summaryrefslogtreecommitdiff
diff options
context:
space:
mode:
Diffstat
-rw-r--r--Master/texmf-dist/source/latex/bez123/README27
-rw-r--r--Master/texmf-dist/source/latex/bez123/bez123.dtx1818
-rw-r--r--Master/texmf-dist/source/latex/bez123/bez123.ins61
3 files changed, 1906 insertions, 0 deletions
diff --git a/Master/texmf-dist/source/latex/bez123/README b/Master/texmf-dist/source/latex/bez123/README
new file mode 100644
index 00000000000..ae7e4100efb
--- /dev/null
+++ b/Master/texmf-dist/source/latex/bez123/README
@@ -0,0 +1,27 @@
+ LaTeX can be used for drawing quadratic Bezier curves. The bez123
+package provides additional facilities for drawing linear, cubic, and
+rational quadratic Bezier curves. The multiply package provides a
+command for multiplication of a length without numerical overflow.
+
+
+ The distribution consists of the following files:
+README (this file)
+bez123.ins
+bez123.dtx
+
+ The distribution is provided under the terms of the LaTeX
+Project Public License (LPPL).
+
+ To install the package:
+o run: latex bez123.ins (which will generate bez123.sty and multiply.sty)
+o run: latex bez123.dtx (Note that bez123.dtx requires bez123.sty.
+ Several Package Warning messages will occur, but
+ these can be ignored.)
+o run: makeindex -s gind.ist bez123
+o run: latex bez123.dtx
+o Print bez123.dvi for a hardcopy of the package manual
+o Move bez123.sty and multiply.sty to a location where LaTeX will find them.
+
+1998/10/14
+Peter Wilson
+peter.r.wilson@boeing.com
diff --git a/Master/texmf-dist/source/latex/bez123/bez123.dtx b/Master/texmf-dist/source/latex/bez123/bez123.dtx
new file mode 100644
index 00000000000..042c208347b
--- /dev/null
+++ b/Master/texmf-dist/source/latex/bez123/bez123.dtx
@@ -0,0 +1,1818 @@
+% \iffalse meta-comment
+%
+% bez123.dtx
+% Author: Peter Wilson (CUA and NIST)
+% peter.r.wilson@boeing.com
+%
+% Copyright 1998 Peter R. Wilson
+%
+% This program is provided under the terms of the
+% LaTeX Project Public License distributed from CTAN
+% archives in directory macros/latex/base/lppl.txt.
+%
+%
+%<*driver>
+\documentclass{ltxdoc}
+\usepackage{bez123}
+\EnableCrossrefs
+\CodelineIndex
+\setcounter{StandardModuleDepth}{1}
+\begin{document}
+ \DocInput{bez123.dtx}
+\end{document}
+%</driver>
+%
+% \fi
+%
+% \CheckSum{990}
+%
+% \DoNotIndex{\',\.,\@M,\@@input,\@addtoreset,\@arabic,\@badmath}
+% \DoNotIndex{\@centercr,\@cite}
+% \DoNotIndex{\@dotsep,\@empty,\@float,\@gobble,\@gobbletwo,\@ignoretrue}
+% \DoNotIndex{\@input,\@ixpt,\@m}
+% \DoNotIndex{\@minus,\@mkboth,\@ne,\@nil,\@nomath,\@plus,\@set@topoint}
+% \DoNotIndex{\@tempboxa,\@tempcnta,\@tempdima,\@tempdimb}
+% \DoNotIndex{\@tempswafalse,\@tempswatrue,\@viipt,\@viiipt,\@vipt}
+% \DoNotIndex{\@vpt,\@warning,\@xiipt,\@xipt,\@xivpt,\@xpt,\@xviipt}
+% \DoNotIndex{\@xxpt,\@xxvpt,\\,\ ,\addpenalty,\addtolength,\addvspace}
+% \DoNotIndex{\advance,\Alph,\alph}
+% \DoNotIndex{\arabic,\ast,\begin,\begingroup,\bfseries,\bgroup,\box}
+% \DoNotIndex{\bullet}
+% \DoNotIndex{\cdot,\cite,\CodelineIndex,\cr,\day,\DeclareOption}
+% \DoNotIndex{\def,\DisableCrossrefs,\divide,\DocInput,\documentclass}
+% \DoNotIndex{\DoNotIndex,\egroup,\ifdim,\else,\fi,\em,\endtrivlist}
+% \DoNotIndex{\EnableCrossrefs,\end,\end@dblfloat,\end@float,\endgroup}
+% \DoNotIndex{\endlist,\everycr,\everypar,\ExecuteOptions,\expandafter}
+% \DoNotIndex{\fbox}
+% \DoNotIndex{\filedate,\filename,\fileversion,\fontsize,\framebox,\gdef}
+% \DoNotIndex{\global,\halign,\hangindent,\hbox,\hfil,\hfill,\hrule}
+% \DoNotIndex{\hsize,\hskip,\hspace,\hss,\if@tempswa,\ifcase,\or,\fi,\fi}
+% \DoNotIndex{\ifhmode,\ifvmode,\ifnum,\iftrue,\ifx,\fi,\fi,\fi,\fi,\fi}
+% \DoNotIndex{\input}
+% \DoNotIndex{\jobname,\kern,\leavevmode,\let,\leftmark}
+% \DoNotIndex{\list,\llap,\long,\m@ne,\m@th,\mark,\markboth,\markright}
+% \DoNotIndex{\month,\newcommand,\newcounter,\newenvironment}
+% \DoNotIndex{\NeedsTeXFormat,\newdimen}
+% \DoNotIndex{\newlength,\newpage,\nobreak,\noindent,\null,\number}
+% \DoNotIndex{\numberline,\OldMakeindex,\OnlyDescription,\p@}
+% \DoNotIndex{\pagestyle,\par,\paragraph,\paragraphmark,\parfillskip}
+% \DoNotIndex{\penalty,\PrintChanges,\PrintIndex,\ProcessOptions}
+% \DoNotIndex{\protect,\ProvidesClass,\raggedbottom,\raggedright}
+% \DoNotIndex{\refstepcounter,\relax,\renewcommand,\reset@font}
+% \DoNotIndex{\rightmargin,\rightmark,\rightskip,\rlap,\rmfamily,\roman}
+% \DoNotIndex{\roman,\secdef,\selectfont,\setbox,\setcounter,\setlength}
+% \DoNotIndex{\settowidth,\sfcode,\skip,\sloppy,\slshape,\space}
+% \DoNotIndex{\symbol,\the,\trivlist,\typeout,\tw@,\undefined,\uppercase}
+% \DoNotIndex{\usecounter,\usefont,\usepackage,\vfil,\vfill,\viiipt}
+% \DoNotIndex{\viipt,\vipt,\vskip,\vspace}
+% \DoNotIndex{\wd,\xiipt,\year,\z@}
+%
+% \def\fileversion{v1.1}
+% \def\filedate{1998/10/14}
+% \newcommand*{\Lpack}[1]{\textsf {#1}} ^^A typest a package
+% \newcommand*{\Lopt}[1]{\textsf {#1}} ^^A typeset an option
+% \newcommand*{\file}[1]{\texttt {#1}} ^^A typeset a file
+% \newcommand*{\Lcount}[1]{\textsl {\small#1}} ^^A typeset a counter
+% \newcommand*{\pstyle}[1]{\textsl {#1}} ^^A typeset a pagestyle
+% \newcommand*{\Lenv}[1]{\texttt {#1}} ^^A typeset an environment
+% \newcommand{\eqref}[1]{equation~(\ref{#1})} ^^A typeset ref to an equation
+%
+% \title{The \Lpack{bez123} and \Lpack{multiply} packages\thanks{This
+% file has version number \fileversion, last revised
+% \filedate.}}
+%
+% \author{%
+% Peter Wilson\\
+% Catholic University of America\thanks{This work was originally
+% performed as
+% a Guest Researcher at the National Institute of Standards and Technology.} \\
+% Now at \texttt{peter.r.wilson@boeing.com}
+% }
+% \date{\filedate}
+% \maketitle
+% \begin{abstract}
+% The \Lpack{bez123} package provides for the drawing of linear, cubic,
+% and rational quadratic Bezier curves. The \Lpack{multiply} package
+% provides a command to multiply a length without numerical overflow.
+% \end{abstract}
+% \tableofcontents
+% \listoftables
+% \listoffigures
+%
+% \StopEventually{}
+%
+%
+%
+% \section{Introduction}
+%
+% This document provides the commented source for a \LaTeX{}
+% package file that extends the \LaTeX{} facilities for drawing
+% Bezier curves. The package was originally developed as part of
+% a suite designed for the typesetting of
+% documents according to the rules for ISO international
+% standards~\cite{PRW96i}.
+% This manual is typeset according to the conventions of the
+% \LaTeX{} \textsc{docstrip} utility which enables the automatic
+% extraction of the \LaTeX{} macro source files~\cite{GOOSSENS94}.
+%
+% Drawing a non-rational quadratic Bezier curve is provided as part
+% of the standard \LaTeX{} system.
+% Section~\ref{sec:usage} provides the user manual for the new commands
+% supplied by this package for drawing a variety of Bezier curves.
+% These include commands for drawing linear and cubic non-rational Bezier
+% curves and rational quadratic curves.
+%
+% Section~\ref{sec:bez} describes the implementation of the package.
+% As a side-effect of the implementation, a facility is also provided
+% for performing multiplication in \TeX{} without overflow. This is
+% described in Section~\ref{sec:mnoflow}.
+%
+%
+%
+% \section{Usage} \label{sec:usage}
+%
+% Leslie Lamport provided the means of drawing a quadratic Bezier curve
+% \emph{via} the \LaTeXe{} |\qbezier|~\cite[pp. 125--126]{LAMPORT94} command.
+% This package
+% extends the Bezier facility by providing commands to draw linear,
+% rational quadratic, and cubic Bezier curves.
+%
+% Bezier curves are named after Pierre Bezier who invented them. They
+% are widely used within Computer Aided Design (CAD) programs and other
+% graphics systems; descriptions can be found in many places, with varying
+% degrees of mathematical complexity, such
+% as~\cite{FandP,MORTENSON85,FARIN90}.
+%
+% The Bezier curve is a parameterized curve of degree $n$ and can
+% therefore be specified by $(n+1)$ points
+% (i.e., point $p_{0}$ through $p_{n}$).
+% Among its other properties, a Bezier curve of degree $n$ passes through
+% through the points $p_{0}$ and $p_{n}$ and passes close to the other
+% defining points. The general equation for a Bezier curve of degree $n$ with
+% parameter $t$ is
+% \begin{equation}
+% p(t) = a_{0} + a_{1}t + a_{2}t^{2} + \cdots + a_{n}t^{n} \label{eq:gen}
+% \end{equation}
+% where the coefficients $a_{i}$ depend on the defining points, and
+% traditionally $0 \leq t \leq 1$.
+%
+% For a linear (degree $1$) curve, the equation is
+% \begin{equation}
+% p(t) = p_{0} + (p_{1} - p_{0})t \label{eq:lin}
+% \end{equation}
+% By inspection, $p(0) = p_{0}$ and $p(1) = p_{1}$.
+%
+% Rearranging \eqref{eq:gen} slightly we get
+% \begin{equation}
+% p(t) - p_{0} = (p_{1} - p_{0})t \label{eq:lin2}
+% \end{equation}
+% In other words, we can march along the curve from the starting point to
+% the ending point by evaluating the right hand side of
+% \eqref{eq:lin2} for increasing values of the parameter $t$.
+%
+% In order to shorten the equations slightly, and also make them more
+% convenient to work with numerically, we will use the notation
+% \begin{displaymath}
+% l_{pq} = p_{p} - p_{q}
+% \end{displaymath}
+% Thus, the final form for the linear Bezier curve is
+% \begin{equation}
+% p(t) - p_{0} = l_{10}t \label{eq:lin3}
+% \end{equation}
+%
+% \DescribeMacro{\lbezier}
+% The command |\lbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)| draws a
+% linear Bezier curve with \meta{N} plotted points from the point \meta{p0}
+% (with coordinates \meta{x0,y0}) to the point \meta{p1} (with
+% coordinates \meta{x1,y1}). \meta{N} is an optional argument. If it is
+% either not given or is given with a value of zero,
+% then the command will calculate the number of points to be
+% plotted, subject to a maximum number.
+% There must be no spaces between the arguments to the
+% |\lbezier| command; this restriction also applies to the other Bezier
+% drawing commands provided by the \Lpack{bez123} package.
+%
+% Figure~\ref{fig:beta} shows an example of a dotted line drawn using
+% the |\lbezier| command. The actual code used is:
+% \begin{verbatim}
+% \lbezier[50](15,30)(30,0)
+% \end{verbatim}
+% thus drawing a straight line consisting of 50 points.
+%
+% \DescribeMacro{\qbeziermax}
+% The standard \LaTeX{} command |\qbeziermax| sets a maximum limit
+% on the number of points used to draw any of the Bezier curves.
+%
+% \DescribeMacro{\thinlines}
+% \DescribeMacro{\thicklines}
+% \DescribeMacro{\linethickness}
+% The `points' used in drawing the Bezier curves are small squares. The
+% size of these squares are controlled by the standard \LaTeX{}
+% |\thinlines|, |\thicklines| and/or |\linethickness| commands.
+% Consult Lamport~\cite{LAMPORT94} for descriptions of these, and
+% |\qbeziermax|, commands.
+%
+%
+% It is convenient to introduce some general properties of Bezier curves
+% at this point.
+% \begin{itemize}
+% \item A degree $n$ Bezier curve is defined by $(n+1)$ points which we
+% will label as $p_{0}$ through $p_{n}$. The lines joining the points
+% $p_{0}, p_{1}, \ldots , p_{n}$ are called the \emph{control polygon}.
+% The Bezier curve is parameterized by a variable we will call $t$, with
+% $0 \leq t \leq 1$.
+% \item A degree $n$ Bezier curve starts at point $p_{0}$ and ends at
+% point $p_{n}$.
+% \item At $t=0$ the curve passes through $p_{0}$ and is tangent to the
+% line $l_{10} = p_{1}-p_{0}$.
+% \item At $t=1$ the curve passes through $p_{n}$ and is tangent to the
+% line $l_{(n)(n-1)} = p_{n}-p_{(n-1)}$.
+% \item A \emph{non-rational} Bezier curve lies within the \emph{convex
+% hull}\footnote{The convex hull can be thought of as the shape that a rubber
+% band will take if it is stretched around pins placed at each point.}
+% of the points $p_{0}$ through $p_{n}$. For examples of convex hulls see
+% figure~\ref{fig:ch}. Note that the shape of a convex hull is independant
+% of the ordering of the points.
+% \end{itemize}
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \begin{picture}(70,80)
+% ^^A degree 3
+% \put(0,5){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0}}
+% \put(10,30){\circle{2}}
+% \put(8,30){\makebox(0,0)[br]{1}}
+% \put(20,0){\circle{2}}
+% \put(22,0){\makebox(0,0)[bl]{2}}
+% \put(30,30){\circle{2}}
+% \put(32,30){\makebox(0,0)[bl]{3}}
+% ^^A convex hull
+% \put(0,0){\line(1,0){20}}
+% \put(20,0){\line(1,3){10}}
+% \put(30,30){\line(-1,0){20}}
+% \put(10,30){\line(-1,-3){10}}
+% ^^A control polygon
+% ^^A \put(0,0){\vector(1,3){10}}
+% ^^A \put(10,30){\vector(1,-3){10}}
+% ^^A \put(20,0){\vector(1,3){10}}
+% \thicklines
+% ^^A \cbezier[30](0,0)(10,30)(20,0)(30,30)
+% \thinlines
+% \end{picture}}
+% ^^A degree 3
+% \put(0,45){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0, 3}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{1}}
+% \put(0,30){\circle{2}}
+% \put(2,30){\makebox(0,0)[bl]{2}}
+% ^^A convex hull
+% \put(0,0){\line(1,0){30}}
+% \put(30,0){\line(-1,1){30}}
+% \put(0,30){\line(0,-1){30}}
+% ^^A control polygon
+% ^^A \put(0,0){\vector(1,0){30}}
+% ^^A \put(30,0){\vector(-1,1){30}}
+% ^^A \put(0,30){\vector(0,-1){30}}
+% \thicklines
+% ^^A \cbezier[30](0,0)(30,0)(0,30)(0,0)
+% \thinlines
+% \end{picture}}
+% ^^A degree 3
+% \put(45,0){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0}}
+% \put(10,30){\circle{2}}
+% \put(8,30){\makebox(0,0)[br]{2}}
+% \put(20,0){\circle{2}}
+% \put(22,0){\makebox(0,0)[bl]{3}}
+% \put(30,30){\circle{2}}
+% \put(32,30){\makebox(0,0)[bl]{1}}
+% ^^A convex hull
+% \put(0,0){\line(1,0){20}}
+% \put(20,0){\line(1,3){10}}
+% \put(30,30){\line(-1,0){20}}
+% \put(10,30){\line(-1,-3){10}}
+% ^^A control polygon
+% ^^A \put(0,0){\vector(1,1){30}}
+% ^^A \put(30,30){\vector(-1,0){20}}
+% ^^A \put(10,30){\vector(1,-3){10}}
+% \thicklines
+% ^^A \cbezier(0,0)(30,30)(10,30)(20,0)
+% \thinlines
+% \end{picture}}
+% \put(45,45){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{1}}
+% \put(0,30){\circle{2}}
+% \put(2,30){\makebox(0,0)[bl]{2}}
+% \put(10,10){\circle{2}}
+% \put(8,10){\makebox(0,0)[br]{3}}
+% ^^A convex hull
+% \put(0,0){\line(1,0){30}}
+% \put(30,0){\line(-1,1){30}}
+% \put(0,30){\line(0,-1){30}}
+% ^^A control polygon
+% ^^A \put(0,0){\vector(1,0){30}}
+% ^^A \put(30,0){\vector(-1,1){30}}
+% ^^A \put(0,30){\vector(1,-2){10}}
+% \thicklines
+% ^^A \cbezier(0,0)(30,0)(0,30)(10,10)
+% \thinlines
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \caption{Four sets of points and their convex hulls} \label{fig:ch}
+% \end{figure}
+%
+% The equation for cubic Bezier curves is
+% \begin{equation}
+% p(t) - p_{0} = 3l_{10}t + 3(l_{21} - l_{10})t^{2} + (l_{30} - 3l_{21})t^{3}
+% \label{eq:cubic}
+% \end{equation}
+%
+% \DescribeMacro{\cbezier}
+% The command
+% |\cbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{p3}|)|
+% draws a cubic Bezier curve, as defined by \eqref{eq:cubic},
+% from point \meta{p0} to point \meta{p3}, where \meta{p1} and \meta{p2}
+% are the intermediate points defining the control polygon.
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \begin{picture}(70,80)
+% ^^A degree 3
+% \put(0,5){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0}}
+% \put(10,30){\circle{2}}
+% \put(8,30){\makebox(0,0)[br]{1}}
+% \put(20,0){\circle{2}}
+% \put(22,0){\makebox(0,0)[bl]{2}}
+% \put(30,30){\circle{2}}
+% \put(32,30){\makebox(0,0)[bl]{3}}
+% ^^A convex hull
+% ^^A \put(0,0){\line(1,0){20}}
+% ^^A \put(20,0){\line(1,3){10}}
+% ^^A \put(30,30){\line(-1,0){20}}
+% ^^A \put(10,30){\line(-1,-3){10}}
+% ^^A control polygon
+% \put(0,0){\vector(1,3){10}}
+% \put(10,30){\vector(1,-3){10}}
+% \put(20,0){\vector(1,3){10}}
+% \thicklines
+% \cbezier[30](0,0)(10,30)(20,0)(30,30)
+% \thinlines
+% \end{picture}}
+% ^^A degree 3
+% \put(0,45){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0, 3}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{1}}
+% \put(0,30){\circle{2}}
+% \put(2,30){\makebox(0,0)[bl]{2}}
+% ^^A convex hull
+% ^^A \put(0,0){\line(1,0){30}}
+% ^^A \put(30,0){\line(-1,1){30}}
+% ^^A \put(0,30){\line(0,-1){30}}
+% ^^A control polygon
+% \put(0,0){\vector(1,0){30}}
+% \put(30,0){\vector(-1,1){30}}
+% \put(0,30){\vector(0,-1){30}}
+% \thicklines
+% \cbezier[30](0,0)(30,0)(0,30)(0,0)
+% \thinlines
+% \end{picture}}
+% ^^A degree 3
+% \put(45,0){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0}}
+% \put(10,30){\circle{2}}
+% \put(8,30){\makebox(0,0)[br]{2}}
+% \put(20,0){\circle{2}}
+% \put(22,0){\makebox(0,0)[bl]{3}}
+% \put(30,30){\circle{2}}
+% \put(32,30){\makebox(0,0)[bl]{1}}
+% ^^A convex hull
+% ^^A \put(0,0){\line(1,0){20}}
+% ^^A \put(20,0){\line(1,3){10}}
+% ^^A \put(30,30){\line(-1,0){20}}
+% ^^A \put(10,30){\line(-1,-3){10}}
+% ^^A control polygon
+% \put(0,0){\vector(1,1){30}}
+% \put(30,30){\vector(-1,0){20}}
+% \put(10,30){\vector(1,-3){10}}
+% \thicklines
+% \cbezier(0,0)(30,30)(10,30)(20,0)
+% \thinlines
+% \end{picture}}
+% \put(45,45){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{1}}
+% \put(0,30){\circle{2}}
+% \put(2,30){\makebox(0,0)[bl]{2}}
+% \put(10,10){\circle{2}}
+% \put(8,10){\makebox(0,0)[br]{3}}
+% ^^A convex hull
+% ^^A \put(0,0){\line(1,0){30}}
+% ^^A \put(30,0){\line(-1,1){30}}
+% ^^A \put(0,30){\line(0,-1){30}}
+% ^^A control polygon
+% \put(0,0){\vector(1,0){30}}
+% \put(30,0){\vector(-1,1){30}}
+% \put(0,30){\vector(1,-2){10}}
+% \thicklines
+% \cbezier(0,0)(30,0)(0,30)(10,10)
+% \thinlines
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \caption{Four sets of points, the cubic Bezier curves and their control
+% polygons. Left --- curves plotted with $N=30$; Right ---
+% curves plotted with $N=0$} \label{fig:cp}
+% \end{figure}
+%
+% Figure~\ref{fig:cp} shows four such cubic Bezier curves, their
+% defining points and their control polygons. These are the same points
+% that were used in figure~\ref{fig:ch} to illustrate convex hulls. It is
+% easy to verify that a cubic Bezier curve does indeed lie within the convex
+% hull of its defining points. The curves on the left of the figure were
+% specified with a value of 30 for the argument \meta{N}, while those
+% on the right had no value given for \meta{N} and thus were drawn with
+% the number of plotted points calculated by the drawing algorithm.
+% The actual drawing commands used were:
+% \begin{verbatim}
+% \cbezier[30](0,0)(10,30)(20,0)(30,30)
+% \cbezier[30](0,0)(30,0)(0,30)(0,0)
+% \cbezier(0,0)(30,30)(10,30)(20,0)
+% \cbezier(0,0)(30,0)(0,30)(10,10)
+% \end{verbatim}
+% Note that points are plotted along the curve at equidistant values of the
+% of the parameter $t$. However, as the relationship between the actual
+% distance in $(x,y)$ coordinate space is a non-linear function of $t$,
+% the seperation between the plotted points is non-uniform.
+%
+% The equation for a \emph{non-rational} quadratic Bezier curve is
+% \begin{equation}
+% p(t) - p_{0} = 2l_{10}t + (l_{20} - 2l_{10})t^{2} \label{eq:quad}
+% \end{equation}
+% Using standard \LaTeX{} this can be drawn by the |\qbezier| command.
+% There is another form of a quadratic Bezier curve called a \emph{rational}
+% quadratic Bezier curve. Its equation is
+% \begin{equation}
+% p(t) - p_{0} = \frac^^A
+% {2w_{1}l_{10}t + (w_{2}l_{20} - 2w_{1}l_{10})t^{2}}^^A
+% {w_{0} + 2\omega_{10}t + (\omega_{20} - \omega_{10})t^{2}}
+% \label{eq:rqfull}
+% \end{equation}
+% where the $w_{i}$ are the \emph{weights} corresponding to the
+% points $p_{i}$ and $\omega_{pq} = w_{p} - w_{q}$. The shape of a
+% non-rational curve can be changed by changing the positions of the defining
+% points. The shape of a rational curve can also be modified by changing
+% the values of the weights. A rational curve
+% has the same general properties, outlined above, as a non-rational curve
+% with the exception that the curve may lie outside the convex hull of the
+% control polygon.
+%
+% For the purposes at hand, we use a more restricted form of a
+% rational quadratic Bezier curve, obtained by putting
+% $W = w_{1}/w_{0}$ and then making
+% $w_{0} = w_{2} = 1$ in \eqref{eq:rqfull}. Performing these
+% substitutions we end up with
+% \begin{equation}
+% p(t) - p_{0} = \frac^^A
+% {2Wl_{10}t + (l_{20} - 2Wl_{10})t^{2}}^^A
+% {1 + 2(1 - W)t + 2(1 - W)t^{2}}
+% \label{eq:rqfinal}
+% \end{equation}
+% Note that when $W=1$, (\ref{eq:rqfinal}) reduces to \eqref{eq:quad}
+% and when $W=0$ it effectively reduces to \eqref{eq:lin3}.
+%
+% It turns out that a non-rational quadratic Bezier curve is an arc of
+% a parabola, which is one of the conic curves. All the other conic curves
+% can be represented by the rational quadratic Bezier curve described
+% by \eqref{eq:rqfinal} by suitable choices for the value of $W$.
+% From now on, we will call $W$ the \emph{weight} of the rational quadratic
+% Bezier curve. Table~\ref{tab:rq} lists the value, or value range,
+% of $W$ for the various forms of the conic curve.\footnote{We do not deal
+% with the degenerate cases.} For the case of a circle, $\beta$ is the
+% angle between the lines $l_{10} = (p_{1} - p_{0})$ and
+% $l_{20} = (p_{2} - p_{0})$, as shown in figure~\ref{fig:beta}.
+%
+% \begin{table}
+% \centering
+% \caption{Conic forms of the rational quadratic Bezier curve} \label{tab:rq}
+% \begin{tabular}{lc} \hline
+% Conic form & Weight ($W$) \\ \hline
+% Hyperbola & $\|W\| > 1$ \\
+% Parabola & $\|W\| = 1$ \\
+% Ellipse & $0 < \|W\| < 1$ \\
+% Circle & $\|l_{10}\| = \|l_{21}\|$ and $W = \cos \beta$ \\
+% Straight line & $W = 0$ \\ \hline
+% \end{tabular}
+% \end{table}
+%
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \begin{picture}(30,40)
+% \put(0,5){\begin{picture}(30,30)
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,30){\circle{2}}
+% \put(13,30){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{2}}
+% ^^A polygon
+% \thicklines
+% \put(15,30){\vector(-1,-2){15}}
+% \put(0,0){\vector(1,0){30}}
+% ^^A dashed line from 0 to 2
+% ^^A \lbezier(15,30)(30,0)
+% \lbezier[50](15,30)(30,0)
+% \thinlines
+% \put(15,26){\makebox(0,0){$\beta$}}
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \caption{The angle $\beta$} \label{fig:beta}
+% \end{figure}
+%
+% \DescribeMacro{\rqbezier}
+% The command
+% |\rqbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{W}|)|
+% draws a rational quadratic Bezier curve from \meta{p0} to \meta{p2} with
+% weight \meta{W}, according to \eqref{eq:rqfinal}. As in the
+% other Bezier commands, \meta{N} is optional and controls the number
+% of plotted points along the curve. Figure~\ref{fig:qrb} shows several
+% rational quadratic curves, all with the same control polygon but with
+% differing values for the weight $W$. The code is:
+% \begin{verbatim}
+% \rqbezier[100](15,30)(0,0)(30,0)(4)
+% \rqbezier[100](15,30)(0,0)(30,0)(2)
+% \rqbezier(15,30)(0,0)(30,0)(1)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.75)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.5)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.25)
+% \rqbezier(15,30)(0,0)(30,0)(0)
+% \end{verbatim}
+% When $W > 1$ the curve is pulled toward the point $p_{1}$. Conversely,
+% when $W < 1$ the curve is pushed away from the point $p_{1}$. In all
+% cases, though, the curve starts and stops at $p_{0}$ and $p_{2}$
+% respectively.
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \begin{picture}(30,40)
+% \put(0,5){\begin{picture}(30,30)
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,30){\circle{2}}
+% \put(13,30){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{2}}
+% ^^A polygon
+% \thinlines
+% \put(15,30){\vector(-1,-2){15}}
+% \put(0,0){\vector(1,0){30}}
+% ^^A dashed line from 0 to 2
+% ^^A \lbezier(15,30)(30,0)
+% ^^A \lbezier[50](15,30)(30,0)
+% ^^A \thinlines
+% ^^A \put(15,26){\makebox(0,0){$\beta$}}
+% \thicklines
+% \rqbezier[100](15,30)(0,0)(30,0)(4)
+% \rqbezier[100](15,30)(0,0)(30,0)(2)
+% \rqbezier(15,30)(0,0)(30,0)(1)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.75)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.5)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.25)
+% \rqbezier(15,30)(0,0)(30,0)(0)
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \caption{The effect of weight variation ($W \geq 0$)
+% on rational quadratic Bezier curves
+% (\texttt{weightscale = \theweightscale} (the default)) }
+% \label{fig:qrb}
+% \end{figure}
+%
+% Like the case of the cubic curve, points are plotted at equidistant
+% values of the parameter $t$. The relationship between parameter value
+% and coordinate positions in the rational case are highly non-linear.
+% Thus the distance between the plotted points can vary quite remarkably.
+% This is an inherent disadvantage with this type of curve. The user's remedy
+% is to increase the number of points to be plotted, but this can lead to
+% \TeX{} running out of memory, not to mention the increased time to
+% generate the drawing.
+%
+% \DescribeMacro{\setweightscale}
+% \DescribeMacro{\resetweightscale}
+% Because of the way in which \TeX{} performs arithmetic, and especially
+% division, it
+% is necessary to perform some scaling operations on the divisor when
+% evaluating \eqref{eq:rqfinal}. The optimum value for the
+% scaling is a complex function of the weight and the size and orientation
+% of the control polygon. The algorithm uses a heuristic approach to
+% calculate a `good' value but is not always successful. The
+% |\setweightscale{|\meta{number}|}| command can be used to specify
+% a scale factor. \meta{number} must be a positive integer. The
+% |\resetweightscale| command resets the scale factor to its default
+% value, which is currently 10000 (ten thousand).
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \begin{picture}(70,80)
+% \put(0,5){\begin{picture}(30,30)
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,30){\circle{2}}
+% \put(13,30){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{2}}
+% ^^A polygon
+% \thinlines
+% ^^A \put(15,30){\vector(-1,-2){15}}
+% ^^A \put(0,0){\vector(1,0){30}}
+% ^^A dashed line from 0 to 2
+% ^^A \lbezier(15,30)(30,0)
+% ^^A \lbezier[50](15,30)(30,0)
+% ^^A \thinlines
+% ^^A \put(15,26){\makebox(0,0){$\beta$}}
+% \thicklines
+% \setweightscale{100}
+% \rqbezier[100](15,30)(0,0)(30,0)(4)
+% \rqbezier[100](15,30)(0,0)(30,0)(2)
+% \rqbezier(15,30)(0,0)(30,0)(1)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.75)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.5)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.25)
+% \rqbezier(15,30)(0,0)(30,0)(0)
+% \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
+% \end{picture}}
+% \put(0,45){\begin{picture}(30,30)
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,30){\circle{2}}
+% \put(13,30){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{2}}
+% ^^A polygon
+% \thinlines
+% ^^A \put(15,30){\vector(-1,-2){15}}
+% ^^A \put(0,0){\vector(1,0){30}}
+% ^^A dashed line from 0 to 2
+% ^^A \lbezier(15,30)(30,0)
+% ^^A \lbezier[50](15,30)(30,0)
+% ^^A \thinlines
+% ^^A \put(15,26){\makebox(0,0){$\beta$}}
+% \thicklines
+% \setweightscale{1000}
+% \rqbezier[100](15,30)(0,0)(30,0)(4)
+% \rqbezier[100](15,30)(0,0)(30,0)(2)
+% \rqbezier(15,30)(0,0)(30,0)(1)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.75)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.5)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.25)
+% \rqbezier(15,30)(0,0)(30,0)(0)
+% \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
+% \end{picture}}
+% \put(45,5){\begin{picture}(30,30)
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,30){\circle{2}}
+% \put(13,30){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{2}}
+% ^^A polygon
+% \thinlines
+% ^^A \put(15,30){\vector(-1,-2){15}}
+% ^^A \put(0,0){\vector(1,0){30}}
+% ^^A dashed line from 0 to 2
+% ^^A \lbezier(15,30)(30,0)
+% ^^A \lbezier[50](15,30)(30,0)
+% ^^A \thinlines
+% ^^A \put(15,26){\makebox(0,0){$\beta$}}
+% \thicklines
+% \resetweightscale
+% \rqbezier[100](15,30)(0,0)(30,0)(4)
+% \rqbezier[100](15,30)(0,0)(30,0)(2)
+% \rqbezier(15,30)(0,0)(30,0)(1)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.75)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.5)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.25)
+% \rqbezier(15,30)(0,0)(30,0)(0)
+% \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
+% \end{picture}}
+% \put(45,45){\begin{picture}(30,30)
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,30){\circle{2}}
+% \put(13,30){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{2}}
+% ^^A polygon
+% \thinlines
+% ^^A \put(15,30){\vector(-1,-2){15}}
+% ^^A \put(0,0){\vector(1,0){30}}
+% ^^A dashed line from 0 to 2
+% ^^A \lbezier(15,30)(30,0)
+% ^^A \lbezier[50](15,30)(30,0)
+% ^^A \thinlines
+% ^^A \put(15,26){\makebox(0,0){$\beta$}}
+% \thicklines
+% \setweightscale{100000}
+% \rqbezier[100](15,30)(0,0)(30,0)(4)
+% \rqbezier[100](15,30)(0,0)(30,0)(2)
+% \rqbezier(15,30)(0,0)(30,0)(1)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.75)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.5)
+% \rqbezier[100](15,30)(0,0)(30,0)(0.25)
+% \rqbezier(15,30)(0,0)(30,0)(0)
+% \put(15,-5){\makebox(0,0)[b]{\texttt{weightscale = \theweightscale}}}
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \resetweightscale
+% \caption{The effect of \texttt{weightscale} on the drawing
+% of rational quadratic Bezier curves}
+% \label{fig:qrw}
+% \end{figure}
+%
+% Figure~\ref{fig:qrw} illustrates the effect on changing the
+% \texttt{weightscale} used for drawing the same curves as shown
+% in figure~\ref{fig:qrb}. Note that the \texttt{weightscale}
+% has no effect when
+% $W = 1$ or $W = 0$ as in these cases the curves are drawn using the
+% algorithms for the |\qbezier| and |\lbezier| commands respectively.
+%
+% It is obvious that some choices give very poorly formed curves. In
+% other cases the curves may be poorly formed but do result in interesting
+% cross-stitch like patterns.
+%
+% Table~\ref{tab:rq} indicates that it is possible to draw circular
+% arcs using a rational quadratic Bezier curves. The two legs of the
+% control polygon define the tangents to the curve at the
+% end points.
+% Therefore, to draw a circular arc the two legs must be equal in length.
+% That is, the convex hull is an isosceles triangle. In the special case
+% when the convex hull forms an equilateral triangle, the required
+% weight ($\cos \beta$, see figure~\ref{fig:beta}) for drawing a circular
+% arc is $\cos \beta = 0.5$. Further,
+% for any given control polygon the the curves drawn with weights of
+% $\pm W$ are complementary. That is, the curve with weight $-W$ is
+% the `remainder' of the curve drawn with weight $W$. Thus, we have a
+% simple means of drawing a complete circle, as shown in figure~\ref{fig:qrc}.
+% The plotting commands of interest were:
+% \begin{verbatim}
+% \lbezier[25](0,0)(15,26)
+% \lbezier[25](0,0)(30,0)
+% \setweightscale{50000}
+% \rqbezier[100](15,26)(0,0)(30,0)(0.5)
+% \rqbezier[200](15,26)(0,0)(30,0)(-0.5)
+% \resetweightscale
+% \end{verbatim}
+% where the |\lbezier| drawing commands were used to draw the dotted outline
+% of the control polygon.
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \setweightscale{50000}
+% \begin{picture}(60,62)
+% \put(0,5){\begin{picture}(30,30)
+% \thinlines
+% \put(0,0){\circle{2}}
+% \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,26){\circle{2}}
+% \put(13,26){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% \put(32,0){\makebox(0,0)[bl]{2}}
+% ^^A polygon
+% \lbezier[25](0,0)(15,26)
+% \lbezier[25](0,0)(30,0)
+% ^^A RQBEZIER
+% \thicklines
+% \rqbezier[100](15,26)(0,0)(30,0)(0.5)
+% \rqbezier[200](15,26)(0,0)(30,0)(-0.5)
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \caption{Rational quadratics with weights of $\pm 0.5$ and an equilateral
+% triangular convex hull
+% (\texttt{weightscale = \theweightscale}) } \label{fig:qrc}
+% \resetweightscale
+% \end{figure}
+%
+% A more robust picture of the same circle is shown in
+% figure~\ref{fig:qr3c} where the complete circle is pieced together from
+% three non-complementary circular arcs. The drawing commands of interest
+% were
+% \begin{verbatim}
+% \rqbezier[100](15,26)(0,0)(30,0)(0.5)
+% \rqbezier[100](30,0)(60,0)(45,26)(0.5)
+% \rqbezier[100](45,26)(30,52)(15,26)(0.5)
+% \end{verbatim}
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \begin{picture}(60,62)
+% \put(0,5){\begin{picture}(60,52)
+% \thinlines
+% \put(0,0){\circle{2}}
+% ^^A \put(-2,0){\makebox(0,0)[br]{1}}
+% \put(15,26){\circle{2}}
+% ^^A \put(13,30){\makebox(0,0)[br]{0}}
+% \put(30,0){\circle{2}}
+% ^^A \put(32,0){\makebox(0,0)[bl]{2}}
+% \put(60,0){\circle{2}}
+% \put(45,26){\circle{2}}
+% \put(30,52){\circle{2}}
+% ^^A polygon
+% \lbezier[50](0,0)(30,52)
+% \lbezier[50](30,52)(60,0)
+% \lbezier[50](0,0)(60,0)
+% ^^A RQBEZIER
+% \thicklines
+% \rqbezier[100](15,26)(0,0)(30,0)(0.5)
+% \rqbezier[100](30,0)(60,0)(45,26)(0.5)
+% \rqbezier[100](45,26)(30,52)(15,26)(0.5)
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \caption{Three rational quadratics with weights of $0.5$
+% (\texttt{weightscale = \theweightscale}) }
+% \label{fig:qr3c}
+% \end{figure}
+%
+% The astute reader will have realised that the divisor in
+% \eqref{eq:rqfinal} can go to zero, and can even be negative.
+% This has interesting consequences, both when trying to do computer
+% arithmetic, and also on the the kind of curve that results. Essentially,
+% the curve tends to $\infty$ as $W \rightarrow +0$. At $W = -0$ the curve
+% is at $-\infty$ and then it tends to $-0$ as $W \rightarrow -\infty$.
+% We will get a curve point at $\infty$ whenever $W = -1$ and a `negative'
+% curve for $W < -1$.
+%
+% \begin{figure}
+% \centering
+% \setlength{\unitlength}{1mm}
+% \begin{picture}(60,60)
+% \put(30,20){\begin{picture}(30,20)
+% \thinlines
+% \put(0,10){\circle{2}}
+% \put(30,0){\circle{2}}
+% \put(30,20){\circle{2}}
+% ^^A polygon
+% \lbezier[25](30,20)(0,10)
+% \lbezier[25](0,10)(30,0)
+% ^^A RQBEZIER
+% \thicklines
+% \rqbezier[100](30,20)(0,10)(30,0)(2)
+% \rqbezier[100](30,20)(0,10)(30,0)(-2)
+% \end{picture}}
+% \end{picture}
+% \setlength{\unitlength}{1pt}
+% \caption{A rational quadratic that has gone negative; weights of $\pm 2$
+% (\texttt{weightscale = \theweightscale}) }
+% \label{fig:neg}
+% \end{figure}
+%
+% This effect is shown in figure~\ref{fig:neg} which draws the two branches
+% of a hyperbola. The basic code for the illustration was
+% \begin{verbatim}
+% \lbezier[25](30,20)(0,10)
+% \lbezier[25](0,10)(30,0)
+% \rqbezier[100](30,20)(0,10)(30,0)(2)
+% \rqbezier[100](30,20)(0,10)(30,0)(-2)
+% \end{verbatim}
+% where the control polygon was drawn using the |\lbezier| commands.
+%
+%
+% \section{The \Lpack{bez123} package implementation} \label{sec:bez}
+%
+% \LaTeX{} provides a facility for drawing quadratic Bezier curves.
+% This package provides additional facilities for drawing linear,
+% rational quadratic, and cubic Bezier curves.
+%
+%
+% Announce the name and version of the package, which requires \LaTeXe.
+% \begin{macrocode}
+%<*bez>
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{bez123}[1998/10/14 v1.1 Bezier curves]
+% \end{macrocode}
+% \changes{v1.1}{1998/10/14}{Added call to include multiply package}
+% The package also requires the \Lpack{multiply} package.
+% \begin{macrocode}
+\RequirePackage{multiply}[1998/10/14]
+%</bez>
+% \end{macrocode}
+%
+%
+% \subsection{Arithmetic in \TeX}
+%
+% All arithmetic in \TeX{} is based on integer arithmetic, with a
+% maximum integer value of $M = \number\maxdimen$. For example,
+% \setcounter{weightscale}{8}\makeatletter\divide\c@weightscale by 3\makeatother
+% $8/3 = \theweightscale$,
+% \setcounter{weightscale}{9}\makeatletter\divide\c@weightscale by 3\makeatother
+% $9/3 = \theweightscale$, and
+% \setcounter{weightscale}{10}\makeatletter\divide\c@weightscale by 3\makeatother
+% $10/3 = \theweightscale$.
+% In other words, division always reduces the absolute value of the dividend,
+% and also possibly truncates the value. One consequence of this is that the
+% ordering of multiplication and division is important. For instance,
+% \setcounter{weightscale}{8}\makeatletter\multiply\c@weightscale by 3 \divide\c@weightscale by 3\makeatother
+% $(8 \times 3)/3 = \theweightscale$ but
+% \setcounter{weightscale}{8}\makeatletter\divide\c@weightscale by 3 \multiply\c@weightscale by 3\makeatother
+% $(8/3) \times 3 = \theweightscale$!
+% Thus, in arithmetic calculations involving both multiplication and
+% division, the dividend should be maximised and the divisor minimised,
+% with multiplication preceeding division; also remembering that there
+% is a limit on the size of an integer. To avoid multiplication overflow
+% when calculating say, $a \times b$, we must ensure that
+% $\|a\| \leq \|M/b\|$.
+%
+% When calculating polynomials, such as that in \eqref{eq:gen},
+% we use a technique called Horner's schema, which is also known as nested
+% multiplication. A general cubic equation, for example, can be written as:
+% \begin{equation}
+% p(t) - a_{0} = t(a_{1} + t(a_{2} + ta_{3})) \label{eq:horn}
+% \end{equation}
+% The following pseudo-code shows one way to implement Horner's schema for
+% plotting $N$ points in the interval $0 \leq t \leq 1$
+% of \eqref{eq:horn} using integer arithmetic.
+% \begin{verbatim}
+% procedure plot_cubic(a0, a1, a2, a3:vector; N:integer);
+% local p:vector; end_local;
+% a3 := a3/N;
+% repeat i := 0 to N by 1;
+% p := a3*i;
+% p := p + a2; p := p/N; p := p*i;
+% p := p + a1; p := p/N; p := p*i;
+% draw(p + a0);
+% end_repeat;
+% return;
+% end_procedure;
+% \end{verbatim}
+% We use the above algorithm, with suitable modifications according to the
+% degree of the polynomial, for plotting the points along Bezier curves.
+%
+% \subsection{Linear Bezier curves}
+%
+% \begin{macrocode}
+%<*bez>
+% \end{macrocode}
+%
+% As a linear curve is simpler than a quadratic curve there is no
+% need to declare extra variables from those used in the kernel by the
+% |\qbezier| macro.
+%
+% \begin{macro}{\lbezier}
+% The user command to draw a linear Bezier curve represented by
+% \eqref{eq:lin3}. The form of the command is:\\
+% |\lbezier[|\meta{N}|]{(|\meta{p0}|)(|\meta{p1}|)| \\
+% where \meta{pN} is the comma seperated X and Y coordinate values of
+% point \textit{pN}.
+%
+% \begin{macrocode}
+\newcommand{\lbezier}[2][0]{\@lbez{#1}#2}
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\@lbez}
+% The drawing macro.
+% \begin{macrocode}
+\gdef\@lbez#1(#2,#3)(#4,#5){%
+%%%%\def\lbezier#1(#2,#3)(#4,#5){%
+ \ifnum #1<\@ne
+% \end{macrocode}
+% When the number of plotting points are not given, then we calculate
+% how many are needed. First determine the X distance between the end points.
+% \begin{macrocode}
+ \@ovxx = #4\unitlength
+ \advance\@ovxx by -#2\unitlength
+ \ifdim \@ovxx < \z@
+ \@ovxx = -\@ovxx
+ \fi
+% \end{macrocode}
+% Similarly calculate the Y distance.
+% \begin{macrocode}
+ \@ovyy = #5\unitlength
+ \advance\@ovyy by -#3\unitlength
+ \ifdim \@ovyy < \z@
+ \@ovyy = -\@ovyy
+ \fi
+% \end{macrocode}
+% Temporarily store the maximum distance in |\@multicnt|.
+% \begin{macrocode}
+ \ifdim \@ovxx > \@ovyy
+ \@multicnt = \@ovxx
+ \else
+ \@multicnt = \@ovyy
+ \fi
+% \end{macrocode}
+% We use a small square as the visual representation of a point.
+% Calculate the number of points required to give 50\% overlap of adjacent
+% squares, making
+% sure that it doesn't exceed the limit. Store the result in |\@multicnt|.
+% \begin{macrocode}
+ \@ovxx = 0.5\@halfwidth
+ \divide\@multicnt by \@ovxx
+ \ifnum \qbeziermax < \@multicnt
+ \@multicnt = \qbeziermax\relax
+ \fi
+ \else
+% \end{macrocode}
+% The number of points is given.
+% \begin{macrocode}
+ \@multicnt = #1\relax
+ \fi
+% \end{macrocode}
+%
+% Now we can prepare the constants for the plotting loop.
+% \begin{macrocode}
+ \@tempcnta = \@multicnt
+ \advance\@tempcnta by \@ne
+ \@ovdx = #4\unitlength
+ \advance\@ovdx by -#2\unitlength
+ \divide\@ovdx by \@multicnt
+ \@ovdy = #5\unitlength
+ \advance\@ovdy by -#3\unitlength
+ \divide\@ovdy by \@multicnt
+% \end{macrocode}
+% The next bit of code defines the size of the square representing a point.
+% \begin{macrocode}
+ \setbox\@tempboxa\hbox{\vrule \@height\@halfwidth
+ \@depth \@halfwidth
+ \@width \@wholewidth}%
+% \end{macrocode}
+% Start the plot at the first point.
+% \begin{macrocode}
+ \put(#2,#3){%
+ \count@ = \z@
+ \@whilenum{\count@ < \@tempcnta}\do
+% \end{macrocode}
+% Evaluate the polynomial (simple in this case) using Horner's schema.
+% \begin{macrocode}
+ {\@xdim = \count@\@ovdx
+ \@ydim = \count@\@ovdy
+% \end{macrocode}
+% Plot this point.
+% \begin{macrocode}
+ \raise \@ydim
+ \hb@xt@\z@{\kern\@xdim
+ \unhcopy\@tempboxa\hss}%
+ \advance\count@\@ne}}%
+% \end{macrocode}
+% The end of the definition of |\@lbez|.
+% \begin{macrocode}
+}
+% \end{macrocode}
+% \end{macro}
+%
+% \subsection{Cubic Bezier curves}
+%
+% As cubic curves are more complex than quadratic curves we need some
+% extra variables.
+% \begin{macro}{\@wxc}
+% \begin{macro}{\@wyc}
+% Lengths.
+% \begin{macrocode}
+\newlength{\@wxc}
+\newlength{\@wyc}
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\cbezier}
+% The user command for drawing a cubic Bezier curve as represented
+% by \eqref{eq:cubic}. It is called as: \\
+% |\cbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{p3}|)|.
+%
+% \begin{macrocode}
+\newcommand{\cbezier}[2][0]{\@cbez{#1}#2}
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\@cbez}
+% The drawing macro for cubic Bezier curves.
+% \begin{macrocode}
+\gdef\@cbez#1(#2,#3)(#4,#5)(#6,#7)(#8,#9){%
+ \ifnum #1<\@ne
+% \end{macrocode}
+% We have to calculate the number of plotting points required. We will
+% use the maximum of the box enclosing the convex hull as a measure.
+% First do the X value, using |\@ovxx| to store the maximum X coordinate
+% and |\@ovdx| the minimum.
+% \begin{macrocode}
+ \@ovxx = #2\unitlength
+ \@ovdx = \@ovxx
+ \@ovdy = #4\unitlength
+ \ifdim \@ovdy > \@ovxx
+ \@ovxx = \@ovdy
+ \fi
+ \ifdim \@ovdy < \@ovdx
+ \@ovdx = \@ovdy
+ \fi
+ \@ovdy = #6\unitlength
+ \ifdim \@ovdy > \@ovxx
+ \@ovxx = \@ovdy
+ \fi
+ \ifdim \@ovdy < \@ovdx
+ \@ovdx = \@ovdy
+ \fi
+ \@ovdy = #8\unitlength
+ \ifdim \@ovdy > \@ovxx
+ \@ovxx = \@ovdy
+ \fi
+ \ifdim \@ovdy < \@ovdx
+ \@ovdx = \@ovdy
+ \fi
+% \end{macrocode}
+% Store the maximum X in |\@ovxx|.
+% \begin{macrocode}
+ \advance\@ovxx by -\@ovdx
+% \end{macrocode}
+% Repeat the process for the maximum Y value, finally storing
+% this in |\@ovyy|.
+% \begin{macrocode}
+ \@ovyy = #3\unitlength
+ \@ovdy = \@ovyy
+ \@ovdx = #5\unitlength
+ \ifdim \@ovdx > \@ovyy
+ \@ovyy = \@ovdx
+ \fi
+ \ifdim \@ovdx < \@ovdy
+ \@ovdy = \@ovdx
+ \fi
+ \@ovdx = #7\unitlength
+ \ifdim \@ovdx > \@ovyy
+ \@ovyy = \@ovdx
+ \fi
+ \ifdim \@ovdx < \@ovdy
+ \@ovdy = \@ovdx
+ \fi
+ \@ovdx = #9\unitlength
+ \ifdim \@ovdx > \@ovyy
+ \@ovyy = \@ovdx
+ \fi
+ \ifdim \@ovdx < \@ovdy
+ \@ovdy = \@ovdx
+ \fi
+ \advance\@ovyy by -\@ovdy
+% \end{macrocode}
+% Temporarily store the max of X and Y in |\@multicnt|.
+% \begin{macrocode}
+ \ifdim \@ovxx > \@ovyy
+ \@multicnt = \@ovxx
+ \else
+ \@multicnt = \@ovyy
+ \fi
+% \end{macrocode}
+% Calculate the number of points required to give 50\% overlap, making
+% sure that it doesn't exceed the limit. Store the number of points in
+% |\@multicnt|.
+% \begin{macrocode}
+ \@ovxx = 0.5\@halfwidth
+ \divide\@multicnt by \@ovxx
+ \ifnum \qbeziermax < \@multicnt
+ \@multicnt = \qbeziermax\relax
+ \fi
+ \else
+% \end{macrocode}
+% The number of points is given.
+% \begin{macrocode}
+ \@multicnt = #1\relax
+ \fi
+% \end{macrocode}
+%
+% Now we can prepare the constants for the plotting loop. First the control
+% counts.
+% \begin{macrocode}
+ \@tempcnta = \@multicnt
+ \advance\@tempcnta by \@ne
+% \end{macrocode}
+% Then the cubic coefficients, firstly for X.
+% \begin{macrocode}
+ \@ovdx = #4\unitlength \advance\@ovdx by -#2\unitlength
+ \@ovxx = #6\unitlength \advance\@ovxx by -\@ovdx
+ \multiply\@ovdx by \thr@@
+ \advance\@ovxx by -#4\unitlength \multiply\@ovxx by \thr@@
+ \@wxc = #4\unitlength \advance\@wxc by -#6\unitlength
+ \multiply\@wxc by \thr@@ \advance\@wxc by #8\unitlength
+ \advance\@wxc by -#2\unitlength \divide\@wxc by \@multicnt
+% \end{macrocode}
+% And similarly for Y.
+% \begin{macrocode}
+ \@ovdy = #5\unitlength \advance\@ovdy by -#3\unitlength
+ \@ovyy = #7\unitlength \advance\@ovyy by -\@ovdy
+ \multiply\@ovdy by \thr@@
+ \advance\@ovyy by -#5\unitlength \multiply\@ovyy by \thr@@
+ \@wyc = #5\unitlength \advance\@wyc by -#7\unitlength
+ \multiply\@wyc by \thr@@ \advance\@wyc by #9\unitlength
+ \advance\@wyc by -#3\unitlength \divide\@wyc by \@multicnt
+% \end{macrocode}
+% Set up the plotting box.
+% \begin{macrocode}
+ \setbox\@tempboxa\hbox{\vrule \@height\@halfwidth
+ \@depth \@halfwidth
+ \@width \@wholewidth}%
+% \end{macrocode}
+% Start the plot at the first point.
+% \begin{macrocode}
+ \put(#2,#3){%
+ \count@ = \z@
+ \@whilenum{\count@ < \@tempcnta}\do
+ {\@xdim = \count@\@wxc
+ \advance\@xdim by \@ovxx
+ \divide\@xdim by \@multicnt
+ \multiply\@xdim by \count@
+ \advance\@xdim by \@ovdx
+ \divide\@xdim by \@multicnt
+ \multiply\@xdim by \count@
+ \@ydim = \count@\@wyc
+ \advance\@ydim by \@ovyy
+ \divide\@ydim by \@multicnt
+ \multiply\@ydim by \count@
+ \advance\@ydim by \@ovdy
+ \divide\@ydim by \@multicnt
+ \multiply\@ydim by \count@
+% \end{macrocode}
+% Plot the point.
+% \begin{macrocode}
+ \raise \@ydim
+ \hb@xt@\z@{\kern\@xdim
+ \unhcopy\@tempboxa\hss}%
+ \advance\count@\@ne}}%
+% \end{macrocode}
+% The end of the definition of |\@cbez|.
+% \begin{macrocode}
+}
+% \end{macrocode}
+% \end{macro}
+%
+%
+% \subsection{Quadratic rational Bezier curve}
+%
+% This is the most complex of the Bezier curves that we deal with.
+% We need yet more variables.
+%
+% \begin{macro}{\@ww}
+% \begin{macro}{\@wwa}
+% \begin{macro}{\@wwb}
+% \begin{macro}{\@wwo}
+% \begin{macro}{\@wwi}
+% Variables for the weight calculations.
+% \begin{macrocode}
+\newlength{\@ww}
+\newlength{\@wwa}
+\newlength{\@wwb}
+\newlength{\@wwo}
+\newlength{\@wwi}
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\c@@pntscale}
+% Scale factor for points.
+% \begin{macrocode}
+\newcounter{@pntscale}
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\c@weightscale}
+% Scale factor for divisor.
+% \begin{macrocode}
+\newcounter{weightscale}
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\botscale}
+% Scale factor for bottom weights.
+% \begin{macrocode}
+\newlength{\botscale}
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\setweightscale}
+% User level command |\setweightscale{|\meta{number}|}| for setting the
+% divisor scaling.
+% \begin{macrocode}
+\newcommand{\setweightscale}[1]{\setcounter{weightscale}{#1}}
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\resetweightscale}
+% User level command for setting the divisor scaling to its default
+% value ($10^{4}$). We also ensure that the scaling is set to this value.
+% \begin{macrocode}
+\newcommand{\resetweightscale}{\setcounter{weightscale}{10000}}
+\resetweightscale
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\rqbezier}
+% The user level command for drawing a rational quadratic Bezier curve
+% as represented by \eqref{eq:rqfinal}. The form of the command is \\
+% |\rqbezier[|\meta{N}|](|\meta{p0}|)(|\meta{p1}|)(|\meta{p2}|)(|\meta{W}|)| \\
+% where the arguments are as per the other Bezier drawing commands, but with
+% the final argument being the weight.
+%
+% \begin{macrocode}
+\newcommand{\rqbezier}[2][0]{\@rqbez{#1}#2}
+% \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\@rqbez}
+% The drawing macro for a rational quadratic Bezier curve. If the weight
+% is such that the curve is either rational quadratic ($W = 1$) or
+% linear ($W = 0$), we use the simpler drawing macro.
+% \begin{macrocode}
+\gdef\@rqbez#1(#2,#3)(#4,#5)(#6,#7)(#8){%
+ \@ovxx = #8\unitlength
+ \ifdim\@ovxx = \unitlength
+ \PackageWarning{bez123}{Rational quadratic denerates to quadratic}
+ \qbezier[#1](#2,#3)(#4,#5)(#6,#7)
+ \else
+ \ifdim\@ovxx = \z@
+ \PackageWarning{bez123}{Rational quadratic degenerates to linear}
+ \lbezier[#1](#2,#3)(#6,#7)
+ \else
+% \end{macrocode}
+% Calculate the maximum length of the control polygon's bounding box.
+% Store the result in |\@wwi|.
+% \begin{macrocode}
+ \@ovxx = #4\unitlength
+ \advance\@ovxx by -#2\unitlength
+ \ifdim \@ovxx < \z@
+ \@ovxx = -\@ovxx
+ \fi
+ \@ovdx = #6\unitlength
+ \advance\@ovdx by -#4\unitlength
+ \ifdim \@ovdx < \z@
+ \@ovdx = -\@ovdx
+ \fi
+ \ifdim \@ovxx < \@ovdx
+ \@ovxx = \@ovdx
+ \fi
+ \@ovyy = #5\unitlength
+ \advance\@ovyy by -#3\unitlength
+ \ifdim \@ovyy < \z@
+ \@ovyy = -\@ovyy
+ \fi
+ \@ovdy = #7\unitlength
+ \advance\@ovdy by -#5\unitlength
+ \ifdim \@ovdy < \z@
+ \@ovdy = -\@ovdy
+ \fi
+ \ifdim \@ovyy < \@ovdy
+ \@ovyy = \@ovdy
+ \fi
+ \ifdim \@ovxx > \@ovyy
+ \@multicnt = \@ovxx
+ \else
+ \@multicnt = \@ovyy
+ \fi
+ \@wwi = \@multicnt sp
+% \end{macrocode}
+% Now determine the number of points to be plotted.
+% \begin{macrocode}
+ \ifnum #1<\@ne
+ \@ovxx = 0.5\@halfwidth
+ \divide\@multicnt by \@ovxx
+ \ifnum\qbeziermax < \@multicnt
+ \@multicnt = \qbeziermax\relax
+ \fi
+ \else
+% \end{macrocode}
+% Number of points is a given.
+% \begin{macrocode}
+ \@multicnt = #1\relax
+ \fi
+% \end{macrocode}
+% We are going to plot the curve in two halves in an attempt to reduce
+% roundoff problems. At
+% a minimum this should at least make a symmetrical curve look symmetric
+% about its mid point.
+% \begin{macrocode}
+ \@tempcnta = \@multicnt
+ \advance\@tempcnta by \@ne
+ \divide\@tempcnta by \tw@
+ \advance\@tempcnta by \@ne
+% \end{macrocode}
+% We now have to deal with a possible multiplication overflow problem due
+% to multiplication by the weight. In \eqref{eq:rqfinal} the potentially
+% largest term is the coefficient of $t^{2}$ (i.e., $(l_{20}-2Wl_{10})$).
+% The maximum length likely to be encountered is, say, 10 inches for a
+% drawing on either A4 or US letterpaper. This is approximately
+% $4.8\times10^{8}$sp. Doing a little arithmetic, and remembering that the
+% maximum length in \TeX{} is $M = \number\maxdimen$sp, it means that we must
+% have $\|W\| \leq 1$ to prevent overflow. However, a typical range for
+% $W$ is $-10 \leq W \leq 10$. Therefore we might have to do some scaling.
+% Being pessimistic, we'll assume that $l_{20} = - l_{10}$ and that $l_{10}$
+% is the largest dimension in the drawing. To prevent overflow we then have to
+% meet the condition $\|W\| \leq (M - l_{20})/2l_{20}$, where all lengths are
+% positive. We will use |\c@@pntscale| as a scale factor on $W$ to meet this
+% condition. Earlier we set |\@wwi| to be the positive value of the largest
+% dimension in the drawing.
+%
+% Set the distance scale factor. First evaluating the test condition.
+% \begin{macrocode}
+ \@wwo = \maxdimen
+ \advance\@wwo by -\@wwi
+ \divide\@wwo by \tw@
+ \divide\@wwo by \@wwi
+% \end{macrocode}
+% Now perform the check and set the scale factor. We have to get a positive
+% integer value for $W$ as it may be a fraction. Actually, we only need to
+% be concerned if $\|W\| > 1$.
+% \begin{macrocode}
+ \@wwi = 10sp
+ \@wwi = #8\@wwi
+ \ifdim\@wwi < \z@
+ \@wwi = -\@wwi
+ \fi
+ \divide\@wwi by 10\relax
+ \ifdim\@wwi < \@wwo
+ \c@@pntscale = \@ne
+ \else
+ \divide\@wwi by \tw@
+ \ifdim\@wwi < \@wwo
+ \c@@pntscale = \tw@
+ \else
+ \divide\@wwi by \tw@
+ \ifdim\@wwi < \@wwo
+ \c@@pntscale = 4\relax
+ \else
+ \divide\@wwi by \tw@
+ \ifdim\@wwi < \@wwo
+ \c@@pntscale = 8\relax
+ \else
+ \c@@pntscale = 16\relax
+ \fi
+ \fi
+ \fi
+ \fi
+% \end{macrocode}
+% Calculate the constants for the top line of the function.
+% \begin{macrocode}
+ \@ovxx = #4\unitlength \advance\@ovxx by -#2\unitlength
+ \multiply\@ovxx by \tw@
+ \divide\@ovxx by \c@@pntscale
+ \@ovdx = #8\@ovxx
+ \@ovxx = #6\unitlength \advance\@ovxx by -#2\unitlength
+ \divide\@ovxx by \c@@pntscale
+ \advance\@ovxx by -\@ovdx
+ \divide\@ovxx by \@multicnt
+ \@ovyy = #5\unitlength \advance\@ovyy by -#3\unitlength
+ \multiply\@ovyy by \tw@
+ \divide\@ovyy by \c@@pntscale
+ \@ovdy = #8\@ovyy
+ \@ovyy = #7\unitlength \advance\@ovyy by -#3\unitlength
+ \divide\@ovyy by \c@@pntscale
+ \advance\@ovyy by -\@ovdy
+ \divide\@ovyy by \@multicnt
+% \end{macrocode}
+% Now the constants for the bottom line. We also need to do some scaling
+% here. This scaling can be set by the user.
+% \begin{macrocode}
+ \setlength{\botscale}{\c@weightscale sp}
+ \@wwo = \botscale
+ \@wwi = #8\@wwo
+ \@wwa = \@wwo \advance\@wwa by -\@wwi
+ \multiply\@wwa by \tw@
+ \@wwb = \@wwa
+ \divide\@wwb by \@multicnt
+% \end{macrocode}
+% Prepare for the drawing.
+% \begin{macrocode}
+ \@wwi = \botscale
+ \setbox\@tempboxa\hbox{\vrule \@height\@halfwidth
+ \@depth \@halfwidth
+ \@width \@wholewidth}%
+% \end{macrocode}
+% Draw the first half of the curve.
+% \begin{macrocode}
+ \put(#2,#3){%
+ \count@ = \z@
+ \@whilenum{\count@ < \@tempcnta}\do
+ {\@xdim = \count@\@ovxx
+ \advance\@xdim by \@ovdx
+ \divide\@xdim by \@multicnt
+ \multiply\@xdim by \count@
+ \@ydim = \count@\@ovyy
+ \advance\@ydim by \@ovdy
+ \divide\@ydim by \@multicnt
+ \multiply\@ydim by \count@
+ \@ww = \count@\@wwb
+ \advance\@ww by -\@wwa
+ \divide\@ww by \@multicnt
+ \multiply\@ww by \count@
+ \advance\@ww by \@wwo
+ \divide\@ww by \c@@pntscale
+ \ifdim\@ww = \z@
+% \end{macrocode}
+% We are about to divide by |\@ww| which is zero. Treat |\@ww| as unity.
+% \begin{macrocode}
+ \else
+ \divide\@xdim by \@ww
+ \divide\@ydim by \@ww
+ \fi
+% \end{macrocode}
+% For reasons I don't understand, the \% signs at the end of the next few
+% lines are important!
+% \begin{macrocode}
+ \multnooverflow{\@xdim}{\botscale}%
+ \multnooverflow{\@ydim}{\botscale}%
+ \raise \@ydim
+ \hb@xt@\z@{\kern\@xdim
+ \unhcopy\@tempboxa\hss}%
+ \advance\count@\@ne}}
+% \end{macrocode}
+%
+% We now repeat the above process for plotting the second half of the
+% curve, starting at the end point.
+%
+% Calculate the constants for the top line of the function.
+% \begin{macrocode}
+ \@ovxx = #4\unitlength \advance\@ovxx by -#6\unitlength
+ \multiply\@ovxx by \tw@
+ \divide\@ovxx by \c@@pntscale
+ \@ovdx = #8\@ovxx
+ \@ovxx = #2\unitlength \advance\@ovxx by -#6\unitlength
+ \divide\@ovxx by \c@@pntscale
+ \advance\@ovxx by -\@ovdx
+ \divide\@ovxx by \@multicnt
+ \@ovyy = #5\unitlength \advance\@ovyy by -#7\unitlength
+ \multiply\@ovyy by \tw@
+ \divide\@ovyy by \c@@pntscale
+ \@ovdy = #8\@ovyy
+ \@ovyy = #3\unitlength \advance\@ovyy by -#7\unitlength
+ \divide\@ovyy by \c@@pntscale
+ \advance\@ovyy by -\@ovdy
+ \divide\@ovyy by \@multicnt
+% \end{macrocode}
+% The constants for the bottom line are the same as before as the function
+% is symmetric. Similarly we don't need to recalculate the size of the
+% rule box.
+%
+% Draw the second half of the curve.
+% \begin{macrocode}
+ \put(#6,#7){%
+ \count@ = \z@
+ \@whilenum{\count@ < \@tempcnta}\do
+ {\@xdim = \count@\@ovxx
+ \advance\@xdim by \@ovdx
+ \divide\@xdim by \@multicnt
+ \multiply\@xdim by \count@
+ \@ydim = \count@\@ovyy
+ \advance\@ydim by \@ovdy
+ \divide\@ydim by \@multicnt
+ \multiply\@ydim by \count@
+ \@ww = \count@\@wwb
+ \advance\@ww by -\@wwa
+ \divide\@ww by \@multicnt
+ \multiply\@ww by \count@
+ \advance\@ww by \@wwo
+ \divide\@ww by \c@@pntscale
+ \ifnum\@ww = \z@
+% \end{macrocode}
+% We are about to divide by |\@ww| which is zero. Treat |\@ww| as unity.
+% \begin{macrocode}
+ \else
+ \divide\@xdim by \@ww
+ \divide\@ydim by \@ww
+ \fi
+% \end{macrocode}
+% For reasons I don't understand, the \% signs at the end of the next few
+% lines are important!
+% \begin{macrocode}
+ \multnooverflow{\@xdim}{\botscale}%
+ \multnooverflow{\@ydim}{\botscale}%
+ \raise \@ydim
+ \hb@xt@\z@{\kern\@xdim
+ \unhcopy\@tempboxa\hss}%
+ \advance\count@\@ne}}
+% End of definition of |\@rqbez|.
+% \begin{macrocode}
+ \fi\fi}
+% \end{macrocode}
+% \end{macro}
+%
+% The end of this package.
+% \begin{macrocode}
+%</bez>
+% \end{macrocode}
+%
+% \section{Multiplication without overflow: The \Lpack{multiply} package} \label{sec:mnoflow}
+%
+% \TeX{} provides for integer arithmetic, subject to an upper limit
+% given by |\maxdim|. For at least the \Lpack{bez123} package we need to
+% be able to multiply without overflow.
+%
+% Announce the name of the package.
+% \changes{v1.1}{1998/10/14}{Put multnooverflow into a seperate package}
+% \begin{macrocode}
+%<*mult>
+\ProvidesPackage{multiply}[1998/10/14 v1.1 Multiplication of lengths without overflow]
+% \end{macrocode}
+%
+% \begin{macro}{\n@fl@wa}
+% \begin{macro}{\n@fl@wb}
+% \begin{macro}{\n@fl@wc}
+% \begin{macro}{\ifch@nge}
+% We need three length variables for this function.
+% We also need a boolean flag for dealing with negative numbers.
+% \begin{macrocode}
+\newlength{\n@fl@wa}
+\newlength{\n@fl@wb}
+\newlength{\n@fl@wc}
+\newif\ifch@nge
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\multnooverflow}
+%
+% The routine |\multnooverflow{|\meta{a}|}{|\meta{b}|}| sets
+% $a$ to the minimum of $ab$ and |\maxdimen|, preserving signs. \meta{a}
+% must be a length; it must not be a number literal.
+%
+% \begin{macrocode}
+\newcommand{\multnooverflow}[2]{%
+ \n@fl@wa = #1\relax%
+ \n@fl@wb = #2\relax%
+ \ch@ngefalse%
+% \end{macrocode}
+% Easy if $-1 \leq b \leq 1$.
+% \begin{macrocode}
+ \ifnum\n@fl@wb = \@ne%
+ \else%
+ \ifnum\n@fl@wb = \z@%
+ \n@fl@wa = \z@%
+ \else%
+ \ifnum\n@fl@wb = \m@ne%
+ \ch@ngetrue%
+ \else%
+% \end{macrocode}
+% Also easy if $-1 \leq a \leq 1$.
+% \begin{macrocode}
+ \ifnum\n@fl@wa = \z@%
+ \else%
+ \ifnum\n@fl@wa = \@ne%
+ \n@fl@wa = \n@fl@wb%
+ \else%
+ \ifnum\n@fl@wa = \m@ne%
+ \n@fl@wa = -\n@fl@wb%
+ \else%
+% \end{macrocode}
+% We have to check for potential overflow. First make sure that we deal
+% only with positive values.
+% \begin{macrocode}
+ \ifnum\n@fl@wa < \z@%
+ \ch@ngetrue%
+ \n@fl@wa = -\n@fl@wa%
+ \fi%
+ \ifnum\n@fl@wb < \z@%
+ \n@fl@wb = -\n@fl@wb%
+ \ifch@nge%
+ \ch@ngefalse%
+ \else%
+ \ch@ngetrue%
+ \fi%
+ \fi%
+% \end{macrocode}
+% Check for overflow.
+% \begin{macrocode}
+ \n@fl@wc = \maxdimen%
+ \divide\n@fl@wc by \n@fl@wb%
+ \advance\n@fl@wc by -1sp% \m@ne
+ \ifnum\n@fl@wa > \n@fl@wc%
+% \end{macrocode}
+% We have overflow. Set the multiplication result to |\maxdimen|.
+% \begin{macrocode}
+ \n@fl@wa = \maxdimen%
+ \PackageWarning{multiply}{Multiplication overflow}%
+ \else%
+% \end{macrocode}
+% It is safe to do the multiplication.
+% \begin{macrocode}
+ \multiply\n@fl@wa by \n@fl@wb%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+ \fi%
+% \end{macrocode}
+% The result of $ab$ is in |\n@fl@wa|. Adjust the sign if necessary.
+% \begin{macrocode}
+ \ifch@nge%
+ \n@fl@wa = -\n@fl@wa%
+ \fi%
+% \end{macrocode}
+% Return the result in the first argument variable.
+% \begin{macrocode}
+ #1 = \n@fl@wa%
+}
+% \end{macrocode}
+% \end{macro}
+%
+%
+% The end of this package.
+% \begin{macrocode}
+%</mult>
+% \end{macrocode}
+%
+%
+% \bibliographystyle{alpha}
+%
+% \begin{thebibliography}{GMS94}
+%
+% \bibitem[Far90]{FARIN90}
+% Gerald Farin.
+% \newblock {\em Curves and Surfaces for Computer Aided Geometric Design --- A
+% Practical Guide}.
+% \newblock Academic Press, Inc., second edition, 1990.
+%
+% \bibitem[FP81]{FandP}
+% I.~D. Faux and M.~J. Pratt.
+% \newblock {\em Computational Geometry for Design and Manufacture}.
+% \newblock Ellis Horwood, 1981.
+%
+% \bibitem[GMS94]{GOOSSENS94}
+% Michel Goossens, Frank Mittelbach, and Alexander Samarin.
+% \newblock {\em The LaTeX Companion}.
+% \newblock Addison-Wesley Publishing Company, 1994.
+%
+% \bibitem[Lam94]{LAMPORT94}
+% Leslie Lamport.
+% \newblock {\em LaTeX: A Document Preparation System}.
+% \newblock Addison-Wesley Publishing Company, second edition, 1994.
+%
+% \bibitem[Mor85]{MORTENSON85}
+% Michael~E. Mortenson.
+% \newblock {\em Geometric Modeling}.
+% \newblock John Wiley \& Sons, Inc., 1985.
+%
+% \bibitem[Wil96]{PRW96i}
+% Peter~R. Wilson.
+% \newblock {\em {LaTeX for standards: The LaTeX package files user manual}}.
+% \newblock NIST Report NISTIR, June 1996.
+%
+% \end{thebibliography}
+%
+%
+% \Finale
+% \PrintIndex
+%
+\endinput
+
+%% \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 \~}
+
+
diff --git a/Master/texmf-dist/source/latex/bez123/bez123.ins b/Master/texmf-dist/source/latex/bez123/bez123.ins
new file mode 100644
index 00000000000..f7983576b0e
--- /dev/null
+++ b/Master/texmf-dist/source/latex/bez123/bez123.ins
@@ -0,0 +1,61 @@
+%% bez123.ins
+%% Copyright 1998 Peter R. Wilson
+%%
+%% This file will generate fast loadable files and documentation
+%% driver files from the doc files in this package when run through
+%% LaTeX or TeX.
+%%
+%% This program is provided under the terms of the
+%% LaTeX Project Public License distributed from CTAN
+%% archives in directory macros/latex/base/lppl.txt.
+%%
+%% Author: Peter Wilson (CUA and NIST)
+%% now at: peter.r.wilson@boeing.com
+%%
+%% --------------- start of docstrip commands ------------------
+%%
+\def\batchfile{bez123.ins}
+\input docstrip.tex
+\preamble
+
+Author: Peter Wilson (CUA and NIST)
+ now at: peter.r.wilson@boeing.com
+
+Copyright 1998 Peter R. Wilson
+
+ This program is provided under the terms of the
+ LaTeX Project Public License distributed from CTAN
+ archives in directory macros/latex/base/lppl.txt.
+
+\endpreamble
+
+\keepsilent
+
+%%% \Msg{*** Generating the driver file ***}
+%%% \generateFile{bez123.drv}{t}{\from{bez123.dtx}{driver}}
+
+\Msg{*** Generating the LaTeX2e package files ***}
+
+\generateFile{bez123.sty}{t}{\from{bez123.dtx}{bez}}
+\generateFile{multiply.sty}{t}{\from{bez123.dtx}{mult}}
+
+\Msg{***********************************************************}
+\Msg{*}
+\Msg{* To finish the installation you have to move the following}
+\Msg{* file(s) into a directory searched by TeX:}
+\Msg{*}
+\Msg{* \space\space bez123.sty}
+\Msg{* \space\space multiply.sty}
+\Msg{*}
+\Msg{* To produce the documentation run the files ending with}
+\Msg{* `.dtx' through LaTeX.}
+\Msg{*}
+\Msg{* Process any name.idx file by:}
+\Msg{* \space\space\space\space makeindex -s gind.ist name}
+\Msg{* Process any name.glo file by:}
+\Msg{* \space\space\space\space makeindex -s gglo.ist -o name.gls name.glo}
+\Msg{*}
+\Msg{* Happy TeXing}
+\Msg{***********************************************************}
+
+\endinput
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-12 18:05:13 +0000