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\preamble
  ______________________________________________________
  The curve2e package for LaTeX and XeLATeX
  Copyright (C) 2010 Claudio Beccari 
  All rights reserved

  License information appended

\endpreamble
\postamble

Copyright 2005-2015 Claudio Beccari

Distributable under the LaTeX Project Public License,
version 1.3c or higher (your choice). The latest version of
this license is at: http://www.latex-project.org/lppl.txt

This work is "author-maintained"

This work consists of this file curve2e.dtx, a README file
and the derived files curve2e.sty and curve2e.pdf.

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%
%%
%% File `curve2e.dtx'.
%% Copyright (C)  2005--2015 Claudio Beccari  all rights reserved.
%%
% What follows is the usual trick that is not typeset in the documentation
% dvi file that is produced by LaTeX. It is used to define the date, the version
% and the short description that characterizes both this file and the package;
% the point is that |\ProvidesFile| is being read only by the driver, while
% |\ProvidePackage| goes to the stripped package file; it must be done before
% starting the documentation otherwise |\GetFileInfo|  can't get the necessary
% information.
% \fi
%
% \iffalse
%<*package>
%<package>\NeedsTeXFormat{LaTeX2e}[2014/05/01]
%</package>
%<*driver>
\ProvidesFile{curve2e.dtx}%
%</driver>
%<+package>\ProvidesPackage{curve2e}%
%<*package>
        [2015/06/06 v.1.42 Extension package for pict2e]
%</package>
%<*driver>
\documentclass{ltxdoc}\errorcontextlines=9
\hfuzz 10pt
\usepackage{multicol,amsmath}
\usepackage[utf8]{inputenc}
\usepackage{lmodern,textcomp}
\usepackage{mflogo}
\usepackage{curve2e}
\GetFileInfo{curve2e.dtx}
\title{The extension package \textsf{curve2e}}
\author{Claudio Beccari}
\date{Version number \fileversion; last revised \filedate.}
\providecommand*\diff{\mathop{}\!\mathrm{d}}
\begin{document}
 \maketitle
\begin{multicols}{2}
 \tableofcontents
 \end{multicols}
 \DocInput{curve2e.dtx}
\end{document}
%</driver>
% \fi
%
% \CheckSum{2484}
% \begin{abstract}
% This file documents the |curve2e| extension package to the recent
% implementation of the |pict2e| bundle that has been described by Lamport
% himself in the second edition of his \LaTeX\ handbook.
%
% Please take notice that in April 2011 a new updated version of the package 
% |pict2e| has been released that incorporates some of the commands defined in
% this package; apparently there are no conflicts, but only the advanced features
% of |curve2e| remain available for extending the above package. Moreover
% the |xetex.def| driver was introduced so that certain commands previously 
% defined in this extension not only become unnecessary, but also would produce
% errors when the program is used under XeLaTeX. Therefore these commands were
% either eliminated or corrected.
%
% This extension redefines a couple of commands and introduces some more drawing
% facilities that allow to draw circular arcs and arbitrary curves with the
% minimum of user intervention. This beta version is open to the contribution of
% other users as well as it may be incorporated in other people's packages.
% Please cite the original author and the chain of contributors.
% \end{abstract}
%
% \section{Package \texttt{pict2e} and this extension \texttt{curve2e}}
% Package \texttt{pict2e} was announced in issue 15 of \texttt{latexnews}
% around December 2003; it was declared that the new package would replace the
% dummy one that has been accompanying every release of \LaTeXe\ since its
% beginnings in 1994. The dummy package was just issuing an info message that
% simply announced the temporary unavailability of the real package.
%
% Eventually Gäßlein and Niepraschk implemented what Lamport himself had already
% documented in the second edition of his \LaTeX\ handbook, that is a \LaTeX\
% package that contained the macros capable of removing all the limitations
% contained in the standard commands of the original \texttt{picture}
% environment; specifically:
% \begin{enumerate}
% \item the line and vector slopes were limited to the ratios of relatively
% prime one-digit integers of magnitude not exceeding 6 for lines and 4 for
% vectors;
% \item filled and unfilled full circles were limited by the necessarily
% limited number of specific glyphs contained in the special \LaTeX\
% \texttt{picture} fonts;
% \item quarter circles were also limited in their radii for the same reason;
% \item ovals (rectangles with rounded corners) could not be too small because
% of the unavailability of small radius quarter circles, nor could be too
% large, in the sense that after a certain radius the rounded corners remained
% the same and would not increase proportionally to the oval size.
% \item vector arrows had only one possible shape and matched the limited
% number of vector slopes;
% \item for circles and inclined lines and vectors just two possible thicknesses
%  were available.
% \end{enumerate}
%
% The package \texttt{pict2e} removes most if not all the above limitations:
% \begin{enumerate}
% \item line and vector slopes are virtually unlimited; the only remaining
% limitation is that the direction coefficients must be three-digit integer
% numbers; they need not be relatively prime; with the 2009 upgrade even this
% limitation was removed and now slope coefficients can be any fractional number
% whose magnitude does not exceed 16\,384, the maximum dimension in points that
% \TeX\ can handle;
% \item filled and unfilled circles can be of any size;
% \item ovals can be designed with  any specified corner curvature and there is
% virtually no limitation to such curvatures; of course corner radii should not
% exceed half the lower value between the base and the height of the oval;
% \item there are two shapes for the arrow tips; the triangular one traditional
% with \LaTeX\ vectors, or the arrow tip with PostScript style.
% \item the |\linethickness| command changes the thickness of all lines, straight,
% curved, vertical, horizontal, arrow tipped, et cetera.
% \end{enumerate}
%
%  This specific extension adds the following features
% \begin{enumerate}
% \item commands for setting the line terminations are introduced; the user can
% chose between square or rounded caps; the default is set to rounded caps (now
% available also with |pict2e|); 
% \item commands for specifying the way two lines or curves join to one nanother;
% ^^A
% \item the |\line| macro is redefined so as to allow integer and fractional
% direction coefficients, but maintaining the same syntax as in the original
% \texttt{picture} environment (now available also with |pict2e|);
% ^^A
% \item a new macro |\Line| was defined so as to avoid the need to specify the
% horizontal projection of inclined lines (now available also with |pict2e|);
% this conflicts with |pict2e| 2009 version; therefore its name is changed to
% |\LIne| and supposedly it will not be used very often, if ever used; 
% ^^A
% \item a new macro |\LINE| was defined in order to join two points specified with
% their coordinates; this is now the normal behavior of the |\Line| macro of
% |pict2e| so that |\LINE| is now renamed |\segment|; of course there is no need
% to use the |\put| command with this line specification;
% ^^A
% \item a new macro |\DLine| is defined in order to draw dashed lines joining any
% two given points; the dash length and gap (equal to one another) must be
% specified;
% ^^A
% \item similar macros are redefined for vectors; |\vector| redefines the
% original macro but with the vector slope limitations removed; |\Vector| gets
% specified with its two horizontal and vertical components; |\VECTOR|
% joins two specified points (without using the |\put| command) with  the arrow
% pointing to the second point;
% \item a new macro |\polyline| for drawing polygonal lines is defined that
% accepts from two vertices up to an arbitrary (reasonably limited) number of
% them (available now also in |pict2e|); here if is redefined so as to allow
% an optional specification of the way segments fo the polyline are join to
% one another.;
% \item a new macro |\Arc| is defined in order to draw an arc with arbitrary
% radius and arbitrary angle amplitude; this amplitude is specified in
% sexagesimal degrees, not in radians; the same functionality is now achieved with
% the |\arc| macro of |pict2e|, which provides also the strar version |\arc*| that
% fills up the interior of the generated circular arc. It must be noticed that
% the syntax is slighltly different, so that it's reasonable that both commands,
% in spite  of producing identical arcs, might be more comfortable with this or
% that syntax.
% \item two new macros are defined in order to draw circular arcs with one
% arrow at one or both ends;
% \item a new macro |\Curve| is defined so as to draw arbitrary curved lines
% by means of third order Bézier splines; the |\Curve| macro requires only the
% curve nodes and the direction of the tangents at each node.
% \end{enumerate}
%
% In order to make the necessary calculations many macros have been defined so
% as to use complex number arithmetics to manipulate point coordinates,
% directions (directional versors), rotations and the like. The trigonometric
% functions have also been defined in a way that the author believes to be
% more efficient than that implied by the \texttt{trig} package; in any case
% the macro names are sufficiently different to accommodate both definitions
% in the same \LaTeX\ run.
%
% Many aspects of this extension could be fine tuned for better performance;
% many new commands could be defined in order to further extend this extension.
% If the new service macros are accepted by other \TeX\ and \LaTeX\ programmers,
% this beta version could become the start for a real extension of the
% \texttt{pict2e} package or even become a part of it. Actually some macros
% have already been included in the \texttt{pict2e} package. Actually the
% \verb|\Curve| algorithm might be redefined so as to use the macros introduced
% in the \texttt{hobby} package, that implements for the typesetting engines
% the same functionalities that John Hobby wrote for \MF\ and \MP\ programs. 
%
% For this reason I suppose that every enhancement should be submitted to
% Gäßlein and Niepraschk  who are the prime maintainers of \texttt{pict2e};
% they only can decide whether or not to incorporate new macros in their package.
%
% \section{Summary of modifications and new commands}
% This package \texttt{curve2e} extends the power of \texttt{pict2e} with the
% following modifications and the following new commands.
% \begin{enumerate}
% \item This package |curve2e| calls directly the \LaTeX\ packages |color| and 
% |pict2e| to whom it passes any possible option that the latter can receive;
% actually the only options that make sense are those concerning the arrow tips,
% either \LaTeX\ or PostScript styled, because it is assumed that if you use this
% package you are not interested in using the original \LaTeX\ commands. See the
% |pict2e| documentation in order to use the correct options |pict2e| can receive.
% \item The user is offered new commands in order to control the line terminators
% and the line joins; specifically:
% \begin{itemize}
% \item |\roundcap|: the line is terminated with a semicircle;
% \item |\squarecap|: the line is terminated with a half square;
% \item |\roundjoin|: two lines are joined with a rounded join;
% \item |\beveljoin|: two lines are joined with a bevel join;
% \item |\miterjoin|: two lines are joined with a miter join.
% \end{itemize}
% All the above commands should respect the intended range; but since they act at
% the PostScript or PDF level, not at \TeX\ level, it might be necessary to issue
% the necessary command in order to restore the previous terminator or join.
% \item The commands |\linethickness|, |\thicklines|, |\thinlines| together with
% |\defaultlinethickness| always redefine the internal |\@wholewidth| and
% |\@halfwidth| so that the latter always refer to a full width and to a half of
% it in this way: if you issue the command |\defaultlinewidth{2pt}| all thin
% lines will be drawn with a thickeness of 1\,pt while if a drawing command
% directly refers to the internal value |\@wholewidth|, its line will be drawn
% with a thickness of 2\,pt.
% If one issues the declaration |\thinlines| all lines will be drawn with a  1\,pt
% width, but if a command refers to the internal value |\@halfwidth| the line will
% be drawn with a thickness of 0.5\,pt. The command |\linethickness| redefines the
% above internals but does not change the default width value; all these width
% specifications apply to all lines, straight ones, curved ones, circles, ovals,
% vectors, dashed, et cetera. It's better to recall that |thinlines| and
% |thicklines| are declarations that do not take arguments; on the opposite the
% other two commands follow the standard syntax:
% \begin{flushleft}
% |\linethickness|\marg{dimensioned value}\\
% |\defaultlinewidth|\marg{dimensioned value}
% \end{flushleft}
% where \meta{dimensioned value} means a length specification complete of its
% units or a dimensional expression.
% \item Straight lines and vectors are redefined in such a way that fractional
% slope coefficients may be specified; the zero length line does not produce
% errors and is ignored; the zero length vectors draw only the arrow tips.
% \item New line and vector macros are defined that avoid the necessity of
% specifying the horizontal component |\put(3,4){\LIne(25,15)}| specifies a
% segment that starts at point $(3,4)$ and goes to point $(3+25,4+15)$; the
% command |\segment(3,4)(28,19)| achieves the same result without the need of
% the using command |\put|.
% The same applies to the vector commands |\Vector| and |\VECTOR|. Experience has
% shown that the commands intended to joint two specified coordinates are
% particularly useful.
% \item The |\polyline| command has been introduced: it accepts an unlimited
% list of  point coordinates enclosed within round parentheses; the command
% draws a sequence of connected segments that joins in sequence the specified
% points; the syntax is:
%	\begin{flushleft}
%	\cs{polyline[}\marg{optional join style}\texttt{](}\meta{$P_1$}\texttt{)(}%
%	\meta{$P_2$}\texttt{)...(}\meta{$P_n$}\texttt{)}
%	\end{flushleft}
% See figure~\ref{fig:polyline} where a pentagon is designed..
%
%	\begin{figure}[!ht]
%	\begin{minipage}{.48\linewidth}
%	\begin{verbatim}
%	\unitlength=.5mm
%	\begin{picture}(40,32)(-20,0)
%	\polyline(0,0)(19.0211,13,8197)(11.7557,36.1803)%
%	(-11.7557,36.1803)(-19.0211,13,8197)(0,0)
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%   \hfill
%	\begin{minipage}{.48\linewidth}\raggedleft
%   \unitlength=.5mm
%	\begin{picture}(40,32)(-20,0)
%	\polyline(0,0)(19.0211,13,8197)(11.7557,36.1803)%
%	(-11.7557,36.1803)(-19.0211,13,8197)(0,0)
%	\end{picture}\hspace*{2em}
%	\end{minipage}
%	\caption{Polygonal line obtained by means of the \texttt{\string\polyline}
%   command}	\label{fig:polyline}
%	\end{figure}
%
% Although you can draw polygons with |\polyline|, as it was done in 
% figure~\ref{fig:polyline}, do not confuse this command with the command
% |\polygon| defined in |pict2e| 2009; the latter automatically joins the
% last specified coordinate to the first one with a straight line, therefore
% closing the path. |pict2e| defines also the  starred command that fills up
% the inside of the generated polygon.
% \item The new command
% \begin{flushleft}
% |\Dline(|\textit{first point}|)(|\textit{second point}|)(|\textit{dash length}|)|
% \end{flushleft}
% draws a dashed line containing as many dashes as possible, long as specified,
% and separated by a gap exactly the same size; actually, in order to make an
% even gap-dash sequence, the desired dash length is used to do some computations
% in order to find a suitable length, close to the one specified, such that the
% distance of the end points is evenly divided in equally sized dashes and gaps.
% The end points may be anywhere in the drawing area, without any constraint on
% the slope of the joining segment. The desired dash length is specified as a
% fractional multiple of |\unitlength|; see
% figure~\ref{fig:dashline}.
%	\begin{figure}[!ht]
%	\begin{minipage}{.48\textwidth}
%	\begin{verbatim}
%	\unitlength.5mm
%	\begin{picture}(40,40)
%	\put(0,0){\GraphGrid(40,40)}
%	\Dline(0,0)(40,10){4}
%	\put(0,0){\circle*{2}}
%	\Dline(40,10)(0,25){4}
%	\put(40,10){\circle*{2}}
%	\Dline(0,25)(20,40){4}
%	\put(0,25){\circle*{2}}
%	\put(20,40){\circle*{2}}
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\centering
%	\unitlength.5mm
%	\begin{picture}(40,40)
%	\put(0,0){\GraphGrid(40,40)}
%	\Dline(0,0)(40,10){4}
%	\put(0,0){\circle*{2}}
%	\Dline(40,10)(0,25){4}
%	\put(40,10){\circle*{2}}
%	\Dline(0,25)(20,40){4}
%	\put(0,25){\circle*{2}}
%	\put(20,40){\circle*{2}}
%	\end{picture}
%	\end{minipage}
%	\caption{Dashed lines and graph grid}\label{fig:dashline}
%	\end{figure}
% \item |\GraphGrid| is a command that draws a red grid over the drawing area
% with lines separated |10\unitlength|s; it is described only with a comma
% separated couple of numbers, representing the base and the height of the grid,
% see figure~\ref{fig:dashline}; it's better to specify multiples of ten and
% the grid can be placed anywhere in the drawing plane by means of |\put|,
% whose coordinates are multiples of 10; nevertheless the grid line distance is
% rounded to the nearest multiple of 10, while the point coordinates specified
% to |\put| are not rounded at all; therefore some care should be used to place
% the working grid in the drawing plane. This grid is intended as an aid in
% drawing; even if you sketch your drawing on millimetre paper, the drawing grid
% turns out to be very useful; one must only delete or comment out the command
% when the drawing is finished.
% \item New trigonometric function macros have been implemented; possibly they
% are not better than the corresponding macros of the |trig| package, but they
% are supposed to be more accurate at least they were intended to be so. The
% other difference is that angles are specified in sexagesimal degrees
% ($360^\circ$ to one revolution), so that reduction to the fundamental quadrant
% is supposed to be more accurate; the tangent of odd multiples of $90^\circ$
% are approximated with a ``\TeX\ infinity'', that is the signed value
% 16383.99999. This will possibly produce computational errors in the
% subsequent calculations, but at least it does not stop the tangent
% computation. In order to avoid overflows or underflows in the computation
% of small angles (reduced to the first quadrant), the sine and the tangent
% of angles smaller than $1^\circ$ are approximated by the first term of the
% McLaurin series, while for the cosine the approximation is given by the first
% two terms of the McLaurin series. In both cases theoretical errors are smaller
% than what \TeX\ arithmetics can handle.
%
% These trigonometric functions are used within the complex number macros; but if
% the user wants to use them the syntax is the following:
%\begin{flushleft}
%      \texttt{\char92SinOf}\meta{angle}\texttt{to}\meta{control sequence}
%\\
%      \texttt{\char92CosOf}\meta{angle}\texttt{to}\meta{control sequence}
%\\
%      \texttt{\char92TanOf}\meta{angle}\texttt{to}\meta{control sequence}
%\end{flushleft}
% The \meta{control sequence} may then be used as a multiplying factor of a length.
% \item Arcs can be drawn as simple circular arcs, or with one or two arrows at
% their ends (curved vectors);  the syntax is:
%\begin{flushleft}
% \texttt{\char92Arc(}\meta{center}\texttt{)(}\meta{starting point}\texttt{)}\marg{angle}\\
% \texttt{\char92VectorArc(}\meta{center}\texttt{)(}\meta{starting point}\texttt{)}\marg{angle}\\
% \texttt{\char92VectorARC(}\meta{center}\texttt{)(}\meta{starting point}\texttt{)}\marg{angle}\\
%\end{flushleft}
% If the angle is specified numerically it must be enclosed in braces, while if it
% is specified with a control sequence the braces (curly brackets) are not
% necessary. The above macro |\Arc| draws a simple circular arc without arrows;
% |\VectorArc| draws an arc with an arrow tip at the ending point; |\VectorARC|
% draws an arc with arrow tips at both ends; see figure~\ref{fig:arcs}.
%	\begin{figure}
%	\begin{minipage}{.48\textwidth}
%	\begin{verbatim}
%	\unitlength=0.5mm
%	\begin{picture}(60,40)
%	\put(0,0){\GraphGrid(60,40)}
%	\Arc(0,20)(30,0){60}
%	\VECTOR(0,20)(30,0)\VECTOR(0,20)(32.5,36)
%	\VectorArc(0,20)(15,10){60}
%	\put(20,20){\makebox(0,0)[l]{$60^\circ$}}
%	\VectorARC(60,20)(60,0){-180}
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\centering
%	\unitlength=0.5mm
%	\begin{picture}(60,40)
%	\put(0,0){\GraphGrid(60,40)}
%	\Arc(0,20)(30,0){60}
%	\VECTOR(0,20)(30,0)\VECTOR(0,20)(32.5,36)
%	\VectorArc(0,20)(15,10){60}
%	\put(20,20){\makebox(0,0)[l]{$60^\circ$}}
%	\VectorARC(60,20)(60,0){-180}
%	\end{picture}
%	\end{minipage}
%	\caption{Arcs and curved vectors}\label{fig:arcs}
%	\end{figure}
% \item A multitude of commands have been defined in order to manage complex
% numbers; actually complex numbers are represented as a comma separated pair of
% fractional numbers. They are used to address to specific points in the drawing
% plane, but also as operators so as to scale and rotate other objects. In the
% following \meta{vector} means a comma separated pair of fractional numbers,
% \meta{vector macro} means a macro the contains a comma separated pair of
% fractional numbers; \meta{angle macro} means a macro that contains the angle
% of a vector in sexagesimal degrees; \meta{argument} means a brace delimited
% numeric value, possibly a macro; \textit{macro} is a valid macro name, that
% is a backslash followed by letters, or anything else that can receive a
% definition. A `direction' of a vector is its versor; the angle of a vector
% is the angle between the vector and the positive $x$ axis, generally directly
% used in the Euler formula $ \vec{v} = Me^{\mathrm{j}\varphi}$.
%
% {\footnotesize\begin{itemize}
% \item |\MakeVectorFrom|\meta{two arguments}|to|\meta{vector macro}
% \item |\CopyVect|\meta{first vector}|to|\meta{second vector macro}
% \item |\ModOfVect|\meta{vector}|to|\meta{macro}
% \item |\DirOfvect|\meta{vector}|to|\meta{versor macro}
% \item |\ModAndDirOfVect|\meta{vector}|to|\meta{1st macro}|and|\meta{2nd macro}
% \item |\DistanceAndDirOfVect|\meta{first vector}|minus|\meta{second vector}|to|\meta{1st macro}|and|\meta{2nd macro}
% \item |\XpartOfVect|\meta{vector}|to|\meta{macro}
% \item |\YpartOfVect|\meta{vector}|to|\meta{macro}
% \item |\DirFromAngle|\meta{angle}|to|\meta{versor macro}
% \item |\ArgOfVect|\meta{vector}|to|\meta{angle macro}
% \item |\ScaleVect|\meta{vector}|by|\meta{scaling factor}|to|\meta{vector macro}
% \item |\ConjVect|\meta{vector}|to|\meta{conjugate vector macro}
% \item |\SubVect|\meta{first vector}|from|\meta{second vector}|to|\meta{vector macro}
% \item |\AddVect|\meta{first vector}|and|\meta{second vector}|to|\meta{vector macro}
% \item |\MultVect|\meta{first vector}|by|\meta{second vector}|to|\meta{vector macro}
% \item |\MultVect|\meta{first vector}|by*|\meta{second vector}|to|\meta{vector macro}
% \item |\DivVect|\meta{first vector}|by|\meta{second vector}|to|\meta{vector macro}
% \end{itemize}}
%
% \item General curves can be drawn with the |pict2e| macro |\curve| but it
% requires the specification of the Bézier third order spline control points;
% sometimes it's better to be very specific with the control points and there
% is no other means to do a decent graph; sometimes the curves to be drawn
% are not so tricky and a general set of macros can be defined so as to compute
% the control points, while letting the user specify only the nodes through
% which the curve must pass, and the tangent direction of the curve in such
% nodes. This macro is |\Curve| and must be followed by an ``unlimited''
% sequence of node-direction coordinates as a quadruple defined as
%\[
% \texttt{(}\meta{node coordinates}\texttt{)<}\meta{direction vector}\texttt{>}
%\]
% Possibly if a sudden change of direction has to be performed (cusp) another item
% can be inserted after one of those quadruples in the form
%\[
% \texttt{...(...)<...>[}\meta{new direction vector}\texttt{](...)<...>...}
%\]
% The |\Curve| macro does not (still) have facilities for cycling the path,
% that is to close the path from the last specified node-direction to the first
% specified node-direction. The tangent direction need not be specified with
% a unit vector, although only its direction is relevant; the scaling of the
% specified direction vector to a unit vector is performed by the macro itself.
% Therefore one cannot specify the fine tuning of the curve convexity as it
% can be done with other programs, as for example with \MF\ or the |pgf/tikz|
% package and environment. See figure~\ref{fig:curve} for an example.
% \end{enumerate}
%	\begin{figure}
%	\begin{minipage}{.48\textwidth}
%   \begin{verbatim}
%	\unitlength=8mm
%	\begin{picture}(5,5)
%	\put(0,0){\framebox(5,5){}}\thicklines\roundcap
%	\Curve(2.5,0)<1,1>(5,3.5)<0,1>%
%	      (2.5,3.5)<-.5,-1.2>[-.5,1.2]%
%	      (0,3.5)<0,-1>(2.5,0)<1,-1>
%	\end{picture}
%   \end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\raggedleft\relax
%	\unitlength=8mm\relax
%	\begin{picture}(5,5)
%	\put(0,0.5){\put(0,0){\framebox(5,5){}}\thicklines\roundcap
%	\Curve(2.5,0)<1,1>(5,3.5)<0,1>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](0,3.5)<0,-1>(2.5,0)<1,-1>}
%	\end{picture}
%	\end{minipage}
%	\caption{A heart shaped curve with cusps drawn with \texttt{\string\Curve}}
%	\label{fig:curve}
%	\end{figure}
%
% In spite of the relative simplicity of the macros contained in this package, the
% described macros, as well as the original macros included in the |pict2e| package,
% allow to produce fine drawings that were inconceivable of with the original \LaTeX\ 
% picture environment. Leslie Lamport himself announced an extension to his
% environment when \LaTeXe\ was first issued in 1994; in the |latexnews| news letter
% of December 2003; the first implementation appeared; the first version of this
% package was issued in 2006. It was time to have a better drawing environment; this
% package is a simple attempt to follow the initial path while extending the drawing
% facilities; but Till Tantau's |pgf| package has gone much farther.
%
% \section{Remark}
% There are other packages in the \textsc{ctan} archives that deal with tracing
% curves of various kinds. |PSTricks| and |tikz/pgf| are the most powerful ones.
% But there are also the package |curves| that is intended to draw almost
% anything by using little dots or other symbols partially superimposed to one
% another. It used only quadratic Bézier curves and the curve tracing is eased
% by specifying only the curve nodes, without specifying the control nodes;
% with a suitable option to the package call it is possible to reduce the
% memory usage by using short straight segments drawn with the PostScript
% facilities offered by the |dvips| driver.
%
% Another package |ebezier| performs about the same as |curve2e| but draws its
% Bézier curves by using little dots partially superimposed to one another. The
% documentation is quite interesting but since it explains very clearly what
% exactly are the Bézier splines, it appears that |ebezier| should be used only
% for dvi output without recourse to PostScript machinery.
%
% \section{Acknowledgements}
% I wish to express my deepest thanks to Michel Goosens who spotted some errors
% and very kindly submitted them to me so that I was able to correct them.
%
% Josef Tkadlec and the author collaborated extensively in order to make a better
% real long division so as to get the fractional part and to avoid as much as
% possible any numeric overflow; many Josef's ideas are incorporated in the macro
% that is implemented in this package, although the macro used by Josef is
% slightly different from this one. Both versions aim at a better accuracy and
% at widening the operand ranges.
%
% Daniele Degiorgi spotted a fault in the kernel definition of |\linethickness|
% that heavily influenced also |curve2e|; see below.
%
% Thanks also to Jin-Hwan Cho and Juho Lee who suggested a small but crucial
% modification in order to have \texttt{curve2e} work smoothly also with XeTeX
% (XeLaTeX). Actually if version 0.2x or later, dated 2009/08/05 or later, of
% |pict2e| is being used, such modification is not necessary, but it's true
% that it becomes imperative if older versions are used.
%
% \StopEventually{%
% \begin{thebibliography}{9}
% \bibitem{pict2e} Gäßlein H.,  Niepraschk R., and Tkadlec J.
% \emph{The \texttt{pict2e}
% package}, 2009, PDF document attached to the ``new'' \texttt{pict2e} bundle; the
% bundle may be downloaded from any CTAN archive or one of their mirrors.
% \end{thebibliography}
% }
%
% \section{Source code}
% \subsection{Some preliminary extensions to the \texttt{pict2e} package}
% The necessary preliminary code has already been introduced. Here we require
% the \texttt{color} package and the \texttt{pict2e} one; for the latter one we
% make sure that a sufficiently recent version is used.
%    \begin{macrocode}
\RequirePackage{color}
\RequirePackageWithOptions{pict2e}[2014/01/01]
%    \end{macrocode}
%
% The next macros are just for debugging. With the \texttt{tracing} package it
% would probably be better to define other macros, but this is not for the
% users, but for the developers.
%    \begin{macrocode}
\def\TRON{\tracingcommands\tw@ \tracingmacros\tw@}%
\def\TROF{\tracingcommands\z@ \tracingmacros\z@}%
%    \end{macrocode}
%
% Next we define some new dimension registers that will be used by the
% subsequent macros; should they be already defined, there will not be any
% redefinition; nevertheless the macros should be sufficiently protected so as
% to avoid overwriting register values loaded by other macro packages.
%    \begin{macrocode}
\ifx\undefined\@tdA \newdimen\@tdA \fi
\ifx\undefined\@tdB \newdimen\@tdB \fi
\ifx\undefined\@tdC \newdimen\@tdC \fi
\ifx\undefined\@tdD \newdimen\@tdD \fi
\ifx\undefined\@tdE \newdimen\@tdE \fi
\ifx\undefined\@tdF \newdimen\@tdF \fi
\ifx\undefined\defaultlinewidth \newdimen\defaultlinewidth \fi
%    \end{macrocode}
%
% It is better to define a macro for setting a different value for the line and
% curve thicknesses; the `|\defaultlinewidth| should contain the
% equivalent of |\@wholewidth|, that is the thickness of thick lines; thin lines
% are half as thick; so when the default line thickness is specified to, say,
% 1pt, thick lines will be 1pt thick and thin lines will be 0.5pt thick.  The
% default whole width of thick lines is 0,8pt, but this is specified in the
% kernel of \LaTeX\ and\slash or in \texttt{pict2e}. On the opposite it is
% necessary to redefine |\linethickness| because the \LaTeX\ kernel global
% definition does not hide the space after the closed brace when you enter
% something such as |\linethickness{1mm}| followed by a space or a new line.
%\footnote{Thanks to Daniele Degiorgi (\texttt{degiorgi@inf.ethz.ch}).}
%    \begin{macrocode}
\gdef\linethickness#1{\@wholewidth#1\@halfwidth.5\@wholewidth\ignorespaces}%
\newcommand\defaultlinethickness[1]{\defaultlinewidth=#1\relax
\def\thicklines{\linethickness{\defaultlinewidth}}%
\def\thinlines{\linethickness{.5\defaultlinewidth}}%
\thinlines\ignorespaces}
%    \end{macrocode}
% The |\ignorespaces| at the end of this and the subsequent macros is for
% avoiding spurious spaces to get into the picture that is being drawn, because
% these spaces introduce picture deformities often difficult to spot and
% eliminate.
%
% \subsubsection{Improved line and vector macros}
% The new macro |\LIne| allows to draw an arbitrary inclination line as if it
% was a polygonal with just two vertices. This line should be set by means of a
% |\put| command so that its starting point is always at a relative 0,0
% coordinate point. The two arguments define the horizontal and the
% vertical component respectively.
%    \begin{macrocode}
\def\LIne(#1,#2){\moveto(0,0)
      \pIIe@lineto{#1\unitlength}{#2\unitlength}\strokepath}%
%    \end{macrocode}
%
% A similar macro |\segment| operates between two explicit points with absolute
% coordinates, instead of relative to the position specified by a |\put|
% command; it resorts to the |\polyline| macro that is to be defined in a while.
% The |\@killglue| command might be unnecessary, but it does not harm; it
% eliminates any explicit or implicit spacing that might precede this command.
%    \begin{macrocode}
\def\segment(#1)(#2){\@killglue\polyline(#1)(#2)}%
%    \end{macrocode}
% By passing its ending points coordinates to the |\polyline| macro, both macro
% arguments are a pair of coordinates, not their components; in other words, if
% $P_1=(x_1, y_2)$ and $P_2=(x_2, y_2)$, then the first argument is the couple
% $x_1, y_1$ and likewise the second argument is $x_2, y_2$. Please remember that
% the decimal separator is the decimal \emph{point}, while the \emph{comma} acts
% as coordinate separator. This recommendation is particularly important for
% non-English speaking users, since the ISO regulations allow the decimal point
% only for English speaking countries, while in all other countries the comma
% must be used as the decimal separator.
%
% The |\line| macro is redefined by making use of a new division routine that
% receives in input two dimensions and yields on output their fractional ratio.
% The beginning of the macro definition is the same as that of \texttt{pict2e}:
%    \begin{macrocode}
\def\line(#1)#2{\begingroup
  \@linelen #2\unitlength
  \ifdim\@linelen<\z@\@badlinearg\else
%    \end{macrocode}
% but as soon as it is verified that the line length is not negative, things
% change remarkably; in facts the machinery for complex numbers is invoked.
% This makes the code much simpler, not necessarily more efficient; nevertheless
% |\DirOfVect| takes the only macro argument (that actually contains a comma
% separated pair of fractional numbers) and copies it to |\Dir@line| (an
% arbitrarily named control sequence) after re-normalizing to unit magnitude;
% this is passed to |GetCoord| that separates the two components into the
% control sequences |\d@mX| and|\d@mY|; these in turn are the values that are
% actually operated upon by the subsequent commands.
%    \begin{macrocode}
    \expandafter\DirOfVect#1to\Dir@line
    \GetCoord(\Dir@line)\d@mX\d@mY
%    \end{macrocode}
% The normalized vector direction is actually formed with the directing cosines
% of the line direction; since the line length is actually the horizontal
% component for non vertical lines, it is necessary to compute the actual line
% length for non vertical lines by dividing the given length by the
% magnitude of horizontal cosine |\d@mX|, and the line length is accordingly
% scaled:
%    \begin{macrocode}
    \ifdim\d@mX\p@=\z@\else
      \DividE\ifdim\d@mX\p@<\z@-\fi\p@ by\d@mX\p@ to\sc@lelen
      \@linelen=\sc@lelen\@linelen
    \fi
%    \end{macrocode}
% Of course, if the line is vertical this division must not take place.
% Finally the \texttt{moveto}, \texttt{lineto} and \texttt{stroke} language
% keywords are invoked by means of the internal \texttt{pict2e} commands in
% order to draw the line. Notice that even vertical lines are drawn with the
% ``PostScript'' commands instead of resorting to the dvi low level language
% that was used both in \texttt{pict2e} and in the original \texttt{picture}
% commands; it had a meaning in the old times, but it certainly does not have
% any when lines are drawn by the driver that drives the output to a visible
% document form, not by \TeX\ the program.
%    \begin{macrocode}
    \moveto(0,0)
    \pIIe@lineto{\d@mX\@linelen}{\d@mY\@linelen}%
    \strokepath
  \fi
\endgroup\ignorespaces}%
%    \end{macrocode}
% The new definition of the command |\line|, besides the ease with which is
% readable, does not do different things from the definition of |pict2e| 2009, but
% it did preform in a better way whith the 2004 version that was limited to
% integer direction coefficients up to 999 in magnitude.
%
% Another useful line-type macro creates a dashed line between two given points
% with a dash length that must be specified; actually the specified dash length
% is a desired dash length; the actual length is computed by integer division
% between the distance of the given points and the desired dash length; this
% integer is tested in order to see if it's odd; if it's not, it is increased by
% one. Then the actual dash length is obtained by dividing the above distance by
% this odd number.
% Another vector is created from $P_1-P_0$ by dividing it by the magic odd number;
% then it is multiplied by two in order to have the increment from one dash to the
% next, and finally the number of patterns is obtained by integer dividing the
% magic odd number by 2 and increasing it by 1. A simple |\multiput| completes the
% job, but in order to use the various vectors and numbers within a group and to
% throw the result outside the group while restoring all the intermediate counters
% and registers, a service macro is created with an expanded definition and then
% this service macro is executed.
%    \begin{macrocode}
\ifx\Dline\undefined
\def\Dline(#1,#2)(#3,#4)#5{%
\begingroup
   \countdef\NumA254\countdef\NumB252\relax
   \MakeVectorFrom{#1}{#2}to\V@ttA 
   \MakeVectorFrom{#3}{#4}to\V@ttB 				
   \SubVect\V@ttA from\V@ttB to\V@ttC 			
   \ModOfVect\V@ttC to\DlineMod 				
   \DividE\DlineMod\p@ by#5\p@ to\NumD 			
   \NumA\expandafter\Integer\NumD?? 	
   \ifodd\NumA\else\advance\NumA\@ne\fi 		
   \NumB=\NumA \divide\NumB\tw@ 				
   \DividE\DlineMod\p@ by\NumA\p@  to\D@shMod 	
   \DividE\p@ by\NumA\p@ to \@tempa 			
   \MultVect\V@ttC by\@tempa,0 to\V@ttB	
   \MultVect\V@ttB by 2,0 to\V@ttC
   \advance\NumB\@ne			
   \edef\@mpt{\noexpand\endgroup
   \noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}{\noexpand\LIne(\V@ttB)}}%
   \@mpt\ignorespaces}%
\fi
%    \end{macrocode}
%
% The new macro |\GetCoord| splits a vector (or complex number) specification
% into its components:
%    \begin{macrocode}
\def\GetCoord(#1)#2#3{%
\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
%    \end{macrocode}
% But the macro that does the real work is |\SplitNod@|:
%    \begin{macrocode}
\def\SplitNod@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
%    \end{macrocode}
%
% The redefinitions and the new definitions for vectors are a little more
% complicated than with segments, because each vector is drawn as a filled
% contour; the original \texttt{pict2e} 2004  macro checks if the slopes are
% corresponding to the limitations specified by Lamport (integer three digit
% signed numbers) and sets up a transformation in order to make it possible to
% draw each vector as an horizontal left-to-right arrow and then to rotate it by
% its angle about its tail point; with |pict2e| 2009, possibly this redefinition
% of |\vector| is not necessary, but we do it as well and for the same reasons
% we had for redefining |\line|; actually there are two macros for tracing the
% contours that are eventually filled by the principal macro; each contour
% macro draws the vector with a \LaTeX\ or a PostScript arrow whose parameters
% are specified by default or may be taken from the parameters taken from the
%\texttt{PSTricks} package if this one is loaded before \texttt{pict2e}; in any
% case we did not change the contour drawing macros because if they are
% modified the same modification is passed on to the arrows drawn with the
% \texttt{curve2e} package redefinitions.
%
% Because of these features the redefinitions and the new macros are different
% from those used for straight lines.
%
% We start with the redefinition of |\vector| and we use the machinery for
% vectors (as complex numbers) we used for |\line|.
%    \begin{macrocode}
  \def\vector(#1)#2{%
    \begingroup
      \GetCoord(#1)\d@mX\d@mY
      \@linelen#2\unitlength
%    \end{macrocode}
% As in \texttt{pict2e} we avoid tracing vectors if the slope parameters are
% both zero.
%    \begin{macrocode}
      \ifdim\d@mX\p@=\z@\ifdim\d@mY\p@=\z@\@badlinearg\fi\fi
%    \end{macrocode}
% But we check only for the positive nature of the $l_x$ component; if it is
% negative, we simply change sign instead of blocking the typesetting process.
% This is useful also for macros |\Vector| and |\VECTOR| to be defined in a
% while.
%    \begin{macrocode}
      \ifdim\@linelen<\z@ \@linelen=-\@linelen\fi
%    \end{macrocode}
% We now make a vector with the slope coefficients even if one or the other is
% zero and we determine its direction; the real and imaginary parts of the
% direction vector are also the values we need for the subsequent rotation.
%    \begin{macrocode}
      \MakeVectorFrom\d@mX\d@mY to\@Vect
      \DirOfVect\@Vect to\Dir@Vect
%    \end{macrocode}
% In order to be compatible with the original \texttt{pict2e} we need to
% transform the components of the vector direction in lengths with the specific
% names |\@xdim| and |\@ydim|
%    \begin{macrocode}
        \YpartOfVect\Dir@Vect to\@ynum \@ydim=\@ynum\p@
        \XpartOfVect\Dir@Vect to\@xnum \@xdim=\@xnum\p@
%    \end{macrocode}
% If the vector is really sloping we need to scale the $l_x$ component in order
% to get the vector total length; we have to divide by the cosine of the vector
% inclination which is the real part of the vector direction. I use my division
% macro; since it yields a ``factor'' I directly use it to scale the length of
% the vector. I finally memorize the true vector length in the internal
% dimension |@tdB|
%    \begin{macrocode}
        \ifdim\d@mX\p@=\z@
        \else\ifdim\d@mY\p@=\z@
          \else
            \DividE\ifdim\@xnum\p@<\z@-\fi\p@ by\@xnum\p@ to\sc@lelen
            \@linelen=\sc@lelen\@linelen
          \fi
        \fi
      \@tdB=\@linelen
%    \end{macrocode}
%  The remaining code is definitely similar to that of \texttt{pict2e}; the
% real difference consists in the fact that the arrow is designed by itself
% without the stem; but it is placed at the vector end; therefore the first
% statement is just the transformation matrix used by the output driver to
% rotate the arrow tip and to displace it the right amount. But in order
% to draw only the arrow tip I have to set the |\@linelen| length to zero.
%    \begin{macrocode}
\pIIe@concat\@xdim\@ydim{-\@ydim}\@xdim{\@xnum\@linelen}{\@ynum\@linelen}%
        \@linelen\z@
        \pIIe@vector
        \fillpath
%    \end{macrocode}
% Now we can restore the stem length that must be shortened by the dimension of
% the arrow; examining the documentation of \texttt{pict2e} we discover that
% we have to shorten it by an approximate amount of $AL$ (with the notations of
% \texttt{pict2e}, figs~10 and~11); the arrow tip parameters are stored in
% certain variables with which we can determine the amount of the stem
% shortening; if the stem was too short and the new length is negative, we
% refrain from designing such stem.
%    \begin{macrocode}
      \@linelen=\@tdB
      \@tdA=\pIIe@FAW\@wholewidth
      \@tdA=\pIIe@FAL\@tdA
      \advance\@linelen-\@tdA
      \ifdim\@linelen>\z@
        \moveto(0,0)
        \pIIe@lineto{\@xnum\@linelen}{\@ynum\@linelen}%
        \strokepath\fi
    \endgroup}
%    \end{macrocode}
%
% Now we define the macro that does not require the specification of the length
% or the $l_x$ length component; the way the new |\vector| macro works does not
% actually  require this specification, because \TeX\ can compute the vector
% length, provided the two direction components are exactly the horizontal and
% vertical vector components. If the horizontal component is zero, the actual
% length% must be specified as the vertical component.
%    \begin{macrocode}
\def\Vector(#1,#2){%
\ifdim#1\p@=\z@\vector(#1,#2){#2}
\else
\vector(#1,#2){#1}\fi}
%    \end{macrocode}
%
% On the opposite the next macro specifies a vector by means of the coordinates
% of its end points; the first point is where the vector starts, and the second
% point is  the arrow tip side. We need the difference of these two coordinates, because % it  represents the actual vector.
%    \begin{macrocode}
\def\VECTOR(#1)(#2){\begingroup
\SubVect#1from#2to\@tempa
\expandafter\put\expandafter(#1){\expandafter\Vector\expandafter(\@tempa)}%
\endgroup\ignorespaces}
%    \end{macrocode}
%
% The \texttt{pict2e} documentation says that if the vector length is zero the
% macro designs only the arrow tip; this may work with macro |\vector|,
% certainly not with |\Vector| and |\VECTOR|. This might be useful for adding
% an arrow tip to a circular arc. See examples in figure~\ref{fig:vectors}.
%
%	\begin{figure}
%	\begin{minipage}{.48\textwidth}
%	\begin{verbatim}
%	\unitlength=.5mm
%	\begin{picture}(60,20)
%	\put(0,0){\GraphGrid(60,20)}
%	\put(0,0){\vector(1.5,2.3){10}}
%	\put(20,0){\Vector(10,15.33333)}
%	\VECTOR(40,0)(50,15.33333)
%	\end{picture}
%	\end{verbatim}
%	\end{minipage}
%	\hfill
%	\begin{minipage}{.48\textwidth}\centering
%	\unitlength=.5mm
%	\begin{picture}(60,20)
%	\put(0,0){\GraphGrid(60,20)}
%	\put(0,0){\vector(1.5,2.3){10}}
%	\put(20,0){\Vector(10,15.33333)}
%	\VECTOR(40,0)(50,15.33333)
%	\end{picture}
%	\end{minipage}
%   \caption{Three (displaced) identical vectors obtained with the three vector
%            macros.}\label{fig:vectors}
%	\end{figure}
%
% \subsubsection{Polylines}
% We now define the polygonal line macro; its syntax is very simple
% \begin{flushleft}
% \cs{polygonal}\texttt{(}$P_0$\texttt{)(}$P_1$\texttt{)(}$P_2$)%
%  \texttt{\dots(}$P_n$\texttt{)}
% \end{flushleft}
% In order to write a recursive macro we need aliases for the parentheses;
% actually we need only the left parenthesis, but some editors complain about
% unmatched delimiters, so we define an alias also for the right parenthesis.
%    \begin{macrocode}
\let\lp@r( \let\rp@r)
%    \end{macrocode}
% The first call to |\polyline| examines the first point coordinates and moves
% the drawing position to this point; afterwards it looks for the second point
% coordinates; they start with a left parenthesis; if this is found the
% coordinates should be there, but if the left parenthesis is missing (possibly
% preceded by spaces that are ignored by the |\@ifnextchar| macro) then a
% warning message is output together with the line number where the missing
% parenthesis causes the warning: beware, this line number might point to
% several lines further on along the source file! In any case it's necessary to
% insert a |\@killglue| command, because |\polyline| refers to absolute coordinates
% not necessarily is put in position through a |\put| command that provides to
% eliminate any spurious spaces preceding this command.
%
% Remember: |\polyline| has been incorporated into |pict2e| 2009, but we
% redefine it so as to allow an optional argument to allow the line join
% specification.
%
% In order to allow a specification for the joints of the various segments of
% a polygonal line it is necessary to allow for an optional parameter; the default
% join is the bevel join.
%    \begin{macrocode}
\renewcommand*\polyline[1][\beveljoin]{\p@lylin@[#1]}

\def\p@lylin@[#1](#2){\@killglue#1\GetCoord(#2)\d@mX\d@mY
    \pIIe@moveto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\p@lyline}{%
    \PackageWarning{curve2e}%
    {Polylines require at least two vertices!\MessageBreak
     Control your polyline specification\MessageBreak}%
    \ignorespaces}}

%    \end{macrocode}
% But if there is a second or further point coordinate the recursive macro
% |\p@lyline| is called; it works on the next point and checks for a further
% point; if such a point exists it calls itself, otherwise it terminates the
% polygonal line by stroking it.
%    \begin{macrocode}
\def\p@lyline(#1){\GetCoord(#1)\d@mX\d@mY
    \pIIe@lineto{\d@mX\unitlength}{\d@mY\unitlength}%
    \@ifnextchar\lp@r{\p@lyline}{\strokepath\ignorespaces}}
%    \end{macrocode}
%
% \subsubsection{The red service grid}
% The next command is very useful for debugging while editing one's drawings;
% it draws a red grid with square meshes that are ten drawing units apart;
% there is no graduation along the grid, since it is supposed to be a debugging
% aid and the user should know what he/she is doing; nevertheless it is
% advisable to displace the grid by means of a |\put| command so that its grid
% lines coincide with the graph coordinates multiples of 10. Missing to do so
% the readings become cumbersome. The |\RoundUp| macro provides to increase the
% grid dimensions to integer multiples of ten.
%    \begin{macrocode}
\def\GraphGrid(#1,#2){\begingroup\textcolor{red}{\linethickness{.1\p@}%
\RoundUp#1modulo10to\@GridWd \RoundUp#2modulo10to\@GridHt
\@tempcnta=\@GridWd \divide\@tempcnta10\relax \advance\@tempcnta\@ne
\multiput(0,0)(10,0){\@tempcnta}{\line(0,1){\@GridHt}}%
\@tempcnta=\@GridHt \divide\@tempcnta10\advance\@tempcnta\@ne
\multiput(0,0)(0,10){\@tempcnta}{\line(1,0){\@GridWd}}\thinlines}%
\endgroup\ignorespaces}
%    \end{macrocode}
%  Rounding up is useful because also the grid margins fall on coordinates
% multiples of 10. It resorts to the |\Integer| macro that will be described in
% a while.
%    \begin{macrocode}
\def\RoundUp#1modulo#2to#3{\expandafter\@tempcnta\Integer#1.??%
\count254\@tempcnta\divide\count254by#2\relax
\multiply\count254by#2\relax
\count252\@tempcnta\advance\count252-\count254
\ifnum\count252>0\advance\count252-#2\relax
\advance\@tempcnta-\count252\fi\edef#3{\number\@tempcnta}\ignorespaces}%
%    \end{macrocode}
% The |\Integer| macro takes a possibly fractional number whose decimal
% separator, if present, \textit{must} be the decimal point and uses the point
% as an argument delimiter  If one has the doubt that the number being passed
%  to |\Integer| might be an integer, he/she should call the macro with a
% further point;
% if the argument is truly integer this point works as the delimiter of the
% integer part; if the argument being passed is fractional this extra point
% gets discarded as well as the fractional part of the number.
%    \begin{macrocode}
\def\Integer#1.#2??{#1}%
%    \end{macrocode}
%
% \subsection{The new division macro}
% Now comes one of the most important macros in the whole package: the division
% macro; it takes two lengths as input values and computes their fractional
% ratio into a control sequence.
% It must take care of the signs, so that it examines the operand signs and
% determines the result sign separately conserving this computed sign in the
% macro |\segno|; this done, we are sure that both operands are or are
% made positive; should the numerator be zero it directly issues the zero
% quotient; should the denominator be zero it outputs 
% ``infinity'' (|\maxdimen| in points), that is the maximum allowable length
% measured in points that \TeX\ can deal with.
% Since the result is assigned a value, the calling statement must pass as the
% third argument either a control sequence or an active character. Of course the
% first operand is the dividend, the second the divisor and the third the
% quotient.
%
% Since |curve2e| is supposed to be an extension of |pict2e| and this macro 
%package already contains a division macro, we do not define any other division
% macro; nevertheless, since the macro in |pict2e| may not be so efficient as it
% might be if the |e-tex| extensions of the interpreter program were available,
% here we check and eventually provide a more efficient macro. The latter exploits
% the scaling mechanism embedded in |pdftex| since 2007, if the extended mode is
% enabled, that is used to scale a dimension by a fraction: $L\times N/D$, where
% $L$ is a dimension, and $N$ and $D$ are the numerator an denominator of the
% scaling factor; these might be integers, but it's better they represent the
% numbers of scaled points another two dimensions correspond to, in the philosophy
% that floating point numbers are represented by the measures of lengths in
% points.
%
% Therefore first we test if the macro is already defined:
%    \begin{macrocode}
\ifdefined\dimexpr
%    \end{macrocode}
% then we test if the extended mode exists and/or is enabled:
%    \begin{macrocode}
    \unless\ifdefined\DividE
%    \end{macrocode}
% Notice that |\dimexpr| is the specific extended mode control sequence we are
% going to use in order to perform our task; if the interpreter program is too
% old and/or it is a recent version, but it was compiled without activating the
% extended mode, the macro |\dimexpr| is undefined.
%
% The macro, creates a group where the names of two counters and a
% dimensional register are defined; the numbers of these integer and dimension
% registers are expressly above the value 255, because one of the extensions is
% the possibility of using a virtually unlimited number of registers; moreover
% even if these registers were used within other macros, their use within a group
% does not damage the other macros; we just have to use a dirty trick to throw
% the result beyond the end-group command.
%
% The efficiency of this macro is contained in the extended command |\dimexpr|;
% both the |\@DimA| and |\Num| registers are program words of 32\,bits; the result
% is stored into an internal register of 64\,bits; the final division by a factor
% stored into a register of 32 bits, so that in terms of scaled points a division
% by 1\,pt = $1\times 2^{16}$, scales down the result by 16 bits, and if the total
% length of the result is smaller than $2^{30}$, the result can be correctly
% assigned to a dimension register. In any other case the extended features imply
% suitable error messages end the termination of the program. During the division
% a scaling down by 16 bits, the result is not simply truncated, but it is rounded
% to the nearest integer (in scaled points). The first two operands are lengths
% and the third is a macro.
%
%    \begin{macrocode}
	\def\DividE#1by#2to#3{\bgroup
		\countdef\Num2254\relax \countdef\Den2252\relax
		\dimendef\@DimA 2254
		\Num=\p@ \@DimA=#2\relax \Den=\@DimA
		\ifnum\Den=\z@
		  \edef\x{\noexpand\endgroup\noexpand\def\noexpand#3{\strip@pt\maxdimen}}%
		\else
		  \@DimA=#1\relax
		  \@DimA=\dimexpr\@DimA*\Num/\Den\relax
		  \edef\x{\noexpand\egroup\noexpand\def\noexpand#3{\strip@pt\@DimA}}%
		\fi
	\x\ignorespaces}%
	\fi
%    \end{macrocode}
%
% We need a similar macro to divide two fractional or integer numbers,
% not dimensions, and produce a macro that contains the fractional result.
%    \begin{macrocode}
    \unless\ifdefined\DivideFN
         \def\DivideFN#1by#2to#3{\DividE#1\p@ by#2\p@ to#3}%
    \fi
%    \end{macrocode}
%
% We do the same in order to multiply two integer o fractional numbers held
% in the first two arguments and the third argument is a definable token that
% will hold the result of multiplication in the form of a fractional number,
% possibly with  a non null fractional part; a null fractional part is
% eliminated by \verb|strip@pt|.
%    \begin{macrocode}
    \unless\ifdefined\MultiplY
        \def\MultiplY#1by#2to#3{\bgroup
        \dimendef\@DimA 2254 \dimendef\@DimB2255
        \@DimA=#1\p@\relax \@DimB=#2\p@\relax
        \@DimA=\dimexpr\@DimA*\@DimB/\p@\relax
        \edef\x{\noexpand\egroup\noexpand\def\noexpand#3{\strip@pt\@DimA}}%
        \x\ignorespaces}%
    \fi
\fi
%    \end{macrocode}

% The next macro uses the \verb|\strip@pt| \LaTeX\ kernel macro to get the
% numerical value of a measure in points. One has to call |\Numero| with
% a control sequence and a dimension; the dimension value in points is
% assigned to the control sequence.
%    \begin{macrocode}
\unless\ifdefined\Numero
  \def\Numero#1#2{\dimen3254#2\relax
  \edef#1{\strip@pt\dimen3254}\ignorespaces}%
\fi
%    \end{macrocode}
% The \verb|\ifdefined| primitive command is provided by the e-\TeX\ extension
% of the typesetting engine; the test does not create any hash table entry;
% it is a different way than the \verb|\ifx\csname ....\endcsname| test,
% because the latter first possibly creates a macro with meaning \verb|relax|
% then executes the test; therefore an undefined macro name is always defined
% to mean \verb|\relax|.
%
% \subsection{Trigonometric functions}
%  We now start with trigonometric functions. We define the macros |\SinOf|,
% |\CosOf| and |\TanOf| (we might define also |\CotOf|, but the cotangent does
% not appear so essential) by means of the parametric formulas that require the
% knowledge of the tangent of the half angle. We want to specify the angles
% in sexagesimal degrees, not in radians, so we can make accurate reductions to
% the main quadrants. We use the formulas
% \begin{eqnarray*}
% \sin\theta &=& \frac{2}{\cot x + \tan x}\\
% \cos\theta &=& \frac{\cot x - \tan x}{\cot x + \tan x}\\
% \tan\theta &=& \frac{2}{\cot x - \tan x}\\
% \noalign{\hbox{where}}
% x          &=& \theta/114.591559
% \end{eqnarray*}
% is the half angle in degrees converted to radians.
%
% We use this slightly modified set of parametric formulas because the cotangent
% of $x$ is a by product of the computation of the tangent of $x$; in this way
% we avoid computing the squares of numbers that might lead to overflows. For
% the same reason we avoid computing the value of the trigonometric functions
% in proximity of the value zero (and the other values that might involve high
% tangent or cotangent values) and in that case we prefer to approximate the
% small angle function value with its first or second order truncation of the
% McLaurin series; in facts for angles whose magnitude is smaller than $1^\circ$
% the magnitude of the independent variable $y=2x$ (the angle in degrees
% converted to radians) is so small (less than 0.017) that the sine and tangent
% can be freely approximated with $y$ itself (the error being smaller than
% approximately $10^{-6}$), while the cosine can be freely approximated with
% the formula $1-0.5y^2$ (the error being smaller than about $4\cdot10^{-9}$).
%
% We keep using grouping so that internal variables are local to these groups
% and do not mess up other things.
%
% The first macro is the service routine that computes the tangent and the
% cotangent of the half angle in radians; since we have to use always the
% reciprocal of this value, we call it |\X@| but in spite of the similarity it
% is the reciprocal of $x$. Notice that parameter \texttt{\#1} must be a length.
%    \begin{macrocode}
\def\g@tTanCotanFrom#1to#2and#3{%
\DividE 114.591559\p@ by#1to\X@ \@tdB=\X@\p@
%    \end{macrocode}
% Computations are done with the help of counter |\I|, of the length |\@tdB|,
% and the auxiliary control sequences |\Tan| and |\Cot| whose meaning is
% transparent. The iterative process controlled by |\@whilenum| implements the
% (truncated) continued fraction expansion of the tangent function.
% \[
% \tan x = \frac{1}{\displaystyle \frac{1\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{3\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{5\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{7\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{9\mathstrut}{\displaystyle x}
%     -\frac{1}{\displaystyle \frac{11\mathstrut}{\displaystyle x}
%     -\cdots}}}}}}
% \]
%    \begin{macrocode}
\countdef\I=2546\def\Tan{0}\I=11\relax
\@whilenum\I>\z@\do{%
   \@tdC=\Tan\p@ \@tdD=\I\@tdB
   \advance\@tdD-\@tdC \DividE\p@ by\@tdD to\Tan
   \advance\I-2\relax}%
\def#2{\Tan}\DividE\p@ by\Tan\p@ to\Cot \def#3{\Cot}%
\ignorespaces}%
%    \end{macrocode}
%
%  Now that we have the macro for computing the tangent and cotangent of the
% half angle, we can compute the real trigonometric functions we are interested
% in. The sine value is computed after reducing the sine argument to the
% interval $0^\circ< \theta<180^\circ$; actually special values such as
% $0^\circ$, $90^\circ$, $180^\circ$, et cetera, are taken care separately, so
% that CPU time is saved for these special cases. The sine sign is taken care
% separately according to the quadrant of the sine argument.
%
% Since all computations are done within a group, a trick is necessary in order to
% extract the sine value from the group; this is done by defining within the group
% a macro (in this case |\endSinOf|) with the expanded definition of the result,
% but in charge of of closing the group, so that when the group is closed the
% auxiliary function is not defined any more, although its expansion keeps getting
% executed so that the expanded result is thrown beyond the group end.
%    \begin{macrocode}
\def\SinOf#1to#2{\bgroup%
\@tdA=#1\p@%
\ifdim\@tdA>\z@%
  \@whiledim\@tdA>180\p@\do{\advance\@tdA -360\p@}%
\else%
  \@whiledim\@tdA<-180\p@\do{\advance\@tdA 360\p@}%
\fi \ifdim\@tdA=\z@
  \def\@tempA{0}%
\else
  \ifdim\@tdA>\z@
    \def\Segno{+}%
  \else
    \def\Segno{-}%
    \@tdA=-\@tdA
  \fi
  \ifdim\@tdA>90\p@
    \@tdA=-\@tdA \advance\@tdA 180\p@
  \fi
  \ifdim\@tdA=90\p@
    \def\@tempA{\Segno1}%
  \else
    \ifdim\@tdA=180\p@
      \def\@tempA{0}%
    \else
      \ifdim\@tdA<\p@
        \@tdA=\Segno0.0174533\@tdA
        \DividE\@tdA by\p@ to \@tempA%
      \else
        \g@tTanCotanFrom\@tdA to\T and\Tp
        \@tdA=\T\p@ \advance\@tdA \Tp\p@
        \DividE \Segno2\p@ by\@tdA to \@tempA%
      \fi
    \fi
  \fi
\fi
\edef\endSinOf{\noexpand\egroup
  \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
\endSinOf}%
%    \end{macrocode}
%
% For the computation of the cosine we behave in a similar way using also the
% identical trick for throwing the result beyond the group end.
%    \begin{macrocode}
\def\CosOf#1to#2{\bgroup%
\@tdA=#1\p@%
\ifdim\@tdA>\z@%
  \@whiledim\@tdA>360\p@\do{\advance\@tdA -360\p@}%
\else%
  \@whiledim\@tdA<\z@\do{\advance\@tdA 360\p@}%
\fi
%
\ifdim\@tdA>180\p@
  \@tdA=-\@tdA \advance\@tdA 360\p@
\fi
%
\ifdim\@tdA<90\p@
  \def\Segno{+}%
\else
  \def\Segno{-}%
  \@tdA=-\@tdA \advance\@tdA 180\p@
\fi
\ifdim\@tdA=\z@
  \def\@tempA{\Segno1}%
\else
  \ifdim\@tdA<\p@
    \@tdA=0.0174533\@tdA \Numero\@tempA\@tdA
    \@tdA=\@tempA\@tdA \@tdA=-.5\@tdA
    \advance\@tdA \p@
    \DividE\@tdA by\p@ to\@tempA%
  \else
    \ifdim\@tdA=90\p@
      \def\@tempA{0}%
    \else
      \g@tTanCotanFrom\@tdA to\T and\Tp
      \@tdA=\Tp\p@ \advance\@tdA-\T\p@
      \@tdB=\Tp\p@ \advance\@tdB\T\p@
      \DividE\Segno\@tdA by\@tdB to\@tempA%
    \fi
  \fi
\fi
\edef\endCosOf{\noexpand\egroup
    \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
\endCosOf}%
%    \end{macrocode}
%
% For the tangent computation we behave in a similar way, except that we
% consider the fundamental interval as $0^\circ<\theta<90^\circ$; for the odd
% multiples of $90^\circ$ we assign the result a \TeX\ infinity value, that is
% the maximum a dimension can be.
%    \begin{macrocode}
\def\TanOf#1to#2{\bgroup%
\@tdA=#1\p@%
\ifdim\@tdA>90\p@%
  \@whiledim\@tdA>90\p@\do{\advance\@tdA -180\p@}%
\else%
  \@whiledim\@tdA<-90\p@\do{\advance\@tdA 180\p@}%
\fi%
\ifdim\@tdA=\z@%
  \def\@tempA{0}%
\else
  \ifdim\@tdA>\z@
    \def\Segno{+}%
  \else
    \def\Segno{-}%
    \@tdA=-\@tdA
  \fi
  \ifdim\@tdA=90\p@
    \def\@tempA{\Segno16383.99999}%
  \else
    \ifdim\@tdA<\p@
        \@tdA=\Segno0.0174533\@tdA
        \DividE\@tdA by\p@ to\@tempA%
    \else
      \g@tTanCotanFrom\@tdA to\T and\Tp
      \@tdA\Tp\p@ \advance\@tdA -\T\p@
      \DividE\Segno2\p@ by\@tdA to\@tempA%
    \fi
  \fi
\fi
\edef\endTanOf{\noexpand\egroup
   \noexpand\def\noexpand#2{\@tempA}\noexpand\ignorespaces}%
\endTanOf}%
%    \end{macrocode}
%
% \subsection{Arcs and curves preliminary information}
% We would like to define now a macro for drawing circular arcs of any radius
% and any angular aperture; the macro should require the arc center, the
% arc starting point and the angular aperture. The arc has its reference point in
% its center, therefore it does not need to be put in place by the command |\put|;
% nevertheless if |\put| is used, it may displace the arc into another position.
% The command should have the following syntax:
% \begin{flushleft}\ttfamily
% \cs{Arc}(\meta{{\rmfamily center}})(\meta{{\rmfamily starting
%                 point}}){\marg{{\rmfamily angle}}}
% \end{flushleft}
% which is totally equivalent to:
% \begin{flushleft}\ttfamily
% \string\put(\meta{\rmfamily center})\string{\string\Arc(0,0)(\meta{\rmfamily starting
% point})\marg{\rmfamily angle}\string}
% \end{flushleft}
%  If the \meta{angle} is positive the arc runs counterclockwise from the
% starting point; clockwise if it's negative.
%
% It's necessary to determine the end point and the control points of the
% Bézier spline(s) that make up the circular arc.
%
% The end point is obtained from the rotation of the starting point around the
% center; but the \texttt{pict2e} command |\pIIe@rotate| is such that the
% pivoting point appears to be non relocatable.
% It is therefore necessary to resort to low level \TeX\ commands and the
% defined trigonometric functions and a set of macros that operate on complex
% numbers used as vector scale-rotate operators.
%
% \subsection{Complex number macros}
% We need therefore macros for summing, subtracting, multiplying, dividing
% complex numbers, for determining they directions (unit vectors); a unit vector
% is the complex number divided by its magnitude so that the result is the
% Cartesian form of the Euler's formula
% \[
% \mathrm{e}^{\mathrm{j}\phi} = \cos\phi+\mathrm{j}\sin\phi
% \]
%
% The magnitude of a vector is determined by taking a clever square root of a
% function of the real and the imaginary parts; see further on.
%
% It's better to represent each complex number with one control sequence; this
% implies frequent assembling and disassembling the pair of real numbers that
% make up a complex number. These real components are assembled into the
% defining control sequence as a couple of coordinates, i.e.\ two comma
% separated integer or fractional signed decimal numbers.
%
% For assembling two real numbers into a complex number we use the following
% elementary macro:
%    \begin{macrocode}
\def\MakeVectorFrom#1#2to#3{\edef#3{#1,#2}\ignorespaces}%
%    \end{macrocode}
% Another elementary macro copies a complex number into another one:
%    \begin{macrocode}
\def\CopyVect#1to#2{\edef#2{#1}\ignorespaces}%
%    \end{macrocode}
% The magnitude is determined with the macro |\ModOfVect| with delimited
% arguments; as usual it is assumed that the results are retrieved by means of
% control sequences, not used directly.
%
%  The magnitude $M$ is determined by taking the moduli of the real and
% imaginary parts, changing their signs if necessary; the larger component is
% then taken as the reference one so that, if $a$ is larger than $b$, the
% square root of the sum of their squares is computed as such:
% \[
% M = \sqrt{a^2+b^2} = \vert a\vert\sqrt{1+(b/a)^2}
% \]
% In this way the radicand never exceeds 2 and it is quite easy to get its
% square root by means of the Newton iterative process; due to the quadratic
% convergence, five iterations are more than sufficient. When one of the
% components is zero, the Newton iterative process is skipped. The overall
% macro is the following:
%    \begin{macrocode}
\def\ModOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\@tempdima=\t@X\p@ \ifdim\@tempdima<\z@ \@tempdima=-\@tempdima\fi
\@tempdimb=\t@Y\p@ \ifdim\@tempdimb<\z@ \@tempdimb=-\@tempdimb\fi
\ifdim\@tempdima>\@tempdimb
   \DividE\@tempdimb by\@tempdima to\@T
   \@tempdimc=\@tempdima
\else
   \DividE\@tempdima by\@tempdimb to\@T
   \@tempdimc=\@tempdimb
\fi
\ifdim\@T\p@=\z@
\else
   \@tempdima=\@T\p@ \@tempdima=\@T\@tempdima
   \advance\@tempdima\p@%
   \@tempdimb=\p@%
   \@tempcnta=5\relax
   \@whilenum\@tempcnta>\z@\do{\DividE\@tempdima by\@tempdimb to\@T
   \advance\@tempdimb \@T\p@ \@tempdimb=.5\@tempdimb
   \advance\@tempcnta\m@ne}%
   \@tempdimc=\@T\@tempdimc
\fi
\Numero#2\@tempdimc
\ignorespaces}%
%    \end{macrocode}
% As a byproduct of the computation the control sequence |\@tempdimc| contains
% the vector or complex number magnitude multiplied by the length of one point.
%
% Since the macro for determining the magnitude of a vector is available, we
% can now normalize the vector to its magnitude, therefore getting the Cartesian
% form of the direction vector. If by any chance the direction of the null
% vector is requested, the output is again the null vector, without
% normalization.
%    \begin{macrocode}
\def\DirOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to\@tempa
\unless\ifdim\@tempdimc=\z@
   \DividE\t@X\p@ by\@tempdimc to\t@X
   \DividE\t@Y\p@ by\@tempdimc to\t@Y
\fi
\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
%    \end{macrocode}
%
% A cumulative macro uses the above ones for determining with one call both the
% magnitude and the direction of a complex number. The first argument is the
% input complex number, the second its magnitude, and the third is again a
% complex number normalized to unit magnitude (unless the input was the null
% complex number); remember always that output quantities must be specified
% with control sequences to be used at a later time.
%    \begin{macrocode}
\def\ModAndDirOfVect#1to#2and#3{%
\GetCoord(#1)\t@X\t@Y
\ModOfVect#1to#2%
\ifdim\@tempdimc=\z@\else
  \DividE\t@X\p@ by\@tempdimc to\t@X 
  \DividE\t@Y\p@ by\@tempdimc to\t@Y
\fi
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
% The next macro computes the magnitude and the direction of the difference of
% two complex numbers; the first input argument is the minuend, the second is
% the subtrahend; the output quantities are the third argument containing the
% magnitude of the difference and the fourth is the direction of the difference.
%  The service macro |\SubVect| executes the difference of two complex numbers
% and is described further on.
%    \begin{macrocode}
\def\DistanceAndDirOfVect#1minus#2to#3and#4{%
\SubVect#2from#1to\@tempa 
\ModAndDirOfVect\@tempa to#3and#4\ignorespaces}%
%    \end{macrocode}
% We now have two macros intended to fetch just the real or, respectively, the
% imaginary part of the input complex number.
%    \begin{macrocode}
\def\XpartOfVect#1to#2{%
\GetCoord(#1)#2\@tempa\ignorespaces}%
%
\def\YpartOfVect#1to#2{%
\GetCoord(#1)\@tempa#2\ignorespaces}%
%    \end{macrocode}
% With the next macro we create a direction vector (second argument) from a
% given angle (first argument).
%    \begin{macrocode}
\def\DirFromAngle#1to#2{%
\CosOf#1to\t@X
\SinOf#1to\t@Y
\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
%    \end{macrocode}
%
% As of today the anomaly (angle) of a complex number may not be necessary, but
% it might become useful in the future; therefore with macro \verb|\ArgOfVect|
% we calculate the four quadrant arctangent (in degrees) of the given vector
% taking into account the sings of the vector components. For the principal
% value of the arctangent we would like to use the continued fraction:
%\begin{equation}
%\arctan x = \cfrac{x}{1+ \cfrac{x^2}{3-x^2 + \cfrac{(3x)^2}{5-3x^2 + 
%            \cfrac{(5x)^2}{7-5x^2 + \cfrac{(7x)^2}{9-7x^2 + \ddots}}}}}
%\label{equ:arctan-fraz-cont}
%\end{equation}
% but after some testing we had to give up due to the slow convergence of 
% continued fraction~\eqref{equ:arctan-fraz-cont}, strictly connected with
% the slow convergence of the McLaurin series from which it is derived.
%
% Waiting for a faster convergence continued fraction, we examined the
% parametric formula and its inverse:
%\begin{equation}
%\begin{subequations}
%\begin{aligned}
%\tan\theta     &= \frac{2\tan(\theta/2))}{1 - \tan^2(\theta/2)}\\
%\tan(\theta/2) &= \frac{\sqrt{\tan^2\theta +1}-1}{\tan\theta}
%\label{equ:tanfimezzi}
%\end{aligned}
%\end{subequations}
%\end{equation}
% If we count the times we use the above formula we can arrive at a point
% where we have to compute the arctangent of a very small value, where the
% arctangent and it argument are approximately equal, so that the angle value
% in radians is equal to its tangent; at that point we multiply by $2^n$,
% where $n$ is the number of bisections and transform the radians in degrees.
% The procedure is pretty good, even if is is very rudimental and based on an
% approximation; the fixed radix computation of the typesetting engine does
% not help, but we get pretty decent results, although we loose some accuracy
% that hopefully would not harm further computations.
%
% The results obtainable with equation~\eqref{equ:tanfimezzi} are possibly
% acceptable, but the square that must be computed in it tends to go in
% underflow if too many iterations are performed and the algorthim crashes;
% therefore it's virtually impossibile to get more than three correct digits
% after the decimal separator.
% 
% It is probably better to refer to the Newton iterations for solving the
% equation:
%\begin{equation}
% \tan\theta -\tan\theta_\infty= 0
%\end{equation}
% in the unknown $\theta$ given the value $t=\tan\theta_\infty$; see
% figure~\ref{fig:tangenti}.
%
%\begin{figure}\centering\unitlength=0.007\textwidth
%\begin{picture}(100,70)
%\put(10,63){\framebox(18,7){$y=\tan\theta$}}
%\put(30,63){\framebox(20,7){$t=\tan\theta_\infty$}}
%\put(0,0){\vector(1,0){100}}\put(100,3){\makebox(0,0)[br]{$\theta$}}
%\put(0,0){\vector(0,1){70}}\put(3,70){\makebox(0,0)[tl]{$y$}}
%\multiput(75,0)(0,5){14}{\line(0,1){2.5}}\put(77,2){\makebox(0,0)[bl]{$\pi/2$}}
%{\linethickness{1pt}\cbezier(0,0)(5,5)(55,40)(60,70)}
%\put(51,50){\circle*{2}}
%\multiput(51,0)(0,5){10}{\line(0,1){2.5}}\put(54,3){\makebox(0,0)[bl]{$\theta_{i-1}$}}
%\multiput(0,50)(5,0){10}{\line(1,0){2.5}}\put(3,53){\makebox(0,0)[bl]{$y_{i-1}$}}
%\put(0,20){\line(1,0){70}}\put(3,23){\makebox(0,0)[bl]{$t$}}
%\Line(34,20)(51,50)
%\put(34,20){\circle*{2}}
%\multiput(34,0)(0,5){4}{%
%  \line(0,1){2.5}}\put(36,3){\makebox(0,0)[bl]{$\theta_{i}$}}
%\put(24,20){\circle*{2}}
%\multiput(24,0)(0,5){4}{\line(0,1){2.5}}\put(21,3){\makebox(0,0)[br]{$\theta_\infty$}}
%\end{picture}
%\caption{Newton method}\label{fig:tangenti}
%\end{figure}
%
% The iterative algorithm with Newton method implies the recurrence
%\begin{equation}\begin{subequations}\begin{aligned}
%y'_{i-1} &= \frac{\diff\tan(\theta_{i-1})}{\diff\theta} 
%         = \frac{1}{\cos^2\theta_{i-1}}\\
%\theta_i &= \theta_{i-1} - y'_{i-1}(\tan \theta_{i-1} - t)
%         =\theta_{i-1} - \cos^2 \theta_{i-1}(\tan \theta_{i-1} - t)
%\end{aligned}
%\label{equ:iterazione}
%\end{subequations}\end{equation}
%
% The algorithm starts with an initial value $\theta_0$, at each iteration
% for $i=1, 2, 3,\dots$ a new value of $\theta_i$ is computed from the data
% of the previous iteration $i-1$. When for a certain $i$, $\tan\theta_i$
% is sufficiently close to $t$, the iterations may be stopped; since we
% already have the algorithms for computing both the tangent and the cosine;
% such Newton iterative method dos not pose any problems, especially if we
% use the properties of the trigonometric functions and we confine the
% computations to the first quadrant.
%    \begin{macrocode}
\def\ArcTanOf#1to#2{\bgroup
\edef\@tF{#1}\@tdF=\@tF\p@
\@tdE=57.295779\p@
\ifdim\@tdF=\z@\def\@tX{0}\else
\edef\@tXX{1}%
\MultiplY57.295779by\@tXX to \@tX
\countdef\I 2323 \I=7\relax
\@whilenum\I>0\do{\TanOf\@tX to\@tG 
\CosOf\@tX to \@tH 
\edef\@tG{\strip@pt\dimexpr\@tG\p@-\@tdF\relax}%
\MultiplY\@tH by\@tH to\@tH
\MultiplY\@tH by\@tG to \@tH
\edef\@tXX{\strip@pt\dimexpr\@tXX\p@ - \@tH\p@\relax}%
\MultiplY57.295779by\@tXX to\@tX
\advance\I\m@ne}\fi
\edef\x{\egroup\noexpand\edef\noexpand#2{\@tX}}\x}%
%    \end{macrocode} 
%
% Now we have the algorithm to compute the arctangent of a number; and
% it should be relatively easy to compute the angle of a complex number.
% We have to pay attention that the algorithm to compute the arctangent
% does not care about the quadrant where the complex number lays in, and
% it yields the principal value of the arctan in the domain $\pi/2 <
% \theta \leq \pi/2$. with complex numbers we have just a sign change in
% their angle when the lay in the first or the fourth quadrants; while
% for the third and second quadrants we have to reflect the complex number
% to its opposite and in the result we have to add a ``flat angle'', that
% is 180°, since we are working in degrees. Even if mathematically it
% is undefined we decided to assign a null angle to a null complex number;
% possibly a warning message would be helpful, but for drawing purposes
% we think that the problem is irrelevant.
%
%    \begin{macrocode}
\def\ArgOfVect#1to#2{\bgroup\GetCoord(#1){\t@X}{\t@Y}%
\def\s@gno{}\def\addflatt@ngle{0}
\ifdim\t@X\p@=\z@
  \ifdim\t@Y\p@=\z@
    \def\ArcTan{0}%
  \else
    \def\ArcTan{90}%
    \ifdim\t@Y\p@<\z@\def\s@gno{-}\fi
  \fi
\else
  \ifdim\t@Y\p@=\z@
    \ifdim\t@X\p@<\z@
      \def\ArcTan{180}%
    \else
      \def\ArcTan{0}%
    \fi
  \else
    \ifdim\t@X\p@<\z@%
      \def\addflatt@ngle{180}%
      \edef\t@X{\strip@pt\dimexpr-\t@X\p@}%
      \edef\t@Y{\strip@pt\dimexpr-\t@Y\p@}%
      \ifdim\t@Y\p@<\z@
        \def\s@gno{-}%
        \edef\t@Y{-\t@Y}%
      \fi
    \fi
  \DivideFN\t@Y by\t@X to \t@A
  \ArcTanOf\t@A to\ArcTan
  \fi
\fi
\edef\ArcTan{\unless\ifx\s@gno\empty\s@gno\fi\ArcTan}%
\unless\ifnum\addflatt@ngle=0\relax 
   \edef\ArcTan{%
   \strip@pt\dimexpr\ArcTan\p@\ifx\s@gno\empty-\else+\fi
   \addflatt@ngle\p@\relax}%
\fi
\edef\x{\noexpand\egroup\noexpand\edef\noexpand#2{\ArcTan}}%
\x\ignorespaces}
%    \end{macrocode}
%^^A \begin{tabular}{ll}
%^^A 0		& \ArcTanOf 0 to\Res \Res\\
%^^A 1		& \ArcTanOf 1 to\Res \Res\\
%^^A 2		& \ArcTanOf 2 to\Res \Res\\
%^^A 0.5	& \ArcTanOf 0.5 to\Res \Res\\
%^^A 0.707	& \ArcTanOf 0.707 to\Res \Res\\
%^^A \end{tabular}
%^^A
%^^A\bigskip
%^^A
%^^A \begin{tabular}{rl}
%^^A 0,0		& \ArgOfVect0,0to\Res \Res\\
%^^A 1,0		& \ArgOfVect1,0to\Res \Res\\
%^^A -1,0		& \ArgOfVect-1,0to\Res \Res\\
%^^A 0,1		& \ArgOfVect0,1to\Res \Res\\
%^^A 0,-1		& \ArgOfVect0,-1to\Res \Res\\
%^^A 1,1		& \ArgOfVect1,1to\Res \Res\\
%^^A 1,-1		& \ArgOfVect1,-1to\Res \Res\\
% ^^A-1,1		& \ArgOfVect-1,1to\Res \Res\\
% ^^A-1,-1		& \ArgOfVect-1,-1to\Res \Res\\
%^^A \end{tabular}
% It is worth noting that the absolute error in these computations is lower
% than 0.001°, that is 0.000017\,rad; pretty satisfactory since the typesetting
% engines work in fixed radix notation with 16 fractional binary digits, and
% an error on the fifth fractional digit is almost the best it can be expected
% from this kind of arithmetics. 
%
% Sometimes it is necessary to scale a vector by an arbitrary real factor; this
% implies scaling both the real and imaginary part of the input given vector.
%    \begin{macrocode}
\def\ScaleVect#1by#2to#3{\GetCoord(#1)\t@X\t@Y
\@tempdima=\t@X\p@ \@tempdima=#2\@tempdima\Numero\t@X\@tempdima
\@tempdima=\t@Y\p@ \@tempdima=#2\@tempdima\Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
% Again, sometimes it is necessary to reverse the direction of rotation; this
% implies changing the sign of the imaginary part of a given complex number;
% this operation produces the complex conjugate of the given number.
%    \begin{macrocode}
\def\ConjVect#1to#2{\GetCoord(#1)\t@X\t@Y
\@tempdima=-\t@Y\p@\Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
%    \end{macrocode}
%
% With all the low level elementary operations we can now proceed to the
% definitions of the binary operations on complex numbers. We start with the
% addition:
%    \begin{macrocode}
\def\AddVect#1and#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y 
\@tempdima\tu@X\p@\advance\@tempdima\td@X\p@ \Numero\t@X\@tempdima 
\@tempdima\tu@Y\p@\advance\@tempdima\td@Y\p@ \Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
% Then the subtraction:
%    \begin{macrocode}
\def\SubVect#1from#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y 
\@tempdima\td@X\p@\advance\@tempdima-\tu@X\p@ \Numero\t@X\@tempdima 
\@tempdima\td@Y\p@\advance\@tempdima-\tu@Y\p@ \Numero\t@Y\@tempdima
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%    \end{macrocode}
%
% For the multiplication we need to split the operation according to the fact
% that we want to multiply by the second operand or by the complex conjugate of
% the second operand; it would be nice if we could use the usual
% postfixed asterisk notation for the complex conjugate, but I could not find
% a simple means for doing so; therefore I use the prefixed notation, that is
% I put the asterisk before the second operand. The first part of the
% multiplication macro just takes care of the multiplicand and then checks for
% the asterisk; if there is no asterisk it calls a second service macro that
% performs a regular complex multiplication, otherwise it calls a third
% service macro that executes the conjugate multiplication.
%    \begin{macrocode}
\def\MultVect#1by{\@ifstar{\@ConjMultVect#1by}{\@MultVect#1by}}%
%
\def\@MultVect#1by#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y 
\@tempdima\tu@X\p@ \@tempdimb\tu@Y\p@
\@tempdimc=\td@X\@tempdima\advance\@tempdimc-\td@Y\@tempdimb
\Numero\t@X\@tempdimc
\@tempdimc=\td@Y\@tempdima\advance\@tempdimc\td@X\@tempdimb
\Numero\t@Y\@tempdimc
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}%
%
\def\@ConjMultVect#1by#2to#3{\GetCoord(#1)\tu@X\tu@Y
\GetCoord(#2)\td@X\td@Y \@tempdima\tu@X\p@ \@tempdimb\tu@Y\p@
\@tempdimc=\td@X\@tempdima\advance\@tempdimc+\td@Y\@tempdimb
\Numero\t@X\@tempdimc
\@tempdimc=\td@X\@tempdimb\advance\@tempdimc-\td@Y\@tempdima
\Numero\t@Y\@tempdimc
\MakeVectorFrom\t@X\t@Y to#3\ignorespaces}
%    \end{macrocode}
%
% The division of two complex numbers implies scaling down the dividend by the
% magnitude of the divisor and by rotating the dividend scaled vector by the
% opposite direction of the divisor; therefore:
%    \begin{macrocode}
\def\DivVect#1by#2to#3{\ModAndDirOfVect#2to\@Mod and\@Dir
\DividE\p@ by\@Mod\p@ to\@Mod \ConjVect\@Dir to\@Dir
\ScaleVect#1by\@Mod to\@tempa
\MultVect\@tempa by\@Dir to#3\ignorespaces}%
%    \end{macrocode}
%
% \subsection{Arcs and curved vectors}
% We are now in the position of really doing graphic work.
% \subsubsection{Arcs}
% We start with tracing
% a circular arc of arbitrary center, arbitrary starting point and arbitrary
% aperture; the first macro checks the aperture; if this is not zero it
% actually proceeds with the necessary computations, otherwise it does
% nothing.
%    \begin{macrocode}
\def\Arc(#1)(#2)#3{\begingroup
\@tdA=#3\p@ 
\ifdim\@tdA=\z@\else
  \@Arc(#1)(#2)%
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
% The aperture is already memorized in |\@tdA|; the |\@Arc| macro receives
% the center coordinates in the first argument and the coordinates of the
% starting point in the second argument.
%    \begin{macrocode}
\def\@Arc(#1)(#2){%
\ifdim\@tdA>\z@
  \let\Segno+%
\else
  \@tdA=-\@tdA \let\Segno-%
\fi
%    \end{macrocode}
% The rotation angle sign is memorized in |\Segno| and |\@tdA| now contains the
% absolute value of the arc aperture.
% If the rotation angle is larger than $360^\circ$ a message is issued that
% informs the user that the angle will be reduced modulo $360^\circ$; this
% operation is performed by successive subtractions rather than with modular
% arithmetics on the assumption that in general one subtraction suffices.
%    \begin{macrocode}
\Numero\@gradi\@tdA
\ifdim\@tdA>360\p@
 \PackageWarning{curve2e}{The arc aperture is \@gradi\space degrees
      and gets reduced\MessageBreak%
      to the range 0--360 taking the sign into consideration}%
 \@whiledim\@tdA>360\p@\do{\advance\@tdA-360\p@}%
\fi
%    \end{macrocode}
% Now the radius is determined and the drawing point is moved to the stating
% point.
%    \begin{macrocode}
\SubVect#2from#1to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
%    \end{macrocode}
% From now on it's better to define a new macro that will be used also in the
% subsequent macros that trace arcs; here we already have the starting point
% coordinates and the angle to draw the arc, therefore we just call the new
% macro, stroke the line and exit.
%    \begin{macrocode}
\@@Arc
\strokepath\ignorespaces}%
%    \end{macrocode}
% And the new macro |\@@Arc| starts with moving the drawing point to the first
% point and does everything needed for tracing the requested arc, except
% stroking it; I leave the \texttt{stroke} command to the completion of the
% calling macro and nobody forbids to use the |\@@Arc| macro for other purposes.
%    \begin{macrocode}
\def\@@Arc{%
\pIIe@moveto{\@pPunX\unitlength}{\@pPunY\unitlength}%
%    \end{macrocode}
% If the aperture is larger than $180^\circ$ it traces a semicircle in the
% right direction and correspondingly reduces the overall aperture.
%    \begin{macrocode}
\ifdim\@tdA>180\p@
  \advance\@tdA-180\p@
  \Numero\@gradi\@tdA
  \SubVect\@pPun from\@Cent to\@V
  \AddVect\@V and\@Cent to\@sPun
  \MultVect\@V by0,-1.3333333to\@V \if\Segno-\ScaleVect\@V by-1to\@V\fi
  \AddVect\@pPun and\@V to\@pcPun
  \AddVect\@sPun and\@V to\@scPun
  \GetCoord(\@pcPun)\@pcPunX\@pcPunY
  \GetCoord(\@scPun)\@scPunX\@scPunY
  \GetCoord(\@sPun)\@sPunX\@sPunY
  \pIIe@curveto{\@pcPunX\unitlength}{\@pcPunY\unitlength}%
               {\@scPunX\unitlength}{\@scPunY\unitlength}%
               {\@sPunX\unitlength}{\@sPunY\unitlength}%
  \CopyVect\@sPun to\@pPun
\fi
%    \end{macrocode}
% If the remaining aperture is not zero it continues tracing the rest of the arc.
% Here we need the extrema of the arc and the coordinates of the control points
% of the Bézier cubic spline that traces the arc. The control points lay on the
% perpendicular to the vectors that join the arc center to the starting
% and end points respectively. Their distance $K$ from the adjacent nodes is
% determined with the formula
% \[
% K= \frac{4}{3}\,\frac{1-\cos\theta}{\sin\theta}R
% \]
% where $\theta$ is half the arc aperture and $R$ is its radius.
%    \begin{macrocode}
\ifdim\@tdA>\z@
  \DirFromAngle\@gradi to\@Dir \if\Segno-\ConjVect\@Dir to\@Dir \fi
  \SubVect\@Cent from\@pPun to\@V
  \MultVect\@V by\@Dir to\@V
  \AddVect\@Cent and\@V to\@sPun
  \@tdA=.5\@tdA \Numero\@gradi\@tdA
  \DirFromAngle\@gradi to\@Phimezzi
  \GetCoord(\@Phimezzi)\@cosphimezzi\@sinphimezzi
  \@tdB=1.3333333\p@ \@tdB=\@Raggio\@tdB
  \@tdC=\p@ \advance\@tdC -\@cosphimezzi\p@ \Numero\@tempa\@tdC
  \@tdB=\@tempa\@tdB
  \DividE\@tdB by\@sinphimezzi\p@ to\@cZ
  \ScaleVect\@Phimezzi by\@cZ to\@Phimezzi
  \ConjVect\@Phimezzi to\@mPhimezzi
  \if\Segno-%
    \let\@tempa\@Phimezzi
    \let\@Phimezzi\@mPhimezzi
    \let\@mPhimezzi\@tempa
  \fi
  \SubVect\@sPun from\@pPun to\@V
  \DirOfVect\@V to\@V
  \MultVect\@Phimezzi by\@V to\@Phimezzi
  \AddVect\@sPun and\@Phimezzi to\@scPun
  \ScaleVect\@V by-1to\@V
  \MultVect\@mPhimezzi by\@V to\@mPhimezzi
  \AddVect\@pPun and\@mPhimezzi to\@pcPun
  \GetCoord(\@pcPun)\@pcPunX\@pcPunY
  \GetCoord(\@scPun)\@scPunX\@scPunY
  \GetCoord(\@sPun)\@sPunX\@sPunY
  \pIIe@curveto{\@pcPunX\unitlength}{\@pcPunY\unitlength}%
               {\@scPunX\unitlength}{\@scPunY\unitlength}%
               {\@sPunX\unitlength}{\@sPunY\unitlength}%
\fi}
%    \end{macrocode}
%
% \subsubsection{Arc vectors}
% We exploit much of the above definitions for the |\Arc| macro for drawing
% circular arcs with an arrow at one or both ends; the first macro
% |\VerctorArc| draws an arrow at the ending point of the arc; the second macro
% |\VectorARC| draws arrows at both ends; the arrows have the same shape as
% those for vectors; actually they are drawn by putting a vector of zero
% length at the proper arc end(s), therefore they are styled as traditional \LaTeX\
% or PostScript arrows according to the option of the \texttt{pict2e} package.
%
% But the specific drawing done here shortens the arc so as not to overlap on
% the arrow(s); the only arrow (or both ones) are also lightly tilted in order to
% avoid the impression of a corner where the arc enters the arrow tip.
%
%  All these operations require a lot of ``playing'' with vector directions,
% but even if the operations are numerous, they do not do anything else but:
% (a) determining the end point and its direction; (b) determining the arrow
% length as an angular quantity, i.e. the arc amplitude that must be subtracted
% from the total arc to be drawn; (c) the direction of the arrow should be
% corresponding to the tangent to the arc at the point where the arrow tip is
% attached;(d) tilting the arrow tip by half its angular amplitude; (e)
% determining the resulting position and direction of the arrow tip so as to
% draw a zero length vector; (f) possibly repeating the same procedure for the
% other end of the arc; (g) shortening the total arc angular amplitude by the
% amount of the  arrow tip(s) already set, and (h) then drawing the final circular
% arc that joins the starting point to the final arrow or one arrow to the other
% one.
%
%  The calling macros are very similar to the |\Arc| macro initial one:
%    \begin{macrocode}
\def\VectorArc(#1)(#2)#3{\begingroup
\@tdA=#3\p@ \ifdim\@tdA=\z@\else
  \@VArc(#1)(#2)%
\fi
\endgroup\ignorespaces}%
%
\def\VectorARC(#1)(#2)#3{\begingroup
\@tdA=#3\p@
\ifdim\@tdA=\z@\else
  \@VARC(#1)(#2)%
\fi
\endgroup\ignorespaces}%
%    \end{macrocode}
%
% The single arrowed arc is defined with the following long macro where all the
% described operations are performed more or less in the described succession;
% probably the macro requires a little cleaning, but since it works fine I did
% not try to optimize it for time or number of tokens. The final part of the
% macro is almost identical to that of the plain arc; the beginning also is
% quite similar. The central part is dedicated to the positioning of the arrow
% tip and to the necessary calculations for determining the tip tilt and the
% reduction of the total arc length; pay attention that the arrow length, stored
% in |\@tdE| is a real length, while the radius stored in |\@Raggio| is just
% a multiple of the |\unitlength|, so that the division (that yields a good
% angular approximation to the arrow length as seen from the center of the arc)
% must be done with real lengths. The already defined |\@@Arc| macro actually
% draws the curved vector stem without stroking it.
%    \begin{macrocode}
\def\@VArc(#1)(#2){%
\ifdim\@tdA>\z@
  \let\Segno+%
\else
  \@tdA=-\@tdA \let\Segno-%
\fi \Numero\@gradi\@tdA
\ifdim\@tdA>360\p@
 \PackageWarning{curve2e}{The arc aperture is \@gradi\space degrees
      and gets reduced\MessageBreak%
      to the range 0--360 taking the sign into consideration}%
 \@whiledim\@tdA>360\p@\do{\advance\@tdA-360\p@}%
\fi
\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
\@tdE=\pIIe@FAW\@wholewidth \@tdE=\pIIe@FAL\@tdE
\DividE\@tdE by \@Raggio\unitlength to\DeltaGradi
\@tdD=\DeltaGradi\p@
\@tdD=57.29578\@tdD \Numero\DeltaGradi\@tdD
\@tdD=\ifx\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
\DirFromAngle\@tempa to\@Dir
\MultVect\@V by\@Dir to\@sPun
\edef\@tempA{\ifx\Segno-\m@ne\else\@ne\fi}%
\MultVect\@sPun by 0,\@tempA to\@vPun
\DirOfVect\@vPun to\@Dir
\AddVect\@sPun and #1 to \@sPun
\GetCoord(\@sPun)\@tdX\@tdY
\@tdD\ifx\Segno--\fi\DeltaGradi\p@
\@tdD=.5\@tdD \Numero\DeltaGradi\@tdD
\DirFromAngle\DeltaGradi to\@Dird
\MultVect\@Dir by*\@Dird to\@Dir
\GetCoord(\@Dir)\@xnum\@ynum
\put(\@tdX,\@tdY){\vector(\@xnum,\@ynum){0}}%
\@tdE =\ifx\Segno--\fi\DeltaGradi\p@
\advance\@tdA -\@tdE \Numero\@gradi\@tdA
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
\@@Arc
\strokepath\ignorespaces}%
%    \end{macrocode}
%
% The macro for the arc terminated with arrow tips at both ends is again very
% similar, except it is necessary to repeat the arrow tip positioning also at
% the starting point. The |\@@Arc| macro draws the curved stem.
%    \begin{macrocode}
\def\@VARC(#1)(#2){%
\ifdim\@tdA>\z@
  \let\Segno+%
\else
  \@tdA=-\@tdA \let\Segno-%
\fi \Numero\@gradi\@tdA
\ifdim\@tdA>360\p@
 \PackageWarning{curve2e}{The arc aperture is \@gradi\space degrees
      and gets reduced\MessageBreak%
      to the range 0--360 taking the sign into consideration}%
 \@whiledim\@tdA>360\p@\do{\advance\@tdA-360\p@}%
\fi
\SubVect#1from#2to\@V \ModOfVect\@V to\@Raggio \CopyVect#2to\@pPun
\@tdE=\pIIe@FAW\@wholewidth \@tdE=0.8\@tdE
\DividE\@tdE by \@Raggio\unitlength to\DeltaGradi
\@tdD=\DeltaGradi\p@ \@tdD=57.29578\@tdD \Numero\DeltaGradi\@tdD
\@tdD=\ifx\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
\DirFromAngle\@tempa to\@Dir
\MultVect\@V by\@Dir to\@sPun% corrects the end point
\edef\@tempA{\ifx\Segno-\m@ne\else\@ne\fi}%
\MultVect\@sPun by 0,\@tempA to\@vPun
\DirOfVect\@vPun to\@Dir
\AddVect\@sPun and #1 to \@sPun
\GetCoord(\@sPun)\@tdX\@tdY
\@tdD\ifx\Segno--\fi\DeltaGradi\p@
\@tdD=.5\@tdD \Numero\@tempB\@tdD
\DirFromAngle\@tempB to\@Dird
\MultVect\@Dir by*\@Dird to\@Dir
\GetCoord(\@Dir)\@xnum\@ynum
\put(\@tdX,\@tdY){\vector(\@xnum,\@ynum){0}}% arrow tip at the end point
\@tdE =\DeltaGradi\p@
\advance\@tdA -2\@tdE \Numero\@gradi\@tdA
\CopyVect#1to\@Cent \GetCoord(\@pPun)\@pPunX\@pPunY
\SubVect\@Cent from\@pPun to \@V
\edef\@tempa{\ifx\Segno-\else-\fi\@ne}%
\MultVect\@V by0,\@tempa to\@vPun
\@tdE\ifx\Segno--\fi\DeltaGradi\p@
\Numero\@tempB{0.5\@tdE}%
\DirFromAngle\@tempB to\@Dird
\MultVect\@vPun by\@Dird to\@vPun% corrects the starting point
\DirOfVect\@vPun to\@Dir\GetCoord(\@Dir)\@xnum\@ynum
\put(\@pPunX,\@pPunY){\vector(\@xnum,\@ynum){0}}% arrow tip at the starting point
\edef\@tempa{\ifx\Segno--\fi\DeltaGradi}%
\DirFromAngle\@tempa to \@Dir
\SubVect\@Cent from\@pPun to\@V
\MultVect\@V by\@Dir to\@V
\AddVect\@Cent and\@V to\@pPun
\GetCoord(\@pPun)\@pPunX\@pPunY
\@@Arc
\strokepath\ignorespaces}%
%    \end{macrocode}
%
% It must be understood that the curved vectors, the above circular arcs
% terminated with an arrow tip at one or both ends, have a nice appearance only
% if the arc radius is not too small, or, said in a different way, if the arrow
% tip  angular width does not exceed a maximum of a dozen degrees (and this is
% probably already too much); the tip does not get curved as the arc is,
% therefore there is not a smooth transition from the curved stem and the
% straight arrow tip if this one is large in comparison to the arc radius.
%
% \subsection{General curves}
% Now we define a macro for tracing a general, not necessarily circular arc.
% This macro resorts to a general triplet of macros with which it is possible
% to draw almost anything. It traces a single Bézier spline from a first point
% where the tangent direction is specified to a second point where again it is
% specified the tangent direction. Actually this is a special (possibly useless)
% case where the general |\curve| macro could do the same or a better job. In
% any case\dots
%    \begin{macrocode}
\def\CurveBetween#1and#2WithDirs#3and#4{%
\StartCurveAt#1WithDir{#3}\relax
\CurveTo#2WithDir{#4}\CurveFinish}%
%    \end{macrocode}
%
% Actually the above macro is a special case of concatenation of the triplet
% formed by macros |\StartCurve|, |\CurveTo| and|\CurveFinish|; the second of
% which can be repeated an arbitrary number of times.
%
% The first macro initializes the drawing and the third one strokes it; the
% real work is done by the second macro. The first macro initializes the
% drawing but also memorizes the starting direction; the second macro traces
% the current Bézier arc reaching the destination point with the specified
% direction, but memorizes this direction as the one with which to start the
% next arc. The overall curve is then always smooth because the various
% Bézier arcs join with continuous tangents. If a cusp is desired it is
% necessary to change the memorized direction at the end of the arc before the
% cusp and before the start of the next arc; this is better than stroking the
% curve before the cusp and then starting another curve, because the curve
% joining point at the cusp is not stroked with the same command, therefore we get
% two superimposed curve terminations. We therefore need another small macro
% |\ChangeDir| to perform this task.
%
% It is necessary to recall that the directions point to the control points,
% but they do not define the control points themselves; they are just
% directions, or, even better, they are simply vectors with the desired
% direction; the macros themselves provide to the normalization and
% memorization.
%
% The next desirable point would be to design a macro that accepts optional node
% directions and computes the missing ones according to a suitable strategy. I
% can think of many such strategies, but none seems to be generally applicable,
% in the sense that one strategy might give good results, say, with sinusoids
% and another one, say, with cardioids, but neither one is suitable for both
% cases.
%
% For the moment we refrain from automatic direction computation, but we design
% the general macro as if directions were optional.
%
%  Here we begin with the first initializing macro that receives in the  first
% argument the starting point and in the second argument the direction of the
% tangent (not necessarily normalized to a unit vector)
%    \begin{macrocode}
\def\StartCurveAt#1WithDir#2{%
\begingroup
\GetCoord(#1)\@tempa\@tempb
\CopyVect\@tempa,\@tempb to\@Pzero
\pIIe@moveto{\@tempa\unitlength}{\@tempb\unitlength}%
\GetCoord(#2)\@tempa\@tempb
\CopyVect\@tempa,\@tempb to\@Dzero
\DirOfVect\@Dzero to\@Dzero}
%    \end{macrocode}
% And this re-initializes the direction after a cusp
%    \begin{macrocode}
\def\ChangeDir<#1>{%
\GetCoord(#1)\@tempa\@tempb
\CopyVect\@tempa,\@tempb to\@Dzero
\DirOfVect\@Dzero to\@Dzero
\ignorespaces}
%    \end{macrocode}
%
% The next macro is the finishing one; it strokes the whole curve and closes the
% group that was opened with |\StartCurve|.
%    \begin{macrocode}
\def\CurveFinish{\strokepath\endgroup\ignorespaces}%
%    \end{macrocode}
%
% The ``real'' curve macro comes next; it is supposed to determine the control
% points for joining the previous point (initial node) with the specified
% direction to the next point with another specified direction (final node).
% Since the control points are along the specified directions, it is necessary
% to determine the distances from the adjacent curve nodes. This must work
% correctly even if nodes and directions imply an inflection point somewhere
% along the arc.
%
% The strategy I devised consists in determining each control point as if it
% were the control point of a circular arc, precisely an arc of an
% osculating circle, a circle tangent to the curve at that node. The ambiguity
% of the stated problem may be solved by establishing that the chord of the
% osculating circle has the same direction as the chord of the arc being drawn,
% and that the curve chord is divided into two parts each of which should be
% interpreted as half the chord of the osculating circle; this curve chord
% division is made proportionally to the projection of the tangent directions
% on the chord itself. Excluding degenerate cases that may be dealt with
% directly, imagine the triangle built with the chord and the two tangents;
% this triangle is straightforward if there is no inflection point; otherwise it
% is necessary to change one of the two directions by reflecting it about the
% chord. This is much simpler to view if a general rotation of the whole
% construction is made so as to bring the curve chord on the $x$ axis, because
% the reflection about the chord amounts to taking the complex conjugate of one
% of the directions. In facts with a concave curve the ``left'' direction
% vector arrow and the ``right'' direction vector tail lay in the same half
% plane, while with an inflected curve, they lay in opposite half plains, so
% that taking the complex conjugate of one of directions re-establishes the
% correct situation for the triangle we are looking for.
%
% This done the perpendicular from the triangle vertex to the cord divides the
% chord in two parts (the foot of this perpendicular may lay outside the chord,
% but this is no problem since we are looking for positive solutions, so that
% if we get negative numbers we just negate them); these two parts are taken as
% the half chords of the osculating circles, therefore there is no problem
% determining the distances $K_{\mathrm{left}}$ and $K_{\mathrm{right}}$ from
% the left and right
% nodes by using the same formula we used with circular arcs. Well\dots\ the
% same formula means that we have to determine the radius from the half chord
% and its inclination with the node tangent; all things we can do with the
% complex number algebra and macros we already have at our disposal. If we look
% carefully at this computation done for the circular arc we discover that in
% practice we used the half chord length instead of the radius; so the coding
% is actually the same, may be just with different variable names.
%
% We therefore start with getting the points and directions and calculating the
% chord and its direction
%    \begin{macrocode}
\def\CurveTo#1WithDir#2{%
\def\@Puno{#1}\def\@Duno{#2}\DirOfVect\@Duno to\@Duno
\DistanceAndDirOfVect\@Puno minus\@Pzero to\@Chord and\@DirChord
%    \end{macrocode}
% Then we rotate everything about the starting point so as to bring the chord on
% the real axis
%    \begin{macrocode}
\MultVect\@Dzero by*\@DirChord to \@Dpzero
\MultVect\@Duno by*\@DirChord to \@Dpuno
\GetCoord(\@Dpzero)\@Xpzero\@Ypzero
\GetCoord(\@Dpuno)\@Xpuno\@Ypuno
%    \end{macrocode}
% The chord needs not be actually rotated because it suffices its length
% along the real axis; the chord length is memorized in |\@Chord|.
%
% We now examine the various degenerate cases, when either tangent is
% perpendicular to the chord, or when it is parallel pointing inward or outward,
% with or without inflection.
%
% We start with the $90^\circ$ case for the ``left'' direction
% separating the cases when the other direction is or is not $90^\circ$~\dots
%    \begin{macrocode}
\ifdim\@Xpzero\p@=\z@
  \ifdim\@Xpuno\p@=\z@
    \@tdA=0.666666\p@
    \Numero\@Mcpzero{\@Chord\@tdA}%
    \edef\@Mcpuno{\@Mcpzero}%
  \else
    \@tdA=0.666666\p@
    \Numero\@Mcpzero{\@Chord\@tdA}%
    \SetCPmodule\@Mcpuno from\@ne\@Chord\@Dpuno%
  \fi
%    \end{macrocode}
% \dots\ from when the ``left'' direction is not perpendicular to the chord; it
% might be  parallel and we must distinguish the cases for the other direction~\dots
%    \begin{macrocode}
\else
  \ifdim\@Xpuno\p@=\z@
    \@tdA=0.666666\p@
    \Numero\@Mcpuno{\@Chord\@tdA}%
    \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero%
  \else
     \ifdim\@Ypzero\p@=\z@
       \@tdA=0.333333\p@
       \Numero\@Mcpzero{\@Chord\@tdA}%
       \edef\@Mcpuno{\@Mcpzero}% 
%    \end{macrocode}
% \dots\ from when the left direction is oblique and the other direction is
% either parallel to the chord~\dots
%    \begin{macrocode}
     \else
       \ifdim\@Ypuno\p@=\z@
         \@tdA=0.333333\p@
         \Numero\@Mcpuno{\@Chord\@tdA}%
         \SetCPmodule\@Mcpzero from\@ne\@Chord\@Dpzero
%    \end{macrocode}
% \dots\ and, finally, from when both directions are oblique with respect to
% the chord; we must see if there is an inflection point; if both direction
% point to the same half plane we have to take the complex conjugate of one
% direction so as to define the triangle we were speaking about above.
%    \begin{macrocode}
       \else
         \@tdA=\@Ypzero\p@ \@tdA=\@Ypuno\@tdA
         \ifdim\@tdA>\z@
           \ConjVect\@Dpuno to\@Dwpuno
         \else
           \edef\@Dwpuno{\@Dpuno}%
         \fi
%    \end{macrocode}
% The control sequence |\@Dwpuno| contains the right direction for forming the
% triangle; we can make the weighed subdivision of the chord according to the
% horizontal components of the directions; we eventually turn negative values
% to positive ones since we are interested in the magnitudes of the control
% vectors.
%    \begin{macrocode}
         \GetCoord(\@Dwpuno)\@Xwpuno\@Ywpuno
         \@tdA=\@Xpzero\p@ \@tdA=\@Ywpuno\@tdA
         \@tdB=\@Xwpuno\p@ \@tdB=\@Ypzero\@tdB
         \DividE\@tdB by\@tdA to\@Fact
         \@tdC=\p@ \advance\@tdC-\@Fact\p@
         \ifdim\@tdC<\z@ \@tdC=-\@tdC\fi
         \DividE\p@ by \@Fact\p@ to\@Fact
         \@tdD=\p@ \advance\@tdD-\@Fact\p@
         \ifdim\@tdD<\z@ \@tdD=-\@tdD\fi
%    \end{macrocode}
% Before dividing by the denominator we have to check the directions, although
% oblique to the chord are not parallel to one another; in this case there is
% no question of a weighed subdivision of the chord
%    \begin{macrocode}
         \ifdim\@tdD<0.0001\p@
           \def\@factzero{1}%
           \def\@factuno{1}%
         \else
           \DividE\p@ by\@tdC to\@factzero
           \DividE\p@ by\@tdD to\@factuno
         \fi
%    \end{macrocode}
% We now have the subdivision factors and we call another macro for determining
% the required magnitudes
%    \begin{macrocode}
         \SetCPmodule\@Mcpzero from\@factzero\@Chord\@Dpzero
         \SetCPmodule\@Mcpuno from\@factuno\@Chord\@Dwpuno
      \fi
     \fi
  \fi
\fi
%    \end{macrocode}
% Now we have all data we need and we determine the positions of the control
% points; we do not work any more on the rotated diagram of the horizontal
% chord, but we operate on the original points and directions; all we had to
% compute, after all, were the distances of the control points along the
% specified directions; remember that the ``left'' control point is along the
% positive ``left'' direction, while the ``right'' control point precedes the
% curve node along the ``right'' direction, so that a vector subtraction must
% be done.
%    \begin{macrocode}
\ScaleVect\@Dzero by\@Mcpzero to\@CPzero
\AddVect\@Pzero and\@CPzero to\@CPzero
\ScaleVect\@Duno by\@Mcpuno to\@CPuno
\SubVect\@CPuno from\@Puno to\@CPuno
%    \end{macrocode}
% Now we have the four points and we can instruct the internal \texttt{pict2e}
% macros to do the path tracing.
%    \begin{macrocode}
\GetCoord(\@Puno)\@XPuno\@YPuno
\GetCoord(\@CPzero)\@XCPzero\@YCPzero
\GetCoord(\@CPuno)\@XCPuno\@YCPuno
\pIIe@curveto{\@XCPzero\unitlength}{\@YCPzero\unitlength}%
             {\@XCPuno\unitlength}{\@YCPuno\unitlength}%
             {\@XPuno\unitlength}{\@YPuno\unitlength}%
%    \end{macrocode}
%  It does not have to stroke the curve because other Bézier splines might still
%  be added to the path. On the opposite it memorizes the final point as the
% initial point of the next spline
%    \begin{macrocode}
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
%    \end{macrocode}
%
%  The next macro is used to determine the control vectors lengths when we have
% the chord fraction, the chord length and the direction along which to compute
% the vector; all the input data (arguments from \#2 to \#4) may be passed as
% control sequences so the calling statement needs not use any curly braces.
%    \begin{macrocode}
\def\SetCPmodule#1from#2#3#4{%
\GetCoord(#4)\t@X\t@Y
\@tdA=#3\p@
\@tdA=#2\@tdA
\@tdA=1.333333\@tdA
\@tdB=\p@ \advance\@tdB +\t@X\p@
\DividE\@tdA by\@tdB to#1\relax
\ignorespaces}%
%    \end{macrocode}
%
% We finally define the overall |\Curve| macro that recursively examines an
% arbitrary list of nodes and directions; node coordinates are grouped within
% regular parentheses while direction components are grouped within angle
% brackets. The first call of the macro initializes the drawing process and
% checks for the next node and direction; if a second node is missing, it issues
% a warning message and does not draw anything. It does not check for a change in
% direction, because it would be meaningless at the beginning of a curve.
% The second macro defines the path to the next point and checks for another node;
% if the next list item is a square bracket delimited argument, it interprets it as
% a change of direction, while if it is another parenthesis delimited argument it
% interprets it as a new node-direction specification; if the node and direction
% list is terminated, it issues the stroking command and exits the recursive
% process. The |@ChangeDir| macro is just an interface for executing the regular
% |\ChangeDir| macro, but also for recursing again by recalling |\@Curve|.
%    \begin{macrocode}
\def\Curve(#1)<#2>{%
    \StartCurveAt#1WithDir{#2}%
    \@ifnextchar\lp@r\@Curve{%
    \PackageWarning{curve2e}{%
    Curve specifications must contain at least two nodes!\Messagebreak
    Please, control your Curve specifications\MessageBreak}}}
\def\@Curve(#1)<#2>{%
    \CurveTo#1WithDir{#2}%
    \@ifnextchar\lp@r\@Curve{%
    \@ifnextchar[\@ChangeDir\CurveFinish}}
\def\@ChangeDir[#1]{\ChangeDir<#1>\@Curve}
%    \end{macrocode}
%
% As a concluding remark, please notice that the |\Curve| macro is certainly the
% most comfortable to use, but it is sort of frozen in its possibilities. The
% user may certainly use the |\StartCurve|, |\CurveTo|, |\ChangeDir|, and
% |\CurveFinish| for a more versatile set of drawing macros; evidently nobody
% forbids to exploit the full power of the |\cbezier| original macro for cubic
% splines.
%
% I believe that the set of new macros can really help the user to draw his/her
% diagrams with more agility; it will be the accumulated experience to decide if
% this is true.
% \Finale
% \endinput