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\begin{document}

\author{Claudio Beccari\\{\small\texttt{claudio(dot)beccari(at)gmail(dot)com}}}
\title{Package \pack{curve2e} user manual}
\GetFileInfo{curve2e.sty}
\date{Version \fileversion~--~Last revised \filedate}
\maketitle
\columnseprule=0.4pt
\begin{multicols}{2}
 \tableofcontents
 \end{multicols}

\begin{abstract}\setfontsize{9.5}
This file contains the user manual of the \pack{curve2e} extension package to the \pack{pict2e} bundle; the latter was described by Lamport himself in the 1994 second edition of his \LaTeX\ handbook.

Please take notice that on April 2011 a package \pack{pict2e} upgraded version has been released that incorporates some of the commands defined in early versions of this package \pack{curve2e}; apparently there are no conflicts, because this package contains only the advanced features that extend the above package.

Since this extension redefines some commands and introduces some more drawing facilities (that allow to draw circular arcs and arbitrary curves with the minimum of user intervention) users need a user manual that contains several actual examples; this auxiliary manual is contained in file 
|curve2e-manual.pdf|. The software available to show the drawing code and its result after compilation is incompatible with the usual \LaTeX\ |ltxdoc| class for code documentation, therefore a separate user manual has been made available. If  users want to explore the \pack{curve2e} code and its documentation they have available the |curve2e.pdf| file. Either file is readable by entering in a terminal window the command |texdoc curve2e-manual| or |texdoc curve2e.pdf|; please do not forget the extension when you want to read the code documentation.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Installation} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If \TeX\ system was installed with a \TeX~Live or a MiK\TeX\ complete and updated distribution this package is already installed; in order to verify open a terminal or command prompt window and use it to enter the \texttt{texdoc curve2e-manual} command; as soon as the command is executed a screen window should open displaying this manual. If this does not happen, either you misspelt the command (it happens more often than not), or your installation is not complete and updated. 

I suggest you to use your computer installation facilities to install this bundle. Otherwise, download this curve2e.zip package from one of the CTAN (Comprehensive TeX Archive Network) archives, to your downloads folder. 
Before doing anything else verify if you have a \emph{personal} \texttt{texmf} tree; if not, create one reading you distribution instruction; let us assume that your personal |texmf| archive is in |HOME/texmf| (on Windows change the slash with a backslash) and |HOME| is a path starting from the root of you hard disk and going though several other folders. On Windows~10 it might be |C:\Users\YourName|; on Linux it might simply be  |~|; on Mac it would be |~/\Library|, but sometimes the |HOME| might be different, especially on Windows platforms. Create the following subfolders:
\begin{enumerate}[noitemsep]
\item |HOME/texmf/source/latex/curve2e/|
\item |HOME/texmf/doc/latex/curve2e/|
\item |HOME/texmf/tex/latex/curve2e/|
\end{enumerate} 

Now move file \file{curve2e.zip} to the  |.../source/latex/curve2e/| folder; then decompress the |.zip| file with the software you have available on your platform. Run |pdflatex| on the |.dtx| file; then compile the |curve2e-manual.tex| file. You might need to repeat these compilations two or three times in order to have the table of contents and all the references correctly connected.

This done, move the |.pdf| files to the |.../doc/latex/curve2e/| folder. and move the |.sty| files to the |.../tex/latex/curve2e/| folder.

Clear the |.../source/latex/curve2e/| folder from the auxiliary files, all those remaining in the folder except those that have the extensions |.zip|, |.dtx|, |.tex|, and |.txt|. Read the \file{README.txt} file.

If your \TeX\ system is correctly set up, your files in your personal tree should be immediately usable; probably you have to create or update the file-name database with MiK\TeX; in this case read the documentation of your MiK\TeX\ installation to discover how to~do~it.

Remember to delete all these subfolders if you decide to install a complete updated version of your favourite distribution, and you'd better keep it updated approximately once every 7 or 10 days. This is much simpler than to struggle with these manual operations. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Package \pack{pict2e} and this extension \pack{curve2e}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 Package \pack{pict2e} was announced in issue 15 of 
 \file{latexnews}  around December 2003; it was declared that 
 the new package would replace the dummy one that had 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, it was 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}[noitemsep]
 \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.
%
 \item Filled and unfilled full circles were limited by the necessarily
 limited number of specific glyphs contained in the special \LaTeX\
 \texttt{picture} fonts.
%
 \item Quarter circles radii  were also limited for the same reason.
%
 \item Ovals (rectangles with rounded corners) could not be too small
 because of the unavailability of small radius quarter circles, nor
 could be too large, in the sense that after a certain radius the
 rounded corners remained the same curvature and would not increase proportionally  to the oval size.
%
 \item Vector tips had only one possible shape and matched the limited
 number of vector slopes.
%
 \item For circles and inclined lines and vectors just two possible
 thicknesses  were available.
 \end{enumerate}

 Package \pack{pict2e} removes most if not all the above
 limitations.
 \begin{enumerate}[noitemsep]
 \item Line and vector slopes are virtually unlimited; the only
 remaining limitation is that the direction coefficients must be
 three-digit integer numbers; they need not be relatively prime; with
 the 2009 upgrade even this limitation was removed and now slope
 coefficients can be any fractional number whose magnitude does not
 exceed 16\,384, the maximum dimension in points that \TeX\ can handle.
%
 \item Filled and unfilled circles can be of any size.
%
 \item Ovals can be designed with any specified corner curvature and
 there is virtually no limitation to such curvatures; of course corner
 radii should not exceed half the lower value between the base and the
 height of the oval.
%
 \item There are two shapes for the arrow tips; the triangular one
 traditional with \LaTeX\ vectors, or the arrow tip with PostScript
 style.
%
 \item The |\linethickness| command changes the thickness of all lines,
 straight, curved, vertical, horizontal, arrow tipped, et cetera.
 \end{enumerate}

 This specific extension package \pack{curve2e} adds the following 
 features.
 \begin{enumerate}[noitemsep]
%
 \item  Point coordinates can be specified with macros; this is similar
 to “naming” points; it eases editing the user's graphic work, because 
 points that are used several times are specified with a single macro; 
 it also eases the transmission of coordinates between different macros 
 and environments. It is also important for the following feature, described in the following entry..
%
 \item  Point coordinates my be specified in both cartesian and polar
 form: internally they are handled as cartesian coordinates, but the
 user can specify  his/her points also in polar form. In order to avoid
 confusion with other graphic packages, \pack{curve2e} uses the usual
 comma separated couple \meta{$x,y$} of integer or fractional numbers for
 cartesian coordinates, and the couple \meta{$\theta$}:\meta{$\rho$} for
 polar coordinates (the angle preceding the radius).
 
 All graphic object commands accept polar or cartesian coordinates at
 the choice of the user who may use for each object the formalism s/he
 prefers. Also the |\put| and |\multiput| commands have been redefined so
 as to accept cartesian or polar coordinates.

 Of course the user should pay attention to the meaning of cartesian
 vs.~polar coordinates. Both imply a displacement with respect to the
 actual origin of the axes. So when a circle center  is placed at 
 coordinates $a,b$ with a normal |\put| command, the circle is placed 
 exactly in that point; with a normal |\put| command the same happens if
 coordinates $\alpha{:}\rho$ are specified.
 
 But if the |\put| command is nested into another |\put| command, the
 current origin of the axes is displaced — this is obvious and the
 purpose of nesting |\put| commands is exactly that. But if a segment
 is specified so that its ending point is at a specific distance and in
 specific direction from its starting point, polar coordinates appear to 
 be the most convenient to use; in this case, though, the origin of the
 axes becomes the starting point of the segment, therefore the segment
 might be drawn in a strange way. Attention has been paid to avoid such 
 misinterpretation, but maybe some unusual situation may not have come to 
 my mind; feedback is very welcome. Meanwhile pay attention when you use 
 polar coordinates.
%
\item At user level 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 s/he 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 as
 roto-amplification operators.
%
 \item Commands for setting the line terminations were introduced; the
 user can chose between square or rounded caps; the default is set to
 rounded caps; now this feature is directly available with \pack{pict2e}.
%
 \item Commands for specifying the way two straight or curves lines 
 join to one another.
 %
 \item Originally the |\line| macro was redefined so as to allow large
 (up to three digits) integer direction coefficients, but maintaining
 the same syntax as in the original \texttt{picture} environment; now 
 \pack{pict2e} removes the integer number limitations and allows 
 fractional values, initially implemented by \pack{curve2e}; now direction
 coefficients may be specified in polar form.
 %
 \item A new macro |\Line| was originally defined by \pack{curve2e} so 
 as to avoid the need to specify the horizontal projection of inclined 
 lines; now this functionality is available directly with \pack{pict2e};
 but this \pack{curve2e} macro name now conflicts with the \pack{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).
 %
 \item A new macro |\LINE| was defined in order to join two points
 specified with their coordinates; this is now the normal behaviour of
 the |\Line| macro of \pack{pict2e}, so that in this package |\LINE| is
 now renamed |\segment|; there is no need to use the |\put| command with
 this segment specification.
 %
 \item A new macro |\DashLine| (alias: |\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. The starting point may be specified in cartesiano or polar
 form; the end point in cartesian format specifies the desired end
 point; but, if the second point is in polar form, it is meant
 \emph{relative to the starting point}, not as an absolute end point.
 See the examples further on.
 %
 \item A similar 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. Polar coordinates for the second
 point have the same relative meaning as specified for the |\Dashline|
 macro. The dot diameter may be specified with an optional argument; by 
 default this diameter equals the |1pt| width.
 %
 \item Similar macros are redefined for vectors; |\vector| redefines the
 original macro but with the vector slope limitations removed and the
 vector direction may be given in polar form; |\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. 
 |\VVECTOR| may be available if used with a sufficiently recent \LaTeX\ 
 kernel version; it draws a vector between two given points, with arrow
 tips at both ends. 

%
 \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 \pack{pict2e}); here it is 
 redefined so as to allow an optional specification of the way segments 
 for the polyline are joined to one another. Vertices may be specified 
 with polar coordinates and are always relative to the preceding point.
%
 \item The \pack{pict2e} |\polygon| macro draws closed polylines (in
 practice general polygons) has been redefined in such a way that it
 can accept the various vertices specified with (relative) polar
 coordinates. The |polygon*| macro produces a color filled polygon; the
 default color is black, but a different color may be specified with the
 usual |\color| command given within the same group where |\polygon*| is
 enclosed.
%
 \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
 \pack{pict2e}, which provides also the starred version |\arc*| that 
 fills up with the current color the sector generated by a 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.
%
 \item Two new macros |\VectorArc| and |\VectorARC| (alias |\VVectorArc|) 
 are defined in order to draw circular arcs with an arrow at one or both 
 ends.
%
 \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
 current color.
%
 \item The above |\Curve| macro is a recursive macro that can draw an
 unlimited (reasonably limited) number of connected Bézier spline arcs
 with specification of the 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.|\Curve| recognises also 
 an optional argument to set a direction change, that is a cusp.
%
 \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 the |\Curve|
 macro.
\end{enumerate}

 The \pack{pict2e} package already defines macros such as |\moveto|,
 |\lineto|, |\curveto|, |\closepath|, |\fillpath|, and |\strokepath|;
 |curve2e| just redefines them so as to accept also polar coordinates;
 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 splines
 control points; in this case the control points are sort of rigid. It
 would be very useful to resort to the \pack{hobby} package, but its 
 macros are compatible with those of the \pack{tikz} and \pack{pgf} 
 packages, not with\pack{curve2e}; an interface should be created in 
 order to deal with  the \pack{hobby} package, but this has not been 
 done yet. 

 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. In the first versions of this package the
 trigonometric functions were also defined in a way that the author
 believed to be more efficient than those defined by the \texttt{trig}
 package; in any case the macro names were sufficiently different to
 accommodate both definition sets in the same \LaTeX\ run. With the
 progress of the \LaTeX3 language, package \pack{xfp} functionalities 
 have  become available, and any sort of calculations can be done floating
 point decimal numbers; therefore the most common algebraic, irrational
 and transcendental functions can be computed in the background with the
 stable internal floating point facilities. We maintain some computation
 with complex number algebra, but use the |xfp| functionalities for
 other computations.

 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 \pack{pict2e} package or even become a part of
 it. Actually some macros have already been included in the \pack{pict2e}
 package. The |\Curve| algorithm, as said before, might be redefined
 so as to use the macros  introduced by the \pack{hobby} package, that
 implements for the \pack{tikz} and \pack{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
 \pack{pict2e}; they are the only ones who can decide whether or not
 to incorporate new macros in their package.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Summary and examples of new commands}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 This package \pack{curve2e} extends the power of \pack{pict2e} with
 the following modifications and the following new commands.
 \begin{enumerate}[noitemsep]
 \item This package \pack{curve2e} calls directly the \LaTeX\ packages
 |color| and \pack{pict2e}; it passes to the latter one any possible 
 option that it can receive; actually the only options that make sense 
 for \pack{pict2e} are those concerning the arrow tip shapes, either 
 \LaTeX\ (default) or PostScript styled, because it is assumed that if
 this package is used, the original \LaTeX\ commands are of no interest: 
 see the \pack{pict2e} documentation in order to find the correct options 
 \pack{pict2e} can receive. If the user wants to use the \pack{xcolor} 
 package, it has to load it \emph{before} \pack{curve2e}.
%%
\item Since they are used very much in the following examples, we recall some commands to label certain drawing elements, such as points, lines, arcs, and similar lines, and to insert legends in some figures. They are \cs{legenda}, \cs{Zbox} and \cs{Pbox}; their syntax is the following:
\begin{Sintassi}[9.75]\ttfamily
|\legenda|\parg{coordinates}\marg{formula}
|\Zbox|\parg{coordinates}\oarg{position}\marg{formula}\oarg{dot diameter}
|\Pbox|\parg{coordinates}\oarg{position}\marg{text}\oarg{dot diameter}\meta{\texttt{*}}\aarg{angle}
\end{Sintassi}
They are described in any up to date documentation of \pack{curve2e}.
%%
 \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 commands in order to restore the
 previous terminator or join; in other words, groups and environments do
 not have any influence on these commands.
%%
 \item The commands 
 \begin{Sintassi}
 |\linethickness|\marg{thickness} 
 |\thicklines| 
 |\thinlines|
 |\defaultlinethickness|\marg{thickness}
 \end{Sintassi} 
 always redefine the internal |\@wholewidth| and |\@halfwidth| so that the 
 latter ones always refer to a full width and to a half of it in this way
 if you issue the command {\setfontsize{8.75}|\defaultlinethickness{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 
 lines, and so on. It's better to recall that |\thinlines| and 
 |\thicklines| are declarations that do not take arguments; on the pposite 
 the other commands, as shown in the above syntax medallion, accept a 
 specific thickness value, that is either a length specification  complete 
 of its units, or a dimensional expression.
%%
 \item Straight lines and vectors are redefined in such a way that
 fractional slope coefficients may be specified; the zero length line and 
 vector does not produce errors and is ignored; the zero length vectors 
 draw only the arrow tips. 
%%
 \item New line and vector macros are defined that avoid the necessity
 of specifying the horizontal component; |\put(3,4){\LIne(25,15)}|
 specifies a segment that starts at point $(3,4)$ and goes to point
 $(3+25,4+15)$; the command |\segment(3,4)(28,19)| achieves the same
 result without the need of using the |\put| command. Therefore |\LIne|
 is just for internal usage, rather than a user command. Now 
 \pack{curve2e} has available also the “arc vectors” with the arrow tips
 at one or at both  ends.  The |\segment| syntax is
 \begin{Sintassi}
 \cs{segment}\parg{starting point}\parg{ending point}
 \end{Sintassi}

 The same applies to the vector commands |\Vector| and |\VECTOR| and
 |\VVECTOR|; the latter command behaves as |\VECTOR| but draws a vector
 with arrow tips at both ends; furthermore this command is available
 only with main versions~2 or higher of \pack{curve2e}.
 \begin{Sintassi}\small
 |\put|\parg{starting point}\Marg{\cs{Vector}\parg{vector cartesian or polar components}}
 |\VECTOR|\parg{starting point}\parg{ending point}
 |\VVECTOR|\parg{starting point}\parg{ending point}
 \end{Sintassi}

 
 Experience has shown that the commands intended 
 to join two specified points are particularly
 useful; see figure~\ref{fig:vectors}.

\begin{figure}[!hb]
\begin{Esempio}[\small](0.60)
\unitlength=0.01\linewidth
\begin{picture}(80,20)
\AutoGrid[red]
\put(0,0){\vector(1.5,2.3){10}}
\put(20,0){\Vector(10,15.33333)}
\VECTOR(40,0)(50,15.33333)
\ifdefined\VVECTOR \VVECTOR(60,0)(80,10)\fi
\end{picture}
\end{Esempio}
\caption{Three (displaced) identical vectors
   obtained with the three
   vector macros\ifdefined\VVECTOR; a double
   tipped vector is also shown\fi. The darker
   cyan color is an usage example optional grid 
   coloring }\label{fig:vectors}
\end{figure}
%
 \item The |\polyline| command has already been introduced in \pack{pict2e}:  in \pack{curve2e} it is redefined so as to accept also polar coordinates; this new version of |\polyline| accepts also an optional argument to specify how two consecutive segments join together; it accepts an unlimited list of point coordinates, possibly stored in macros, enclosed within round parentheses; the command draws a sequence of connected segments that join in order the specified points; the syntax is:
	\begin{Sintassi}
\cs{polyline}\oarg{optional join style}\parg{$P_1$}\parg{$P_2$}\texttt{...}\parg{$P_n$}
	\end{Sintassi}
 See figure~\ref{fig:polyline} where a regular pentagon is drawn; usage
 of polar coordinates is also shown; please notice how  polar
 coordinates act in this figure.

	\begin{figure}[!ht]
	\begin{Esempio}[\small](0.55)
	\unitlength=0.5mm
	\begin{picture}(40,32)(-20,-17)
	\polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
	\end{picture}
\end{Esempio}
	\caption{Polygonal line obtained by means of the \texttt{\string\polyline}
   command; vertex coordinates are in polar form.}
	\label{fig:polyline}
	\end{figure}

Examples of using polar and cartesian coordinates are shown in figure~\ref{fig:polar}. Notice the 
|\AutoGrid| macro that draws the grid of mesh lines that are very useful to set objects at the right positions. The main lines are 
\verb|10\unitlength| apart so that the main squares are clearly visible; if the squares are too small only the median lines are traced with thinner lines; if the squares are not so small thin lines are traced \verb|2\unitlength| apart; otherwise the thin median lines and the thinner \verb|1\unitlength| apart are traced as in regular millimetre drawing paper. The rendering on the screen depends very much on its pixel density, nevertheless we think that e nicely thick grid is very helpful while drawing geometric graphs. When the drawing is completed the \verb|\AutoGrid| command may be commented out so that the grid does not appear in the final document.

\begin{figure}[htb]
\begin{Esempio}[\normalsize]%
\unitlength=0.02\textwidth
\begin{picture}(40,30)
\AutoGrid
\Zbox(40,0)[l]{40,0}[1]
\Zbox(90:30)[bc]{90{:}30}[1]
\Zbox(45:30)[bc]{45{:}30}[1]
\Zbox(30,30)[bc]{30,30}[1]
\multiput(0,0)(20:10){5}%
   {\makebox(0,0){\rule{1.5mm}{1.5mm}}}
\end{picture}
\end{Esempio}
\caption[Use of cartesian and absolute polar coordinates]{Use of
 cartesian and absolute polar coordinates. The \texttt{\string\Zbox}
 macro is just a shortcut to set a small dot with a (math) legend close
 to it.}
\label{fig:polar}
\end{figure}

 A similar example may be obtained with the |\polygon| macro that does
 not require to terminate the polyline at the starting point.
 Figure~\ref{fig:polygon} shows how to get a coloured filled pentagon.

\begin{figure}[!ht]
\begin{Esempio}[\normalsize]%
\unitlength=.5mm
\begin{picture}(40,32)(-20,-20)
\color{magenta}
\polygon*(90:20)(162:20)(234:20)(306:20)(378:20)
\end{picture}
\end{Esempio}
	\caption{A pentagon obtained by means of the \texttt{\string\polygon*}
   command; vertex coordinates are in relative polar form.}
	\label{fig:polygon}
	\end{figure}
%
 \item The new command |\Dashline| (alias: |\Dline| for backwards
 compatibility):
 \begin{Sintassi}
 |\Dashline|\parg{first point}\parg{second point}\marg{dash and gap length}
 \end{Sintassi}
 draws a dashed line containing as many dashes as possible, just as 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{Esempio}[\normalsize]%
\unitlength=1mm
\begin{picture}(40,40)
\AutoGrid(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}[0.75mm]
\put(40,40){\circle*{2}}
\end{picture}
\end{Esempio}
\caption{Dashed lines and graph grid}\label{fig:dashline}
\end{figure}

 Another example of usage of cartesian and polar coordinates usage is 
 shown in figure~\ref{fig:polar} together with its code.
 
\begin{figure}
\begin{Esempio}[\normalsize]%
\unitlength=0.025\textwidth
\begin{picture}(40,30)
\noinnerlines % <-------- !
\AutoGrid(40,30)\thicklines
\color{red}%
\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}
\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{Esempio}
\caption{Different length dashed lines with the same nominal dash
 length; notice the relative polar coordinates used for the dashed
 lines starting at the grid lower right vertex.}
\label{fig:dashedlines}
\end{figure}
%%
\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.

\begin{figure}[htb]
\begin{Esempio}[\normalsize]%
\unitlength=0.025\textwidth
\begin{picture}(40,30)
\AutoGrid(40,30)
\Dotline(0,0)(40,10){1.5}[2pt]
\Dotline(0,0)(40,20){1.5}[2pt]
\Dotline(0,0)(40,30){1.5}[2pt]
\Dotline(0,0)(30,30){1.5}[2pt]
\Dotline(0,0)(20,30){1.5}[2pt]
\Dotline(0,0)(10,30){1.5}[2pt]
{\color{red}\relax
\Dotline(40,0)(108:30){1.5}
\Dotline(40,0)(126:30){1.5}[2pt]
\Dotline(40,0)(144:30){1.5}[2pt]
\Dotline(40,0)(162:30){1.5}[2pt]}%
\end{picture}
\end{Esempio}
\caption{Different length dotted lines with the same nominal dot gap;
 again notice the relative polar coordinates for the dotted lines
 starting at the grid lower right vertex.}
\label{fig:dottedlines}
\end{figure}
%%
 \item |\GraphGrid| and |\AutoGrid| are commands that draw 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 on the drawing canvas. 
 \begin{Sintassi}
 |\GraphicGrid|\parg{grad base, grid height}
 |\AutoGrid|
 \end{Sintassi}
 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. Several  examples of usage of such grid are shown in several 
 figures. 
 
 |\Autogrid|  does not require arguments, but requires the canvas 
 dimensions and  offsets to be specified as multiples of~10; if the latter 
 are  specified they are simply ignored.
%%
 \item New trigonometric function macros have been  computed by means of
 the functionalities of the |xfp| included in the \LaTeX\ kernel. The 
 difference with the other  existing macros is that angles are specified 
 in sexagesimal degrees, so that the users need not transform to radians. 
 The computations are done taking into account that “abnormal” values that 
 can occasionally be avoided, for example $\tan 90^\circ$ must be avoided 
 and replaced with a suitably large number, because the \TeX\ system does 
 not handle “infinity”.

 These trigonometric functions are used within the complex number
 macros; but if the user wants to use them the syntax is the following:
\begin{Sintassi}
      \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{Sintassi}
 The \meta{control sequence} may then be used, for example, as a
 multiplying factor of a length.
%%
 \item Arcs can be drawn as simple circular arcs, or with one or two
 arrows at their ends (curved vectors);  the syntax is:
\begin{Sintassi}
       \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}
\cs{VVectorArc}\parg{center}\parg{starting point}\marg{angle}
\end{Sintassi}
 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| (alias |\VVectorArc|) draws an 
 arc with arrow tips at both ends; see figure~\ref{fig:arcs}.
 
 Notice that the starting point may be specified with polar coordinates; differently to cartesian coordinates, that are absolute with respect with the drawing axes, the polar ones are relative to the center of the arcs with or without vector tips.
	\begin{figure}
	\begin{Esempio}[\small]%
	\unitlength=0.5mm
	\begin{picture}(60,40)
	\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}
	\VVectorArc(60,20)(135:10){90}
	\end{picture}
	\end{Esempio}
	\caption{Arcs and curved vectors}\label{fig:arcs}
	\end{figure}
%%
 \item The available commands allow to create the drawings necessary to prove some geometrical theorems; for example let us prove the Pitagora's theorem. Figure~\ref{fig:pitagora} displays a right triangle with its hypothenuse laying horizontally; its vertices are labeled \textsf{A}, \textsf{B}, and \textsf{C}, being \textsf{A} the right angle vertex. The height relative to the hypothenuse intersects this side in point \textsf{H}, and divides the whole triangle \textsf{ABC},  in two similar smaller ones \textsf{AHC}, and\textsf{ABH}. Segments \textsf{CH}, and\textsf{HB}, add to the whole hypothenuse length. Figure~\ref{fig:pitagora} displays also the square \textsf{CBDE}, built on the hypothenuse and divided in two rectangles \textsf{CHFE}, and \textsf{HBDF}, where \textsf{HF} is the continuation of the height line.

The lengths of the original right triangle sides are $a, b, c$ as marked on the figure; Point \textsf{H} divides the hypothenuse of length $c$ in two shorter segments of lengths $d, e$, respectively,  as marked in the figure
  
Since the three triangles are similar, we can set up the relationships between their sides:
\begin{align*}
d : a = a : c &\Longrightarrow d = a^2/c\\
e : b = b : c &\Longrightarrow e = b^2/c
\end{align*}

Therefore rectangle \textsf{CHFE} area equals $ c\cdot d = c\cdot a^2/c = a^2$. Similarly rectangle \textsf{HBDF} area equals $ c\cdot e = c\cdot b^2/c = b^2$. The square built on the hypothenuse has an area equal to $c^2$; therefore it is 
\begin{equation*}
c^2 = a^2 + b^2
\end{equation*}
which proves the Pitagora's theorem.

\begin{figure}
\centering
\begin{Esempio}
\unitlength=1mm
\begin{picture}(70,80)(-10,-50)
\AutoGrid
\thicklines
\polygon(0,0)(50,0)(18,24)
\thinlines
\polyline(0,0)(0,-50)(50,-50)(50,0)
\Pbox(0,0)[r]{C}[0.75ex]
\Pbox(18,0)[bl]{H}[0.75ex]
\Pbox(50,0)[l]{B}[0.75ex]
\Pbox(18,24)[b]{A}[0.75ex]
\Pbox(0,-50)[br]{E}[0.75ex]
\Pbox(50,-50)[bl]{D}[0.75ex]
\Pbox(18,-50)[br]{F}[0.75ex]
\Zbox(10,12)[br]{a}[0]
\Zbox(33,12)[bl]{b}[0]
\Zbox(25,0)[t]{c}[0]
\Zbox(9,0)[b]{d}[0]
\Zbox(30,0)[b]{e}[0]
\Zbox(50,-25)[r]{c}[0]
\Zbox(25,-50)[b]{c}[0]
\Zbox(0,-25)[l]{c}[0]
\Dashline(18,24)(18,-50){1.5}
\end{picture}
\end{Esempio}
\caption{Geometrical construction to prove Pitagora's theorem}
\label{fig:pitagora}
\end{figure}
%%
 \item A multitude of commands have been defined in order to manage
 complex numbers; actually complex numbers are represented as a comma
 separated pair of fractional numbers; internally these macros use 
 only the cartesian form, and output, unless 
 differently specified is also in cartesian form; but input can be in 
 polar form. They are used to address 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 that 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;
 \meta{numeric macro} means a macro that contains a fractional number;
 \textit{macro} is a valid macro name, i.e.~a backslash followed by
 letters, or anything else that can receive a definition. A
 \emph{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, as it is used in the
 Euler formula $ \vec{v} = M\eu^{\mathrm{j}\varphi}$.

\begin{Sintassi}
  |\MakeVectorFrom|\meta{numeric macro}\meta{numeric macro}|to|\meta{vector macro}
  |\CopyVect|\meta{first vector}|to|\meta{second vector macro}
  |\ModOfVect|\meta{vector}|to|\meta{modulus macro}
  |\DirOfvect|\meta{vector}|to|\meta{versor macro}
  |\ModAndDirOfVect|\meta{vector}|to|\meta{modulus macro}|and|\meta{versor macro}
  |\ModAndAngleOfVect|\meta{vector}|to|\meta{modulus macro}|and|\meta{angle macro}
  {\setfontsize{10}|\DistanceAndDirOfVect|\meta{1st vector} |minus|\meta{2nd vector} 
  \qquad\qquad|to|\meta{distance macro} |and|\meta{versor macro}}
  |\XpartOfVect|\meta{vector}|to|\meta{macro}
  |\YpartOfVect|\meta{vector}|to|\meta{macro}
\end{Sintassi}

 
\begin{Sintassi}
  |\DirFromAngle|\meta{angle}|to|\meta{versor macro}
  |\ArgOfVect|\meta{vector}|to|\meta{angle macro}
  |\ScaleVect|\meta{vector}|by|\meta{scaling factor}|to|\meta{vector macro}
  |\ConjVect|\meta{vector}|to|\meta{conjugate vector macro}
  |\SubVect|\meta{subtrahend vector}|from|\meta{minuend vector}|to|\meta{vector macro}
  |\AddVect|\meta{first vector}|and|\meta{second vector}|to|\meta{vector macro}
  |\Multvect|\marg{first vector}*\marg{second vector}*\marg{vector macro} (\textcolor{blue}{the 
\qquad\qquad  asterisks are optional; either one changes the second vector
\qquad\qquad  into its complex conjugate})
  |\MultVect|\meta{first vector}|by|\meta{second vector}|to|\meta{vector macro}
\qquad\qquad (\textcolor{red}{discouraged; maintained for backwards compatibility})
  |\MultVect|\meta{first vector}|by*|\meta{second vector}|to|\meta{vector macro}
\qquad\qquad  (\textcolor{red}{discouraged; maintained for backwards compatibility})
  |\Divvect|\marg{dividend vector}\marg{divisor vector}\marg{vector macro} {\color{blue}notice 
  \qquad\qquad that this new command warns the user if the divisor is zero}
  |\DivVect|\meta{dividend vector}|by|\meta{divisor vector}|to|\meta{vector macro} 
\qquad\qquad  (\textcolor{red}{discouraged: maintained for backwards compatibility})
\end{Sintassi}
%%
 \item General curves can be drawn with the \pack{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. Such commands are the following:
\begin{itemize}[noitemsep]

\item \cs{Curve}  draws a sequence of arcs as explained above, using
 third order (cubic) Bézier splines. The starred version of this command
 fills the internal part of the curve with the current color; if the
 last arc finishes where the fist arc starts, it is clear what is the
 interior; if it does not, the driver (not the code of this package,
 but the driver between this code and the physical representation on
 paper or screen) assumes a straight line closure of the whole path.
 The syntax offers several variants but it is substantially the following:
 \begin{Sintassi}
 |\Curve|\parg{node}\aarg{direction}\dots\parg{node}\aarg{direction}
 |\Curve*|\parg{node}\aarg{direction}\dots\parg{node}\aarg{direction}
 \qquad\qquad\dots\aarg{direction}\oarg{new direction}\parg{node}\dots
 \end{Sintassi}
 See some more explanation below.

\item \cs{Qurve} is similar to |\Curve|, but with second order 
(quadratic) Bézier splines. The starred version fills the interior 
with the current color. Its syntax si similar to that of |\Curve|.

\item \cs{CurveBetween} draws a single cubic Bézier spline between two
 given nodes and with two given direction vectors. This macro is similar to |\Curve|, but it is used to draw a third order Bézier curve between just two nodes and require just two directions

\item \cs{CBezierBetween}  draws a single cubic Bézier spline between
 two given nodes, with two given direction versors along which the
 control node distances are specified. This is the most general macro
 (rather difficult to use) with which not only the arc end points are
 specified but also the control nodes coordinates are given. It is similar to |\CurveBetween| but the contol points of the single arc can be specified; the arc is perfect, but the syntax is more complicated. See below for examples.

\end{itemize}

% le equazioni parametriche del cuore, da stackexchange, sono
%\def\x(#1){sin(#1)^3}
%\def\y(#1){(13*cos(#1)-5*cos(2*#1)-2*cos(3*#1)-cos(4*#1))/16}
% Varrebbe a penda di provarle con curve2e
%
	\begin{figure}[!ht]
\begin{Esempio}[\small]%
	\unitlength=8mm\relax
	\begin{picture}(5,5)
	\put(0,0){\framebox(5,5){}}\thicklines\roundcap
	\Curve(2.5,0)<0.1,1>(5,3.5)<0,1>%
	  (4,5)<-1,0>(2.5,3.5)<-0.1,-1.2>[-0.1,1.2]%
	  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<0.1,-1>
	\end{picture}
\end{Esempio}
	\caption{A heart shaped curve with cusps drawn with \texttt{\string\Curve}}
	\label{fig:curve}

\vspace*{2\baselineskip}

\begin{Esempio}[\small]%
	\unitlength=8mm\relax
	\begin{picture}(5,5)
	\put(0,0){\framebox(5,5){}}\thicklines\roundcap
   \color{green}\relax
	\Curve*(2.5,0)<0.1,1>(5,3.5)<0,1>%
	  (4,5)<-1,0>(2.5,3.5)<-0.1,-1.2>[-0.1,1.2]%
	  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<0.1,-1>
	\end{picture}
\end{Esempio}
\caption{Coloring the inside of a closed path drawn with \texttt{\string\Curve*}}
\label{fig:colored-curve}
	\end{figure}

 The main 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
\[
 \mbox{\dots\parg{...}\aarg{...}\oarg{new direction vector}\parg{...}\aarg{...}\dots}
\]

 Sometimes it is necessary to specify the “tension” or the “looseness”
 of a specific Bézier arc; such tension parameters range from 0 (zero)
 to~4; the zero value implies a very stiff arc, as if it was a string
 subject to a high tension (i.e. with zero looseness); a value of~4
 implies a very low tension (very high looseness), almost as if the
 string was not subject to any tension. In \MF\ or \MP\ language such a
 concept is used very often; in this package, where the Hobby
 algorithms are not used, the parameter value appears to mean the
 opposite of tension.
 A couple of comma separated tension values may be optionally used, they
 are separated with a semicolon from the direction vector,
 and they apply to the arc terminating with the last node; their
 specification  must precede any possible change of tangent according to
 this syntax\footnote{The tension may be specified only for cubic
 splines, because the quadratic ones do not use enough parameters to
 control the tension; not all commands for drawing cubic splines accept
 this optional tension specification.}:
\[
\makebox[\linewidth]{\small\dots\parg{node}\Aarg{\meta{direction vector};\meta{start tension},\meta{end tension}}\parg{node}\aarg{dirextion}\dots}
\]

 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; but, as already mentioned, if
 the ending node of the last arc does not coincide with the starting
 node of the first arc, a straight line is assumed to join such nodes;
 this line does not get drawn, but with starred commands no lines are
 drawn because only the interior is coloured.
 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 or commands, as, for example, with
 \MF\  or the |pgf/tikz| package and environment.
 See figure~\ref{fig:curve} for an example.

 With the starred version of |\Curve|, instead of stroking the contour,
 the macro fills up the contour with the selected current color, see
 figure~\ref{fig:colored-curve}.

 Figure~\ref{fig:arcspline} shows a geometric construction that
 contains the geometric elements and symbols used to determine the
 parameters of a cubic spline required to draw a quarter circle. This
 construction contains many of the commands described so far.

\begin{figure}[p]
  \begin{minipage}{\linewidth}\small
  \begin{verbatim}
  \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]{$C$}}%
  \segment(0,-40)(-40,0)\segment(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}}%
  \segment(-40,0)(-20,20)\put(-20,20){\circle*{2}}%
  \put(-20,21.5){\makebox(0,0)[b]{$C_1$}}%
  \segment(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}
  \end{verbatim}
  \end{minipage}
\par
\vspace*{\stretch{1}}
\par
  \begin{minipage}{\linewidth}\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]{$C$}}%
%  \segment(0,-40)(-40,0)\segment(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}}%
%  \segment(-40,0)(-20,20)\put(-20,20){\circle*{2}}%
%  \put(-20,21.5){\makebox(0,0)[b]{$C_1$}}%
%  \segment(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}
  \end{minipage}
  \caption{The code to display the nodes and 
  control points for an arc to be approximated 
  with a cubic Bézier spline}
  \label{fig:arcspline}
\end{figure}


 To show what you can do with |\CurveBetween| see the code and result
 shown in figure~\ref{fig:curva-due-punti}. Notice the effect of
 changing the directions at both or at the end nodes of a single cubic
 spline. The directions are conveniently expressed with unit vectors
 described by polar coordinates. The |\CurveBetween| macro is built on 
 |\CBezierBetween|; this latter command is very complicated to describe and its use is reserved to experienced users; its syntax is described in the code documentation file |cureve2e.pdf|; the reader is encouraged to examine it in case s/he is willing to use it.

\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 with different start and end slopes}\label{fig:curva-due-punti}
\end{figure}

 A little more complicated is the use of the
 |\CBezierBetween| macro, figure~\ref{fig:Cbezier}. The 
 directions are specified with unit vectors in polar form; 
 the control points are specified by adding their distances 
 from their neighbouring nodes; actually the right distance
 is maintained to the value~1, while the left one increases 
 from~4 to~10.
 The black line corresponds to the standard |\CurveBetween| 
 where the default distance is computed to trace an arc of a 
 circle and is approximately~3.5. 

\begin{figure}[!tb]
\begin{minipage}[t]{0.52\textwidth}\small
\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} with the default tension values (black)
 and with several tension values specified (red)}
\label{fig:Cbezier}
\end{figure}

 In figure~\ref{fig:tensions} the effect of tension 
 specification is shown. The red line corresponds to the 
 default tension, since the tension values are not specified. 
 The black lines correspond to the various values used in the 
 various commands to the |\Curve| macro.
 With a tension of zero, the spline is almost coincident 
 with the horizontal base line of the frame. Increasing the 
 parameter value to~4.5, the curved becomes taller and 
 taller, until it wraps itself displaying an evident loop. 
 We would say that the value of ~2 is a reasonable maximum 
 one and that increasing that value is just to obtain special 
 effects.

\begin{figure}[!htb]\centering
\begin{Esempio}[\setfontsize{8.5}]%
\unitlength=0.01\textwidth
\begin{picture}(70,70)
\put(0,0){\color{black}\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{Esempio}
\caption{The effects of tension factors}\label{fig:tensions}
\end{figure}

 Figure~\ref{fig:sinewave} displays two approximations of 
 a sine wave; Bézier splines can approximate transcendental 
 curves, but the approximation may be a poor one, depending 
 on the approximated curve, when few arcs are used to draw 
 it. With arcs specified with more complicated macros the 
 approximation is better even with a lower number of arcs. 
 With many arcs it is possible to approximate almost nything.
 On the left side of figure~\ref{fig:sinewave} a modest 
 approximation is obtained with just three standard arcs 
 obtained with |\Curve| and four node specifications; on the 
 right we have just two arcs created with |\CBezierBetween| 
 with tension specification and control point  distances;  
 this drawing is almost undistinguishable from a real 
 sinusoid.

\begin{figure}[!htb]
\begin{minipage}{\linewidth}\small
\begin{verbatim}
\unitlength=0.01\textwidth
\begin{picture}(100,50)(0,-25)
\put(0,0){%
\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
\Pbox(45,0)[b]{x}[0]\Pbox(0,26)[tl]{y[0]}
\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)
\Pbox(45,0)[b]{x}[0]\Pbox(0,26)[tl]{y}[0]
\CbezierBetween0,0And20,0WithDirs77:1And-77:1UsingDists28And{28}
\CbezierBetween20,0And40,0WithDirs-77:1And77:1UsingDists28And{28}}
\end{picture}
\end{verbatim}
\end{minipage}\vspace{\baselineskip}

\begin{minipage}{\linewidth}
\unitlength=0.01\textwidth
\begin{picture}(100,50)(0,-25)
\put(0,0){\VECTOR(0,0)(45,0)\VECTOR(0,-25)(0,25)
\Pbox(45,0)[b]{x}[0]\Pbox(0,26)[tl]{y}[0]
\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)
\Pbox(45,0)[b]{x}[0]\Pbox(0,26)[tl]{y}[0]
\CbezierBetween0,0And20,0WithDirs77:1And-77:1UsingDists28And{28}
\CbezierBetween20,0And40,0WithDirs-77:1And77:1UsingDists28And{28}}
\end{picture}
\end{minipage}
\caption{A sequence of arcs; the left figure has been drawn with the
 \cs{Curve} command with a sequence of four couples of node-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}

 In figure~\ref{fig:quadratic-arcs} some lines are shown; 
 they are drawn  with quadratic splines by means of the 
 |\Qurve| macro. In the left there are some open and closed 
 curves inscribed within a square.
 On the right a “real" circle is compared to a quadratic 
 spline circle; the word “real” is emphasised because it 
 actually is an approximation with four quarter-circle cubic 
 splines that, in spite of being drawn with third degree 
 parametric polynomials, approximate very well a real circle; 
 on the opposite the quadratic spline circle is clearly a 
 poor approximation even if the maximum radial error amounts 
 just to about 6\% of the radius.
 
\begin{figure}[!htb]
\begin{minipage}{\linewidth}
\begin{Verbatim}[fontsize=\setfontsize{7.75}]
\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)[tc]{0,0}\Zbox(100,0)[tc]{100,0}
\Zbox(100,100)[bc]{100,100}\Zbox(0,100)[bc]{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}
\end{Verbatim}
\end{minipage}\vspace{2\baselineskip}

\begin{minipage}{\linewidth}
\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>}
\put(50,50){\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)[tc]{0,0}\Zbox(100,0)[tc]{100,0}
\Zbox(100,100)[bc]{100,100}\Zbox(0,100)[bc]{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}[0]
\end{picture}
\end{minipage}

\caption{\rule{0pt}{4ex}Several graphs drawn with quadratic 
 Bézier splines. On the right a quadratic spline circle is 
 compared with a cubic line circle.}
\label{fig:quadratic-arcs}
\end{figure}

 Notice that the previous version of \pack{curve2e} contained 
 an error and would color the outside of the green four-pointed 
 star. The |curve2e-v161| package, attached to this bundle, has 
 been corrected; therefore it is not actually identical to the 
 previous version, although the latter one performed correctly 
 for everything else except for color filled quadratic paths.

The “real” circle, even if rendered with quarter circles, is visually very similar to a true circumference, but any Bézier circle arc that approximates a circumference arc of the same amplitude, displays a small error at an angle of 1/4 and 3/4 the total arc amplitude; this error diminishes as the total arc amplitude diminishes, nevertheless the error is there. If the arc amplitude amounts to 180°, the maximum error is at 45° and at 135° angular distances from the staring point. It amounts to 2\% of the arc radius; see figure~\ref{fig:halfcircle}, where a half circle is compared with a sequence of four arcs of 45° each, drawn by means of the \cs{Curve} macro; the code is the following:
\begin{verbatim}
\unitlength=0.005\linewidth
\begin{picture}(100,50)(-50,0)
  \AutoGrid
 \Zbox(0,0)[t]{\textsf{O}}[1]
 \Arc(0,0)(50,0){180}
 \VECTOR(0,0)(45:50)\Zbox(45:25)[br]{R}[0]
 \Curve(50,0)<0,1>(45:50)<135:1>(0,50)<-1,0>%
   (135:50)<225:1>(-50,0)<0,-1>
% \Arc(0,0)(50,0){90}\Arc(0,0)(0,50){90}
% \Curve(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>
\end{picture}
\end{verbatim}
 
\begin{figure}\centering
 \unitlength=0.005\linewidth
 \begin{picture}(100,50)(-50,0)
 \AutoGrid
 \Zbox(0,0)[t]{\textsf{O}}[1]
 \Arc(0,0)(50,0){180}
 \VECTOR(0,0)(45:50)\Zbox(45:25)[br]{R}[0]
 \Curve(50,0)<0,1>(45:50)<135:1>(0,50)<-1,0>%
   (135:50)<225:1>(-50,0)<0,-1>
% \Arc(0,0)(50,0){90}\Arc(0,0)(0,50){90}
% \Curve(50,0)<0,1>(0,50)<-1,0>(-50,0)<0,-1>
 \end{picture}
\caption{Comparison between a 180° arc (red) drawn with the \cs{Arc} macro, and a comparable half circle (blaxk) drawn with four 45° \cs{Curve} arcs}\label{fig:halfcircle}
\end{figure}

The circle radius amounts to $R=50$\cs{unitlength}, while the external line, drawn with \cs{Arc}, at 45° displays the maximum distance from the circle center \textsf{O}; it amounts to $51$\cs{unitlength}, 2\% larger than the correct value. In most circumstances this error is negligible, but sometimes the drawing by means of \cs{Curve} is more appropriate. 

An alternative solution consists in tracing two consecutive arcs of 90° each drawn with \cs{Arc}, instead of four arcs of 45° each drawn with \cs{Curve} or with \cs{Arc}. The code above contains also the other solutions; they are commented out but the user can copy the code and paste it in a personal \texttt{.tex} file, so that each solution can be tested individually, so as to remark the (invisible) differences.
 
%
\item The new version of |\multiput| is backwards compatibile with
 the original version contained in the \LaTeX\ kernel. The new macro
 adds the handling of the coordinate increments from one position to
 the next for the \meta{object} to include in the drawing.
 The syntax is the following:
 \begin{Sintassi}\setfontsize{8}
 |\multiput|\parg{initial point}\parg{increment}\marg{number of items}\marg{object}\oarg{handler}
 \end{Sintassi}
The only small change is the addition of the last optional argument that allows to do several interesting actions on the sequence o objects to be repeated, for example, to set them on a curved line, instead of on a straight line, as shown in figure~\ref{pag:multiput}.

 In this figure we show the code for the picture shown there. The red grid 
 is nothing new, except that it displays the traditional |\multiput| used 
 in this code, shown in a previous example, produces exactly the same 
 result. But for what concerns the four “graphs” on the grid, it displays 
 an alignment of black dots along the diagonal of the grid (again 
 traditional |\multiput| rendered with the new version); a number of blue 
 dots along a parabola; another number of magenta coloured dots alined 
 along a half sine wave; a number of little  green squares aligned along 
 a $-15~\circ$ sloping line starting from the center of the grid; notice 
 the polar values that are used as polar relative coordinate increments.

\noindent\begin{figure}[!htb]
\begin{minipage}{0.45\linewidth}
\begin{Verbatim}[fontsize=\setfontsize{8}]
\unitlength=0.01\linewidth
\begin{picture}(100,100)
\GraphGrid(100,100)
\multiput(0,0)(10,10){11}{\circle*{2}}
\color{blue!70!white}
\multiput(0,0)(10,0){11}{\circle*{2}}%
  [\GetCoord(\R)\X\Y
  \edef\X{\fpeval{\X+10}}
  \edef\Y{\fpeval{(\X/10)**2}}
  \CopyVect\X,\Y to\R]
\color{magenta}
\multiput(0,0)(10,1){11}{\circle*{2}}%
  [\GetCoord(\R)\X\Y
   \edef\X{\fpeval{\X+10}}
   \edef\Y{\fpeval{sind(\X*1.8)*100}}
   \CopyVect\X\Y to\R]
\color{green!80!black}
\multiput(50,50)(-15:5){11}}{%
\polygon*(-1,-1)(1,-1)(1,1)(-1,1)}
\end{picture}
\end{Verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.45\linewidth}
\unitlength=0.01\linewidth
\begin{picture}(100,100)
\GraphGrid(100,100)
\multiput(0,0)(10,10){11}{\circle*{2}}
\color{blue!70!white}
\multiput(0,0)(10,0){11}{\circle*{2}}%
  [\GetCoord(\R)\X\Y
  \edef\X{\fpeval{\X+10}}
  \edef\Y{\fpeval{(\X/10)**2}}
  \CopyVect\X,\Y to\R]
\color{magenta}
\multiput(0,0)(10,1){11}{\circle*{2}}%
  [\GetCoord(\R)\X\Y
   \edef\X{\fpeval{\X+10}}
   \edef\Y{\fpeval{sind(\X*1.8)*100}}
   \CopyVect\X,\Y to\R]
\color{green!60!black}
\multiput(50,50)(-15:5){11}{\polygon*(-1,-1)(1,-1)(1,1)(-1,1)}
\end{picture}
\end{minipage}
\caption{Some examples of the \meta{handler} optional argument}\label{pag:multiput}
\end{figure}

 A new command |\xmultiput| (not available with the previous versions
 of \pack{curve2e}) is extended with respect to the original 
 |\multiput|; it is  defined by using some L3 functions; in particular the 
 cycling counter is accessible to the \LaTeX\ commands and it is stepped
 up from~1 to the value specified in the proper command argument
 (in the original command it starts from that value and is stepped down
 to zero). See the figure on page~\ref{pag:orologio} to inspect its
 usage. It is important to notice that if the command|\rotatebox|
 has to be used, as in the example of figure~\ref{pag:orologio}, the
 package |graphicx| should be also loaded, because \pack{curve2e} does 
 not load it.
 
 Th |\xmultiput| syntax is the is similar to that of |\multiput| but besides the stepping up or down of the iteration counter, it can access and modify certain internal with the commands that appear in the \meta{handler} argument. Actually this \meta{handler} is available also with |\multiput|. In both cases the handler can be defined to modify some internals, including the iteration counter only for |\xmultipot|, but also the |\R| and |\D| internals; |\R| contains the coordinates where to put the \meta{object}, while |\D|, if set, contains the angle of rotation of the object. The code and picture examples in figures~\ref{pag:multiput} and~\ref{pag:orologio} show some examples of usage through tis \meta{handler} code.
.
\begin{Sintassi}\small
|\xmultiput|\parg{initial point}\parg{increment}\marg{iterations}\marg{object}\oarg{handler}
\end{Sintassi}
 

\begin{figure}[!htb]
\begin{minipage}{0.45\textwidth}\setfontsize{9.5}%
\begin{verbatim}
\unitlength=0.0095\linewidth
\begin{picture}(100,100)
\GraphGrid(100,100)
\put(50,50){\thicklines\circle{100}}
\xmultiput[50,50](60:40)(-30:1){12}%
  {\makebox(0,0){\circle*{2}}}%
  [\MultVect\R by\D to\R]% 
\xmultiput[50,50](60:46)(-30:1){12}%
  {\ArgOfVect\R to\Ang
  \rotatebox{\fpeval{\Ang-90}}%
  {\makebox(0,0)[b]{\Roman{multicnt}}}}%
  [\Multvect{\R}{\D}\R]
\end{picture}
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.40\textwidth}\raggedleft
\unitlength=0.0095\linewidth
\begin{picture}(100,100)
\GraphGrid(100,100)
\put(50,50){\thicklines\circle{100}}
\xmultiput[50,50](60:40)(-30:1){12}%
  {\makebox(0,0){\circle*{2}}}[\Multvect{\R}{\D}{\R}]% 
\xmultiput[50,50](60:43)(-30:1){12}%
  {\ArgOfVect\R to\Ang\rotatebox{\fpeval{\Ang-90}}%
    {\makebox(0,0)[b]{\Roman{multicnt}}}}%
      [\Multvect{\R}{\D}\R]
\end{picture}
\end{minipage}
\caption{Usage example of the \texttt{\string\xmultiput} command}
\label{pag:orologio}\hfill
\end{figure}
 
 \item This implementation of \pack{curve2e} includes an extension
 to package |xfp|, in the sense that adds three more L3 commands:
 |\fptest|, |\fpdowhile|, |\fpwhiledo| to the two already contained and
 documented  in the latter package. The syntax of such new commands
 is the following
  \begin{Sintassi}
  \cs{fptest}\marg{test}\marg{true}\marg{false}
  \cs{fpdowhile}\marg{test}\marg{operations to be repeated}
  \cs{fpwhiledo}\marg{test}\marg{operations to be repeated}
  \end{Sintassi}
 The macro |\fptest| requires two further arguments that contain
 what to do if the \meta{test} is true, and what to do if the
 \meta{test} is false.
 The \meta{test} is a logical expressions that connects math relation
 expressions, even floating point ones, by means of \emph{logical
 operators}; such operators are \verb+||+, \verb|&&|, and \verb|!|,
 respectively for OR, AND, NOT; math relation expressions contain
 relation operators, even negated ones: for example \verb|!<| means 
 “not lower than”, which is equivalent to “equal or grater than”, i.e. 
 \verb|=>|. The logical expression is parsed left to right and normal 
 parentheses may be used to alter this sequence. The logical operators 
 work also between logical variables, therefore the \meta{test} may 
 contain an interesting mixture of relation and logical operators.

 Before using |\fpdowhile| and |\fpwhiledo|, the arguments of \meta{test} 
 depends-on must be set so that the test is true; during the execution
 of the \meta{operations to be repeated} there must be some setting
 that eventually renders the \meta{test} false. The user should
 pay attention to set the elements that \meta{test} depends on,
 because the risk is to enter an infinite loop and end up with some
 error message stating that the working memory of the program is full.
 Notice that |\fpdowhile| first puts the \meta{operations to be repeated} 
 into the work flow then checks the \meta{test} and possibly repeats the 
 cycle; on the opposite, |\fpwhiledo| first checks the \meta{test} then 
 possibly inserts the \meta{operations to be repeated} and cycles. 
 Evidently with the same \meta{test} the two while cycles produce 
 different results with a little, but important difference: if the input 
 data are macros defined by previous computations, there is no guarantee 
 that the \meta{test} is initially true; if it sis false, |fpwhiledo| 
 does not do anything, while |\fpdowhile| executes one cycle and produces 
 in the output stream something that might be nonsense. Cycles done 
 with |\fpwhiledo| should be safer and should be preferred.
 
 Nevertheless such commands are very useful also for drawing graphics;
 the |xmultiput| command already makes use of such L3~functions.

 As an example of use, we show how to plot a mathematical function
 expressed in parametric form:
\[\begin{cases}
x = f_1(t)\\
y = f_2(t)
\end{cases}\]
 The plot is executed with a piecewise linear approximation of the
 curve; if the $t$ steps are sufficiently small, the plot turns out to
 be very nice; here we show an example where we plot a Lissajous curve
 with two sinusoids of different periods.

 We start by defining the Lissajous function with arguments
 to specify the parameter $t$, the sinusoid amplitudes $A_1, A_2$,
 the respective “frequencies”, by means of integer multiples of
 a unit pulsation, $N_1, N_2$, and the initial phases
 $\phi_1, \phi_2$ of such sinusoids. It is better to keep
 apart the input of the curve coefficients from the actual curve
 argument/parameter and output point coordinates:
\begin{verbatim}
\def\LissajousCoefs#1,#2,#3,#4,#5,#6!{%
\edef\LAu{#1}\edef\LNu{#2}\edef\LFu{#3}%
\edef\LAd{#4}\edef\LNd{#5}\edef\LFd{#6}}

\def\LissajousCode#1#2{%
\edef\X{\fpeval{\LAu*cosd(\LNu*#1+\LFu)}}%
\edef\Y{\fpeval{\LAd*cosd(\LNd*#1+\LFd)}}%
\CopyVect\X,\Y to#2\ignorespaces}
\end{verbatim}
 As it is shown, the coefficients are specified as a comma separated
 list; the  \verb|!| list terminator is taken care by the actual
 drawing command.

 Then the curve drawing command requires the coefficient specification
 only the first time it is used; some messages\footnote{Here it would
 be better to have available a “Macro Error” message; such a macro
 is not available, but it would be possible to define it by means
 of the \cs{GenericError} macro provided by the \LaTeX2e\ kernel.
 Here we skip this definition in order to avoid overloading this
 documentation with such details.} are output if the coefficients
 have been “forgotten”.
\begin{verbatim}
\NewDocumentCommand\Lissajous{m o m}{%
\IfValueTF{#2}{\LissajousCoefs#2!\relax
\LissajousCode{#1}{#3}}%
{\ifcsname LAu\endcsname
\LissajousCode{#1}{#3}%
\else\PackageError{curve2e}%
  {This Lissajous' curve coefficients\MessageBreak
   are missing}{Nothing done}\fi}%
\ignorespaces}
\end{verbatim}

 The syntax is the following:
\begin{Sintassi}
\cs{Lissajous}\marg{in}\Oarg{\meta{$A_1$},\meta{$N_1$},\meta{$\phi_1$},\meta{$A_2$},\meta{$N_2$},\meta{$\phi_2$}}\meta{$P_{\mathrm{out}}$}
\end{Sintassi}
 where \meta{$P_{\mathrm{out}}$} is a macro that gets
 defined with the cartesian  coordinates of the computed output 
 point. Arguments \meta{in} (the $t$ parameter) and \meta{$P_\mathrm{out}$} (the comput coordinates) need not be enclosed within braces if they are given as macros; actually the code shown in figure~\ref{fig:lissajous} shows such procedure that renders the input code simpler to read. 

 After this definition the diagram is plotted in
 figure~\ref{fig:lissajous}.
%
\begin{figure}[!htb]\centering
\begin{Esempio}[\setfontsize{8.5}](0.55)
\unitlength=0.01\linewidth
\begin{picture}(100,100)(-50,-50)
\AutoGrid
\VECTOR(-50,0)(50,0)\Pbox(50,0)[tr]{x}[0]
\VECTOR(0,-50)(0,50)\Pbox(0,50)[tr]{y}[0]
\Pbox(0,0)[tr]{O}[2]
\thicklines
{\countdef\I=2560 \I=0
\fpdowhile{\I !> 360}{%
  \fptest{\I=0}%
     {\Lissajous\I[40,2,90,40,3,0]\Pout
       \moveto(\Pout)}%
     {\Lissajous\I\Pout
       \lineto(\Pout)}%
  \advance\I by1}\strokepath}%
\end{picture}
\end{Esempio}
\caption{A Lissajous diagram}\label{fig:lissajous}
\end{figure}

 For the independent variable $t$, the parameter of the Lissajous
 parametric equations, it is better to work with degrees instead
 of radians, and with integer numbers, so that the whole range
 from $0^\circ$ to $360^\circ$ is certainly spanned. Notice the
 braces that include the code for the Lissajous diagram; they may
 be useful to render that group suitable to be |\put| somewhere
 else than with its center at the origin of the canvas axes,
 and/or to be used as the second argument of a 
 \cs{rotatebox}\marg{angle} command so as to rotate the whole
 diagram.
%
\item Another useful application of |\fpdowhile| is the following: when making a diagram the axes should get suitable labeled ticks in order to show the graduations; the label of each tick should lay close to the axis on the other side than the tick. It is also necessary to know if the axis to be marked is horizontal or vertical, since in the former case each tick is vertical, while in the latter case it is horizontal. The example in figure~\ref{fig:ticks} shows both the code and its usage. The syntax of the |\Tbox| macro is the following.
\begin{Sintassi}
|\Tbox|\parg{coordinates}\oarg{reference}\marg{label}\oarg{size}\aarg{direction}
\end{Sintassi}
where \meta{coordinates} indicates the position of the tick base along its axis; \meta{reference} is a letter, either |t| (if the axis is on top of the label) or |b|(if the axis is at the bottom of the label) for vertical ticks or either |r| (if the axis is at the right of the label) or |l| (if the axis is at the left of the label) for horizontal ticks; \meta{label} is the value of the scale or a literal label that by default is typeset in math mode, so that a math symbol may be used without the need of entering math mode, while in the unusual circumstance that a textual label is to be used, users should use the |\text| macro; the \meta{size} argument is the size of the tick: if such size is zero, just the label is set, but in this case the |\Tbox| macro behaves as the the |\Zbox| one, and accepts two reference codes in order to print the label the same as |\Zbox| would do when the dot size is zero; the \meta{direction} argument is the letter |V| for vertical ticks, and |H| for horizontal ticks. There is some redundancy because |\Tbox| may behave as |\Zbox|, but experience shows that this is not a problem.

The source preamble should contain the following code.
{\small\begin{verbatim}
\makeatletter
\NewDocumentCommand\Tbox{D(){0,0} O{cc} m O{0pt} D<>{Z}}{\bgroup
\edef\TBoxCode{#5}\dimen0=#4\relax %
\edef\tempE{\fpeval{round(#4/\unitlength,3)}}%
\put(#1){%
\ifdim\dimen0=\z@
    \Zbox(0,0)[#2]{#3}[\z@]%
\else
  \if\TBoxCode V\relax
    \let\tempC=b\relax
    \if\tempC #2\edef\tempD{0,-\tempE}\else\edef\tempD{0,\tempE}\fi
    \segment(0,0)(\tempD)\Zbox(0,0)[#2]{#3}[\z@]%
  \else
    \if\TBoxCode H\relax
      \let\tempC=l\relax%
      \if\tempC #2\edef\tempD{-\tempE,0}\else\edef\tempD{\tempE,0}\fi
      \segment(0,0)(\tempD)\Zbox(0,0)[#2]{#3}[\z@]%
    \else
      \typeout{The specified code\space #5\space is invalid!}%
      \typeout{\string\Tbox\space command ignored}% 
    \fi
  \fi
\fi
}\egroup\ignorespaces}%
\end{verbatim}
}
If the code is contained in a personal |.sty| file the 
|\makeatletter| should be omitted.

As it can be seen, the fifth argument code is preset to to |Z| so that if users forget to specify it, error messages pop op and can be seen in both the console window and in the log file, but no tick and label are typeset. This kind of commands to label in one way or another may be very useful depending on the users' kind of drawings. Figure~\ref{fig:labeled diagram} contains also the iterations to label the axes and are done with  the 
|\fpdowhile| macro; it can be seen that the actual coordinate of each tick is transformed into the actual label that is put in its position by the 
|\Zbox| command; of course the  diagram scale and the actual label are related to one another but do not have the same value.

\begin{figure}
{\small\begin{verbatim}
\centering\unitlength=0.007\linewidth
\begin{picture}(100,70)(-50,-10)%
\AutoGrid
\thicklines
\VECTOR(-50,0)(50,0)\Zbox(50,0)[br]{x}[0pt]%
\VECTOR(0,0)(0,60)\Zbox(0,60)[tl]{y}[0pt]%
%
{\def\Coord{-40}%
\fpdowhile{\Coord<=40}{\edef\X{\fpeval{round(\Coord/20,1)}}%
\Tbox(\Coord,0)[t]{\X}[1.5mm]<V>\edef\Coord{\fpeval{\Coord+10}}}%
%
\def\Coord{10}%
\fpdowhile{\Coord<=50}{\edef\Y{\fpeval{round(\Coord/20,1)}}%
\Tbox(0,\Coord)[r]{\Y}[1.5mm]<H>\edef\Coord{\fpeval{\Coord+10}}}%
}%
%
{\color{blue}\linethickness{1.5pt}%
\Curve(-40,0)<1,0>(0,20)<45:1>%      Parabola: y=0.25(x_2)^2
\Curve(0,20)<1,0>(20,0)<0,-1>%       Quarter circle: x^2+y^2=1
\Curve(20,0)<0,1>(40,34.64)<49:1>}%  Hyperbola:x^2-y^=1
\thinlines
\Dashline(0,0)(45,45){2}% asymptote
\end{picture}
\end{verbatim}
}
\bigskip

\centering\unitlength=0.007\linewidth
\begin{picture}(100,70)(-50,-10)%
\AutoGrid
\thicklines
\VECTOR(-50,0)(50,0)\Zbox(50,0)[br]{x}[0pt]%
\VECTOR(0,0)(0,60)\Zbox(0,60)[tl]{y}[0pt]%
%
{\def\Coord{-40}%
\fpdowhile{\Coord<=40}{\edef\X{\fpeval{round(\Coord/20,1)}}%
\Tbox(\Coord,0)[t]{\X}[1ex]<V>\edef\Coord{\fpeval{\Coord+10}}}%
%
\def\Coord{10}%
\fpdowhile{\Coord<=50}{\edef\Y{\fpeval{round(\Coord/20,1)}}%
\Tbox(0,\Coord)[r]{\Y}[1ex]<H>\edef\Coord{\fpeval{\Coord+10}}}%
}%
{\color{red}\linethickness{1.5pt}%
\Curve(-40,0)<1,0>(0,20)<45:1>%     Parabola: y=0.25(x_2)^2
\Curve(0,20)<1,0>(20,0)<0,-1>%      Quarter circle: x^2+y^2=1
\Curve(20,0)<0,1>(40,34.64)<49:1>}% Hyperbola:x^2-y^2=1
\thinlines
\Dashline(0,0)(45,45){2}% asymptote
\end{picture}
\caption{The code to draw a diagram with labeled axes}\label{fig:labeled diagram}
\end{figure}
%
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Remarks}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 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 \pack{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 released in 1994; in the |ltnews| news-letter of December
 2003, the first implementation of Lamport's extension was announced;
 the first version of this package \pack{curve2e} 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 further extending 
 the drawing facilities.

 There are other packages in the \textsc{ctan} archives that deal with
 tracing curves of various kinds. \pack{PSTricks} and \pack{pgf/tikz} are 
 the most powerful ones. And they are becoming the standard for  computer 
 drawing. Their documentation is huge and the multitude of extra modules 
 to perform special tasks is countless. Therefore they are difficult to 
 use; when the user gets used to their particular syntax and got 
 sufficient familiarity  with several modules, s/he can use these bundles 
 very comfortably. 
 
 This difficulty in becoming a TikZ or PS expert is why I think a simpler 
 drawing machinery should be appreciated. I admit it: I like the 
 \env{picture} environment; and I like to deal with simple codes so as to 
 create my own macros.
 
 But there is also \pack{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 package option, 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, \pack{ebezier} performs about the same as 
 \pack{curve2e}  but draws its Bézier curves by using little dots 
 partially superimposed to one another. The documentation is quite 
 interesting since it explains very clearly what exactly are the Bézier 
 splines. Apparently \pack{ebezier} should be used only for DVI output 
 without recourse to PostScript or PDF machinery.

 The \pack{picture} package extends the performance of the \env{picture}
 environment (extended with \pack{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 may be very useful.

 Package \pack{xpicture} builds over the \env{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 
 rather comfortable, although it is not a simple extension of the 
 \env{picture} own syntax; for all these curves it uses the quadratic 
 Bézier splines.
 
 Package \pack{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 \env{picture} environment extended with \pack{pict2e}, and,
 why not, with \pack{curve2e}.
 
If you are interested in further extensions of |curve2e|, besides creating yourself the macros you need, examine the |euclideangeometry| package; I created it in order to have available further functionalities useful to deal with more complicated geometrical constructions. Another small extension is |graphpaper| by which users can produce themselves drawing paper with bilinear coordinates, semi-logarithmic and bi-logarithmic coordinates, polar linear or semilogarithmic coordinates, and Smith charts.  They are an extension chain of the packages that extend the picture environment: |graphpaper| calls |euclideangeometry| which calls 
|curve2e| which calls |pict2e|. I find all this very useful. Of corse these packages and their documentation are all contained in \TeX~Live.

I am about to complete a book on electromagnetism and electronic circuit theory: it contains some 300 drawings, diagrams, electronic circuits. I created them  with |curve2e| and some macros based on this |picture| extension package. The graphics of my book are pretty nice, and I am sure that a professional technical artist can draw better ones, but such an artist would be very expensive and the process would require a lot of time to correct, modify, add details, and so on, while doing things by oneself is certainly a better solution. Packages |tikz|/|pgfplots| and |PStricks|? Certainly they are very good and much more powerful, but, besides being sort of difficult to learn to use, they require a lot of working memory and more often than not I ran out of computer memory.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

I wish to express my deepest thanks to the many persons who submitted notices about bugs or inconsistencies. 
 
 Michel Goosens spotted some errors and very kindly submitted 
 them to me so that I was able to correct them.

 I collaborated extensively with Josef Tkadlec in order to 
 implement a better real long division so as to get correctly
 the quotient fractional part and to avoid as much as 
 possible any numeric overflow; many Josef's ideas were 
 incorporated in the long division macro that was implemented
 in the previous version of this package, although the macro 
 used  by Josef  was slightly  different.  Both  versions 
 aim/aimed at a better accuracy and at widening the operand 
 ranges. 
 In this version of \pack{curve2e} I abandoned our long 
 division macro, and substituted it with the floating point 
 division provided by the |xfp| package functionalities that 
 are now part of the \LaTeX\ kernel.

Daniele Degiorgi spotted a fault in the kernel definition of 
|\linethickness| that heavily influenced also \pack{curve2e}; see the code documentation \file{curve2e.pdf} file.
 
Domenicus van der Wijst  spotted a sneaky bug in the updated |\Arc| macro; for arcs not wider than $180^\circ$ that macro worked properly, while for larger angle apertures it produced syntax errors. It was just a typo, but very sneaky.

Jin-Hwan Cho and Juho Lee suggested a small but crucial modification in order to have \pack{curve2e} work smoothly also with XeTeX (XeLaTeX). Actually if \pack{pict2e}, version 0.2x or  later, dated 2009/08/05  or later, is being used, such modification is not necessary any more, but it is true that it was imperative when legacy versions were used. 
 
Ashish Kumar Das spotted an inconsistency in the design of vectors with PostScript style arrow tips with large line width settings, that did not show up with \LaTeX\ styled ones.


\end{document}