path: root/macros/latex/contrib/ebezier/ebezier.dtx

% \iffalse meta comment
%
% Copyright (c) Gerhard A. Bachmaier 2001-2005
%
% This program can be redistributed and/or modified under the terms
% of the LaTeX Project Public License Distributed from CTAN
% archives in directory macros/latex/base/ as file lppl.txt; either
% version 1 of the License, or (at your option) any later version.
% 
% Gerhard A. Bachmaier
% Institute for Medical Informatics, Statistics, and Documentation
% Medical University of Graz
% send bugs to: gerhard.bachmaier@meduni-graz.at
%
% \fi
%
% \iffalse
%
%<*driver>
\ProvidesFile{ebezier.drv}
%</driver>
%<package>\ProvidesPackage{ebezier}
  [2005/03/01 v4]  
%
%<*driver> 
\documentclass{article}
\usepackage{ebezier}
\usepackage{calc}
\usepackage{doc}
\EnableCrossrefs
\CodelineIndex
%\DisableCrossrefs 
\begin{document}
 \DocInput{ebezier.dtx}
\end{document}
%</driver>
% \fi
%
% \CheckSum{1955}
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%
% \title{Using ebezier}
% \author{Gerhard A. Bachmaier}
% \date{March 1, 2005}
% 
% \renewcommand{\topfraction}{.6}
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% \setcounter{totalnumber}{5}
% \renewcommand{\textfraction}{.1}
% \setlength{\unitlength}{1pt}
% \setlength{\parskip}{3pt}
% \font \logo=logo10 scaled \magstep1
% \newcommand{\formstrut}{\rule{0mm}{2mm}}
% \providecommand{\Metafont}{%
% {\logo META}\discretionary{}{-}{}{\logo FONT}}
% \renewcommand{\thefootnote}{\fnsymbol{footnote}}
% \renewcommand{\arraystretch}{1.2}
% \newcommand\SB[2]{\setbox1=\hbox{#1#2}}
% \newcommand*{\Lpack}[1]{\textsf {#1}}
% 
% \maketitle
% 
% \begin{abstract}
% The package \Lpack{ebezier} is an extension of the (old)
% \Lpack{bezier.sty} which is now part of \LaTeXe. It defines
% linear and cubic Bernste\u\i{}n polynomials together with some
% plotting macros for arcs. 
% 
% With the aid of the \Lpack{calc} package also the calculation of
% square roots and henceforward lengths is supplied.
% \end{abstract}
% 
% 
% \StopEventually{}
%
% \section*{Preamble}
% 
% If you want to draw complicated and/or lots of pictures, you should use
% \textsc{PostScript} for generating your plots and \Lpack{dvips} to include
% them in \TeX\ documents. \textsc{PostScript} can plot lines with arbitrary
% slope and unlimited length and circles with arbitrary radius just by using
% one command. See also the \LaTeX\ Graphics Companion\cite{T1} for further
% possibilities. There is also a new package \Lpack{pict2e}\cite{pict2e} a\-vail\-able
% which is preferrable for PDF and \textsc{PostScript}.
% 
% This package will support also lines with arbitrary slopes and unlimited
% length, but each line has to be generated as a sample of points. Each
% point reduces \TeX's memory and you will very likely have to overcome some
% \texttt{TeX capacity excxeeded...} messages.
% 
% Exact circles would involve trigonometric functions or square roots 
% to be evaluated by \TeX.
% Even with some tricks for reducing the effort of the calculation algorithm
% there would be hundreds of calculations for each point.\footnote{%
% To use \TeX\ for complex computations is as satisfactory as using your desk 
% calculator for writing tasks. But if you really want to do it e.g.\ the digits 7353 
% can be read (rotating by $180^0$) as 
% \texttt{ESEL}, the german word for ``donkey''.}
% But they may be quite
% well approximated by cubic bezier curves, also supplied in this package
% (The quality of interpolation is discussed in some detail in the Section
% \textit{Fitting Arcs}.) In fact, the small circles in the \LaTeX-\texttt{lcircle} fonts
% are also generated by the same method.
% 
% For draft papers use all kind of bezier curves with small number of points,
% just for the final run increase the numbers. \TeX\ memory can be set free
% again with {\verb+\clearpage+} at the end of complicated pictures. It's
% also a good idea to have them at an extra page (option \verb+[p]+ for
% \texttt{figure} environments).
% 
% For optical constructions the software LaTeXPiX\cite{PiX} may be a starting point. 
% This software supports cubic bezier curves defined in this package or from 
% \Lpack{bez123}\cite{T5}.
%
% \section{Mathematical Definitions}
% 
% A Bernste\u\i{}n polynomial of degree $n-1\: (n\ge 2)$ is defined by
% $n$ points $z_1, z_2,\ldots,z_n$
% 
% \[ \mathcal{B}_{n-1} [t] = \sum_{i=0}^{n-1} {n-1\choose i}
% (1-t)^{n-1-i} t^i z_{i+1}\quad t\in[0,1].\footnotemark\]%
% \footnotetext{There are also
% variants of this definitions with all coefficients $\equiv 1$.}
% 
% The points $z_i, \: i\in\lbrace1,\ldots,n\rbrace$, may be considered as
% real numbers, then $\mathcal{B}$ is really a polynomial in $t$. Or they
% denote points in a plane, which notation we will use further. In this case
% both \emph{components} are polynominials and the graph for $\mathcal{B}$ 
% is---part of---an algebraic curve.
% 
% \bigskip
% 
% All these graphs have in common:
% \begin{itemize}
% \item The graph is contained in the convex hull of the defining points.
% \item The graph starts at $z_1$ and stops at $z_n$.
% \item At the endpoints the tangents coincident with
% the directions $z_1-z_2$ and $z_{n-1}-z_n$ correspondingly.
% \end{itemize}
% 
% For $n=2$ the Bernste\u\i{}n polynomial $\mathcal{B}_1$ reduces to the
% linear form spanned by $z_1$ and $z_2$. The parametrization in $t$
% 
% \[ \mathcal{B}_1 [t] =(1-t) z_1 + t z_2=: t[z_1,z_2]\]
% 
% is also known as \emph{convex coordinates} for the segment
% $\overline{\formstrut z_1z_2}$.
% 
% \begin{figure}[htb]
% \begin{center}
% \begin{picture}(100,60)
% \put(-25,-10){\framebox(155,70){}}
% \put(20,10){\line(2,1){60}}
% \put(20,10){\makebox(0,0){$\bullet$}}
% \put(80,40){\makebox(0,0){$\bullet$}}
% \put(40,20){\makebox(0,0){$\bullet$}}
% \put(-17,15){$z_1 (t=0)$}
% \put(75,47){$z_2 (t=1)$}
% \put(40,10){$t=1/3$}
% \end{picture}
% \end{center}
% \caption{Line defined by two points}
% \end{figure}
% 
% \bigskip
% 
% For $n=3$ the result is a (quadratic) parabola which can also
% be constructed as the convex hull of all tangents in the
% triangle $\Delta\,z_1 z_2 z_3$ (examplified in Fig.\ 2b). 
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(250,120)
% \put(-15,-10){\framebox(280,130){}}
% \Qbezier[300](0,0)(50,100)(100,0)
% \put(0,0){\makebox(0,0){$\bullet$}}
% \put(100,0){\makebox(0,0){$\bullet$}}
% \put(50,100){\makebox(0,0){$\bullet$}}
% \put(-6,10){$z_1$}
% \put(50,105){$z_2$}
% \put(99,10){$z_3$}
% \put(0,0){\line(1,2){50}}
% \put(100,0){\line(-1,2){50}}
%%
% \put(150,0){\makebox(0,0){$\bullet$}}
% \put(250,0){\makebox(0,0){$\bullet$}}
% \put(200,100){\makebox(0,0){$\bullet$}}
% \put(144,10){$z_1$}
% \put(200,105){$z_2$}
% \put(249,10){$z_3$}
% \put(150,0){\line(1,2){50}}
% \put(250,0){\line(-1,2){50}}
% \lbezier[50](160,20)(210,80)
% \lbezier[30](170,40)(220,60)
% \lbezier[30](180,60)(230,40)
% \lbezier[50](190,80)(240,20)
% \lbezier[50](155,10)(205,90)
% \lbezier[50](165,30)(215,70)
% \lbezier[30](175,50)(225,50)
% \lbezier[50](185,70)(235,30)
% \lbezier[50](195,90)(245,10)
% \end{picture}
% \end{center}
% \caption{Quadratic parabola (a) as Bernste\u\i{}n polynom of degree 2
% and (b) as hull of tangents}
% \end{figure}
% 
% For $n=4$ finally we arrive at the cubic curves used e.g.\ in the \Metafont\
% book\cite{T3}.
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(100,100)
% \put(-10,-10){\framebox(130,110){}}
% \Cbezier[500](0,0)(10,80)(70,40)(100,0)
% \put(5,0){$z_1$}
% \put(10,85){$z_2$}
% \put(70,50){$z_3$}
% \put(105,0){$z_4$}
% \end{picture}
% \end{center}
% \caption{A simple cubic parabola.}
% \end{figure}
% 
% 
% We will not use more complicated polynomials for several reasons:
% 
% \begin{itemize}
% \item Higher degree polynomials require more operations to calculate
% just one point of the graph.
% \item For sketches (and \textbf{not} exact graphs!) cubic splines are
% sufficient to scope with all kind of different curvature requirements.
% \item \TeX\ can handle integers up to $2^{28}$, and ``real number'' lengths 
% are transformed to integers (multiples of scaled points: 1\,pt=$2^{16}$ sp) \cite{T2}. To stay
% within this restricted range even for cubic beziers we have to do calculations 
% in the right order. Changing the order of multiplication and divisions will 
% result very soon in arithmetic overflows. Also multiplication with these pseudo-real
% numbers is not an associative operation (due to the range limits!).
% \item The maximum number of arguments for commands in \TeX\ is limited to nine, 
% which is just enough for four points and a number.
% \end{itemize}
% 
% 
% %
% \section{The Plotting Macros}
% 
% \subsection{Simple Beziers}
% 
% There are two first level plot commands to be used in a
% \LaTeXe\ \texttt{picture} environment:
% 
% \begin{verbatim}
% \lbezier[n](x1,y1)(x2,y2)
% \cbezier[n](x1,y1)(x2,y2)(x3,y3)(x4,y4)
% \end{verbatim}
% 
% The arguments in square brackets are optional! If they are omitted or $n=0$ an adequate number
% will be calculated (cf. Section 8).
% 
% \DescribeMacro{\qbezier}
% \verb+\lbezier+ draws line segments from point $(x_1,y_1)$
% to $(x_2,y_2)$, or more exactly, $n+1$ intermediate points, while
% \verb+\cbezier+ is an implementation of the cubic variant. Just for
% completeness let me remind you that the quadratic
% variant---\verb+\qbezier[n](x1,y1)(x2,y2)(x3,y3)+---is part of \LaTeXe.
% 
% \DescribeMacro{\qbeziermax}
% $n$ is always limited by the number \verb+\qbeziermax+ (=500).
% 
% You may change \verb+\qbeziermax+ by a command like (it is not a counter!)
% \verb+\renewcommand{\qbeziermax}{1000}+.
% 
% \subsubsection{lbezier}
% \DescribeMacro{\lbezier}
% \verb+\lbezier+ is straightforward defined as
% linear polynomial. It produces equally spaced points.
% 
% \begin{verbatim}
% ...
% \put(0,25){\line(1,0){90}}
% \lbezier[20](0,10)(90,10)
% \lbezier[200](0,-5)(90,-5)
% ...
% \end{verbatim}
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(150,30)
% \put(-20,-15){\framebox(230,50){}}
% \put(0,25){\line(1,0){90}}
% \put(95,25){\makebox(100,0){\protect\LaTeXe\ \protect\texttt{line}}}
% \lbezier[20](0,10)(90,10)
% \put(95,10){\makebox(100,0){\protect\texttt{lbezier} (21 points)}}
% \lbezier[200](0,-5)(90,-5)
% \put(95,-5){\makebox(100,-5){\protect\texttt{lbezier} (201 points)}}
% \end{picture}
% \end{center}
% \caption{Different line commands}
% \end{figure}
% 
% Use \verb+\lbezier+ only in cases where the line you want to plot is not within
% the scope of the \verb+\line+ command, i.e. the slope is not a small rational number
% and/or the length is too small.
% 
% \subsubsection{cbezier}
% 
% \DescribeMacro{\cbezier}
% Just like the \verb+\lbezier+ macro \verb+\cbezier+ uses no tricks to generate
% the third order polynomial. The examples are from the \Metafont\ book
% (pp. 13)\cite{T3}, where the influence of changing the order of the
% controlling points ($z_1$ up to $z_4$) is also demonstrated.
% 
% \begin{verbatim}
% ...
% % z1=(0,16) z2=(40,84) z3=(136,96) z4=(250,0)
% % z12=(20,50) z23=(88,90) z34=(193,48) z123=(54,70) 
% % z234=(140.5,69)
% \cbezier[400](0,16)(40,84)(136,96)(250,0)
% \lbezier[30](0,16)(40,84)
% \lbezier[30](40,84)(136,96)
% \lbezier[30](136,96)(250,0)
% \lbezier[30](20,50)(88,90)
% \lbezier[30](88,90)(193,48)
% \lbezier[30](54,70)(140.5,69)
% ...
% \end{verbatim}
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(250,100)
% \put(-10,-10){\framebox(270,115){}}
% \cbezier[400](0,16)(40,84)(136,96)(250,0)
% \lbezier[30](0,16)(40,84)
% \lbezier[30](40,84)(136,96)
% \lbezier[30](136,96)(250,0)
% \lbezier[30](20,50)(88,90)
% \lbezier[30](88,90)(193,48)
% \lbezier[30](54,70)(140.5,69)
% \end{picture}
% \end{center}
% \caption{Iteration scheme for one point}
% \end{figure}
% 
% 
% \DescribeMacro{\Cbezier}
% The variant \verb+\Cbezier+ draws also dots and lines for the controlling points (see
% Fig.\ 6)\footnote{It resets also the plot symbol to the standard one; cf. Section 7}.
% 
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(184,100)
% \put(-10,-10){\framebox(204,100){}}
% \Cbezier[200](0,50)(12,72)(43,78)(84,50)
% \Cbezier[200](100,50)(143,78)(112,72)(184,50)
% \Cbezier[200](12,22)(0,0)(43,28)(84,0)
% \Cbezier[200](100,0)(184,0)(112,22)(143,28)
% \end{picture}
% \end{center}
% \caption{Examples for cubic curves with varying the order of the controlling points}
% \end{figure}
% 
% \subsection{Circles and Arcs}
% 
% All complex plotting commands in this package
% use a variant of \verb+\cbezier+ as building block. As
% in the \Metafont\ book circles and arcs may be represented by
% \verb+\cbezier+. 
% 
% To illustrate the procedure of the macro
% we do one calculation explicitely.
% 
% E.g. we want to draw the upper right quarter of a circle with end points $z_1=(0,r)$
% and $z_4=(r,0)$. $z_2$ and $z_3$ determine the tangents. So we may introduce
% them as $z_2=(h,r)$ and $z_3=(r,h)$ with a---so far unspecified---parameter
% $h$. 
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(50,50)
% \put(20,10){\line(1,0){30}}
% \put(20,10){\line(0,1){30}}
% \put(20,40){\line(1,0){10}}
% \put(50,10){\line(0,1){10}}
% \put(30,40){\line(1,-1){20}}
% \put(20,40){\makebox(0,0){$\bullet$}}
% \put(50,20){\makebox(0,0){$\bullet$}}
% \put(30,40){\makebox(0,0){$\bullet$}}
% \put(50,10){\makebox(0,0){$\bullet$}}
% \put(5,35){$z_1$}
% \put(45,0){$z_4$}
% \put(52,18){$z_3$}
% \put(25,45){$z_2$}
% \put(-5,-10){\framebox(70,70){}}
% \end{picture}
% \end{center}
% \caption{Sketch for the geometrical configuration}
% \end{figure}
% 
% If we substitute all points in the formula for the Bernste\u\i{}n
% polynomial for both components, we end at (for $t=1/2$)
% \[ x[\frac{1}{2}]=y[\frac{1}{2}]=\frac{r}{2}+\frac{3h}{8}\]
% These values should be $r/\sqrt{2}$ for a circle.
% So we arrive at
% \[ h=\frac{4}{3}\left(\sqrt{2}-1\right).\]
% 
% \DescribeMacro{\cArc}
% \DescribeMacro{\cCircle}
% The plot commands are:
% \begin{verbatim}
% \cArc[n](xm,ym)(x1,y1)
% \cCircle[n](xm,ym){r}[loc]
% \end{verbatim}
% 
% The optional qualifier $n$ determines the number
% of plotted points (There are as before $n+1$ plotted points for arcs; for circles the
% number depends on the specifier \textit{loc} and may be $n+1$, $2n+2$, or $4n+4$.).
% 
% \verb+\cArc+ plots a half circle with centre $(x_m,y_m)$ and $x$-axis through
% $(x_1,y_1)$ counterclockwise.
% 
% $r$ is the radius of the circle, specified as decimal constant in terms of 
% \verb+\unitlength+.
% 
% \verb+\cCircle+ plots full, halves and quarters of circles by specifying 
% \textit{loc} (see the corresponding table). 
% 
% \begin{table}[hbtp]
% \caption{Location specifiers for \texttt{cCircle}s}
% \begin{center}
% \begin{tabular}{|l|l|}
% \hline
% \textit{loc} & specifies \dots\\
% \hline
% \texttt{f} & full circle\\
% \texttt{l} & left half circle\\
% \texttt{r} & right half circle\\
% \texttt{b} & bottom half circle\\
% \texttt{t} & top half circle\\
% \texttt{lb} or \texttt{bl} & left bottom quarter of the circle\\
% \texttt{lt} or \texttt{tl} & left top quarter of the circle\\
% \texttt{rb} or \texttt{br} & right bottom quarter of the circle\\
% \texttt{rt} or \texttt{tr} & right top quarter of the circle\\
% \hline
% \end{tabular}
% \end{center}
% \end{table}
% 
% \begin{figure}[hbtp]
% \begin{center}
% \begin{picture}(300,100)
% \cCircle[1600](150,50){50}[f]
% \cCircle[150](100,0){50}[tr]
% \cCircle[150](100,100){50}[br]
% \cCircle[150](200,0){50}[tl]
% \cCircle[150](200,100){50}[bl]
% \cCircle[200](50,90){45}[b]
% \cCircle[200](50,0){45}[t]
% \cCircle[200](5,45){45}[r]
% \cCircle[200](95,45){45}[l]
% \cArc[200](250,70)(280,50)
% \cArc[200](250,60)(280,50)
% \cArc[200](250,50)(280,50)
% \cArc[200](250,40)(280,50)
% \cArc[200](250,30)(280,50)
% \put(-10,-10){\framebox(320,120){}}
% \end{picture}
% \end{center}
% \caption{Examples for \texttt{cCircle} and \texttt{cArc}}
% \end{figure}
% %
% 
% \section{Fitting Arcs}
% 
% The quality of representating arcs by cubic bezier curves is quite
% satisfactory. The differences between circles and beziers may be
% estimated in two ways.
% 
% \begin{enumerate}
% \item If we test the overall fit the area enclosed by
% the curves is a good metric: The area of \texttt{Carc} for the quarter circle
% is $1/30 (-33+40\sqrt{2})r^2$ to be conferred with $\pi/4\:r^2$. This is an overshot
% by just 0.028\%!
% \item The pointwise fit is measured by the radial difference.
% The maximum is $\cong 0.00025\,r$ (at odd multiples of $\pi/8$),
% it is zero for all multiples of $\pi/4$.
% \end{enumerate}
% 
% \section{Some \TeX{}nical Notes}
% 
% For the macros therein a lot of counters and lengths have to
% be declared.\footnote{Although I reuse some internal lengths I had to 
% declare some more to be used in function calls.} 
% Counters represent integer numbers, lengths are 
% ``real'' numbers (actually they are just integer multiples of
% $1/65536=2^{-16}$). \TeX\ has just a limited number of these
% stacks and therefore I use the same counters/lengths in all the macros.
% 
% One cannot store a real number for further use in these internal stacks just a
% multiplication of a \textit{decimal constant} with a length is possible (counters
% may be multiplied also with real numbers but just the integer part of the decimal
% constant is used!)
% 
% The package \Lpack{calc} introduced in the \LaTeX\ Companion\cite{T4} adds a
% new possiblity for multiplying lengths with the ratio of two lengths. This feature will be
% utilized furthermore.
% 
% \section{Calculating Lengths}
% 
% If I define lengths with respect to some \verb+\unitlength+ I can now define a 
% product or fraction of two lengths:
% 
% \verb+\lengthc = \lengtha*\ratio{\lengthb}{\unitlength}+
% 
% and
% 
% \verb+\lengthc = \unitlenght*\ratio{\lengtha}{\lengthb}+
% 
% The dimension of \verb+\lengthc+ \textit{in terms of} \verb+\unitlength+ (!) is the
% product, or factor respectively, of the two other lengths. 
% 
% With these operations it is even possible to
% calculate square roots. Simply use the iteration scheme ($m$ integer)
% \[ \xi_{m+1}=\frac{1}{2}\left( \xi_m + \frac{a}{\xi_m} \right) \]
% which will converge fast (with accuracy \verb+\eps+=1\,sp) to $\sqrt a$ (starting with 
% $\xi_0=a>0$).
% 
% 
% Lengths (in a \texttt{picture} environment) are easily calculated too, one just has to
% care for the upper limits (the maximum length for \TeX\ is roughly 16384\,pt!).
% 
% \DescribeMacro{\LenMult}
% \DescribeMacro{\LenDiv}
% \DescribeMacro{\AbsLen}
% \DescribeMacro{\LenSqrt}
% \DescribeMacro{\Length}
% \DescribeMacro{\LenNorm}
% The macros are:
% \begin{itemize}
% \item \verb+\LenMult#1#2#3+ and \verb+\LenDiv#1#2#3+ with two input and one output length
% (\verb+#3+).
% \item \verb+\AbsLen#1+ which returns the input length as positive length 
% (\TeX\ lengths can be negative!).
% \item \verb+\LenSqrt#1#2+ returns in the length \verb+#2+ the square root of length \verb+#1+
% (to say it again: measured in terms of \verb+\unitlength+).
% \item \verb+\Length(#1,#2)(#3,#4)#5+ stores in \verb+#5+ the length of the line
% segment between points \verb+(#1,#2)+ and \verb+(#3,#4)+ (coordinates may be decimal
% constants as in the \texttt{picture} commands).
% \item \verb+\LenNorm#1#2#3+ returns in \verb+#3+ the length of the hypothenuse of the
% rectangular triangle with catheti \verb+#1+ and \verb+#2+.
% \end{itemize}
% 
% \DescribeMacro{\eps}
% \textbf{All calculations} can be only exact up to the smallest length in \TeX\ which is 
% \verb+\eps+=1\,sp=$2^{-16}$\,pt=0.000015\,pt.
% 
% Examples (\verb+\unitlength+=1\,pt):
% \begin{verbatim}
% Mult: \LenMult{3pt}{4.333333pt}{\PathLength}\the\PathLength
% Div:  \LenDiv{3pt}{4.3333333pt}{\PathLength}\the\PathLength
% Abs:  \setlength{\PathLength}{-10pt}\the\PathLength\
%       \AbsLen{\PathLength}\the\PathLength
% Sqrt: \LenSqrt{16pt}{\PathLength}\the\PathLength\
%       \LenSqrt{2pt}{\PathLength}\the\PathLength\
%       \Length(1.5,4.3)(2.7,5){\PathLength}\the\PathLength\
%       \LenNorm{3pt}{4pt}{\PathLength}\the\PathLength
% \end{verbatim}
% 
% Mult: \LenMult{3pt}{4.333333pt}{\PathLength}\the\PathLength\ (exact: 13\,pt)
% 
% Div:  \LenDiv{3pt}{4.333333pt}{\PathLength}\the\PathLength\ (exact: 0.692308\,pt)
% 
% Abs:  \setlength{\PathLength}{-10pt}\the\PathLength\ 
%       \AbsLen{\PathLength}\the\PathLength
% 
% Sqrt: \LenSqrt{16pt}{\PathLength}\the\PathLength\ (exact: 4\,pt)\ 
%       \LenSqrt{2pt}{\PathLength}\the\PathLength\  (exact: 1.414213\,pt)
% 
% \hspace*{10mm}  \Length(1.5,4.3)(2.7,5){\PathLength}\the\PathLength\ (exact: 1.389244\,pt)
%       \LenNorm{3pt}{4pt}{\PathLength}\the\PathLength (exact: 5\,pt)
% 
% \DescribeMacro{\PathLengthQ}
% \DescribeMacro{\PathLengthC}
% \DescribeMacro{\PathLength}
% \DescribeMacro{\pathmax}
% Furthermore you can use these macros to evaluate the length of linear interpolations 
% of the curves displayed by \verb+\qbezier+ and \verb+\cbezier+. The syntax is
% \verb+\PathLengthQ[n](x1,y1)(x2,y2)(x3,y3)+ and\\
% \verb+\PathLengthC[n](x1,y1)(x2,y2)(x3,y3)(x4,y4)+ respectively. $n$ is the 
% number of interpolation points which is bounded by \verb+\pathmax+=50. The length 
% is stored in the% 
% ---already defined and used---length \verb+\PathLength+. Note: $n$ is \emph{not} optional
% for these two macros.
% 
% Example: For the cubic spline\\
% \verb+\cbezier[200](0,0)(50,100)(50,0)(100,100)+ 
% shown in Fig.~9 the results of the \verb+\PathLength+ \\
% for $n$=2,5,10,20,30,40,50 
% are displayed below. You may increase the value of \verb+\pathmax+ as for 
% \verb+\qbeziermax+ but the result will due to the internal calculation problems
% not become sigificant better.
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(100,80)
% \put(0,0){\framebox(100,100){}}
% \cbezier[300](0,0)(50,100)(50,0)(100,100)
% \end{picture}
% \end{center}
% \caption{A nice cubic curve}
% \end{figure}
% 
% The results are: \PathLengthC[2](0,0)(50,100)(50,0)(100,100)\the\PathLength, 
% \PathLengthC[5](0,0)(50,100)(50,0)(100,100)\the\PathLength,
% \PathLengthC[10](0,0)(50,100)(50,0)(100,100)\the\PathLength,
% \PathLengthC[20](0,0)(50,100)(50,0)(100,100)\the\PathLength,
% \PathLengthC[30](0,0)(50,100)(50,0)(100,100)\the\PathLength,
% \PathLengthC[40](0,0)(50,100)(50,0)(100,100)\the\PathLength,
% \PathLengthC[50](0,0)(50,100)(50,0)(100,100)\the\PathLength.
% (An good numercial integration program will yield more accurate 149.999.)
% 
% \section{More general arcs}
% 
% \DescribeMacro{\cArcs}
% Finally you can plot an arc (i.e.\ a cubic approximation to the circle arc) between
% two points with given centre of the circle:\\
% \verb+\cArcs[n](xm,ym)(x1,y1)(x2,y2)+\\
% with $n+1$ number of points (limited by 
% \verb+\qbeziermax+ again) and centre $(x_m,y_m)$.
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(200,200)
% \put(0,0){\framebox(200,200){}}
% \put(100,100){\makebox(0,0){$\bullet$}}
% \cArcs[300](100,100)(120,130)(130,120)
% \cArcs[100](100,100)(150,110)(110,150)
% \cArcs[300](100,100)(130,180)(130,20)
% \cArcs[300](100,100)(120,170)(80,170)
% \cArcs[300](100,100)(60,150)(60,50)
% \cArcs[100](100,100)(90,90)(110,90)
% \cArcs[200](100,100)(60,70)(130,60)
% \end{picture}
% \end{center}
% \caption{Some examples for arcs; the centre is marked by $\bullet$}
% \end{figure}
% 
% Limitations: 
% \begin{itemize} 
% \item The arc should be smaller than the half of a circle (The limit is
% handled by \verb+\cArc+ and is built-in again in \verb+\cArcs+.) Otherwise the shape
% will become ``elliptic'' and ly in the wrong half plane.
% \item There is no check for consistency if $r_1^2=(x_1-x_m)^2+(y_1-y_m)^2$ and
% $r_2^2=(x_2-x_m)^2+(y_2-y_m)^2$ are really equal. The graph will contain in any case 
% both points as border points.
% \end{itemize}
% I will shortly derive the formulas used in the code. The code is even more tricky 
% due to the fact that I had just a limited number of lengths and the code reuses 
% some lengths explicitely and implicitely by calling routines.
% 
% \begin{figure}[hbt]
% \begin{center}
% \begin{picture}(200,200)
% \put(0,0){\framebox(200,200){}}
% \put(80,20){\makebox(0,0){$\bullet$}}
% \put(50,120){\makebox(0,0){$\bullet$}}
% \put(171.65,70){\makebox(0,0){$\bullet$}}
% \put(100,135){\makebox(0,0){$\bullet$}}
% \put(146.65,115.825){\makebox(0,0){$\bullet$}}
% \put(120.3,115.6){\makebox(0,0){$\bullet$}}
% \lbezier[150](80,20)(50,120)
% \lbezier[150](80,20)(171.65,70)
% \cArcs[200](80,20)(50,120)(171.65,70)
% \lbezier[150](50,120)(150,150)
% \lbezier[150](171.65,70)(121.65,161.65)
% \lbezier[30](80,20)(141.65,170)
% \put(66,21){$M$}
% \put(41,115){4}
% \put(174,65){1}
% \put(90,135){3}
% \put(150,112){2}
% \put(116,120){5}
% \end{picture}
% \end{center}
% \caption{Sketch for the geometric situation}
% \end{figure}
% 
% We know the coordinates for the points $M$, 1, and 4. The tangents $\overline{43}$ and 
% $\overline{12}$ are normals to the radius in the corresponding points. The distances 
% $\overline{43}$ and $\overline{12}$ should be equal. 5 lies on the symmetry axis (dotted
% line) with distance $r$ from $M$.
% 
% \noindent Normal vectors: $\vec n_1=(y_m-y_1,x_1-x_m)$ and $\vec n_2=(y_4-y_m,x_m-x_4)$
% 
% \noindent Coordinate vectors: $\vec 2 = \vec 1 + \lambda \vec n_1$ and 
% $\vec 3 = \vec 4 + \lambda \vec n_2$ ($\lambda$ is the same because both normal 
% vectors have length $r$)
% 
% \noindent Furthermore $\vec 5={\cal B}_4 [1/2]$ (the cubic spline
% should also be symmetric and contain 5)
% 
% Now we have:
% \begin{eqnarray}
%  x[ t] & = & (1-t)^3 x_1 + 3 t (1-t)^2 x_2 + 3 t^2 (1-t) x_3 +t^3 x_4\\
%  y[ t] & = & (1-t)^3 y_1 + 3 t (1-t)^2 y_2 + 3 t^2 (1-t) y_3 +t^3 y_4
% \end{eqnarray}
% 
% Substituting for $x_2$, $y_2$, $x_3$, and $y_3$ and $t\to1/2$:
% \begin{eqnarray}
%  x_5=x\left[ \frac{1}{2}\right] & = & \frac{1}{2}
% (x_1 +x_4) + \frac{3}{8} \lambda(y_4-y_1) \\
%  y_5=y\left[ \frac{1}{2}\right] & = & \frac{1}{2}
% (y_1 +y_4) + \frac{3}{8} \lambda(x_1-x_4 )
% \end{eqnarray}
% 
% We could now calculate the norm of this point and set it equal to the radius $r^2=
%  (x_m-x_1)^2+(y_m-y_1)^2$. This gives a quadratic equation for 
% $\lambda$. But the result is a rather complex term with respect to our input parameters.
% 
% A nicer term can be found if we define 
% \begin{equation}
%  x_5=x_m+\kappa (x_1+x_4-2x_m) \quad y_5=y_m+\kappa (y_1+y_4-2y_m)
% \end{equation}
% with aid of the symmetry vector. $\kappa$ is simply $r$ divided by the norm of the
% symmetry vector.
% 
% The resulting $\lambda$ is now (using just the $x$-equation)
% \begin{equation}
%  \lambda=\frac{4}{3} (-1+2 \kappa)\frac{x_1+x_4-2x_m}{y_4-y_1}
% \end{equation}
% 
% Special cases: 
% \begin{itemize}
% \item The symmetry vector is the null vector if $\overline{14}$ is a diameter of the
%  circle. But this case is already solved by \verb+\cArc+.
% \item For $y_4=y_1$ one needs the equation for the $y$-component, i.e.\ we have as factor
% $(y_1+y_4-2y_m)/(x_1-x_4)$ in $\lambda$.
% \end{itemize}
% 
% 
% \section{Varying the line thickness}
% 
% There is another package, \Lpack{bez123}\cite{T5}, which introduces also linear and cubic 
% bezier curves, even variants which plot exactly all kind of conic curves (ellipses,
% parabolas, and hyperbolas). There are two features in \Lpack{bez123}, which I added in the 
% third version of \Lpack{ebezier}:
% 
% \DescribeMacro{\thinlines}
% \DescribeMacro{\thicklines}
% \DescribeMacro{\linethickness}
% \DescribeMacro{\qbeziermax}
% \begin{enumerate}
% \item Changing the size of the plot squares by the \LaTeX\ commands\\
% \verb+\thinlines+, \verb+\thicklines+, and/or \verb+\linethickness+. 
% \item Calulation of an optimal number of plot points if $n$=0 instead of using the
% maximum \verb+\qbeziermax+ (see next section).
% \end{enumerate}
% 
% If you look close to lines you will note some peculiarity. For instance the original 
% \LaTeX\ \verb+\line+ is in horizontal/vertical mode a simple \verb+\ruler+.
% 
% \begin{figure}[htbp]
% \begin{center}
% \begin{picture}(100,100)
% \thinlines
% \put(-5,-5){\framebox(110,110){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,10){\line(1,0){100}}
% \put(10,0){\line(0,1){100}}
% \setlength{\linethickness}{10pt}
% \put(10,10){\line(1,0){60}}
% \put(10,10){\line(0,1){60}}
% \end{picture}
% \end{center}
% \caption{Axes with standard lines}
% \end{figure}
% 
% Remark: The \textit{line} is exactly as long as specified.
% 
% \DescribeMacro{\@wholewidth}
% But the plot point used by \verb+\qbezier+, \Lpack{bez123} and \Lpack{ebezier} 
% (until version 2!) is a small square which is not centered at the control points
% (dimension \verb+\@wholewidth+)
% 
% \begin{figure}[htbp]
% \begin{center}
% \begin{picture}(50,50)
% \thinlines
% \put(-5,-5){\framebox(60,60){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,25){\line(1,0){50}}
% \put(15,0){\line(0,1){50}}
% \DefOldPlotSymbol
% \setlength{\linethickness}{10pt}
% \lbezier[1](15,25)(15,25)
% \end{picture}
% \end{center}
% \caption{Old plot symbol}
% \end{figure}
% 
% which results in a shifted $y$-axis and \textit{lines} which are actually longer 
% by an amount of one square (i.e. \verb+\@wholewidth+)
% 
% \begin{figure}[htbp]
% \begin{center}
% \begin{picture}(100,100)
% \thinlines
% \put(-5,-5){\framebox(110,110){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,10){\line(1,0){100}}
% \put(10,0){\line(0,1){100}}
% \setlength{\linethickness}{10pt}
% \DefOldPlotSymbol
% \setlength{\linethickness}{10pt}
% \lbezier[10](10,10)(80,10)
% \lbezier[10](10,10)(10,80)
% \end{picture}
% \end{center}
% \caption{Axes with old plot symbol}
% \end{figure}
%  
% or with hollow squares ($\bullet$ references to the end points).
% 
% \begin{figure}[htbp]
% \begin{center}
% \begin{picture}(100,100)
% \thinlines
% \put(-5,-5){\framebox(110,110){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,10){\line(1,0){100}}
% \put(10,0){\line(0,1){100}}
% \thinlines
% \put(10,5){\framebox(70,10){}}
% \put(10,5){\framebox(10,70){}}
% \put(10,5){\framebox(10,10){}}
% \put(70,5){\framebox(10,10){}}
% \put(10,65){\framebox(10,10){}}
% \put(10,10){\makebox(0,0){$\bullet$}}
% \put(70,10){\makebox(0,0){$\bullet$}}
% \put(10,70){\makebox(0,0){$\bullet$}}
% \end{picture}
% \end{center}
% \caption{Axes with old plot symbol (hollow)}
% \end{figure}
% 
% 
% This version uses centered plot symbols (standard is again a square)
% 
% \begin{figure}[htbp]
% \begin{center}
% \begin{picture}(50,50)
% \put(-5,-5){\framebox(60,60){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,25){\line(1,0){50}}
% \put(25,0){\line(0,1){50}}
% \DefStandardPlotSymbol
% \setlength{\linethickness}{10pt}
% \lbezier[1](25,25,)(25,25)
% \end{picture}
% \end{center}
% \caption{New standard plot symbol}
% \end{figure}
% 
% which corrects the shift of the $y$-axis. The line is again longer but this 
% time the excess is symmetrically on both ends
% 
% \begin{figure}[htbp]
% \begin{center}
% \begin{picture}(100,100)
% \thinlines
% \put(-5,-5){\framebox(110,110){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,10){\line(1,0){100}}
% \put(10,0){\line(0,1){100}}
% \setlength{\linethickness}{10pt}
% \lbezier[10](10,10)(80,10)
% \lbezier[10](10,10)(10,80)
% \end{picture}
% \end{center}
% \caption{Axes with new standard plot symbol}
% \end{figure}
% 
% or again with hollow squares.
% 
% \begin{figure}[htbp]
% \begin{center}
% \begin{picture}(100,100)
% \thinlines
% \put(-5,-5){\framebox(110,110){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,10){\line(1,0){100}}
% \put(10,0){\line(0,1){100}}
% \thinlines
% \put(5,5){\framebox(70,10){}}
% \put(5,5){\framebox(10,70){}}
% \put(5,5){\framebox(10,10){}}
% \put(65,5){\framebox(10,10){}}
% \put(5,65){\framebox(10,10){}}
% \put(10,10){\makebox(0,0){$\bullet$}}
% \put(70,10){\makebox(0,0){$\bullet$}}
% \put(10,70){\makebox(0,0){$\bullet$}}
% \end{picture}
% \end{center}
% \caption{Axes with new standard plot symbol (hollow)}
% \end{figure}
% 
% \DescribeMacro{\DefOldPlotSymbol}
% \DescribeMacro{\Qbezier}
% To be consistent with the old version the command \verb+\DefOldPlotSymbol+
% is supplied which uses the old form. Also a variant \verb+\Qbezier+ for 
% \verb+\qbezier+ is
% defined which can use the new plot symbol.\footnote{This command is just for convenience. 
% A quadratic bezier can be plotted as cubic bezier as follows. If you want to plot
% $\backslash$\texttt{qbezier[100](z1)(zm)(z4)} with $(z)=(x,y)$ you may calulate points 
% $z_2=2/3[z_m,z_1]$ and $z_3=2/3[z_m,z_4]$. The cubic bezier $\backslash$%
% \texttt{cbezier[100](z1)(z2)(z3)(z4)} is exactly the same as the quadratic one!}%
% \footnote{It can also use the other new symbols defined later.}
% 
% The next point of consideration is the handling of slanted lines.
% In the ordinary \LaTeX-\texttt{picture} environment
% \verb+\linethickness+ has no effect on slanted lines. Now the change applies
% but a new problem occurs. If you plot a slanted line (slope angle $\varphi$) 
% with squares
% 
% \begin{figure}[htb]
% \begin{center} 
% \begin{picture}(100,80)
% \thinlines
% \put(-5,-5){\framebox(110,90){}}
% \put(5,5){\framebox(30,30){}}
% \put(25,15){\framebox(30,30){}}
% \put(45,25){\framebox(30,30){}}
% \put(65,35){\framebox(30,30){}}
% \put(35,5){\line(2,1){70}}
% \put(5,35){\line(2,1){70}}
% \put(60,5){\vector(-1,2){5}}
% \put(55,15){\line(-1,2){23}}
% \put(32.5,60){\vector(1,-2){5}}
% \put(61,22){$\scriptscriptstyle\varphi$}
% \put(32,47){$\scriptscriptstyle\varphi$}
% \put(65,10){$d$}
% \end{picture}
% \end{center}
% \caption{Effective thickness for slanted lines}
% \end{figure}
% 
% your line gets effective thicker! The factor of enlargement is $\sin \varphi
% +\cos \varphi$ which has its maximum $\sqrt 2$ with slope $\varphi_0=45^0$.
% 
% 
% There are two possiblities to correct the thickness
% \begin{itemize}
% \item correct the line thickness of each line or
% \item use other plot symbols which behave better.
% \end{itemize}
% 
% \DescribeMacro{\Lbezier}
% The first possibilitiy can be realized just for \verb+\lbezier+ and not 
% \verb+\cbezier+ because the slope changes from point to point in the latter case.
% The solution is established by internally changing the \verb+\linethickness+
% by the factor $\ell/(\Delta x+\Delta y)$ where $\ell$ denotes the length of the 
% line ($=\sqrt{\Delta^2 x +\Delta^2 y}$)
% and $\Delta x$ is the horizontal difference of the the points 
% ($\Delta y$ respectivelly for the vertical difference).
% 
% To use this line type call \verb+\Lbezier[n](x1,y1)(x2,y2)+.
% 
% The second chance is to change the plot symbol to a disc. The smallest disk 
% available is the character ``.'' at 5pt. Unfortunately this method will
% implicitely restrict the \verb+\linethickness+ to some definite values (see the 
% following table for the numbers in question).
% 
% \begin{table}[hbtp]
% \caption{Dimensions for various plot symbols}
% \begin{center}
% \begin{tabular}{|ll|rr|l|}
% \hline
% Font&Size for (10pt) & Width & Heigth & Rule \\
% \hline
% \verb+\vrm+ &tiny& \SB{\vrm}{.}\the\wd1 & \SB{\vrm}{.}\the\ht1 & 
% \SB{\vrm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\virm+ &tiny for 11/12pt& \SB{\virm}{.}\the\wd1 & \SB{\virm}{.}\the\ht1 & 
% \SB{\virm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\viirm+ & scriptsize &\SB{\viirm}{.}\the\wd1 & \SB{\viirm}{.}\the\ht1 & 
% \SB{\viirm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\viiirm+ & footnotesize & \SB{\viiirm}{.}\the\wd1 & \SB{\viiirm}{.}\the\ht1 & 
% \SB{\viiirm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\ixrm+ & small &\SB{\ixrm}{.}\the\wd1 & \SB{\ixrm}{.}\the\ht1 & 
% \SB{\ixrm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\xrm+ & normalsize &\SB{\xrm}{.}\the\wd1 & \SB{\xrm}{.}\the\ht1 & 
% \SB{\xrm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\xirm+ & normalsize 11pt& \SB{\xirm}{.}\the\wd1 & \SB{\xirm}{.}\the\ht1 & 
% \SB{\xirm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\xiirm+ & large &\SB{\xiirm}{.}\the\wd1 & \SB{\xiirm}{.}\the\ht1 & 
% \SB{\xiirm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\xivrm+ & Large & \SB{\xivrm}{.}\the\wd1 & \SB{\xivrm}{.}\the\ht1 & 
% \SB{\xivrm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\xviirm+ & LARGE &\SB{\xviirm}{.}\the\wd1 & \SB{\xviirm}{.}\the\ht1 & 
% \SB{\xviirm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\xxrm+ & huge &\SB{\xxrm}{.}\the\wd1 & \SB{\xxrm}{.}\the\ht1 & 
% \SB{\xxrm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+\xxvrm+ & Huge &\SB{\xxvrm}{.}\the\wd1 & \SB{\xxvrm}{.}\the\ht1 & 
% \SB{\xxvrm}{.}\rule{1 cm}{\ht1} \copy1\\
% \verb+$\bullet$+ & &\SB{$\bullet$}{}\the\wd1 &\SB{$\bullet$}{}\the\ht1& 
% \SB{$\bullet$}{}\rule{1cm}{\ht1} \copy1\\
% \hline
% \end{tabular}
% \end{center}
% \end{table}
% 
% \DescribeMacro{\DefPlotSymbol}
% With the aim of the command \verb+\DefPlotSymbol{item}+ you may define any 
% \textit{item} as your plot symbol\footnote{A similar approach with centered 
% symbols can be found in the packages 
% \Lpack{epic}\cite{epic} and PiC\TeX\cite{pictex}.}.
%  It will be centered as the default plot square
% (otherwise an even larger shift of the $y$-axis would occur). Use explicit font 
% selection with the names supplied in the table to ensure style independence 
% (otherwise e.g.\
% \verb+\DefPlotSymbol{\tiny .}+ would be different in 10pt and 11pt context).
% 
% \DescribeMacro{\DefShiftedPlotSymbol}
%  This works
% for all \textit{items} which have a vertical symmetry axis with respect to their
% defining bounding box (defined by \Metafont) and which ly on the baseline 
% (or beyond if they have some defined depth). It will not work otherwise. For example the
% circles from the circle font have heigth and depth zero and their reference point is 
% already the centre (i.e. the symbol extends backward). Or consider the ``*''-sign.
% It does not ly on the baseline. For these cases a generalized command is supplied:\\
% \verb+\DefShiftedPlotSymbol{item}{x-shift}{y-shift}{height}+. 
% 
% The shifts are applied 
% to the left and downward. The supplied heigth will only have effect if you specify
% $n=0$ for the number of plotting points.
% 
% Examples: 
% \begin{verbatim}
% \DefShiftedPlotSymbol{\tencirc n}{0pt}{0pt}{1pt}
% \DefShiftedPlotSymbol{\tencirc \char'176}{0pt}{0pt}{15pt}
% \DefShiftedPlotSymbol{\fbox{\Huge *}}{0pt}{0pt}{25pt}       
% %with bounding box
% \setbox0=\hbox{*}
% \DefShiftedPlotSymbol{*}{.5\wd0}{.7\ht0}{.6\ht0}
% \lbezier[1](100,30)(100,30)
% \lbezier[0](0,20)(125,20)
% \DefShiftedPlotSymbol{*}{.5\wd0}{.7\ht0}{10\ht0}
% \lbezier[0](0,10)(125,10)
% \end{verbatim}
% 
% \setlength{\fboxsep}{0pt}
% \setlength{\fboxrule}{0.1pt}
% 
% \begin{figure}[hbtp]
% \begin{center}
% \begin{picture}(125,50)
% \put(-5,-5){\framebox(135,60){}}
% \setlength{\linethickness}{0.1pt}
% \put(0,30){\line(1,0){125}}
% \multiput(25,20)(25,0){4}{\line(0,1){20}}
% \DefShiftedPlotSymbol{\tencirc n}{0pt}{0pt}{1pt}
% \lbezier[1](25,30)(25,30)
% \DefShiftedPlotSymbol{\tencirc \char'176}{0pt}{0pt}{15pt}
% \lbezier[1](50,30)(50,30)
% \DefShiftedPlotSymbol{\fbox{\Huge *}}{0pt}{0pt}{25pt}
% \lbezier[1](75,30)(75,30)
% \setbox0=\hbox{*}
% \DefShiftedPlotSymbol{*}{.5\wd0}{.7\ht0}{.6\ht0}
% \lbezier[1](100,30)(100,30)
% \lbezier[0](0,20)(125,20)
% \DefShiftedPlotSymbol{*}{.5\wd0}{.7\ht0}{10\ht0}
% \lbezier[0](0,10)(125,10)
% \end{picture}
% \end{center}
% \caption{Examples for other plot symbols}
% \end{figure}
% 
% \emph{Caution:} The commands for changing the line thickness have 
% implicit effects for plot symbols
% defined with  \verb+\DefPlotSymbol{item}+ or \\
% \verb+\DefShiftedPlotSymbol+. The implicit or
% explicit defined height is redefined. But the effect is only visible in case $n=0$. 
% 
%
% \DescribeMacro{\DefStandardPlotSymbol} 
% In any case you may restore \textbf{default values} by stating
% \begin{verbatim}
% \DefStandardPlotSymbol
% \thinlines
% \end{verbatim}
% 
% \section{Estimation for the number of plotting points}
% 
% As mentioned in the last section all plotting macros will calculate the number
% of plotting points if the value $n=0$ is active. All calculations will 
% use the actual length of the object which can
% be calculated with the aim of the calculation macros in Section \textit{Calculating Lengths}.
% For all these calculations \verb+\eps+ is temporarily increased by a factor of 10 and
% for higher bezier curves just 5 intermediate points will be used. If the calculated number 
% exceeds the specified maximum \verb+\qbeziermax+ an info in the log-file will be generated.
% 
% All macros for circles and arcs will use a simpler estimate due to their construction  
% by an intrinsic call of the cubic bezier. It uses the length of the chord  and 
% the maximal deviation factor $\pi/2$ from the  arc length.
% 
% 
% \section{Joining linear beziers}
% 
% \DescribeMacro{\ljoin}
% A further command has been supplied to ease the drawing of polygon paths. Instead of 
% writing a sequence of \verb+\lbezier+s with common vertices you can write compactly
% \verb+\ljoin[n](x1,y1)(x2,y2)(x3,y3)...(xm,ym)+
% 
% Caution: There should be no spaces in the command, so break lines with \% if
% necessary. There should be at least 2 points. The parameter $n$ is optional, internally
% \verb-\lbezier[n](xk,yk)(xk+1,yk+1)- will be executed.
% 
% \DescribeMacro{\Ljoin}
% There is also a variant \verb+\Ljoin+ which uses \verb+\Lbezier+.
% 
% \renewcommand{\qbeziermax}{1200}
% \begin{figure}
% \begin{center}
% \begin{picture}(200,100)
% \put(-5,-5){\framebox(210,110){}}
% \begin{picture}(100,100)
% \ljoin(0,0)(20,100)(20,0)(40,50)(40,0)(60,25)(60,0)(80,12.5)(80,0)(100,6.25)(100,0)
% \end{picture}
% \begin{picture}(100,100)
% \Ljoin(0,0)(20,100)(20,0)(40,50)(40,0)(60,25)(60,0)%
% (80,12.5)(80,0)(100,6.25)(100,0)
% \end{picture}
% \end{picture}
% \end{center}
% \caption{$\backslash$\texttt{ljoin} versus $\backslash$\texttt{Ljoin}}
% \end{figure}
% 
% \changes{Version 1}{2000/07/28}{original version}
% \changes{Version 2}{2001/12/04}{major bug fix for cCircle}
% \changes{Version 3}{2002/10/23}{major changes}
% \changes{Version 4}{2005/03/01}{minor changes}
% 
% \section*{Versions}
%
% This is Version 4 from March 1, 2004.
% 
%  Changes with regard to version 3:
% \begin{itemize}
%  \item Bug-address changed.
%  \item Error in defining (first) equation corrected (thanks to \verb+jens.schwaiger@uni-graz.at+).
%  \item Marginal corrections with regard to \Lpack{l2tabu} (v1.8).
%  \item Documentaion as pdf supplied.
%  \end{itemize}
% 
%  Changes with regard to version 2:
% \begin{itemize}
% \item Implementing line thickness (\verb+\thinlines+, \verb+\thicklines+, and\\
% \verb+\setlength{\linethickness}{dimen}+.
% \item Different plot symbols.
% \item \verb+\Lbezier+ for equally thick lines in all directions.
% \item \verb+\Qbezier+ implementation to be used with new plot symbols.
% \item Calculation of an optimal number of plot symbols (as default number for case $n$=0).
% \item Parameter $n$ is for all \emph{plot} commands optional.
% \item New macro for polygon paths.
% \item Style supplied in dtx-format.
% \item Minor style changes regarding numbers and lengths.
% \end{itemize}
% 
%  Changes with regard to Version 1:
% \begin{itemize} 
% \item \verb+\@tempa+ replaced by \verb+\@TempDim+. \verb+\@tempa+ was also
% used by other packages.
% \item Additionaly supplied \verb+\RequirePackage{calc}+.
% \item Bug fixed for circles. The original macros did actually not support changes in
%  \verb+\unitlength+.
% \end{itemize}
% 
% \begin{thebibliography}{9}
% \bibitem{T2} D.\ E.\ Knuth: \textit{The} \TeX\ \textit{Book}, Addison-Wesley,
% Reading MA, 1986.
% \bibitem{T3} D.\ E.\ Knuth: \textit{The} \Metafont\ \textit{Book}, Addison-Wesley,
% Reading MA, 1986.
% \bibitem{T4} M.\ Goossens, F.\ Mittelbach, A.\ Samarin: \textit{The} \LaTeX\ \textit{Companion},
% Addison-Wesley, Reading MA, 1994.
% \bibitem{T1} M.\ Goossens,  S.\ Rahtz, F.\ Mittelbach: \textit{The} \LaTeX\ 
% \textit{Graphics Companion}, Addison-Wesley, Reading MA, 1997.
% \bibitem{T5} P.\ Wilson: \textit{The} \Lpack{bez123} \textit{and} \Lpack{multiply}
% \textit{packages}, 1998;\\ packages at CTAN/macros/latex/contrib/supported/bez123.
% \bibitem{epic} S.\ Podar: \textit{Enhancements to the Picture Environment
% in }\LaTeX, 1986;\\ package at CTAN/macros/latex/other/epic.
% \bibitem{pictex} M.\ J.\ Wichura: \textit{The PiC}\TeX\ \textit{Manual}, 1992;\\
%  package at CTAN/graphics/pictex.
% \bibitem{pict2e} R.\ Niepraschk, H.\ Gaesslein: The \Lpack{pict2e} Package, 2003;\\
% package at CTAN/macros/latex/contrib/pict2e.
% \bibitem{PiX}  N.\ J.\ H.\ M.\ van Beurden: A \LaTeX\ picture editor for Windows, 2003;\\
% package at CTAN/systems/win32/latexpix.
% \end{thebibliography}
% 
% \OnlyDescription
%
% \section{Implementation}
%  
%  The macros \verb+\lbezier+ and \verb+\cbezier+ are rather old, they existed since
%  I realized the existence of \Lpack{bezier.sty} more then ten years ago. Therefore 
%  the macros  are written rather in pure \TeX\ than in \LaTeX. Only the calculation 
%  macros demand for \LaTeX\ notation to use the package \Lpack{calc}. But with this 
%  version the macros interact more and some \LaTeX\ part occurrs also in the plot macros.
%
%    \begin{macrocode}
%<*package>
\NeedsTeXFormat{LaTeX2e}
\RequirePackage{calc}
%%
%    \end{macrocode}
%    I define new font names because \texttt{cmr} may not be the standard font. They
%    may be needed for plotting symbols.
%    \begin{macrocode}   
\newfont{\vrm}{cmr5}
\newfont{\virm}{cmr6}
\newfont{\viirm}{cmr7}
\newfont{\viiirm}{cmr8}
\newfont{\ixrm}{cmr9}
\newfont{\xrm}{cmr10}
\newfont{\xiirm}{cmr12}
\newfont{\xviirm}{cmr17}
\newfont{\xirm}{cmr10 scaled \magstephalf}
\newfont{\xivrm}{cmr10 scaled \magstep2}
\newfont{\xxrm}{cmr10 scaled \magstep4}
\newfont{\xxvrm}{cmr10 scaled \magstep5}
%%
%    \end{macrocode}
%
%    I need only three new counters,
%    \begin{macrocode}
\newcounter{@cnta}\newcounter{@cntb}\newcounter{@cntc}\newcounter{@cntd}
%%
%    \end{macrocode}
%    but a lot of lengths. Packages like PiC\TeX\ have problems by defining too many
%    lengths, so I try to use as many already defined lengths (defined for usage
%    in a plotting context).
%    \begin{macrocode}
%% \@TempDim#1#2#3{"count"|"dimen"|"box"|"skip"}{\myname}{\realname}
%%  allocate new one or alias is defined, so use it
%%
\def\@TempDim#1#2#3{%  
  \ifx\@und@fined#3\csname new#1\endcsname#2%  
  \else\let#2#3\fi}
%%
\@TempDim{dimen}\@X\@ovxx
\@TempDim{dimen}\@Xa\@ovdx
\@TempDim{dimen}\@Xb\@ovyy
\@TempDim{dimen}\@Xc\@ovdy
\@TempDim{dimen}\@Y\@ovro
\@TempDim{dimen}\@Ya\@ovri
\@TempDim{dimen}\@Yb\@xdim
\@TempDim{dimen}\@Yc\@ydim
\@TempDim{dimen}\@Z\@clnht
\@TempDim{dimen}\@Za\@clnwd
\@TempDim{dimen}\@Zb\@dashdim
\@TempDim{dimen}\@Zc\@tempdima
\@TempDim{dimen}\@Zd\@tempdimb
\@TempDim{dimen}\@Ze\@tempdimc
%%
\newlength{\@Zf}\newlength{\@Zg}\newlength{\@Zh}
\newlength{\@Zi}\newlength{\@Zj}
%    \end{macrocode}
%
%    This special length will be used for the circle macros. The magic number is
%    $0.55228474983=4/3 (\sqrt{2}-1)$.
%
%    \begin{macrocode}
\newlength{\magicnum}
\newcommand\set@magic{%
 \setlength{\magicnum}{0.55228474983\unitlength}}
%%
%    \end{macrocode}
%
%    Another special one is \verb+\eps+. It could be initialized by \verb+\eps\@ne+
%    but due to its context to the calculation part 1sp=1/65536pt is used.
%
%    \begin{macrocode}
\newlength{\eps}
\setlength{\eps}{1sp}
%%
%    \end{macrocode}
%
%    The last one is \verb+\PathLength+. It stores lengths which the user may need for 
%    further use.
%
%    \begin{macrocode}
\newlength{\PathLength}
%%
%    \end{macrocode}
%
%    This two constants are needed in calculations, but I did not want to waste
%    any additional counter. \verb+\pathmax+ may be redefined to exceed 256, so it
%    is not defined by \verb+\chardef+.
%
%    \begin{macrocode}
\chardef\x@=10
\newcommand{\pathmax}{50}
%%
%    \end{macrocode}
%
%    This fundamental box will keep the plotting symbol.
%
%    \begin{macrocode}
\newsavebox{\@pt}
%%
%    \end{macrocode}
%
%    I have to distinguish three cases: standard plot symbol, old standard plot symbol,
%    or any new one. For this purpose I need two logicals.
%
%    \begin{macrocode}
\newif\if@other@symbol
\newif\if@standard@symbol
%    \end{macrocode}
%
%    All plot symbols may be defined by the most general one,\\
%    \verb+\DefShiftedPlotSymbol+, but this way may be faster. The 
%    other important macro is \verb+\set@width+ which redefines the plot
%    box due to changes which may have occurred (line thickness).
%
%    \begin{macrocode}
\newcommand{\DefStandardPlotSymbol}{%
 \@other@symbolfalse\@standard@symboltrue
 \setbox\@pt\hbox{\hskip -.5\wd0\vrule height\@halfwidth  
 depth\@halfwidth width\@wholewidth}}
\newcommand{\DefOldPlotSymbol}{%
 \@other@symbolfalse\@standard@symbolfalse
 \setbox\@pt\hbox{\vrule height\@halfwidth  
 depth\@halfwidth width\@wholewidth}}
\newcommand{\DefPlotSymbol}[1]{\setbox0=\hbox{#1}\@X\ht0\advance\@X-\dp0
 \@halfwidth.5\ht0\@wholewidth\ht0
 \@other@symboltrue\@standard@symbolfalse
 \setbox\@pt\hbox{\hskip -.5\wd0\lower.5\@X\copy0}}
\newcommand{\DefShiftedPlotSymbol}[4]{\setbox0=\hbox{#1}\@X #2\@Y #3
 \@wholewidth #4\@halfwidth.5\@wholewidth
 \@other@symboltrue\@standard@symbolfalse
 \setbox\@pt\hbox{\hskip-\@X\lower\@Y\copy0}}
\newcommand{\set@width}{%
 \if@other@symbol 
  \relax
 \else 
  \if@standard@symbol 
    \@X-.5\@wholewidth
  \else
    \@X\z@
  \fi
  \setbox\@pt\hbox{\hskip\@X\vrule height\@halfwidth  
  depth\@halfwidth width\@wholewidth}%
 \fi}
%%
%    \end{macrocode}
%
%    The initialization is done here. Note that \verb+\thinlines+
%    is already default and needs not be specified here.
%
%    \begin{macrocode}
\DefStandardPlotSymbol
%%
%    \end{macrocode}
%
%    All plot macros have an optional number. Therefore an additional internal macro
%    is needed (it will have the same name with an extra  @ in front of it.
%
%    Here is the simpliest one, the linear case.
%
%    \begin{macrocode}
\def\lbezier{\@ifnextchar [{\@lbezier}{\@lbezier[0]}}
\def\@lbezier[#1](#2,#3)(#4,#5){%
 \c@@cntc#1\relax
 \ifnum \c@@cntc<\@ne
%    \end{macrocode}
%
%    I decrease the precision locally to speed up calculations. We need just
%    an estimate.
%
%    \begin{macrocode}
  \multiply\eps\x@
  \Length(#2,#3)(#4,#5){\PathLength}%
  \divide\eps\x@
  \c@@cntc\PathLength
  \@X.5\@halfwidth \divide\c@@cntc\@X 
  \ifnum \c@@cntc>\qbeziermax%
    \PackageInfo{ebezier}{\the\c@@cntc\space points needed exceeding %
                          qbeziermax=\qbeziermax!}\fi 
 \fi 
 \ifnum \c@@cntc>\qbeziermax
 \c@@cntc\qbeziermax\relax
   \PackageWarning{ebezier}{Counter reset to qbeziermax=\qbeziermax!}\fi 
 \c@@cnta\c@@cntc\relax\advance\c@@cnta\@ne
 \@Xa #4\unitlength \advance\@Xa-#2\unitlength \divide\@Xa\c@@cntc
 \@Ya #5\unitlength \advance\@Ya-#3\unitlength \divide\@Ya\c@@cntc
 \c@@cntb\z@\relax
 \set@width
 \put(#2,#3){\@whilenum{\c@@cntb<\c@@cnta}\do
    {\@X\c@@cntb \@Xa\@Y \c@@cntb\@Ya 
    \raise\@Y\hbox to\z@{\hskip\@X\unhcopy\@pt\hss}%
    \advance\c@@cntb\@ne}}}
%%
%    \end{macrocode}
%
%    \verb+\Lbezier+ changes the line thickness. It is stored in \verb+\@Xb+.
%
%    \begin{macrocode}
\def\Lbezier{\@ifnextchar [{\@Lbezier}{\@Lbezier[0]}}
\def\@Lbezier[#1](#2,#3)(#4,#5){\c@@cntc#1\relax
 \@Xb\@wholewidth
 \@X #4\unitlength \advance\@X-#2\unitlength \AbsLen{\@X}%
 \@Y #5\unitlength \advance\@Y-#3\unitlength \AbsLen{\@Y}%
 \LenNorm{\@X}{\@Y}{\@Xc}\LenMult{\@Xc}{\@wholewidth}{\@Yb}%
 \LenDiv{\@Yb}{\@X+\@Y}{\@wholewidth}\@halfwidth .5\@wholewidth
 \ifnum \c@@cntc<\@ne
  \multiply\eps\x@
  \Length(#2,#3)(#4,#5){\PathLength}%
  \divide\eps\x@
  \c@@cntc\PathLength
  \@X.5\@halfwidth \divide\c@@cntc\@X 
  \ifnum \c@@cntc>\qbeziermax%
    \PackageInfo{ebezier}{\the\c@@cntc\space points needed exceeding %
                          qbeziermax=\qbeziermax!}\fi 
 \fi 
 \ifnum \c@@cntc>\qbeziermax
 \c@@cntc\qbeziermax\relax
    \PackageWarning{ebezier}{Counter reset to qbeziermax=\qbeziermax!}\fi 
 \c@@cnta\c@@cntc\relax \advance\c@@cnta\@ne
 \@Xa #4\unitlength \advance\@Xa-#2\unitlength \divide\@Xa\c@@cntc
 \@Ya #5\unitlength \advance\@Ya-#3\unitlength \divide\@Ya\c@@cntc
 \c@@cntb\z@\relax
 \set@width
 \put(#2,#3){\@whilenum{\c@@cntb<\c@@cnta}\do
    {\@X\c@@cntb \@Xa\@Y \c@@cntb\@Ya 
    \raise\@Y\hbox to\z@{\hskip\@X\unhcopy\@pt\hss}%
    \advance\c@@cntb\@ne}}
 \@wholewidth\@Xb \@halfwidth .5\@Xb}
%%
%    \end{macrocode}
%
%    The two joining macros need two internal steps to process an implicit list.
%
%    \begin{macrocode}
\def\ljoin{\@ifnextchar [{\@ljoin}{\@ljoin[0]}}
\def\@ljoin[#1](#2,#3){\@ifnextchar ({\l@join[#1](#2,#3)}{\relax}}
\def\l@join[#1](#2,#3)(#4,#5){%
 \lbezier[#1](#2,#3)(#4,#5)%
 \ljoin[#1](#4,#5)}
%%
\def\Ljoin{\@ifnextchar [{\@Ljoin}{\@Ljoin[0]}}
\def\@Ljoin[#1](#2,#3){\@ifnextchar ({\L@join[#1](#2,#3)}{\relax}}
\def\L@join[#1](#2,#3)(#4,#5){%
 \Lbezier[#1](#2,#3)(#4,#5)%
 \Ljoin[#1](#4,#5)}
%%
%    \end{macrocode}
%
%    \verb+\Qbezier+ is defined, because \verb+\qbezier+ uses an other plot box.
%    The original macro is a little bit more complicated to handle extra spaces
%    but I hope this will suffice.
%
%    \begin{macrocode}
\def\Qbezier{\@ifnextchar [{\@Qbezier}{\@Qbezier[0]}}
\def\@Qbezier[#1](#2,#3)(#4,#5)(#6,#7){\c@@cntc#1\relax
 \ifnum \c@@cntc<\@ne 
  \multiply\eps\x@
  \PathLengthQ[5](#2,#3)(#4,#5)(#6,#7)%
  \divide\eps\x@
  \c@@cntc\PathLength
  \@X.5\@halfwidth \divide\c@@cntc\@X 
  \ifnum \c@@cntc>\qbeziermax%
    \PackageInfo{ebezier}{\the\c@@cntc\space points needed exceeding %
                          qbeziermax=\qbeziermax!}\fi 
 \fi 
 \ifnum \c@@cntc>\qbeziermax
 \c@@cntc\qbeziermax\relax
    \PackageWarning{ebezier}{Counter reset to qbeziermax=\qbeziermax!}\fi 
 \c@@cnta\c@@cntc\relax \advance\c@@cnta\@ne
 \@Xa #4\unitlength \advance\@Xa-#2\unitlength \multiply\@Xa\tw@
 \@Xb #6\unitlength \advance\@Xb-#2\unitlength 
 \advance\@Xb-\@Xa \divide\@Xb\c@@cntc
 \@Ya #5\unitlength \advance\@Ya-#3\unitlength \multiply\@Ya\tw@
 \@Yb #7\unitlength \advance\@Yb-#3\unitlength 
 \advance\@Yb-\@Ya \divide\@Yb\c@@cntc
 \c@@cntb\z@\relax
 \set@width
 \put(#2,#3){\@whilenum{\c@@cntb<\c@@cnta}\do
     {\@X\c@@cntb \@Xb\@Y \c@@cntb\@Yb
      \advance\@X\@Xa \advance\@Y\@Ya
      \divide\@X\c@@cntc \divide\@Y\c@@cntc
      \multiply\@X\c@@cntb \multiply\@Y\c@@cntb 
      \raise \@Y \hb@xt@\z@{\kern\@X\unhcopy\@pt\hss}%
      \advance\c@@cntb\@ne}}}
%%
%    \end{macrocode}
%
%    \verb+\cbezier+ is the most complex command. All calculations have to be
%    done in the correct order to minimize overflow conditions.
%
%    \begin{macrocode}
\def\cbezier{\@ifnextchar [{\@cbezier}{\@cbezier[0]}}
\def\@cbezier[#1](#2,#3)(#4,#5)(#6,#7)(#8,#9){%
 \c@@cntc#1\relax
 \ifnum \c@@cntc<\@ne
  \multiply\eps\x@
  \PathLengthC[5](#2,#3)(#4,#5)(#6,#7)(#8,#9)%
  \divide\eps\x@
  \c@@cntc\PathLength
  \@X = 0.5\@halfwidth
  \divide\c@@cntc\@X 
  \ifnum \c@@cntc>\qbeziermax%
    \PackageInfo{ebezier}{\the\c@@cntc\space  points needed exceeding %
                          qbeziermax=\qbeziermax!}\fi 
 \fi 
 \ifnum \c@@cntc>\qbeziermax
 \c@@cntc\qbeziermax\relax
   \PackageWarning{ebezier}{Counter reset to qbeziermax=\qbeziermax!}\fi 
  \c@@cnta\c@@cntc\relax \advance\c@@cnta\@ne
  \@Xa #4\unitlength \advance\@Xa-#2\unitlength \multiply\@Xa\thr@@
  \@Xb #6\unitlength \advance\@Xb-#2\unitlength \multiply\@Xb\thr@@
  \advance\@Xb -2\@Xa 
  \@Xc #8\unitlength \advance\@Xc-#2\unitlength
  \advance\@Xc-\@Xa \advance\@Xc-\@Xb 
  \@Ya #5\unitlength \advance\@Ya-#3\unitlength \multiply\@Ya\thr@@
  \@Yb #7\unitlength \advance\@Yb-#3\unitlength \multiply\@Yb\thr@@
  \advance\@Yb-2\@Ya 
  \@Yc #9\unitlength \advance\@Yc-#3\unitlength
  \advance\@Yc-\@Ya \advance\@Yc-\@Yb 
  \divide\@Xc\c@@cntc \divide\@Yc\c@@cntc
  \c@@cntb\z@\relax
 \set@width
  \put(#2,#3){\@whilenum{\c@@cntb<\c@@cnta}\do
      {\@X\c@@cntb \@Xc\@Y \c@@cntb\@Yc
      \advance\@X\@Xb \advance\@Y\@Yb 
      \divide\@X\c@@cntc \divide\@Y\c@@cntc
      \multiply\@X\c@@cntb \multiply\@Y\c@@cntb 
      \advance\@X\@Xa  \advance\@Y\@Ya
      \divide\@X\c@@cntc \divide\@Y\c@@cntc
      \multiply\@X\c@@cntb \multiply\@Y\c@@cntb
      \raise \@Y \hbox to \z@{\hskip \@X\unhcopy\@pt\hss}%
      \advance\c@@cntb\@ne}}}
%%
%    \end{macrocode}
%
%    \verb+\Cbezier+ changes the plot symbol so a restore is needed. But it will
%     not keep the original one!
%
%    \begin{macrocode}
\def\Cbezier{\@ifnextchar [{\@Cbezier}{\@Cbezier[0]}}
\def\@Cbezier[#1](#2,#3)(#4,#5)(#6,#7)(#8,#9){%
 \cbezier[#1](#2,#3)(#4,#5)(#6,#7)(#8,#9)%
 \c@@cntc#1\relax\divide\c@@cntc\thr@@
 \lbezier[\c@@cntc](#2,#3)(#4,#5)%
 \lbezier[\c@@cntc](#4,#5)(#6,#7)%
 \lbezier[\c@@cntc](#6,#7)(#8,#9)%
 \DefPlotSymbol{$\bullet$}
 \lbezier[1](#2,#3)(#2,#3)
 \lbezier[1](#4,#5)(#4,#5)
 \lbezier[1](#6,#7)(#6,#7)
 \lbezier[1](#8,#9)(#8,#9)
 \DefStandardPlotSymbol
 \thinlines}
%%
%    \end{macrocode}
%
%    \verb+\l@put+ is like \verb+\put+ but its arguments are lengths and not
%    decimal constants. It will be used in \verb+\l@cbezier+ which also has
%    lengths as arguments. All complex plotting commands use this form.
%    Just for the calculation of plotting points four more lengths are needed. 
%    I use the ``scratch'' dimens from \TeX.
%
%    \begin{macrocode}
\long\gdef\l@put(#1,#2)#3{%
 \@killglue\raise#2\hb@xt@\z@{\kern#1#3\hss}\ignorespaces}
%%
\long\gdef\l@cbezier[#1](#2,#3)(#4,#5)(#6,#7)(#8,#9){%
 \c@@cntc#1\relax
 \dimen1#2\dimen3#3
%%
 \@Xa #4 \advance\@Xa-#2 \multiply\@Xa\thr@@
 \@Xb #6 \advance\@Xb-#2 \multiply\@Xb\thr@@
 \advance\@Xb-2\@Xa 
 \@Xc #8 \advance\@Xc-#2
 \advance\@Xc-\@Xa \advance\@Xc-\@Xb 
 \@Ya #5 \advance\@Ya-#3 \multiply\@Ya\thr@@
 \@Yb #7 \advance\@Yb-#3 \multiply\@Yb\thr@@
 \advance\@Yb-2\@Ya 
 \@Yc #9 \advance\@Yc-#3
 \advance\@Yc-\@Ya \advance\@Yc-\@Yb 
%%
%% assume half arc
%%
 \ifnum \c@@cntc <\@ne
  \multiply\eps\x@
  \dimen5#2 \advance\dimen5-#8 \AbsLen{\dimen5}%
  \dimen7#3 \advance\dimen7-#9 \AbsLen{\dimen7}%
  \LenNorm{\dimen5}{\dimen7}{\PathLength}%
  \divide\eps\x@
  \c@@cntc\PathLength
  \dimen5.5\@halfwidth
  \divide\c@@cntc\dimen5
%%
%%  11/7 \approx \pi/2
%%
  \divide\c@@cntc 7 \multiply\c@@cntc 11                       
  \ifnum \c@@cntc>\qbeziermax
    \PackageInfo{ebezier}{\the\c@@cntc\space points needed exceeding %
                          qbeziermax=\qbeziermax!}\fi 
 \fi 
 \ifnum\c@@cntc>\qbeziermax
   \c@@cntc\qbeziermax\relax
     \PackageWarning{ebezier}{Counter reset to qbeziermax=\qbeziermax!}\fi 
 \c@@cnta\c@@cntc\relax\advance\c@@cnta\@ne%
 \divide\@Xc\c@@cntc \divide\@Yc\c@@cntc
 \c@@cntb\z@\relax
 \set@width
 \l@put(\dimen1,\dimen3){\@whilenum{\c@@cntb<\c@@cnta}\do
      {\@X\c@@cntb \@Xc\@Y \c@@cntb\@Yc
      \advance\@X\@Xb \advance\@Y\@Yb 
      \divide\@X\c@@cntc \divide\@Y\c@@cntc
      \multiply\@X\c@@cntb \multiply\@Y\c@@cntb 
      \advance\@X\@Xa \advance\@Y\@Ya
      \divide\@X\c@@cntc \divide\@Y\c@@cntc
      \multiply\@X\c@@cntb \multiply\@Y\c@@cntb
      \raise\@Y\hbox to\z@{\hskip\@X\unhcopy\@pt\hss}%
      \advance\c@@cntb\@ne}}}
%%
%    \end{macrocode}
%
%    The building blocks for the circles are the four quarters. Each is defined 
%    separately and will be combined by the \verb+\cCircle+ macro.
%
%    \begin{macrocode}
\def\@circle@rt[#1](#2,#3)#4{%
 \set@magic
 \@Z #4\magicnum\@Za #2\unitlength\@Zb #3\unitlength
 \@Zc #2\unitlength \advance\@Zc\@Z
 \@Zd #3\unitlength \advance\@Zd\@Z
 \@Ze #4\unitlength \advance\@Ze\@Za
 \@Zf #4\unitlength \advance\@Zf\@Zb
 \l@cbezier[#1](\@Ze,\@Zb)(\@Ze,\@Zd)(\@Zc,\@Zf)(\@Za,\@Zf)}
%%
\def\@circle@lt[#1](#2,#3)#4{%
 \set@magic
 \@Z #4\magicnum\@Za #2\unitlength\@Zb #3\unitlength
 \@Zc #2\unitlength \advance\@Zc-\@Z
 \@Zd #3\unitlength \advance\@Zd\@Z
 \@Ze -#4\unitlength \advance\@Ze\@Za
 \@Zf #4\unitlength \advance\@Zf\@Zb
 \l@cbezier[#1](\@Za,\@Zf)(\@Zc,\@Zf)(\@Ze,\@Zd)(\@Ze,\@Zb)}
%%
\def\@circle@rb[#1](#2,#3)#4{%
 \set@magic
 \@Z #4\magicnum\@Za #2\unitlength\@Zb #3\unitlength
 \@Zc #2\unitlength \advance\@Zc\@Z
 \@Zd #3\unitlength \advance\@Zd-\@Z
 \@Ze #4\unitlength \advance\@Ze\@Za
 \@Zf -#4\unitlength \advance\@Zf\@Zb
 \l@cbezier[#1](\@Za,\@Zf)(\@Zc,\@Zf)(\@Ze,\@Zd)(\@Ze,\@Zb)}
%%
\def\@circle@lb[#1](#2,#3)#4{%
 \set@magic
 \@Z #4\magicnum\@Za #2\unitlength\@Zb #3\unitlength
 \@Zc #2\unitlength \advance\@Zc-\@Z
 \@Zd #3\unitlength \advance\@Zd-\@Z
 \@Ze -#4\unitlength \advance\@Ze\@Za
 \@Zf -#4\unitlength \advance\@Zf\@Zb
 \l@cbezier[#1](\@Ze,\@Zb)(\@Ze,\@Zd)(\@Zc,\@Zf)(\@Za,\@Zf)}
%%
%    \end{macrocode}
%
%    I use the logicals from the \verb+\oval+ defined in \LaTeX. So I need just
%    one more logical \verb+\if@ovf+.
%
%    \begin{macrocode}
\newif\if@ovf
\def\cCircle{\@ifnextchar [{\@cCircle}{\@cCircle[0]}}
\def\@cCircle[#1](#2,#3)#4[#5]{%
 \@ovtfalse\@ovbfalse\@ovlfalse\@ovrfalse\@ovffalse
 \c@@cnta#1\relax
 \@tfor\reserved@a:=#5\do{\csname @ov\reserved@a true\endcsname}%
 \if@ovf\@ovttrue \divide\c@@cnta\tw@\fi
 \if@ovt 
  \if@ovr
     \@circle@rt[\c@@cnta](#2,#3){#4}
  \else\if@ovl
     \@circle@lt[\c@@cnta](#2,#3){#4}
  \else\divide\c@@cnta\tw@
     \@circle@rt[\c@@cnta](#2,#3){#4}
     \@circle@lt[\c@@cnta](#2,#3){#4}
  \fi\fi
  \if@ovf 
     \@circle@rb[\c@@cnta](#2,#3){#4}
     \@circle@lb[\c@@cnta](#2,#3){#4}
  \fi
  \else\if@ovb 
  \if@ovr
     \@circle@rb[\c@@cnta](#2,#3){#4}
  \else\if@ovl
     \@circle@lb[\c@@cnta](#2,#3){#4}
  \else\divide\c@@cnta\tw@
     \@circle@rb[\c@@cnta](#2,#3){#4}
     \@circle@lb[\c@@cnta](#2,#3){#4}
  \fi\fi
 \else
   \divide\c@@cnta\tw@
   \if@ovr 
     \@circle@rb[\c@@cnta](#2,#3){#4}
     \@circle@rt[\c@@cnta](#2,#3){#4}
  \else
  \if@ovl 
     \@circle@lb[\c@@cnta](#2,#3){#4}
     \@circle@lt[\c@@cnta](#2,#3){#4}
  \else
   \PackageError{Ebezier}{Missing or illegal position specifier in cCircle}
  \fi\fi\fi\fi}
%%
\def\cArc{\@ifnextchar [{\@cArc}{\@cArc[0]}}
\def\@cArc[#1](#2,#3)(#4,#5){%
 \c@@cntc#1\relax
 \@X #2\unitlength \@Y #3\unitlength
 \@Za #4\unitlength \@Zb #5\unitlength
 \@Zc 2\@X \advance\@Zc-\@Za \@Zd 2\@Y \advance\@Zd-\@Zb
 \@Xa\@Y \advance\@Xa-\@Zb \@Ya\@Za \advance\@Ya-\@X
 \multiply\@Xa 4 \divide\@Xa\thr@@ \multiply\@Ya 4 \divide\@Ya\thr@@
 \@Ze\@Za \advance\@Ze\@Xa \@Zf\@Zb \advance\@Zf\@Ya 
 \@Zg\@Zc \advance\@Zg\@Xa \@Zh\@Zd \advance\@Zh\@Ya
 \l@cbezier[#1](\@Za,\@Zb)(\@Ze,\@Zf)(\@Zg,\@Zh)(\@Zc,\@Zd)}
%%
%    \end{macrocode}
%
%    Historically from this point starts the calculation part. The notation 
%    will be more \LaTeX\ convenient.
%
%    All square roots are calculated by the same iteration. To keep numbers
%    small enough some scaling has to be done (factor \verb+\c@@cntd+).
%
%    \begin{macrocode}
\def\LenMult#1#2#3{\setlength{#3}{#1*\ratio{#2}{\unitlength}}}
%%
\def\LenDiv#1#2#3{\setlength{#3}{\unitlength*\ratio{#1}{#2}}}
%%
\def\AbsLen#1{\ifdim#1<\z@\setlength{#1}{-#1}\fi}
%%
\def\LenSqrt#1#2{%
 \setlength{\@Za}{#1}%
 \ifdim\@Za>\eps\loop\setlength{\@Zb}{(\@Za+\unitlength*\ratio{#1}{\@Za})/2}%
 \setlength{\@Zc}{\@Za-\@Zb}\AbsLen{\@Zc}% 
 \ifdim\@Zc>\eps\setlength{\@Za}{\@Zb}\repeat\fi%
 \setlength{#2}{\@Za}}
%%
\def\Length(#1,#2)(#3,#4)#5{%
 \setlength{\@Zd}{#3\unitlength-#1\unitlength}%
 \setlength{\@Ze}{#4\unitlength-#2\unitlength}%
 \setcounter{@cntd}{1}%
 \setlength{\@Zf}{\@Zd}\ifdim\@Ze>\@Zd\setlength{\@Zf}{\@Ze}\fi
 \loop\setlength{\@Zd}{\@Zd/2}\setlength{\@Ze}{\@Ze/2}\setlength{\@Zf}{\@Zf/2}%
  \multiply\c@@cntd\tw@\ifdim\@Zf>\x@ pt\repeat
 \LenMult{\@Zd}{\@Zd}{\@Zg}\LenMult{\@Ze}{\@Ze}{\@Zh}\setlength{\@Zf}{\@Zg+\@Zh}%
 \LenSqrt{\@Zf}{\@Zg}\setlength{#5}{\@Zg*\value{@cntd}}}
%%
\def\LenNorm#1#2#3{%
 \setlength{\@Zd}{#1}\setlength{\@Ze}{#2}\setcounter{@cntd}{1}%
 \setlength{\@Zf}{\@Zd}\ifdim\@Ze>\@Zd\setlength{\@Zf}{\@Ze}\fi
 \loop\setlength{\@Zd}{\@Zd/2}\setlength{\@Ze}{\@Ze/2}\setlength{\@Zf}{\@Zf/2}%
  \multiply\c@@cntd\tw@\ifdim\@Zf>\x@ pt\repeat
 \LenMult{\@Zd}{\@Zd}{\@Zg}\LenMult{\@Ze}{\@Ze}{\@Zh}\setlength{\@Zf}{\@Zg+\@Zh}%
 \LenSqrt{\@Zf}{\@Zg}\setlength{#3}{\@Zg*\value{@cntd}}}
%%
\def\PathLengthQ[#1](#2,#3)(#4,#5)(#6,#7){%
 \PathLength\z@\c@@cntc#1\relax
 \ifnum \c@@cntc<\@ne \c@@cntc\pathmax\relax\fi
 \ifnum \c@@cntc>\pathmax \c@@cntc\pathmax\relax
    \PackageWarning{ebezier}{Counter reset to pathmax=\pathmax!}\fi
 \@Za\z@ \@Zb\z@ \c@@cntb\c@@cntc\relax \advance\c@@cntb\@ne
 \@Xb #4\unitlength \advance\@Xb-#2\unitlength \multiply\@Xb\tw@
 \@Yb #5\unitlength \advance\@Yb-#3\unitlength \multiply\@Yb\tw@
 \@Xa #6\unitlength \advance\@Xa-#2\unitlength
 \advance\@Xa-\@Xb \divide\@Xa\c@@cntc
 \@Ya #7\unitlength \advance\@Ya-#3\unitlength
 \advance\@Ya-\@Yb \divide\@Ya\c@@cntc \c@@cnta\@ne\relax
 \@whilenum{\c@@cnta<\c@@cntb}\do
  {\@X\c@@cnta\@Xa \advance\@X\@Xb \divide\@X\c@@cntc \multiply\@X\c@@cnta
  \@Y\c@@cnta\@Ya \advance\@Y\@Yb \divide\@Y\c@@cntc \multiply\@Y\c@@cnta
  \@Zi\@X\@Zj\@Y
  \advance\@X-\@Za \advance\@Y-\@Zb \LenNorm{\@X}{\@Y}{\@Z}%
  \advance\PathLength\@Z
  \@Za\@Zi\@Zb\@Zj \advance\c@@cnta\@ne}}
%%
\def\PathLengthC[#1](#2,#3)(#4,#5)(#6,#7)(#8,#9){%
  \PathLength\z@ \c@@cntc#1\relax
 \ifnum \c@@cntc<\@ne \c@@cntc\pathmax\relax\fi 
 \ifnum \c@@cntc>\pathmax \c@@cntc\pathmax\relax
    \PackageWarning{ebezier}{Counter reset to pathmax=\pathmax!}\fi
 \@Za\z@ \@Zb\z@ \c@@cnta\c@@cntc\relax \advance\c@@cnta\@ne
 \@Xa #4\unitlength \advance\@Xa-#2\unitlength \multiply\@Xa\thr@@
 \@Xb #6\unitlength \advance\@Xb-#2\unitlength \multiply\@Xb\thr@@
 \advance\@Xb-2\@Xa 
 \@Xc #8\unitlength \advance\@Xc-#2\unitlength
 \advance\@Xc-\@Xa \advance\@Xc-\@Xb 
 \@Ya #5\unitlength \advance\@Ya-#3\unitlength \multiply\@Ya\thr@@
 \@Yb #7\unitlength \advance\@Yb-#3\unitlength \multiply\@Yb\thr@@
 \advance\@Yb-2\@Ya 
 \@Yc #9\unitlength \advance\@Yc-#3\unitlength
 \advance\@Yc-\@Ya \advance\@Yc-\@Yb 
 \divide\@Xc\c@@cntc \divide\@Yc\c@@cntc
 \c@@cntb\@ne\relax
 \@whilenum{\c@@cntb<\c@@cnta}\do
  {\@X\c@@cntb\@Xc \@Y\c@@cntb\@Yc \advance\@X\@Xb \advance\@Y\@Yb 
  \divide\@X\c@@cntc \divide\@Y\c@@cntc
  \multiply\@X\c@@cntb \multiply\@Y\c@@cntb 
  \advance\@X\@Xa \advance\@Y\@Ya
  \divide\@X\c@@cntc \divide\@Y\c@@cntc
  \multiply\@X\c@@cntb \multiply\@Y\c@@cntb
  \@Zi\@X\@Zj\@Y
  \advance\@X-\@Za \advance\@Y-\@Zb \LenNorm{\@X}{\@Y}{\@Z}%
  \advance\PathLength\@Z
  \@Za\@Zi\@Zb\@Zj\advance\c@@cntb\@ne}}
%% 
%    \end{macrocode}
%
%    The most complex macro is explained in the text. The exception is
%    handled by the logical \verb+\if@ovf+.
%
%    \begin{macrocode}
\def\cArcs{\@ifnextchar [{\@cArcs}{\@cArcs[0]}}
\def\@cArcs[#1](#2,#3)(#4,#5)(#6,#7){%
 \c@@cntc#1\relax
 \@ovffalse
 \@X#2\unitlength\@Y#3\unitlength
 \@Zi#6\unitlength\@Zj#7\unitlength
 \setlength{\@Xa}{\@X-\@Zi}\setlength{\@Ya}{\@Y-\@Zj}%
 \LenNorm{\@Xa}{\@Ya}{\@Xb}%
 \@Xa#4\unitlength \advance\@Xa\@Zi \advance\@Xa-2\@X
 \@Ya#5\unitlength \advance\@Ya\@Zj \advance\@Ya-2\@Y
 \@Xc\@Xa\AbsLen{\@Xc}\@Yc\@Ya\AbsLen{\@Yc}%
 \ifdim\@Xc<\eps\ifdim\@Yc<\eps\@ovftrue\fi\fi
 \if@ovf
  \cArc[#1](#2,#3)(#4,#5)%
 \else
  \LenNorm{\@Xa}{\@Ya}{\@Yb}%
  \setlength{\@Xc}{\unitlength*\ratio{\@Xb}{\@Yb}}%
  \setlength{\@Yc}{(-\unitlength+\@Xc*2)*4/3}%
  \@Xb-#5\unitlength \advance\@Xb\@Zj
  \@Z\@Xb\AbsLen{\@Z}%
  \ifdim\@Z<100\eps \@Xb#4\unitlength \advance\@Xb-\@Zi \@Xa\@Ya\fi
  \setlength{\@Z}{\@Yc*\ratio{\@Xa}{\@Xb}}%
  \@Xa#4\unitlength\@Ya#5\unitlength
  \setlength{\@Za}{\@Y-\@Ya}\setlength{\@Zb}{\@Xa-\@X}%
  \setlength{\@Zc}{\@Zj-\@Y}\setlength{\@Zd}{\@X-\@Zi}%
  \@Xb\@Xa \LenMult{\@Z}{\@Za}{\@Zh}\advance\@Xb\@Zh 
  \@Yb\@Ya\LenMult{\@Z}{\@Zb}{\@Zh}\advance\@Yb\@Zh 
  \@Xc\@Zi\LenMult{\@Z}{\@Zc}{\@Zh}\advance\@Xc\@Zh 
  \@Yc\@Zj\LenMult{\@Z}{\@Zd}{\@Zh}\advance\@Yc\@Zh 
  \@Z\@Xa\@Za\@Ya\@Zb\@Xb\@Zc\@Yb\@Zd\@Xc\@Ze\@Yc
  \l@cbezier[#1](\@Z,\@Za)(\@Zb,\@Zc)(\@Zd,\@Ze)(\@Zi,\@Zj)%
 \fi}
%</package>
%    \end{macrocode}
% \Finale \PrintIndex \PrintChanges
\endinput