
+% appropriate command to process them.
+% \begin{macrocode}
+def compute_fcnspline (suffix points, dx, sl) =
+ % Get delta_x:
+ for j = 1 upto points - 1: dx[j] := xpart (points[j+1]-points[j]);
+ endfor
+ for j=2 upto points - 1:
+ sl[j + 1] * dx[j] + 2sl[j]*(dx[j] + dx[j-1]) + sl[j-1]*dx[j-1]
+ = 3*ypart(points[j+1] - points[j-1]);
+ endfor
+enddef;
+
+vardef mkfcnsplinepath (suffix points, dx, sl) =
+ points1..controls (points1 + (1, sl1)*dx1/3) and
+ for j = 2 upto points - 1:
+ (points[j] - (1, sl[j])*dx[j-1]/3) ..points[j]..
+ controls (points[j] + (1,sl[j])*dx[j]/3) and
+ endfor
+ (points[points] - (1,sl[points])*dx[points-1]/3)..points[points]
+enddef;
+
+vardef mkperiodicfcnspline (suffix pnts) =
+ save _sl, _dx; numeric _dx[], _sl[];
+ compute_fcnspline (pnts, _dx, _sl);
+ % periodicity equations:
+ _sl 1 = _sl[pnts];
+ _sl 2 * _dx 1 + 2 _sl 1 * _dx 1 + 2 _sl[pnts] * _dx[pnts-1]
+ + _sl[pnts-1] * _dx[pnts-1]
+ = 3 * ypart(pnts[2] - pnts[pnts-1]);
+ mkfcnsplinepath (pnts, _dx, _sl)
+enddef;
+
+vardef mkrelaxedfcnspline (suffix pnts) =
+ save _sl, _dx; numeric _dx[], _sl[];
+ compute_fcnspline (pnts, _dx, _sl);
+ % relaxation equations.
+ _sl 2 * _dx 1 + 2 _sl1 * _dx 1 = 3 * ypart(pnts2 - pnts1);
+ _sl[pnts-1] * _dx[pnts-1] + 2 _sl[pnts] * _dx[pnts-1]
+ = 3 * ypart(pnts[pnts] - pnts[pnts-1]);
+ mkfcnsplinepath (pnts, _dx, _sl)
+enddef;
+
+vardef fcnspline (expr periodic) (text the_list) =
+ save _fs; pair _fs[];
+ list_to_array (_fs) (the_list);
+ if periodic:
+ mkperiodicfcnspline (_fs)
+ else:
+ mkrelaxedfcnspline (_fs)
+ fi
+enddef;
+
+%</package>
+% \end{macrocode}
+% \clearpage
+%\Finale
diff --git a/Master/texmf-dist/source/metapost/splines/splines.ins b/Master/texmf-dist/source/metapost/splines/splines.ins
new file mode 100644
index 00000000000..ac047affa34
--- /dev/null
+++ b/Master/texmf-dist/source/metapost/splines/splines.ins
@@ -0,0 +1,48 @@
+\input docstrip
+\keepsilent
+\preamble
+
+-------------------------------------------------------------------
+
+Copyright 2002--2005, Daniel H. Luecking
+
+Splines.mp may be distributed and/or modified under the conditions of the
+LaTeX Project Public License, either version 1.3 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 1.3 or later is part of all distributions of LaTeX version
+2003/12/01 or later.
+
+Splines has maintenance status "author-maintained". The Current Maintainer
+is Daniel H. Luecking. The Base Interpreter is MetaPost (or Metafont).
+
+\endpreamble
+\edef\zyx{%
+endinput.^^J%
+\DoubleperCent^^J%
+\DoubleperCent\space End of file `\outFileName'.}
+
+\askforoverwritefalse
+\usepostamble\zyx
+\generate{\file{splines.mp}{\from{splines.dtx}{package}}}
+
+\def\sJ{^^J \space\space\space}
+\immediate\write16{%
+^^J*******************************************************************
+^^J
+^^J To finish the installation:
+^^J
+^^J -- Copy the file splines.mp to a location where MetaPost
+\sJ will find it, for example TEXMF/metapost/splines/ .
+^^J
+^^J -- If desired, copy splines.mp to a location where Metafont will
+\sJ find it, e.g., TEXMF/metafont/splines/ . Rename it splines.mf.
+^^J
+^^J -- To generate the documentation, run latex on splines.dtx (requires
+\sJ the package docmfp).
+^^J
+^^J -- Refresh the filename database if your TeX system requires it.
+^^J
+^^J*******************************************************************}
+
+\endbatchfile
diff --git a/Master/texmf-dist/tpm/splines.tpm b/Master/texmf-dist/tpm/splines.tpm
new file mode 100644
index 00000000000..4468aa5efd0
--- /dev/null
+++ b/Master/texmf-dist/tpm/splines.tpm
@@ -0,0 +1,30 @@
+<!DOCTYPE rdf:RDF SYSTEM "../../support/tpm.dtd">
+<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:TPM="http://texlive.dante.de/">
+ <rdf:Description about="http://texlive.dante.de/texlive/Package/splines.zip">
+ <TPM:Name>splines</TPM:Name>
+ <TPM:Type>Package</TPM:Type>
+ <TPM:Date>2006/09/25 21:39:00</TPM:Date>
+ <TPM:Version></TPM:Version>
+ <TPM:Creator>karl</TPM:Creator>
+ <TPM:Title>The splines package.</TPM:Title>
+ <TPM:Description></TPM:Description>
+ <TPM:Author></TPM:Author>
+ <TPM:Size>128780</TPM:Size>
+ <TPM:Build/>
+ <TPM:RunFiles size="8100">
+texmf-dist/metapost/splines/splines.mp
+texmf-dist/metapost/splines/testsplines.mp
+texmf-dist/tpm/splines.tpm
+ </TPM:RunFiles>
+ <TPM:DocFiles size="105600">
+texmf-dist/doc/metapost/splines/README
+texmf-dist/doc/metapost/splines/splines.pdf
+ </TPM:DocFiles>
+ <TPM:SourceFiles size="16159">
+texmf-dist/source/metapost/splines/splines.dtx
+texmf-dist/source/metapost/splines/splines.ins
+ </TPM:SourceFiles>
+ <TPM:Provides>Package/splines</TPM:Provides>
+ </rdf:Description>
+</rdf:RDF>
+
diff --git a/Master/texmf/lists/splines b/Master/texmf/lists/splines
new file mode 100644
index 00000000000..eb317003008
--- /dev/null
+++ b/Master/texmf/lists/splines
@@ -0,0 +1,9 @@
+texmf-dist/doc/metapost/splines/README
+texmf-dist/doc/metapost/splines/splines.pdf
+texmf-dist/source/metapost/splines/splines.dtx
+texmf-dist/source/metapost/splines/splines.ins
+texmf-dist/metapost/splines/splines.mp
+texmf-dist/metapost/splines/testsplines.mp
+texmf-dist/tpm/splines.tpm
+
+texmf/lists/splines
diff --git a/Master/texmf/tpm/collection-metapost.tpm b/Master/texmf/tpm/collection-metapost.tpm
index f1179a4eacb..95bb2123edb 100644
--- a/Master/texmf/tpm/collection-metapost.tpm
+++ b/Master/texmf/tpm/collection-metapost.tpm
@@ -32,6 +32,7 @@
<TPM:Package name="piechartmp"/>
<TPM:Package name="roex"/>
<TPM:Package name="slideshow"/>
+ <TPM:Package name="splines"/>
<TPM:TLCore name="bin-metapost"/>
<TPM:TLCore name="collection-basic"/>
</TPM:Requires>
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-01 21:00:05 +0000