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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.
\endpostamble
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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.
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%<*package>
%\NeedsTeXFormat{LaTeX2e}[2014/05/01]
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\ProvidesFile{curve2e.dtx}%
%
%<+package>\ProvidesPackage{curve2e}%
%<*package>
[2015/06/27 v.1.54 Extension package for pict2e]
%
%<*driver>
\documentclass{ltxdoc}\errorcontextlines=9
\hfuzz 10pt
\usepackage[utf8]{inputenc}
\usepackage{lmodern,textcomp}
\usepackage{mflogo}
\usepackage{multicol,amsmath,trace}
\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}}
\renewcommand\meta[1]{{\normalfont\textlangle\textit{#1}\textrangle}}
\renewcommand\marg[1]{\texttt{\{\meta{#1}\}}}
\providecommand\oarg{}
\renewcommand\oarg[1]{\texttt{[\meta{#1}]}}
\providecommand\aarg{}
\renewcommand*\aarg[1]{\texttt{<\meta{#1}>}}
\providecommand\parg{}
\renewcommand\parg[1]{\texttt{(\meta{#1})}}
\makeatletter
\newcommand*\Pall[1][1.5]{\def\circdiam{#1}\@Pall}
\def\@Pall(#1){\put(#1){\circle*{\circdiam}}}
\def\legenda(#1,#2)#3{\put(#1,#2){\setbox3333\hbox{$#3$}%
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\edef\@tempA{\strip@pt\dimen3333}%
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\def\Zbox(#1){\bgroup\edef\@tempA{#1}\@Zbox}
\newcommand*\@Zbox[2][]{\fboxrule\z@\fboxsep=0.75ex\relax
\setbox2575\hbox{\fbox{$\relax\rule[-0.5ex]{0pt}{2.5ex}#2\relax$}}\relax
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\begin{document}
\maketitle
\columnseprule=0.4pt
\begin{multicols}{2}
\tableofcontents
\end{multicols}
\DocInput{curve2e.dtx}
\end{document}
%
% \fi
%
% \CheckSum{3078}
% \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.
%
% 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 what follows.
% \begin{enumerate}
% \item The line and vector slopes were limited to the ratios of relative
% prime one-digit integers of magnitude not exceeding 6 for lines and 4 for
% vectors.
%^^A
% \item Filled and unfilled full circles were limited by the necessarily
% limited number of specific glyphs contained in the special \LaTeX\
% \texttt{picture} fonts.
%^^A
% \item Quarter circles were also limited in their radii for the same reason.
%^^A
% \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.
%^^A
% \item Vector arrows had only one possible shape and matched the limited
% number of vector slopes.
%^^A
% \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.
%^^A
% \item Filled and unfilled circles can be of any size.
%^^A
% \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.
%^^A
% \item There are two shapes for the arrow tips; the triangular one traditional
% with \LaTeX\ vectors, or the arrow tip with PostScript style.
%^^A
% \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 Most if not all coordinate pairs and slope pairs are treated as \emph{ordered pairs}, that is \emph{complex numbers}; in practice the user
% does not notice any difference from what he/she was used to, but all the
% mathematical treatment to be applied to these entities is coded as complex
% number operations, since complex numbers may be viewed non only as ordered
% pairs, but also as vectors or roto-amplification operators.
%^^A
% \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|).
%^^A
% \item Commands for specifying the way two lines or curves join to one another.
% ^^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 macro name now conflicts with |pict2e| 2009 version; therefore its name
% is changed to |\LIne| and supposedly it will not be used very often, if ever,
% by the end user (but it is used within this package macros).
% ^^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|; 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) get
% specified through one of the macro arguments.
% ^^A
% \item A new macro |\Dotline| is defined in order to draw dotted straight
% lines as a sequence of equally spaced dots, where the gap can be specified
% by the user; such straight line may have any inclination, as well as the
% above dashed lines.
% ^^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 in analogy with
% |\LIne|; |\VECTOR| joins two specified points (without using the |\put|
% command) with the arrow pointing to the second point.
%^^A
% \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 it is redefined so as to allow
% an optional specification of the way segments for the polyline are joined to
% one another.
%^^A
% \item A new macro |\Arc| is defined in order to draw an arc with arbitrary
% radius and arbitrary aperture (angle amplitude); this amplitude is specified in
% sexagesimal degrees, not in radians; a similar functionality is now achieved
% with the |\arc| macro of |pict2e|, which provides also the starred version
% |\arc*| that fills up the interior of the generated circular arc. It must be
% noticed that the syntax is slightly different, so that it's reasonable that
% these commands, in spite of producing identical arcs, might be more comfortable
% with this or that syntax.
%^^A
% \item Two new macros |\VectorArc| and |\VectorARC| are defined in order to
% draw circular arcs with an
% arrow at one or both ends.
%^^A
% \item A new macro |\Curve| is defined so as to draw arbitrary curved lines
% by means of cubic Bézier splines; the |\Curve| macro requires only the
% curve nodes and the directions of the tangents at each node.The starred
% version fills up the interior of the curve with the currently specified color.
%^^A
% \item |\Curve| is a recursive macro that can draw an unlimited (reasonably
% low) number of connected Bézier spline arcs with continuos tangents except
% for cusps; these arcs require only the specification of te tangent
% direction at the interpolation nodes. It is possible to use a lower level
% macro |\CbezierTo| that does the same but lets the user specify the control
% points of each arc; it is more difficult to use but it is more performant.
%^^A
% \item Last but not least, all these commands accept polar coordinates or
% cartesian ones at the choice of the user who may use for each object the
% formalism he/she prefers. Also the |put| and |\multiput| commands have been
% redefined so as to accept the cartesian or the polar coordinates.
%^^A
% \item The basic macros used within the cumulative |\Curve| macro can be
% used individually in order to draw any curve, one cubic arc at the time;
% but they are intended for internal use, even if it is not prohibited to use
% them; by themselves such arcs are not different form those used by |Curve|,
% but the final command, |\FillCurve|, should be used in place of
% |\CurveFinish|, so as to fill up the closed path with the locally
% specified color; see figure~\ref{fig:colored-curve}. It is much more
% convenient to use the starred version of |\Curve| macro.
% \end{enumerate}
%
% The |pict2e| package already defines macros such as |\moveto|,
% |\lineto|, |\curveto|, |\closepath|, |\fillpath|, and |\strokepath|; of
% course these macros can be used by the end user, and sometimes they perform
% better than the macros defined in this package, because the user has a better
% control on the position of the Bézier control points, while here the control
% points are sort of rigid. It would be very useful to resort to the |hobby|
% package, but its macros are conforming with those of the |tikz| and |pgf|
% packages, not with |curve2e|; an interface should be created in order to
% deal with the |hobby| package, but this has not yet been done.
%
% In order to make the necessary calculations many macros have been defined so
% as to use complex number arithmetics to manipulate point coordinates,
% directions (unit vectors, also known as `versors'), rotations and the like.
% The trigonometric functions have also been defined in a way that the author
% believes to be more efficient than those defined by the \texttt{trig} package;
% in any case the macro names are sufficiently different to accommodate both
% definition sets 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 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. The |\Curve|
% algorithm, as I said before, might be redefined so as to use the macros
% introduced in the \texttt{hobby} package, that implements for the |tikz| and
% |pgf| packages the same functionalities that John Hobby implemented for the
% \MF\ and \MP\ programs.
%
% For these reasons I suppose that every enhancement should be submitted to
% Gäßlein, Niepraschk, and Tkadlec who are the prime maintainers of
% \texttt{pict2e}; they are the only ones who 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 which 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.
%^^A
% \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.
%^^A
% \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 thickness 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.
%^^A
% \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.
%^^A
% \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
% using command |\put|.
% The same applies to the vector commands |\Vector| and |\VECTOR|. Experience has
% shown that the commands intended to join two specified coordinates are
% particularly useful.
%^^A
% \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 order the specified
% points; the syntax is:
% \begin{flushleft}
%\cs{polyline}\oarg{optional join style}\parg{$P_1$}\parg{$P_2$}\texttt{...}\parg{$P_n$}
% \end{flushleft}
% See figure~\ref{fig:polyline} where a regular pentagon is drawn; usage of polar
% coordinates is also shown.
%
% \begin{figure}[!ht]
% \begin{minipage}{.48\linewidth}
% \begin{verbatim}
% \unitlength=.5mm
% \begin{picture}(40,32)(-20,-20)
% \polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
% \end{picture}
% \end{verbatim}
% \end{minipage}
% \hfill
% \begin{minipage}{.48\linewidth}\raggedleft
% \unitlength=.5mm
% \begin{picture}(40,32)(-20,-20)
% \polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
% \end{picture}\hspace*{2em}
% \end{minipage}
% \caption{Polygonal line obtained by means of the \texttt{\string\polyline}
% command; coordinates are in polar form.}
% \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.
%^^A
% \item The new command |\Dashline| (alias: |\Dline| for backwards compatibility)
% \begin{flushleft}
% |\Dashline|\parg{first point}\parg{second point}\marg{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=1mm
% \begin{picture}(40,40)
% \put(0,0){\GraphGrid(40,40)}
% \Dashline(0,0)(40,10){4}
% \put(0,0){\circle*{2}}
% \Dashline(40,10)(0,25){4}
% \put(40,10){\circle*{2}}
% \Dashline(0,25)(20,40){4}
% \put(0,25){\circle*{2}}
% \put(20,40){\circle*{2}}
% \Dotline(0,0)(40,40){2}
% \end{picture}
% \end{verbatim}
% \end{minipage}
% \hfill
% \begin{minipage}{.48\textwidth}\centering
% \unitlength=1mm
% \begin{picture}(40,40)
% \put(0,0){\GraphGrid(40,40)}
% \Dashline(0,0)(40,10){4}
% \put(0,0){\circle*{2}}
% \Dashline(40,10)(0,25){4}
% \put(40,10){\circle*{2}}
% \Dashline(0,25)(20,40){4}
% \put(0,25){\circle*{2}}
% \put(20,40){\circle*{2}}
% \Dotline(0,0)(40,40){2}
% \end{picture}
% \end{minipage}
% \caption{Dashed lines and graph grid}\label{fig:dashline}
% \end{figure}
%^^A
%\item Analogous to |\Dashline|, a new command |\Dotline| draws a dotted line with
% the syntax:
% \begin{flushleft}
% |\Dotline|\parg{first point}\parg{end point}\marg{dot gap}
% \end{flushleft}
% See figures~\ref{fig:dashline} and~\ref{fig:dottedlines} for examples.
%^^A
% \item |\GraphGrid| is a command that draws a red grid under the drawing
% with lines separated |10\unitlength|s apart; 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 canvas by means of |\put|,
% whose cartesian 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 canvas. This grid is intended as an
% aid while 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.
%^^A
% \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° to one revolution), so that reduction to the fundamental quadrant
% is supposed to be more accurate; the tangent of odd multiples of 90°
% 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° 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}
% \cs{SinOf}\meta{angle}\texttt{to}\meta{control sequence}
%\\
% \cs{CosOf}\meta{angle}\texttt{to}\meta{control sequence}
%\\
% \cs{TanOf}\meta{angle}\texttt{to}\meta{control sequence}
%\end{flushleft}
% The \meta{control sequence} may then be used as a multiplying factor of a
% length.
%^^A
% \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}
% \cs{Arc}\parg{center}\parg{starting point}\marg{angle}\\
% \cs{VectorArc}\parg{center}\parg{starting point}\marg{angle}\\
% \cs{VectorARC}\parg{center}\parg{starting point}\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}
%^^A
% \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, even 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 in counterclockwise
% direction, 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{1st vector}|minus|\meta{2nd 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}}
%^^A
% \item General curves can be drawn with the |pict2e| macro |\curve| but it
% requires the specification of the third-order Bézier-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
%\[
% \parg{node coordinates}\aarg{direction vector}
%\]
% 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
%\[
% \mbox{\dots\parg{...}\aarg{...}\oarg{new direction vector}\parg{...}\aarg{...}\dots}
%\]
% 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}[htb]
% \begin{minipage}{.48\textwidth}
% \begin{verbatim}
% \unitlength=8mm\relax
% \begin{picture}(5,5)
% \put(0,0){\framebox(5,5){}}\thicklines\roundcap
% \Curve(2.5,0)<1,1>(5,3.5)<0,1>%
% (4,5)<-1,0>(2.5,3.5)<-.5,-1.2>[-.5,1.2]%
% (1,5)<-1,0>(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>(4,5)<-1,0>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](1,5)<-1,0>(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}
%\vspace*{2\baselineskip}
%
% \begin{minipage}{.48\textwidth}
% \begin{verbatim}
% \unitlength=8mm\relax
% \begin{picture}(5,5)
% \put(0,0){\framebox(5,5){}}\thicklines\roundcap
% \color{green}\relax
% \Curve*(2.5,0)<1,1>(5,3.5)<0,1>%
% (4,5)<-1,0>(2.5,3.5)<-.5,-1.2>[-.5,1.2]%
% (1,5)<-1,0>(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
% \color{green}\relax
% \Curve*(2.5,0)<1,1>(5,3.5)<0,1>(4,5)<-1,0>(2.5,3.5)<-0.5,-1.2>[-0.5,1.2](1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<1,-1>}
% \end{picture}
% \end{minipage}
%\caption{Coloring the inside of a closeded path drawn with \texttt{\string\Curve*}}
%\label{fig:colored-curve}
% \end{figure}
%
% With the starred version of |\Curve|, instead of stroking the contour,
% the macro fills up the contour with the selected current color,
% figure~\ref{fig:colored-curve}.
%
% In spite of the relative simplicity of the macros contained in this package,
% the described macros, as well as the original ones included in the |pict2e|
% package, allow to produce fine drawings that were unconceivable 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 announced;
% 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 is also the package |curves| that is intended to draw almost
% anything by using little dots or other symbols partially superimposed to one
% another. It uses 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.
%
% The |picture| package extends the performance of the |picture| environment
% (extended with \texttt{pict2e}) by accepting coordinates and lengths in real
% absolute dimensions, not only as multiples of |\unitlength|; it provides
% commands to extend that functionality to other packages. In certain
% circumstances it is very useful.
%
% Package \texttt{xpicture} builds over the |picture| \LaTeX\ environment so
% as to allow to draw the usual curves that are part of an introductory
% analytic geometry course; lines, circles, parabolas, ellipses, hyperbolas, and
% polynomials; the syntax is very comfortable; for all these curves it uses
% the quadratic Bézier splines.
%
% Package |hobby| extends the cubic Bézier spline handling with the algorithms
% John Hobby created for \MF\ and \MP. But by now this package interfaces very
% well with |tikz|; it has not (yet) been adapted to the common |picture|
% environment, even extended with |pict2e|, and, why not, with |curve2e|.
%
% \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}, 2014, PDF documentation of \texttt{pict2e} is part of any modern complete distribution of the \TeX\ system. In case of a basic or partial system installation, the package may be installed by means of the specific facilities of the distribution. It may be read by means of the line command \texttt{texdoc pict2e}.
% \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.
%\iffalse
%<*package>
%\fi
% \begin{macrocode}
\RequirePackage{color}
\RequirePackageWithOptions{pict2e}[2014/01/01]
% \end{macrocode}
%
% The next macros are just for debugging. With the \texttt{trace} 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}
%
% \subsection{Line thickness macros}
% 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.
%
% \subsection{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 inside the box created with |\put|. The two arguments
% define the horizontal and the vertical component respectively.
% \begin{macrocode}
\def\LIne(#1){{\GetCoord(#1)\@tX\@tY
\moveto(0,0)
\pIIe@lineto{\@tX\unitlength}{\@tY\unitlength}\strokepath}\ignorespaces}%
% \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 shall 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 in all other languages 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 with the 2004 version that was limited to
% integer direction coefficients up to 999 in magnitude.
%
% \subsection{Dashed and dotted lines}
% Dashed and dotted lines are very useful in technical drawings; here we
% introduce four macros that help drawing them in the proper way; besides
% the obvious difference between the use of dashes or dots, they may refer
% in a different way to the end points that must be specified to the various
% macros.
%
% The coordinates of the first point $P_1$, where le line starts, are always
% referred to the origin of the coordinate axes; the end point $P_2$
% coordinates with the first macro type are referred to the origin of the
% axes, while with the second macro type they are referred to $P_1$; both
% macro types have their usefulness and figures~\ref{fig:dashedlines}
% and~\ref{fig:dottedlines} show how to use these macro types.
%
% We distinguish these macro types with an asterisk; the unstarred version is
% the first macro type, while the starred one refers to the second macro type.
%
% The above mentioned macros create dashed lines between two given
% points, with a dash length that must be specified, or dotted lines, with a
% dot gap that can be specified; actually the specified dash length or dot gap
% is a desired one; the actual length or gap is computed by integer division
% between the distance of the given points and the desired dash length or dot
% gap; when dashes are involved,this integer is tested in order to see if it
% is an odd number; if it's not, it is increased by one. Then the actual
% dash length or dot gap is obtained by dividing the above distance by this
% number.
%
% Another vector $P_2-P_1$ is created by dividing it by this number;
% then, when dashes are involved, it is multiplied by two in order to have
% the increment from one dash to the next; finally the number of patterns
% is obtained by integer division of this 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.
% Figure~\ref{fig:dashedlines} shows the effect of the slight changing
% of the dash length in order to maintain approximately the same dash-space
% pattern along the line, irrespective o the line length.
% \begin{macrocode}
\ifx\Dashline\undefined
\def\Dashline{\@ifstar{\Dashline@@}{\Dashline@}}
\def\Dashline@(#1)(#2)#3{%
\bgroup
\countdef\NumA3254\countdef\NumB3252\relax
\GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
\GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
\SubVect\V@ttA from\V@ttB to\V@ttC
\ModOfVect\V@ttC to\DlineMod
\DivideFN\DlineMod by#3 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\egroup
\noexpand\multiput(\V@ttA)(\V@ttC){\number\NumB}%
{\noexpand\LIne(\V@ttB)}}%
\@mpt\ignorespaces}%
\let\Dline\Dashline
\def\Dashline@@(#1)(#2)#3{\put(#1){\Dashline@(0,0)(#2){#3}}}
\fi
% \end{macrocode}
%
%\begin{figure}\unitlength=0.007\textwidth
%\begin{minipage}{0.55\textwidth}
%\begin{verbatim}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dashline(0,0)(40,10){2}\Dashline(0,0)(40,20){2}
%\Dashline(0,0)(40,30){2}\Dashline(0,0)(30,30){2}
%\Dashline(0,0)(20,30){2}\Dashline(0,0)(10,30){2}
%{\color{red}\Dashline*(40,0)(108:30){2}
%\Dashline*(40,0)(126:30){2}
%\Dashline*(40,0)(144:30){2}
%\Dashline*(40,0)(162:30){2}}
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.4\textwidth}\raggedleft
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dashline(0,0)(40,10){2}
%\Dashline(0,0)(40,20){2}
%\Dashline(0,0)(40,30){2}
%\Dashline(0,0)(30,30){2}
%\Dashline(0,0)(20,30){2}
%\Dashline(0,0)(10,30){2}
%{\color{red}\Dashline*(40,0)(108:30){2}
%\Dashline*(40,0)(126:30){2}
%\Dashline*(40,0)(144:30){2}
%\Dashline*(40,0)(162:30){2}}%
%\end{picture}
%\end{minipage}
%\caption{Different length dashed lines with the same nominal dash length}
%\label{fig:dashedlines}
%\end{figure}
%
% A simpler |\Dotline| macro can draw a dotted line between two given points;
% the dots are rather small, therefore the inter dot distance is computed in
% such a way as to have the first and the last dot at the exact position of
% the dotted-line end-points; again the specified dot distance is nominal in
% the sense that it is recalculated in such a way that the first and last
% dots coincide with the line end points. The syntax is as follows:
%\begin{flushleft}
%\cs{Dotline}\texttt{(}\meta{start point}\texttt{)(}\meta{end point}\texttt{)\{}\meta{dot distance}\texttt{\}}
%\end{flushleft}
% \begin{macrocode}
\ifx\Dotline\undefined
\def\Dotline{\@ifstar{\Dotline@@}{\Dotline@}}
\def\Dotline@(#1)(#2)#3{%
\bgroup
\countdef\NumA 3254\relax \countdef\NumB 3255\relax
\GetCoord(#1)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttA
\GetCoord(#2)\@tA\@tB \MakeVectorFrom\@tA\@tB to\V@ttB
\SubVect\V@ttA from\V@ttB to\V@ttC
\ModOfVect\V@ttC to\DotlineMod
\DivideFN\DotlineMod by#3 to\NumD
\NumA=\expandafter\Integer\NumD.??
\DivVect\V@ttC by\NumA,0 to\V@ttB
\advance\NumA\@ne
\edef\@mpt{\noexpand\egroup
\noexpand\multiput(\V@ttA)(\V@ttB){\number\NumA}%
{\noexpand\makebox(0,0){\noexpand\circle*{0.5}}}}%
\@mpt\ignorespaces}%
\def\Dotline@@(#1)(#2)#3{\put(#1){\Dotline@(0,0)(#2){#3}}}
\fi
% \end{macrocode}
%
%\begin{figure}[htb]\unitlength=0.007\textwidth
%\begin{minipage}{0.55\textwidth}
%\begin{verbatim}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dotline(0,0)(40,10){1.5}\Dotline(0,0)(40,20){1.5}
%\Dotline(0,0)(40,30){1.5}\Dotline(0,0)(30,30){1.5}
%\Dotline(0,0)(20,30){1.5}\Dotline(0,0)(10,30){1.5}
%{\color{red}\Dotline*(40,0)(108:30){1.5}
%\Dotline*(40,0)(126:30){1.5}
%\Dotline*(40,0)(144:30){1.5}
%\Dotline*(40,0)(162:30){1.5}}%
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.4\textwidth}\raggedleft
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\Dotline(0,0)(40,10){1.5}
%\Dotline(0,0)(40,20){1.5}
%\Dotline(0,0)(40,30){1.5}
%\Dotline(0,0)(30,30){1.5}
%\Dotline(0,0)(20,30){1.5}
%\Dotline(0,0)(10,30){1.5}
%{\color{red}%
%\Dotline*(40,0)(108:30){1.5}
%\Dotline*(40,0)(126:30){1.5}
%\Dotline*(40,0)(144:30){1.5}
%\Dotline*(40,0)(162:30){1.5}}%
%\end{picture}
%\end{minipage}
%\caption{Different length dotted lines with the same nominal dot gap}
%\label{fig:dottedlines}
%\end{figure}
%
% Notice that vectors as complex numbers in their cartesian and polar forms
% always represent a point position referred to the origin of the axes; this is
% why in figures~\ref{fig:dashedlines} and~\ref{fig:dottedlines} the dashed
% and dotted line that depart from the lower right corner of the graph grid,
% and that use polar coordinates, have to be put at the proper position with
% the starred version of the commands that take care of the relative
% specification made with the polar coordinates.
%
% \subsection{Coordinate handling}
% The new macro |\GetCoord| splits a vector (or complex number) specification
% into its components; in particular it distinguishes the polar from the
% cartesian form of the coordinates. The latter have the usual syntax
% \meta{x\texttt{,}y}, while the former have the syntax
% \meta{angle\texttt{:}radius}. The |\put| command is redefined to accept
% the same syntax; the whole work is done by |\SplitNod@|
% and its subsidiaries.
% \begin{macrocode}
\def\GetCoord(#1)#2#3{%
\expandafter\SplitNod@\expandafter(#1)#2#3\ignorespaces}
% \end{macrocode}
% But the macro that detects the form of the coordinates is |\isnot@polar|,
% that examines the parameter syntax in order to see if it contains a colon;
% if it does the coordinates are in polar form, otherwise they are in cartesian
% form:
% \begin{macrocode}
\def\isnot@polar#1:#2!!{\def\@tempOne{#2}\ifx\@tempOne\empty
\expandafter\@firstoftwo\else
\expandafter\@secondoftwo\fi
{\SplitNod@@}{\SplitPolar@@}}
\def\SplitNod@(#1)#2#3{\isnot@polar#1:!!(#1)#2#3}%
\def\SplitNod@@(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}%
\def\SplitPolar@@(#1:#2)#3#4{\DirFromAngle#1to\@DirA
\ScaleVect\@DirA by#2to\@DirA
\expandafter\SplitNod@@\expandafter(\@DirA)#3#4}
\let\originalput\put
\def\put(#1){\bgroup\GetCoord(#1)\@tX\@tY
\edef\x{\noexpand\egroup\noexpand\originalput(\@tX,\@tY)}\x}
\let\originalmultiput\multiput
\let\original@multiput\@multiput
\long\def\@multiput(#1)#2#3{\bgroup\GetCoord(#1)\@mptX\@mptY
\edef\x{\noexpand\egroup\noexpand\original@multiput(\@mptX,\@mptY)}%
\x{#2}{#3}\ignorespaces}
\gdef\multiput(#1)#2{\bgroup\GetCoord(#1)\@mptX\@mptY
\edef\x{\noexpand\egroup\noexpand\originalmultiput(\@mptX,\@mptY)}\x(}%)
% \end{macrocode}
% Examples of using polar and cartesian coordinates are shown in
% figure~\ref{fig:polar}.
%
%\begin{figure}[htb]\unitlength=0.01\textwidth
%\begin{minipage}{0.55\textwidth}
%\begin{verbatim}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\put(40,0){\circle*{1.5}}
% \put(41,0){\makebox(0,0)[bl]{40,0}}
%\put(90:30){\circle*{1.5}}
% \put(90:31){\makebox(0,0)[bl]{90:30}}
%\put(60:30){\circle*{1.5}}
% \put(60:31){\makebox(0,0)[bl]{60:30}}
%\put(30,30){\circle*{1.5}}
% \put(30.7,30.7){\makebox(0,0)[bl]{30,30}}
%\multiput(0,0)(30:10){5}%
% {\makebox(0,0){\rule{1.5mm}{1.5mm}}}
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.4\textwidth}
%\begin{picture}(40,30)
%\put(0,0){\GraphGrid(40,30)}
%\put(40,0){\circle*{1.5}}\put(41,0){\makebox(0,0)[bl]{40,0}}
%\put(90:30){\circle*{1.5}}\put(90:31){\makebox(0,0)[bl]{90:30}}
%\put(60:30){\circle*{1.5}}\put(60:31){\makebox(0,0)[bl]{60:30}}
%\put(30,30){\circle*{1.5}}\put(30.7,30.7){\makebox(0,0)[bl]{30,30}}
%\multiput(0,0)(30:10){5}{\makebox(0,0){\rule{1.5mm}{1.5mm}}}
%\end{picture}
%\end{minipage}
%\caption{Use of cartesian and polar coordinates}
%\label{fig:polar}
%\end{figure}
%
% \subsection{Vectors}
% 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
%|PSTricks| package if this one is loaded before |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
% |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){{%
\GetCoord(#1)\@tX\@tY
\ifdim\@tX\p@=\z@\vector(\@tX,\@tY){\@tY}
\else
\vector(\@tX,\@tY){\@tX}\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}
%
% \subsection{Polylines}
% We now define the polygonal line macro; its syntax is very simple
% \begin{flushleft}
% \cs{polygonal}\oarg{join}\texttt{(}$P_0$\texttt{)(}$P_1$\texttt{)(}$P_2$)%
% \texttt{\dots(}$P_n$\texttt{)}
% \end{flushleft}
% Remember: |\polyline| has been incorporated into |pict2e| 2009, but we
% redefine it so as to allow an optional argument to specify the line join type.
%
% 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|, besides setting the line joints, 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.
%
% 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}
%
% \subsection{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){\bgroup\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}%
\egroup\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}
%
% \section{Math operations on fixed radix operands}
% This is not the place to complain about the fact that all programs of the
% \TeX\ system use only integer arithmetics; LuaTeX can do floating point
% arithmetics through the Lua language that it partially incorporates. But
% this |curve2e| package is supposed to work also with pdfTeX and XeTeX.
% Therefore the Lua language should not be used.
%
% The only possibility to fake fractional arithmetics is to use fractional
% numbers as multipliers of the unit length |\p@| that is 1\,pt long;
% calculations are performed on lengths, and eventually their value,
% extracted from the length registers with the |\the| command is stripped
% off the ``pt'' component. The \LaTeX\ kernel macro does this in one step.
% At the same time the dimensional expressions made available by the |e-TeX|
% extension to all the \TeX\ system engines, allows to perform all operations
% directly on suitable length registers.
%
% The drawback of working with \TeX\ arithmetics for dimensions is that they
% are saved in binary form in computer words of 32 bits; the sixteen less
% significant bits are reserved for the fractional part; the two more
% significant bits are reserved for the sign and the type of dimension.
% There remain in total 30 bits available for the entire number; just to
% simplify this representation the \TeX\-book explains that the computer
% 32-bit word contains the dimension in \emph{scaled points}, where 1\,pt
% equals $2^{16}$\,sp.
%
% Since the number of bits of the fractional part is constant (16) it is said
% that the number representation is in \emph{fixed radix}. This is much
% different form the scientific approach to fractional numbers where
% a 32-bit word reserves 24 bits to the significant digits, one bit for the sign,
% and a signed exponent of 2 that has 7 significant bits and represents the
% number of binary digits that is necessary to move the binary fractional
% sign to the right or to the left in order to remain with a number greater
% or equal to 1, but lower than 2; this way of coding numbers is called
% \emph{floating point} representation (of course special numbers, such as
% zero, require special codes); \TeX\ fixed radix representation may code
% numbers with absolute value not exceeding ($2^{30}-1$)\,sp =1073741823\,sp
% =16383.99998\,pt; a floating point 32-bit number cannot exceed in magnitude
% the value of approximately $1.8446744\times 10^{19}$; with fixed radix
% numbers it is possible to evaluate the absolute value of the imprecision
% of the results by summing the absolute imprecision of the terms of
% summation and subtraction; with floating point numbers it is possible to
% estimate the relative imprecision by summing the relative imprecisions of
% the terms of multiplication and division.
%
% Working with fixed radix numbers one must keep in mind that 16 fractional
% binary digits are more or less equivalent to 5 decimal fractional
% digits; and that 16383,99998\,pt are a little less than six meters (5,75832\,m).
% These limits appear completely sufficient to do most computations necessary
% for typography, but when we pretend to make computations of mathematical
% functions with such a poor ``calculator'', we must expect poorly approximated
% results. Nevertheless using the proper iterative algorithms the results are
% not too bad, but certainly it is necessary to accept the situation.
%
% Then why not using the |fp| package that allows to do computations in \TeX\
% with the floating point representation of numbers? Simply because the results
% would require a lot of time for their execution; this is a serious problem
% with package |pgfplots| with which it is possible to draw beautiful 2D and
% 3D color diagrams, but at the expense of even dozens of seconds of computation
% time instead of milliseconds.
%
% \subsection{The new division macro}
% The most important macro in the whole package is 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 might 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,
% when the extended mode is enabled; it 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 are both represented by dimension registers, that contain
% two lengths expressed in the same units, possibly the fractional scaling
% factor numerator and denominator that `scale'' the unit length |\p@|.
%
% Therefore first we test if the extended mode exists and/or is enabled:
% \begin{macrocode}
\ifdefined\dimexpr
% \end{macrocode}
% then we test if the macro is already defined:
% \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 |\DividE|, 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 Knuthian 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 but not the termination of the program. During the
% division and 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
\dimendef\Num2254\relax \dimendef\Den2252\relax
\dimendef\@DimA 2250
\Num=\p@ \Den=#2\relax
\ifdim\Den=\z@
\edef\x{\noexpand\endgroup\noexpand\def\noexpand#3{\strip@pt\maxdimen}}%
\else
\@DimA=#1\relax
\edef\x{%
\noexpand\egroup\noexpand\def\noexpand#3{%
\strip@pt\dimexpr\@DimA*\Num/\Den\relax}}%
\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
\edef\x{%
\noexpand\egroup\noexpand\def\noexpand#3{%
\strip@pt\dimexpr\@DimA*\@DimB/\p@\relax}}%
\x\ignorespaces}%
\let\MultiplyFN\MultiplY
\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{\bgroup\dimen3254=#2\relax
\edef\x{\noexpand\egroup\noexpand\edef\noexpand#1{%
\strip@pt\dimen3254}}\x\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 meaning \verb|\relax|
% then executes the test; therefore an undefined macro name is always defined
% to mean |\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°
% the magnitude of the independent variable $y=2x$ (the angle in degrees
% converted to radians) is so small (about 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 $10^{-6}$).
%
% 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°, 90°, 180°, 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° we assign the result a \TeX\ infinity value, i.e. |\maxdimen|,
% the maximum dimension \TeX\ can handle.
% \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}
%
% 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{subequations}
%\begin{align}
%\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{align}
%\end{subequations}
% 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 its 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 algorithm crashes;
% therefore it's virtually impossibile to get an absolute error lower than 0.0005.
%
% 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}[!htb]\centering\unitlength=0.01\textwidth
%\begin{picture}(100,80)
%\legenda(15,73){y=\tan\theta}
%\legenda(37,73){t=\tan\theta_\infty}
%\put(0,0){\vector(1,0){100}}\Zbox(100,0)[br]{\theta}
%\put(0,0){\vector(0,1){80}}\Zbox(0,80)[tl]{y}
%\Dashline(90,0)(90,80){2.5}
%\Zbox(90,0)[br]{\pi/2}
%\Pall[2](28.8,11)\Line(0,11)(80,11)%\Dashline(28.8,0)(28.8,11){2.5}
%\Zbox(0,11)[bl]{t}\Zbox(28.8,0)[br]{\theta_\infty}
%
%\Dashline(34,0)(34,11){1.5}\Zbox(34,0)[bl]{\theta_{i+1}}
%
%\Pall[2](47.5,21.83)\Dashline(47.5,0)(47.5,21.83){2.5}
%\Dashline(0,21.83)(47.5,21.83){2.5}
%\Zbox(0,20.83)[bl]{y_i}\Zbox(47.5,0)[bl]{\theta_{i}}
%
%\Pall[2](61.2,36.38)\Dashline(61.2,0)(61.2,36.38){2.5}
%\Dashline(0,36.38)(61.2,36.38){2.5}
%\Zbox(0,36.38)[bl]{y_{i-1}}\Zbox(61.2,0)[bl]{\theta_{i-1}}
%
%\Pall[2](72,61.55367)\Dashline(72,0)(72,61.55367){2.5}
%\Dashline(0,61.55367)(72,61.55367){2.5}
%\Zbox(0,61.55367)[bl]{y_{i-2}}\Zbox(72,0)[bl]{\theta_{i-2}}
%
%\put(0,0){\linethickness{1pt}
%\Curve%(0,0)<1,0.34907>(28.8,11)<1,0.45456>(47.5,21.83)<1,0.76479>%
%(61.2,36.38)<1,1.91499>(72,61.55367)<1,4.65428>(74.4,71.6)<1,6.14571>
%}%
%
%\Line(34,11)(47.5,21.83)
%\Line(61.2,36.38)(47.5,11)
%\Line(72,61.55367)(61.2,11)
%\end{picture}
%\caption{Newton's method of tangents}\label{fig:tangenti}
%\end{figure}
%
% The iterative algorithm with Newton method implies the recurrence
%\begin{subequations}\begin{align}
%y'_{i-1} &= \frac{\diff\tan(\theta_{i-1})}{\diff\theta}
% = \frac{1}{\cos^2\theta_{i-1}}\\
%\theta_i &= \theta_{i-1} - \frac{\tan \theta_{i-1} - t}{y'_{i-1}}
% =\theta_{i-1} - \cos^2 \theta_{i-1}(\tan \theta_{i-1} - t)
%\label{equ:iterazione}
%\end{align}
%\end{subequations}
%
% 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 does not set forth any problem, especially if we
% use the properties of the trigonometric functions and we confine the
% computations to the first quadrant; possibly we limit the computations
% to the first octant and we resort to the cotangent when the tangent exceeds
% one; in this case we use the same algorithm, but we have to get the
% complementary angle; in order to make computations with positive numbers,
% we save the initial tangent sign and we restore it to the result.
% \begin{macrocode}
\def\ArcTanOf#1to#2{\bgroup
\countdef\Inverti 4444\Inverti=0
\def\Segno{}
\edef\@tF{#1}\@tdF=\@tF\p@ \@tdE=57.295778\p@
\@tdD=\ifdim\@tdF<\z@ -\@tdF\def\Segno{-}\else\@tdF\fi
\ifdim\@tdD>\p@
\Inverti=\@ne
\@tdD=\dimexpr\p@*\p@/\@tdD\relax
\fi
\unless\ifdim\@tdD>0.02\p@
\def\@tX{\strip@pt\dimexpr57.295778\@tdD\relax}%
\else
\edef\@tX{45}\relax
\countdef\I 2523 \I=9\relax
\@whilenum\I>0\do{\TanOf\@tX to\@tG
\edef\@tG{\strip@pt\dimexpr\@tG\p@-\@tdD\relax}\relax
\MultiplY\@tG by57.295778to\@tG
\CosOf\@tX to\@tH
\MultiplY\@tH by\@tH to\@tH
\MultiplY\@tH by\@tG to \@tH
\edef\@tX{\strip@pt\dimexpr\@tX\p@ - \@tH\p@\relax}\relax
\advance\I\m@ne}%
\fi
\ifnum\Inverti=\@ne
\edef\@tX{\strip@pt\dimexpr90\p@-\@tX\p@\relax}
\fi
\edef\x{\egroup\noexpand\edef\noexpand#2{\Segno\@tX}}\x\ignorespaces}%
% \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{center})(\meta{starting point})\marg{angle}
% \end{flushleft}
% which is totally equivalent to:
% \begin{flushleft}\ttfamily
% \cs{put}(\meta{center})\marg{\upshape\cs{Arc}(0,0)(\meta{starting point})\marg{angle}}
% \end{flushleft}
% If the \meta{angle}, i.e. the arc angular aperture, is positive the arc
% runs counterclockwise from the starting point; clockwise if it's negative.
% Notice that since the \meta{starting point} is relative to the \meta{center}
% point, its polar coordinates are very convenient, since they become
% \parg{\meta{start angle}:\meta{radius}}, where the
% \meta{start angle} is relative to the arc center. Therefore you can think
% about a syntax such as this one:
%\begin{flushleft}
%\cs{Arc}\parg{\meta{center}}\parg{\meta{start angle}:\meta{radius}}\marg{angle}
%\end{flushleft}
%
% The difference between the |pict2e| |\arc| definition consists in a very
% different syntax:
%\begin{flushleft}
%\cs{arc}\texttt{[}\meta{start angle}\texttt{,}\meta{end angle}\texttt{]}\marg{radius}
%\end{flushleft}
% and the center is assumed to be at the coordinate established with a
% required |\put| command; moreover the difference in specifying angles
% is that \meta{end angle} equals the sum of \meta{start angle} and
% \meta{angle}. With the definition of this |curve2e| package
% use of a |\put| command is not prohibited, but it may be used for fine
% tuning the arc position by means of a simple displacement; moreover the
% \meta{starting point} may be specified with polar coordinates (that are
% relative to the arc center).
%
% 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 roto-amplification operators.
%
% \subsection{Complex number macros}
% In this package \emph{complex number} is a vague phrase; it may be used
% in the mathematical sense of an ordered pair of real numbers; it can be
% viewed as a vector joining the origin of the coordinate axes to the
% coordinates indicated by the ordered pair; it can be interpreted as a
% roto-amplification operator that scales its operand and rotates it about
% a pivot point; besides the usual conventional representation used by the
% mathematicians where the ordered pair is enclosed in round parentheses
% (which is in perfect agreement with the standard code used by the |picture|
% environment) there is the other conventional representation used by the
% engineers that stress the roto-amplification nature of a complex number:
%\[
%(x, y) = x + \mathrm{j}y =M \mathrm{e}^{\mathrm{j}\theta}
%\]
% Even the imaginary unit is indicated with $\mathrm{i}$ by the mathematicians
% and with $\mathrm{j}$ by the engineers. In spite of these differences,
% these objects, the \emph{complex numbers}, are used without any problem by
% both mathematicians and engineers.
%
%The important point is that these objects can be summed, subtracted,
% multiplied, divided, raised to any power (integer, fractional, positive
% or negative), be the argument of transcendental functions according to
% rules that are agreed upon by everybody. We do not need all these properties, but we need some and we must create the suitable macros for doing some of
% these operations.
%
% In facts wee need macros for summing, subtracting, multiplying, dividing
% complex numbers, for determining their directions (unit vectors); a unit vector
% is the complex number divided by its magnitude so that the result is the
% cartesian or polar 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=\z@
\ifdim\@tempdimb=\z@
\def\@T{0}\@tempdimc=\z@
\else
\def\@T{0}\@tempdimc=\@tempdimb
\fi
\else
\ifdim\@tempdima>\@tempdimb
\DividE\@tempdimb by\@tempdima to\@T
\@tempdimc=\@tempdima
\else
\DividE\@tempdima by\@tempdimb to\@T
\@tempdimc=\@tempdimb
\fi
\fi
\unless\ifdim\@tempdimc=\z@
\unless\ifdim\@T\p@=\z@
\@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
\fi
\Numero#2\@tempdimc
\ignorespaces}%
% \end{macrocode}
% As a byproduct of the computation the control sequence |\@tempdimc| contains
% a length the value in points of which is the computed root.
%
% 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 to determine 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}
%
% Since we have the algorithm to compute the arctangent of a number,
% it should be relatively easy to compute the angle of a complex number.
% We just 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 they 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}
%
% It is worth noting that the absolute average error in these computations is
% much lower than 0.0001°; pretty satisfactory since the typesetting engines work
% in fixed radix notation with 16 fractional binary digits, and an error on
% the fourth or fifth fractional decimal 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@
\unless\ifdim\@tdA=\z@
\@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 draw 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 drawing 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.
%
%\begin{figure}\centering\unitlength=0.007\textwidth
%\begin{picture}(100,90)(-50,-50)
%\put(-50,0){\vector(1,0){100}}\put(50,1){\makebox(0,0)[br]{$x$}}
%\put(20,-1){\makebox(0,0)[t]{$s$}}
%\put(0,0){\circle*{2}}\put(-1,-1){\makebox(0,0)[tr]{$M$}}
%\legenda(12,-45){s=\overline{MP_2}=R\sin\theta}
%\put(0,-50){\vector(0,1){90}}
%\put(1,40){\makebox(0,0)[tl]{$y$}}
%\put(0,-40){\circle*{2}}\put(1,-41){\makebox(0,0)[lt]{$C$}}
%\Line(0,-40)(-40,0)\Line(0,-40)(40,0)
%\put(-41,1){\makebox(0,0)[br]{$P_1$}}\put(-40,0){\circle*{2}}
%\put(41,1){\makebox(0,0)[bl]{$P_2$}}\put(40,0){\circle*{2}}
%\put(0,0){\linethickness{1pt}\Arc(0,-40)(40,0){90}}
%\Line(-40,0)(-20,20)\put(-20,20){\circle*{2}}
%\put(-20,21.5){\makebox(0,0)[b]{$C_1$}}
%\Line(40,0)(20,20)\put(20,20){\circle*{2}}
%\put(20,21.5){\makebox(0,0)[b]{$C_2$}}
%\put(0,-40){\put(0,56.5685){\circle*{2}}\put(1,58){\makebox(0,0)[bl]{$P$}}}
%\VectorARC(0,-40)(15,-25){45}\put(10,-18){\makebox(0,0)[c]{$\theta$}}
%\VectorARC(40,0)(20,0){-45}\put(19,5){\makebox(0,0)[r]{$\theta$}}
%\VectorARC(-40,0)(-20,0){45}\put(-19,5){\makebox(0,0)[l]{$\theta$}}
%\put(-20,-18){\makebox(0,0)[bl]{$R$}}
%\put(-32,13){\makebox(0,0)[bl]{$K$}}
%\put(32,13){\makebox(0,0)[br]{$K$}}
%\end{picture}
%\caption{Nodes and control points for an arc to be approximated with a cubic Bézier spline}
%\label{fig:arcspline}
%\end{figure}
%
% With reference to figure~\ref{fig:arcspline},
% the points $P_1$ and $P_2$ are the arc end-points; $C_1$ and $C_2$ are the
% Bézier-spline control-points; $P$ is the arc mid-point, that should be
% distant from the center of the arc the same as $P_1$ and $P_2$. Choosing a
% convenient orientation of the arc relative to the coordinate axes, the
% coordinates of these five points are:
%\begin{align*}
%P_1 &= (-R\sin\theta, 0)\\
%P_2 &= (R\sin\theta, 0)\\
%C_1 &= (-R\sin\theta+K\cos\theta, K\sin\theta)\\
%C_2 &= (R\sin\theta-K\cos\theta, K\sin\theta)\\
%P &= (0, R(1-\cos\theta))
%\end{align*}
% The Bézier cubic spline interpolating the end and mid points is given by
% the parametric equation:
%\begin{equation*}
%P= P_1(1-t)^3 + C_1 3(1-t)^2t + C_2 3(1-t)t^2 + P_2t^3
%\end{equation*}
% where the mid point is obtained for $t=0.5$; the four coefficients then become $1/8, 3/8, 3/8, 1/8$ and the only unknown remains $K$. Solving for $K$ we obtain the formula
% \begin{equation}\label{equ:corda}
% K= \frac{4}{3}\,\frac{1-\cos\theta}{\sin\theta}R
%= \frac{4}{3}\,\frac{1-\cos\theta}{\sin^2\theta}s
% \end{equation}
% where $\theta$ is half the arc aperture, $R$ is its radius, and $s$ is
% half the arc chord.
% \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 specific option to the
% \texttt{pict2e} package.
%
% But the arc drawing done here shortens it 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 finally (h) drawing the 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=\if\Segno--\fi\@gradi\p@ \Numero\@tempa\@tdD
\DirFromAngle\@tempa to\@Dir
\MultVect\@V by\@Dir to\@sPun% corrects the end point
\edef\@tempA{\if\Segno--\fi1}%
\MultVect\@sPun by 0,\@tempA to\@vPun
\DirOfVect\@vPun to\@Dir
\AddVect\@sPun and #1 to \@sPun
\GetCoord(\@sPun)\@tdX\@tdY
\@tdD\if\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}}% end point arrowt ip
\@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{\if\Segno-\else-\fi\@ne}%
\MultVect\@V by0,\@tempa to\@vPun
\@tdE\if\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}}% starting point arrow tip
\edef\@tempa{\if\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}
% The most used method to draw curved lines with computer programs is to
% connect several simple curved lines, general ``arcs'', one to another
% generally maintaining the same tangent at the junction. I the direction
% changes we are dealing with a cusp.
%
% The simple general arcs that are directly implemented in every program that
% display typeset documents, are those drawn with the parametri curves called
% \emph{Béźier splines}; given a sequence of points in the $x,y$ plane, say
% $P_0, P_1, P_2, p_3, \dots$ (represented as coordinate pairs, i.e. by complex
% numbers), the most common Bézier splines are the following ones:
% \begin{align}
% \mathcal{B}_1 &= P_0(1-t) + P_1t \label{equ:B-1} \\
% \mathcal{B}_2 &= P_0(1-t)^2 + P_1 2(1-t)t + P_2t^2 \label{equ:B-2} \\
% \mathcal{B}_3 &= P_0(1-t)^3 + P_1 3(1-t)^2t +P_2 3(1-t)t^2 +P_3t^3
% \label{equ:B-3}
% \end{align}
%
% All these splines depend on parameter $t$; they have the property that for
% $t=0$ each line starts at the first point, while for $t=1$ they reach the
% last point; in each case the generic point $P$ on each curve takes off
% with a direction that points to the next point, while it reaches the
% destination point with a direction coming from the penultimate point;
% moreover, when $t$ varies from 0 to 1, the curve arc is completely
% contained within the convex hull formed by the polygon that has the
% spline points as vertices. Last but not least first order splines implement
% just straight lines and they are out of question for what concerns maxima,
% minima, inflection points and the like. Quadratic splines draw just
% parabolas, therefore they draw arcs that have the concavity just on one
% side of the path; therefore no inflection points. Cubic splines are
% extremely versatile and can draw lines with maxima, minima and inflection
% points. Virtually a multi-arc curve may be drawn by a set of cubic splines
% as well as a set of quadratic splines (fonts are a good example: Adobe
% Type~1 fonts have their contours described by cubic splines, while TrueType
% fonts have their contours described with quadratic splines; with a naked
% eye it is impossible to notice the difference).
%
% Each program that processes the file to be displayed is capable of drawing
% first order Bézier splines (segments) and third order Bézier splines, for
% no other reason, at least, because they have to draw vector fonts whose
% contours are described by Bézier splines; sometimes they have also the
% program commands to draw second order Bézier splines, but not always these
% machine code routines are available to the user for general use. For what
% concerns |pdftex|, |xetex| and |luatex|, they have the user commands for
% straight lines and cubic arcs. At least with |pdftex|, quadratic arcs must
% be simulated with a clever use of third order Bézier splines.
%
% Notice that \LaTeXe\ environment |picture| by itself is capable of drawing
% both cubic and quadratic Bézier splines as single arcs; but it resorts to
% ``poor man'' solutions. The |pict2e| package removes all the old limitations
% and implements the interface macros for sending the driver the
% necessary drawing information, including the transformation from
% typographical points (72.27\,pt/inch) to PostScript big points (72\,bp/inch).
% But for what concerns the quadratic spline it resorts to the clever use of a
% cubic spline.
%
% Therefore here we treat first the drawings that can be made with cubic
% splines; then we describe the approach to quadratic splines.
%
%\subsection{Cubic splines}
% 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 of |pict2e| 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\ignorespaces}%
% \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.
%In any case the directions specified with the direction arguments, both here
% and with the more general macro|\Curve|, the angle between the indicated
% tangent and the arc chord may give raise to some little problems when they
% are very close to 90° in absolute value. Some control is exercised on these
% values, but some tests might fail if the angle derives from computations;
% this is a good place to use polar forms for the direction vectors.
%
%\begin{figure}\centering\unitlength=0.004\textwidth
%\begin{picture}(220,120)(-50,-20)
%\put(0,60){\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs15:1and{-15:1}
%\CurveBetween-50,0and50,0WithDirs30:1and{-30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-45:1}
%\CurveBetween-50,0and50,0WithDirs60:1and{-60:1}
%\CurveBetween-50,0and50,0WithDirs75:1and{-75:1}
%\CurveBetween-50,0and50,0WithDirs90:1and{-90:1}}
%\put(120,60){%
%\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs15:1and{15:1}
%\CurveBetween-50,0and50,0WithDirs30:1and{30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween-50,0and50,0WithDirs60:1and{60:1}
%\CurveBetween-50,0and50,0WithDirs75:1and{75:1}
%\CurveBetween-50,0and50,0WithDirs90:1and{90:1}}
%\put(0,0){%
%\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs45:1and{-15:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-45:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-60:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-75:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{-90:1}}
%\put(120,0){%
%\Line(-50,0)(50,0)
%\CurveBetween-50,0and50,0WithDirs45:1and{15:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{30:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{45:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{60:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{75:1}
%\CurveBetween-50,0and50,0WithDirs45:1and{90:1}}
%\end{picture}
%\caption{Curves between two points}\label{fig:curva-due-punti}
%\end{figure}
%
% 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
\ignorespaces}
% \end{macrocode}
% And this re-initializes the direction to create 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 macros are the finishing ones; the first strokes the whole curve,
% while the second fills the (closed) curve with the default color; both close
% the group that was opened with |\StartCurve|. The third macro is explained
% in a while; we anticipate it is functional to chose between the first two
% macros when a star is possibly used to switch between stroking and filling.
% \begin{macrocode}
\def\CurveFinish{\strokepath\endgroup\ignorespaces}%
\def\FillCurve{\fillpath\endgroup\ignorespaces}
\def\CurveEnd{\fillstroke\endgroup\ignorespaces}
% \end{macrocode}
%
% In order to draw the internal arcs it would be desirable to have a single
% macro that, given the destination point, computes the control points that
% produce a cubic Bézier spline that joins the starting point with the
% destination point in the best possible way. The problem is strongly ill
% defined and has an infinity of solutions; here we give two solutions:
% $(a)$ a supposedly smart one that resorts to osculating circles and
% requires only the direction at the destination point; and $(b)$ a less
% smart solution that requires the control points to be specified in a
% certain format.
%
% We start with solution $(b)$, |\CbezierTo|, the code of which is simpler
% than that of solution $(a)$; then we will produce the solution $(a)$,
% |\CurveTo|, that will become the main building block for a general path
% construction macro, |\Curve|.
%
% The ``naïve'' macro |\CBezierTo| simply uses the previous point direction saved in |\@Dzero| as a unit vector by the starting macro; specifies
% a destination point, the distance of the first control point from the
% starting point, the destination point direction that will save also for the
% next arc drawing macro as a unit vector, and the distance of the second
% control point from the destination point along this last direction. Both
% distances must be positive possibly fractional numbers. The syntax will
% be therefore:
%\begin{flushleft}
%\cs{CbezierTo}\meta{end
% point}|WithDir|\meta{direction}|AndDists|\meta{$K_0$}|And|\meta{$K_1$}
%\end{flushleft}
% where \meta{end point} is a vector macro or a comma separated pair of values;
% again \meta{direction} is another vector macro or a comma separated pair of
% values, that not necessarily indicate a unit vector, since the macro provides
% to normalise it to unity; \meta{$K_0$} and\meta{$K_1$} are the distances of
% the control point from their respective node points; they must be positive
% integers or fractional numbers. If \meta{$K_1$} is a number must be enclosed
% in curly braces, while if it is a macro name (containing the desired fractional
% or integer value) there is no need for braces.
%
% This macro uses the input information to use the internal |pict2e| macro
% |\pIIe@curveto| with the proper arguments, and to save the final direction
% into the same |\@Dzero| macro for successive use of other macros.
% \begin{macrocode}
\def\CbezierTo#1WithDir#2AndDists#3And#4{%
\GetCoord(#1)\@tX\@tY \MakeVectorFrom\@tX\@tY to\@Puno
\GetCoord(#2)\@tX\@tY \MakeVectorFrom\@tX\@tY to \@Duno
\DirOfVect\@Duno to\@Duno
\ScaleVect\@Dzero by#3to\@Czero \AddVect\@Pzero and\@Czero to\@Czero
\ScaleVect\@Duno by-#4to \@Cuno \AddVect\@Puno and\@Cuno to \@Cuno
\GetCoord(\@Czero)\@XCzero\@YCzero
\GetCoord(\@Cuno)\@XCuno\@YCuno
\GetCoord(\@Puno)\@XPuno\@YPuno
\pIIe@curveto{\@XCzero\unitlength}{\@YCzero\unitlength}%
{\@XCuno\unitlength}{\@YCuno\unitlength}%
{\@XPuno\unitlength}{\@YPuno\unitlength}%
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
% \end{macrocode}
%
% With this building block it is not difficult to set up a macro that draws
% a Bézier arc between two given points, similarly as the other macro
% |\CurveBetween| described previously.
%
% \begin{macrocode}
\def\CbezierBetween#1And#2WithDirs#3And#4UsingDists#5And#6{%
\StartCurveAt#1WithDir{#3}\relax
\CbezierTo#2WithDir#4AndDists#5And{#6}\CurveFinish}
% \end{macrocode}
%
% An example of use is shown in figure~\ref{fig:Cbezier}; notice that the
% tangents at the end points are the same for the black curve drawn with
% |\CurveBetween| and the five red curves drawn with |\CbezierBetween|; the
% five red curves differ only for the distance of their control point $C_0$
% from the starting point; the differences are remarkable and the topmost
% curve even presents a slight inflection close to the end point. These
% effects cannot be obtained with the ``smarter'' macro |\CurveBetween|. But
% certainly this simpler macro is more difficult to use because the
% distances of the control point are difficult to estimate and require a
% number of cut-and-try experiments.
%
%\begin{figure}[!tb]
%\begin{minipage}[t]{0.52\textwidth}
%\begin{verbatim}
%\unitlength=0.1\textwidth
%\begin{picture}(10,3)
%\CurveBetween0,0and10,0WithDirs1,1and{1,-1}
%\color{red}%
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists4And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists6And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists8And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists10And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists12And{1}
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.40\textwidth}\raggedleft
%\unitlength=0.1\textwidth
%\begin{picture}(10,3)(0,1.25)
%\CurveBetween0,0and10,0WithDirs1,1and{1,-1}
%\color{red}%
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists4And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists6And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists8And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists10And{1}
%\CbezierBetween0,0And10,0 WithDirs45:1And-45:1UsingDists12And{1}
%\end{picture}
%\end{minipage}
%\caption{Comparison between similar arcs drawn with \cs{CurveBetween} (black)
% and \cs{CbezierTo} (red)}
%\label{fig:Cbezier}
%\end{figure}
%
%
% The ``smarter'' 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, i.e. 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 equal parts each of which should be
% interpreted as half the chord of the osculating circle.
% This makes the algorithm a little rigid; sometimes the path drawn is very
% pleasant, while in other circumstances the determined curvatures are too
% large or too small. We therefore add some optional information that lets
% us have some control over the curvatures; the idea is based on the concept
% of \emph{tension}, similar but not identical to the one used in the drawing
% programs \MF\ and \MP. We add to the direction information, with which the
% control nodes of the osculating circle arcs are determined, a scaling factor
% that should be intuitively related to the tension of the arc: the smaller
% this number, the closer the arc resembles a straight line as a rope subjected
% to a high tension; value zero is allowed, while a value of 4 is close to
% ``infinity'' and turns a quarter circle into a line with an unusual loop;
% a value of 2 turns a quarter circle almost into a polygonal line
% with rounded corner. Therefore these tension factors should
% be used only for fine tuning the arcs, not as the first time a path is drawn.
%
% We devised a syntax for specifying direction and tensions:
%\begin{flushleft}
% \meta{direction\texttt{\upshape;}tension factors}
%\end{flushleft}
% where \emph{direction} contains a pair of fractional number that not
% necessarily refer to the components of a unit vector direction, but simply
% to a vector with the desired orientation; the information contained from
% the semicolon (included) to the rest of the specification is optional; if
% it is present, the \emph{tension factors} is simply a comma separated pair
% of fractional or integer numbers that represent respectively the tension
% at the starting or the ending node of a path arc.
%
% We therefor need a macro to extract the mandatory and optional parts:
% \begin{macrocode}
\def\@isTension#1;#2!!{\def\@tempA{#1}%
\def\@tempB{#2}\unless\ifx\@tempB\empty\strip@semicolon#2\fi}
\def\strip@semicolon#1;{\def\@tempB{#1}}
% \end{macrocode}
% By changing the tension values we can achieve different results: see
% figure~\ref{fig:tensions}.
%\begin{figure}[!htb]\centering
%\begin{minipage}{0.48\textwidth}\small
%\begin{verbatim}
%\raggedleft\unitlength=0.01\textwidth
%\begin{picture}(70,70)
%\put(0,0){\color{blue}\frame(70,70){}}
%\put(0,0){\color{red}\Curve(0,0)<1,1>(70,0)<1,-1>}
%\Curve(0,0)<1,1>(70,0)<1,-1;0,0>
%\Curve(0,0)<1,1>(70,0)<1,-1;0.2,0.2>
%\Curve(0,0)<1,1>(70,0)<1,-1;2,2>
%\Curve(0,0)<1,1>(70,0)<1,-1;4.5,4.5>
%\Curve(0,0)<1,1>(70,0)<1,-1;0,3>
%\Curve(0,0)<1,1>(70,0)<1,-1;3,0>
%\end{picture}
%\end{verbatim}
%\end{minipage}
%\hfill
%\begin{minipage}{0.46\textwidth}
%\raggedleft\unitlength=0.01\textwidth
%\begin{picture}(70,70)
%\put(0,0){\color{blue}\framebox(70,70){}}
%\put(0,0){\color{red}\Curve(0,0)<1,1>(70,0)<1,-1>}
%\Curve(0,0)<1,1>(70,0)<1,-1;0,0>
%\Curve(0,0)<1,1>(70,0)<1,-1;0.2,0.2>
%\Curve(0,0)<1,1>(70,0)<1,-1;2,2>
%\Curve(0,0)<1,1>(70,0)<1,-1;4.5,4.5>
%\Curve(0,0)<1,1>(70,0)<1,-1;0,3>
%\Curve(0,0)<1,1>(70,0)<1,-1;3,0>
%\end{picture}
%\end{minipage}
%\caption{The effects of tension factors}\label{fig:tensions}
%\end{figure}
%
%
% We use the formula we got for arcs~\eqref{equ:corda}, where the half chord is
% indicated with $s$, and we derive the necessary distances:
%\begin{subequations}\label{equ:Kzero-Kuno}
%\begin{align}
%K_0 &= \frac{4}{3} s\frac{1-\cos\theta_0}{\sin^2\theta_0}\\
%K_1 &=\frac{4}{3}s\frac{1-\cos\theta_1}{\sin^2\theta_1}
%\end{align}
%\end{subequations}
%
% We therefore start with getting the points and directions and calculating the
% chord and its direction:
% \begin{macrocode}
\def\CurveTo#1WithDir#2{%
\def\@Tuno{1}\def\@Tzero{1}\relax
\edef\@Puno{#1}\@isTension#2;!!%
\expandafter\DirOfVect\@tempA to\@Duno
\bgroup\unless\ifx\@tempB\empty\GetCoord(\@tempB)\@Tzero\@Tuno\fi
\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)\@DXpzero\@DYpzero
\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
\DivideFN\@Chord by2 to\@semichord
% \end{macrocode}
% The chord needs not be actually rotated because it suffices its length
% along the real axis; the chord length is memorised in |\@Chord| and
% its half is saved in |\@semichord|.
%
% We now examine the various degenerate cases, when either tangent is
% perpendicular or parallel to the chord. Notice that we are calculating
% the distances of the control points from the adjacent nodes using the
% half chord length, not the full length. We also distinguish between the
% computations relative to the arc starting point and those relative to
% the end point.
%
% \begin{macrocode}
\ifdim\@DXpzero\p@=\z@
\@tdA=1.333333\p@
\Numero\@KCzero{\@semichord\@tdA}%
\fi
\ifdim\@DYpzero\p@=\z@
\@tdA=1.333333\p@
\Numero\@Kpzero{\@semichord\@tdA}%
\fi
% \end{macrocode}
% The distances we are looking for are positive generally fractional numbers;
% so if the components are negative, we take the absolute values. Eventually
% we determine the absolute control point coordinates.
% \begin{macrocode}
\unless\ifdim\@DXpzero\p@=\z@
\unless\ifdim\@DYpzero\p@=\z@
\edef\@CosDzero{\ifdim\@DXpzero\p@<\z@ -\fi\@DXpzero}%
\edef\@SinDzero{\ifdim\@DYpzero\p@<\z@ -\fi\@DYpzero}%
\@tdA=\@semichord\p@ \@tdA=1.333333\@tdA
\DividE\@tdA by\@SinDzero\p@ to \@KCzero
\@tdA=\dimexpr(\p@-\@CosDzero\p@)\relax
\DividE\@KCzero\@tdA by\@SinDzero\p@ to \@KCzero
\fi
\fi
\MultiplyFN\@KCzero by \@Tzero to \@KCzero
\ScaleVect\@Dzero by\@KCzero to\@CPzero
\AddVect\@Pzero and\@CPzero to\@CPzero
% \end{macrocode}
% We now repeat the calculations for the arc end point, taking into
% consideration that the end point direction points outwards, so that in
% computing the end point control point we have to take this fact into
% consideration by using a negative sign for the distance; in this way
% the displacement of the control point from the end point takes place
% in a backwards direction.
% \begin{macrocode}
\ifdim\@DXpuno\p@=\z@
\@tdA=-1.333333\p@
\Numero\@KCuno{\@semichord\@tdA}%
\fi
\ifdim\@DYpuno\p@=\z@
\@tdA=-1.333333\p@
\Numero\@KCuno{\@semichord\@tdA}%
\fi
\unless\ifdim\@DXpuno\p@=\z@
\unless\ifdim\@DYpuno\p@=\z@
\edef\@CosDuno{\ifdim\@DXpuno\p@<\z@ -\fi\@DXpuno}%
\edef\@SinDuno{\ifdim\@DYpuno\p@<\z@ -\fi\@DYpuno}%
\@tdA=\@semichord\p@ \@tdA=-1.333333\@tdA
\DividE\@tdA by \@SinDuno\p@ to \@KCuno
\@tdA=\dimexpr(\p@-\@CosDuno\p@)\relax
\DividE\@KCuno\@tdA by\@SinDuno\p@ to \@KCuno
\fi
\fi
\MultiplyFN\@KCuno by \@Tuno to \@KCuno
\ScaleVect\@Duno by\@KCuno to\@CPuno
\AddVect\@Puno and\@CPuno to\@CPuno
% \end{macrocode}
% Now we have the four points and we can instruct the internal \texttt{pict2e}
% macros to do the path drawing.
% \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}\egroup
% \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 memorises the final point as the
% initial point of the next spline
% \begin{macrocode}
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}%
% \end{macrocode}
%
%
% We finally define the overall |\Curve| macro that has two flavors: starred
% and unstarred; the former fills the curve path with the locally selected
% color, while the latter just strokes the path. Both recursively examine 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 initialises 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 or filling command through
% |\CurveEnd|, and exits the recursive process. The |\CurveEnd| control
% sequence has a different meaning depending on the fact that the main macro
% was starred or unstarred. The |@ChangeDir| macro is just an interface to
% execute the regular |\ChangeDir| macro, but also for recursing again by
% recalling |\@Curve|.
% \begin{macrocode}
\def\Curve{\@ifstar{\let\fillstroke\fillpath\Curve@}%
{\let\fillstroke\strokepath\Curve@}}
\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\CurveEnd}}
\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| or |\FillCurve| for a more versatile set of drawing macros;
% evidently nobody forbids to exploit the full power of the |\cbezier| original
% macro for cubic splines; we made available macros |\CbezierTo| and the
% isolated arc macro |\CbezierBetween| in order to use the general internal
% cubic Bézier splines in a more comfortable way.
%
%\begin{figure}[!htb]
%\unitlength=0.01\textwidth
%\begin{picture}(100,50)(0,-25)
%\put(0,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
%\Zbox(45,0)[br]{x}\Zbox(0,26)[tl]{y}
%\Curve(0,0)<77:1>(10,20)<1,0;2,0.4>(30,-20)<1,0;0.4,0.4>(40,0)<77:1;0.4,2>
%}
%\put(55,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
%\Zbox(45,0)[br]{x}\Zbox(0,26)[tl]{y}
%\CbezierBetween0,0And20,0WithDirs77:1And-77:1UsingDists28And{28}
%\CbezierBetween20,0And40,0WithDirs-77:1And77:1UsingDists28And{28}}
%\end{picture}
%\caption{A sequence of arcs; the left figure has been drawn with the
% \cs{Curve} command with a sequence of four couples of point-direction
% arguments; the right figure has been drawn with two commands
% \cs{CbezierBetween} that include also the specification of the control
% points}
%\label{fig:sinewave}
%\end{figure}
%
% As it can be seen in figure~\ref{fig:sinewave} the two diagrams should
% approximately represent a sine wave. With Bézier curves, that resort on
% polynomials, it is impossible to represent a transcendental function, but
% it is only possible to approximate it. It is evident that the approximation
% obtained with full control on the control points requires less arcs and
% it is more accurate than the approximation obtained with the recursive
% |\Curve| macro; this macro requires almost two times as many pieces of
% information in order to minimise the effects of the lack of control on the
% control points, and even with this added information the macro approaches
% the sine wave with less accuracy. At the same time for many applications
% the |\Curve| recursive macro proves to be much easier to use than with
% single arcs drawn with the |\CbezierBetween| macro.
%
% \subsection{Quadratic splines}
% We want to create a recursive macro with the same properties as the above
% described |\Curve| macro, but that uses quadratic splines; we call it
% |\Qurve| so that the initial macro name letter reminds us of the nature
% of the splines being used. For the rest they have an almost identical
% syntax; with quadratic spline it is not possible to specify the distance
% of the control points from the extrema, since quadratic spline have just
% one control point that must lay at the intersection of the two tangent
% directions therefore with quadratic splines the tangents at each point
% cannot have the optional part that starts with a semicolon. The syntax, therefore, is just:
%\begin{flushleft}
%\cs{Qurve}\parg{first point}\aarg{direction}...\parg{any point}\aarg{direction}...\parg{last point}\aarg{direction}
%\end{flushleft}
% As with |\Curve|, also with |\Qurve| there is no limitation on the number
% of points, except for the computer memory size; it is advisable not to use
% many arcs otherwise it might become very difficult to find errors.
%
% The first macros that set up the recursion are very similar to those we
% wrote for |\Curve|:
% \begin{macrocode}
\def\Qurve{\@ifstar{\let\fillstroke\fillpath\Qurve@}%
{\let\fillstroke\strokepath\Qurve@}}
\def\Qurve@(#1)<#2>{%
\StartCurveAt#1WithDir{#2}%
\@ifnextchar\lp@r\@Qurve{%
\PackageWarning{curve2e}{%
Quadratic curve specifications must contain at least
two nodes!\Messagebreak
Please, control your Qurve specifications\MessageBreak}}}%
\def\@Qurve(#1)<#2>{\QurveTo#1WithDir{#2}%
\@ifnextchar\lp@r\@Qurve{%
\@ifnextchar[\@ChangeQDir\CurveEnd}}%
\def\@ChangeQDir[#1]{\ChangeDir<#1>\@Qurve}%
% \end{macrocode}
% Notice that in case of long paths it might be better to use the single
% macros |\StartCurveAt|, |\QurveTo|, |\ChangeDir| and |\CurveFinish|
% (or |\FillCurve|), with their respective syntax, in such a way that a long list
% of node-direction specifications passed to |\Qurve| may be split into
% shorter input lines in order to edit the input data in a more comfortable way.
%
%
% The macro that does everything is |\QurveTo|. it start with reading its
% arguments received through the calling macro |\@Qurve|
% \begin{macrocode}
\def\QurveTo#1WithDir#2{%
\edef\@Puno{#1}\DirOfVect#2to\@Duno\bgroup
\DistanceAndDirOfVect\@Puno minus\@Pzero to\@Chord and\@DirChord
% \end{macrocode}
% It verifies if |\@Dpzero| and |\@Dpuno|, the directions at the two extrema
% of the arc, are parallel or anti-parallel by taking their
% ``scalar'' product (|\@Dpzero| times |\@Dpuno*|); if the imaginary
% component of the scalar product vanishes the two directions are parallel;
% in this case we produce an error message, but we continue skipping this arc
% destination point; evidently the drawing will not be the desired one, but
% the job should not abort.
% \begin{macrocode}
\MultVect\@Dzero by*\@Duno to \@Scalar
\YpartOfVect\@Scalar to \@YScalar
\ifdim\@YScalar\p@=\z@
\PackageWarning{curve2e}%
{Quadratic Bezier arcs cannot have their starting\MessageBreak
and ending directions parallel or antiparallel with\MessageBreak
each other. This arc is skipped and replaced with
a dotted line.\MessageBreak}%
\Dotline(\@Pzero)(\@Puno){2}\relax
\else
% \end{macrocode}
% Otherwise we rotate everything about the starting point so as to bring the
% chord on the real axis; we get also the components of the two directions that,
% we should remember, are unit vectors, not generic vectors, although the user
% can use the vector specifications that are more understandable to him/her:
% \begin{macrocode}
\MultVect\@Dzero by*\@DirChord to \@Dpzero
\MultVect\@Duno by*\@DirChord to \@Dpuno
\GetCoord(\@Dpzero)\@DXpzero\@DYpzero
\GetCoord(\@Dpuno)\@DXpuno\@DYpuno
% \end{macrocode}
% We check if the two directions point to the same half plane; this implies
% that these rotated directions point to different sides of the chord vector;
% all this is equivalent that the two direction Y components have opposite
% signs, and therefore their product is strictly negative, and that the two
% X components product is not negative.
% \begin{macrocode}
\MultiplyFN\@DXpzero by\@DXpuno to\@XXD
\MultiplyFN\@DYpzero by\@DYpuno to\@YYD
\unless\ifdim\@YYD\p@<\z@\ifdim\@XXD\p@<\z@
\PackageWarning{curve2e}%
{Quadratic Bezier arcs cannot have inflection points\MessageBreak
Therefore the tangents to the starting and ending arc\MessageBreak
points cannot be directed to the same half plane.\MessageBreak
This arc is skipped and replaced by a dotted line\MessageBreak}%
\Dotline(\@Pzero)(\@Puno){2}\fi
\else
% \end{macrocode}
%
% After these tests we should be in a ``normal'' situation.We first copy
% the expanded input information into new macros that have more explicit
% names: macros stating wit `S' denote the sine of the direction angle,
% while those starting with `C' denote the cosine of that angle. We will
% use these expanded definitions as we know we are working with the actual
% values. These directions are those relative to the arc chord.
% \begin{macrocode}
\edef\@CDzero{\@DXpzero}\relax
\edef\@SDzero{\@DYpzero}\relax
\edef\@CDuno{\@DXpuno}\relax
\edef\@SDuno{\@DYpuno}\relax
% \end{macrocode}
% Suppose we write the parametric equations of a straight line that departs
% from the beginning of the chord with direction angle $\phi_0$ and the
% corresponding equation of the straight line departing from the end of the
% chord (of length $c$) with direction angle $\phi_1$. We have to find the
% coordinates of the intersection point of these two straight lines.
%\begin{subequations}
%\begin{align}
% t \cos\phi_0 - s \cos\phi_1 &= c\\
% t \sin\phi_0 - s \sin\phi_1 &= 0
%\end{align}
%\end{subequations}
% The parameters $t$ and $s$ are just the running parameters; we have
% to solve those simultaneous equations in the unknown variables $t$ and $s$;
% these values let us comupte the coordinates of the intersection point:
%\begin{subequations}\begin{align}
% X_C &=\dfrac{c\cos\phi_0\sin\phi_1}{\sin\phi_0\cos\phi_1 - \cos\phi_0\sin\phi_1} \\
% Y_C &=\dfrac{c\sin\phi_0\sin\phi_1}{\sin\phi_0\cos\phi_1 - \cos\phi_0\sin\phi_1}
%\end{align}\end{subequations}
%
% Having performed the previous tests we are sure that the denominator is not
% vanishing (direction are not parallel or anti-parallel) and that it lays at
% the same side as the direction with angle $\phi_0$ with respect to the chord.
% The coding then goes on like this:
% \begin{macrocode}
\MultiplyFN\@SDzero by\@CDuno to\@tempA
\MultiplyFN\@SDuno by\@CDzero to\@tempB
\edef\@tempA{\strip@pt\dimexpr\@tempA\p@-\@tempB\p@}\relax
\@tdA=\@SDuno\p@ \@tdB=\@Chord\p@ \@tdC=\@tempA\p@
\edef\@tempC{\strip@pt\dimexpr \@tdA*\@tdB/\@tdC}\relax
\MultiplyFN\@tempC by\@CDzero to \@XC
\MultiplyFN\@tempC by\@SDzero to \@YC
\ModOfVect\@XC,\@YC to\@KC
% \end{macrocode}
% We eventually computed the coordinates and the module of the intersection
% point vector taking into account the rotation of the real axis; getting
% back to the original coordinates before rotation we get:
% \begin{macrocode}
\ScaleVect\@Dzero by\@KC to\@CP
\AddVect\@Pzero and\@CP to\@CP
\GetCoord(\@Pzero)\@XPzero\@YPzero
\GetCoord(\@Puno)\@XPuno\@YPuno
\GetCoord(\@CP)\@XCP\@YCP
% \end{macrocode}
% We have now the coordinates of the two extrema point of the quadratic arc
% and of the control point. Keeping in mind that the symbols $P_0$, $P_1$
% and $C$ denote geometrical points but also their coordinates as ordered
% pairs of real numbers (i.e. they are complex numbers) we have to determine
% the smart cubic arc nodes and control points; we should determine the
% values of $P_a$ and $P_b$ such that
%\[
% P_0(1-t)^3 +3P_a(1-t)^2t +3P_b(1-t)t^2 +P_1t^3
%\]
% is equivalent to
%\[
% P_0(1-t)^2 + 2C(1-t)t + P_1t^2
%\]
% It turns out that the solution is given by
%\begin{equation}
%P_a= C+(P_0-C)/3 \qquad \text{and}\qquad P_b = C+(P_1- C)/3
%\label{equ:spline3}
%\end{equation}
%
% The transformations implied by equations~\eqref{equ:spline3} are performed
% by the following macros already available from the |pict2e| package; we
% use them here with the actual arguments used for this task:
% \begin{macrocode}
\@ovxx=\@XPzero\unitlength \@ovyy=\@YPzero\unitlength
\@ovdx=\@XCP\unitlength \@ovdy=\@YCP\unitlength
\@xdim=\@XPuno\unitlength \@ydim=\@YPuno\unitlength
\pIIe@bezier@QtoC\@ovxx\@ovdx\@ovro
\pIIe@bezier@QtoC\@ovyy\@ovdy\@ovri
\pIIe@bezier@QtoC\@xdim\@ovdx\@clnwd
\pIIe@bezier@QtoC\@ydim\@ovdy\@clnht
\pIIe@moveto\@ovxx\@ovyy
% \end{macrocode}
%
% We call the basic |pict2e| macro to draw a cubic spline and we finish
% the conditional statements with which we started these calculations;
% eventually we close the group we opened at the beginning and we copy
% the terminal node information (position and direction) into the
% 0-labelled macros that indicate the starting point of the next arc.
% \begin{macrocode}
\pIIe@curveto\@ovro\@ovri\@clnwd\@clnht\@xdim\@ydim
\fi\fi\egroup
\CopyVect\@Puno to\@Pzero
\CopyVect\@Duno to\@Dzero
\ignorespaces}
% \end{macrocode}
%
% An example of usage is shown at the left in figure~\ref{fig:quadratic-arcs}\footnote{The commands \cs{legenda}, \cs{Pall} and
% \cs{Zbox} are specifically defined in the preamble of this document; they must
% be used within a \texttt{picture} environment. \cs{legenda} draws a framed
% legend made up of a single (short) math formula; \cs{Pall} is just a shorthand
% to put a filled small circle at a specified position' \cs{Zbox} puts a
% symbol in math mode a little displaced in the proper direction relative to
% a specified position. They are just handy to label certain objects in a
% \texttt{picture} diagram, but they are not part of the \texttt{curve2e}
% package.}.
% created with the following code:
%\begin{verbatim}
%\begin{figure}[!htp]
%\unitlength=0.0045\textwidth
%\begin{picture}(100,100)
%\put(0,0){\framebox(100,100){}}
%\put(50,50){\Qurve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>%
%(-50,0)<0,-1>(0,-50)<1,0>\color{green}%
%\Qurve*(0,-50)<0,1>(50,0)<1,0>[-1,0](0,50)<0,1>[0,-1]
%(-50,0)<-1,0>[1,0](0,-50)<0,-1>}
%\Qurve(0,0)<1,4>(50,50)<1,0>(100,100)<1,4>
%\put(5,50){\Qurve(0,0)<1,1.5>(22.5,20)<1,0>(45,0)<1,-1.5>%
%(67.5,-20)<1,0>(90,0)<1,1.5>}
%\Zbox(0,0)[tl]{0,0}\Zbox(100,0)[tr]{100,0}
%\Zbox(100,100)[br]{100,100}\Zbox(0,100)[bl]{0,100}
%\end{picture}
%\end{figure}
%\end{verbatim}
%
%\begin{figure}[!tb]
%\unitlength=0.0045\textwidth
%\begin{picture}(100,100)
%\put(0,0){\framebox(100,100){}}
%\put(50,50){\Qurve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>\color{green}%
%\Qurve*(0,-50)<0,1>(50,0)<1,0>[-1,0](0,50)<0,1>[0,-1](-50,0)<-1,0>[1,0](0,-50)<0,-1>}
%\Qurve(0,0)<1,4>(50,50)<1,0>(100,100)<1,4>
%\put(5,50){\Qurve(0,0)<1,1.5>(22.5,20)<1,0>(45,0)<1,-1.5>(67.5,-20)<1,0>(90,0)<1,1.5>}
%\Zbox(0,0)[tl]{0,0}\Zbox(100,0)[tr]{100,0}
%\Zbox(100,100)[br]{100,100}\Zbox(0,100)[bl]{0,100}
%\Pall[2](0,0)\Pall[2](100,0)\Pall[2](100,100)\Pall[2](0,100)
%\end{picture}
%\hfill
%\begin{picture}(100,100)
%\put(0,0){\framebox(100,100){}}
%\put(50,50){\Qurve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>
%\Curve(0,-50)<1,0>(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>(0,-50)<1,0>}
%\Zbox(50,50)[t]{O}\Pall[2](50,50)\put(50,50){\Vector(45:50)}\Zbox(67,70)[tl]{R}
%\end{picture}
%
%\caption{\rule{0pt}{4ex}Several graphs drawn with quadratic Bézier splines}
%\label{fig:quadratic-arcs}
%\end{figure}
%
% Notice the green filled path: that result is not expected, but the filling
% operation is controlled by the inner workings of the typesetting program,
% where it fills what \emph{it} considers the interior of a path, not what
% \emph{we} think it is the interior of a path.Knowing this feature it is not
% difficult to fill the external lozenge with green, and then fill the internal
% path with white; the result would be to cover with white the external part of
% the interior path. Sort of odd way of getting the result, but this is not due
% to the quadratic splines but to the internal workings of |pdftex| and its
% companion typesetting engines, that consider ``interior'' the concave side of
% the closed path, not the convex one.
%
% Notice also that the inflexed line is made with two arcs that meet at the
% inflection point; the same is true for the line that resembles a sine wave.
% The cusps of the inner border of the green area are obtained with the usual
% optional argument already used also with the |\Curve| recursive macro.
%
% The ``circle'' inside the square frame is visibly different from a real
% circle, in spite of the fact that the maximum deviation from the true
% circle is just about 6\% relative to the radius; a quarter circle obtained
% with a single parabola is definitely a poor approximation of a real quarter
% circle; possibly by splitting each quarter circle in three or four partial
% arcs the approximation of a real quarter circle would be much better. On the
% right of figure~\ref{fig:quadratic-arcs} it is possible to compare a
% ``circle'' obtained with quadratic arcs with the the internal circle
% obtained with cubic arcs; the difference is easily seen even with a naked eye.
%
% With quadratic arcs we decided to avoid defining specific macros similar
% to |\CurveBetween| and |\CbezierBetween|; the first macro would not save
% any typing to the operator; furthermore it may be questionable if it was
% really useful even with cubic splines; the second macro with quadratic
% arcs is meaningless, since with quadratic arcs there is just one control
% point and there is no choice on its position.
%
% \section{Conclusion}
% I believe that the set of new macros provided by this package 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.
%\iffalse
%
%\fi
%
% \Finale
% \endinput