<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass{article}

\usepackage[a4paper,margin=2cm]{geometry}
\usepackage[T1]{fontenc}

\usepackage{xpicture}

\usepackage{ifthen}
\usepackage{array}
\usepackage{fancyvrb}
\usepackage[colorlinks]{hyperref}

\usepackage{amsmath}
\usepackage{paralist}
\usepackage{graphicx}
\usepackage{makeidx}
\makeindex
\renewcommand{\today}{2012/12/17}

\newcommand{\TIT}{\textit}
\newcommand{\TTT}{\texttt}
\newcommand{\TTTit}[1]{\TTT{\TIT{#1}}}
\newcommand{\cs}[1]{\mbox{\textnormal{\TTT{\textbackslash #1}}}}
\newcommand{\environ}[1]{\textnormal{\TTT{#1}}}
\newcommand{\package}[1]{\textnormal{\TTT{#1}}}
\newcommand{\ttindex}[1]{\index{#1@\texttt{#1}}}
\newcommand{\ttslashindex}[1]{\index{#1@\texttt{\textbackslash #1}}}
\newcommand{\csdef}[1]{\cs{#1}\ttslashindex{#1}}
\newcommand{\packagedef}[1]{%
   \package{#1}\index{#1@\texttt{#1} (package)}}
\newcommand{\environdef}[1]{%
   \package{#1}\index{#1@\texttt{#1} (environment)}}
\newcommand{\optiondef}[1]{%
   \textnormal{\TTT{#1}}\index{#1@\texttt{#1} (package option)}}
\newcounter{exem}\stepcounter{exem}
\newenvironment{exemple}{%
   \VerbatimEnvironment\begin{VerbatimOut}{./xpicture\theexem.tex}}{%
   \end{VerbatimOut}
   \par\medskip\noindent
   \marginpar{\fbox{Ex. \theexem}}\begin{minipage}{\linewidth}
      \begin{minipage}{0.45\linewidth}
         \setlength{\parindent}{2ex}
         \catcode`\%=14
         \input{./xpicture\theexem}
      \end{minipage}\hfill
      \begin{minipage}{0.45\linewidth}
         \small
         \VerbatimInput{./xpicture\theexem.tex}
      \end{minipage}
   \end{minipage}
   \stepcounter{exem}\par\bigskip\noindent}
\newenvironment{Exemple}{%
   \VerbatimEnvironment\begin{VerbatimOut}{./xpicture\theexem.tex}}{%
   \end{VerbatimOut}
   \par\noindent
   \marginpar{\fbox{Ex. \theexem}}\fbox{\begin{minipage}{\linewidth}
      \begin{minipage}{\linewidth}
         \setlength{\parindent}{2ex}
         \bigskip\par
         \catcode`\%=14
         \input{./xpicture\theexem}
      \end{minipage}\medskip\par
      \hspace*{0.125\linewidth}\rule{0.75\linewidth}{0.4pt}\par\medskip
      \small
      \VerbatimInput{./xpicture\theexem.tex}
   \end{minipage}}\stepcounter{exem}\par\bigskip\noindent}

\begin{document}
\begin{titlepage}
   \centering
   \bfseries\Large Robert Fuster

   \rule{\textwidth}{1pt}

   The \textsf{xpicture} package 

   (\Verb+http://www.upv.es/~rfuster/xpicture+)

   Several extensions of the \textsf{picture} standard environment

   User Manual
   \vspace{\stretch{1}}
   \begin{Exemple}
      \setlength{\unitlength}{1cm}
      \footnotesize
      \DIVIDE{1}{12}{\invXII}
      \MULTIPLY{12}{\numberTWOPI}{\phione}
      \MULTIPLY{12}{64}{\divisions}

      \COMPOSITIONfunction{\EXPfunction}{\COSfunction}{\Afunction}
      \SCALEVARIABLEfunction{4}{\COSfunction}{\Bfunction}
      \SCALEVARIABLEfunction{\invXII}{\SINfunction}{\cfunction}
      \POWERfunction{\cfunction}{5}{\Cfunction}
      \LINEARCOMBINATIONfunction{1}{\Afunction}{-2}{\Bfunction}{\ABfunction}
      \SUMfunction{\ABfunction}{\Cfunction}{\ABCfunction}
      \PRODUCTfunction{\SINfunction}{\ABCfunction}{\Xfunction}
         % x=(sin t)(exp(cos t)-2 cos 4t + (sin(t/12))^5)
      \PRODUCTfunction{\COSfunction}{\ABCfunction}{\Yfunction}
         % y=(cos t)(exp(cos t)-2 cos 4t + (sin(t/12))^5)
      \PARAMETRICfunction{\Xfunction}{\Yfunction}{\butterfly}
      \centering
      \begin{Picture}(-4,-3)(4,4)
         \PlotParametricFunction[\divisions]\butterfly{0}{\phione}
      \end{Picture}
      \begin{gather*}
         x=\sin t\left(\mathrm e^{\cos t}-2\cos 4t 
                      +\sin^5\left(\frac t{12}\right)\right) \\
         y=\cos t\left(\mathrm e^{\cos t}-2\cos 4t 
                      +\sin^5\left(\frac t{12}\right)\right)
     \end{gather*}
   \end{Exemple}
    \footnotesize\today
\end{titlepage}
\stepcounter{page}

\tableofcontents
\newpage

   The \package{xpicture} package extends the 
   \environ{picture} standard environment
   and packages \package{pict2e} and \package{curve2e}, 
   adding the ability to work with arbitrary 
   reference systems and with Cartesian or polar coordinates. 
   In addition to other utilities,
   the greater interest of \package{xpicture} 
   lies in its capacity to draw function graphs,
   conic sections and arcs, and parametrically defined curves.
    
   This is the user manual of \package{xpicture}.
   Technical documentation and reference manual are contained 
   in file \texttt{xpicture.pdf}, distributed together with the package.

\section{Introduction. New graphical instructions}
The \package{xpicture} package introduces several new graphical 
instructions, and some enriched versions of standard
instructions used inside the \environ{picture} environment.
All these new instructions can be classified as follows:
\begin{itemize}
 \item Reference systems and coordinates:
\begin{itemize}
\item Declaration and use of different reference systems,
with Cartesian or polar coordinates.
\item Instructions to show Cartesian or polar reference systems.
\end{itemize}
 \item  An alternative to the \environ{picture} environment,
compatible with the new reference systems.
 \item Alternative instructions or extensions of the standard
 \environ{picture} commands and those defined by the packages
 \packagedef{pict2e} and \packagedef{curve2e}:
\begin{itemize}
 \item Enriched versions of marks \cs{put} and \cs{multiput},
 providing an adequate control of the precise position 
in which objects are composed
(this functionality is especially useful in the composition
of not strictly graphical objects, such as formulas or labels).
\item Instructions for drawing straight segments, vectors
(in any direction and using any reference system), polygonal lines,
and regular and arbitrary polygons.
\end{itemize}
\item Regular curves:
\begin{itemize}
\item Instructions for drawing conic sections (circles, ellipses,
hyperbolas and parabolas) and arcs of these curves.
\item Instructions to graph functions and parametrically defined curves
(this is the most interesting feature of this package).
\end{itemize}
\end{itemize}

The only requeriments for \package{xpicture} are packages 
\packagedef{calculator}, \packagedef{calculus}, 
\packagedef{curve2e} and \packagedef{xcolor}.
Therefore, it works with any \TeX{}
extension compatible with these packages. You can compile a document 
including \package{xpicture} pictures directly with 
\TTT{pdflatex},\ttindex{pdflatex}
\TTT{lualatex},\ttindex{lualatex} 
\TTT{xelatex}\ttindex{xelatex} 
or indirectly, via \TTT{latex/dvips}\ttindex{latex},\ttindex{dvips}
 \TTT{latex/dvips/dvipdfm},\ttindex{dvipdfm} \ldots
Pure \TTT{dvi} files are not supported, but some \TTT{dvi} previewers
may show partially \package{xpicture} draws included in \TTT{dvi} files. 

\section{A preliminary observation.
         Compatibility with text composition in color}
The \package{xpicture} package automatically loads the
\packagedef{xcolor} package.
So, we can compose our
pictures (and the whole document) in various colors. However,
when used in the body of the \textsf{picture} environment,
marks \cs{color} and \cs{colortext}
often introduce spurious spaces.
For this reason, the \package{xpicture} package introduces the new command
\csdef{pictcolor}.
\begin{Verbatim}[commandchars=\|\[\]]
\pictcolor{|TIT[color]}
\end{Verbatim}
This mark behaves like the \cs{color} command, but does not produces these
inappropriate spaces.
To change colors inside a picture, instead of \cs{color} or \cs{colortext},
use always the \cs{pictcolor} declaration.

\section{Coordinate systems and the \environ{Picture} environment}
\subsection{Coordinates}
The standard \environ{picture} environment establishes
a rectangular coordinate system, so that all
graphic objects are placed in the picture using the canonical
coordinates of the plane. From now on, we will call
this reference system \emph{the standard reference system}.
Loading the \package{xpicture} package, we can use any other affine
reference system and combine it with the use of polar coordinates.

\subsubsection{Reference systems}
The \package{xpicture} package allows us to use other reference systems.
For the purpose we are interested, a reference system consists
of an origin of coordinates and a pair of linearly independent vectors.
Typing\ttslashindex{referencesystem}
\begin{Verbatim}[commandchars=\|\[\]]
\referencesystem(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])(|TIT[x2],|TIT[y2])
\end{Verbatim}
we declare the new reference system  with origin at point
$(\TTT{\TIT{x0}},\TTT{\TIT{y0})} $ and coordinate vectors
$(\TTT{\TIT{x1}},\TTT{\TIT{y1}})$ and
$(\TTT{\TIT{x2}},\TTT{\TIT{y2}})$.
If the coordinates of the point $P$ with respect to this reference system
are $(\bar{\TTT{\TIT{x}}},\bar{\TTT{\TIT{y}}})$, then the
coordinates of $ P $ with respect to the standard system,
$(\TTT{\TIT{x}},\TTT{\TIT{y}})$, are calculated with the formula
\newenvironment{qmatrix}{\left[\begin{matrix}}{\end{matrix}\right]}
\[
 \begin{qmatrix}
          \TTT{\TIT{x}} \\ \TTT{\TIT{y}}
     \end{qmatrix}=\begin{qmatrix}
         \TTT{\TIT{x0}} \\ \TTT{\TIT{y0}}
    \end{qmatrix} +
    \begin{qmatrix}
         \TTT{\TIT{x1}} &amp; \TTT{\TIT{x2}} \\
         \TTT{\TIT{y1}} &amp; \TTT{\TIT{y2}}
    \end{qmatrix}    \begin{qmatrix}
         \bar{\TTT{\TIT{x}}} \\ \bar{\TTT{\TIT{y}}}
    \end{qmatrix}
\]

For example,
\begin{Verbatim}[commandchars=\|\[\]]
\referencesystem(1,2)(1,0)(0.5,0.5)
\end{Verbatim}
sets a new reference system that has its origin in the point $O(1,2)$
and the coordinate vectors $\vec u_1=(1,0)$ and $\vec u_2=(1/2,1/2)$.
The following pictures show this coordinate system built on the standard
reference system
and a Cartesian grid refered to the new reference system.

\noindent
\setlength\unitlength{1cm}%
 \renewcommand{\Pictlabelsep}{0.2}
\begin{Picture}(-3.1,-3.1)(3.1,3.1)
\put(-1.5,0){\line(1,0){3}}
\put(0,-1.5){\line(0,1){3}}
{\makenolabels
\cartesianaxes(-3,-3)(3,3)}
\thicklines
 \xVECTOR(0,0)(1,2)
\pictcolor{red}
\referencesystem(1,2)(1,0)(0.5,0.5)
\Put[-45](0,0){$O$}
\renewcommand\axescolor{red}
\renewcommand\axeslabelcolor{red}
\cartesianaxes(-2.1,-2.1)(2.1,2.1)
\linethickness{1pt}
\xVECTOR(0,0)(1,0)
\xVECTOR(0,0)(0,1)
\rPut{SE}(1,0){$\vec u_1$}
\Put[SE](0,1){$\vec u_2$}
\end{Picture}
\hfill%
{\referencesystem(1,2)(1,0)(0.5,0.5)
\begin{Picture}(-3.6,-3.6)(3.5,3.5)
\thinlines
\cartesiangrid(-3,-3)(3,3)
\pictcolor{red}
\linethickness{1pt}
\xVECTOR(0,0)(1,0)
\xVECTOR(0,0)(0,1)
\end{Picture}}

Alternatively, you can use the \csdef{changereferencesystem} declaration:
in the instruction
\begin{Verbatim}[commandchars=\|\[\]]
\changereferencesystem(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])(|TIT[x2],|TIT[y2])
\end{Verbatim}
point $(\TTT{\TIT{x0}},\TTT{\TIT{y0})}$ and vectors
$(\TTT{\TIT{x1}},\TTT{\TIT{y1}})$ i $(\TTT{\TIT{x2}},\TTT{\TIT{y2}})$
are not refered to the standard system,
but to the \emph{active} reference system.\footnote{%
In other words, the instruction
\cs{referencesystem} changes from the standard reference system
to the new one, while
\cs{changereferencesystem} changes from the active system.}
Moreover, as the more interesting (and frequent) reference system changes
consist of translations of the origin, rotations of the axes
and symmetries, \package{xpicture}
introduces three specific commands to these special cases:
\ttslashindex{translateorigin}
\begin{Verbatim}[commandchars=\|\[\]]
\translateorigin(|TIT[x0],|TIT[y0])
\end{Verbatim}
moves the origin to the specified coordinates.
\ttslashindex{rotateaxes}
\begin{Verbatim}[commandchars=\|\[\]]
\rotateaxes{|TIT[angle]}
\end{Verbatim}
rotates the axes. The \TTT{\TIT{angle}}  parameter is interpreted
as the rotation angle in radians
(if the \csdef{radiansangles} declaration is active) or in
sexagesimal degrees (if the \csdef{degreesangles} declaration is active).
And\ttslashindex{symmetrize}
\begin{Verbatim}[commandchars=\|\[\]]
\symmetrize{|TIT[angle]}
\end{Verbatim}
performs a symmetry, being \TTT{\TIT{angle}}
the angle between the $x$ axis and  the symmetry axis.
Also here, the \csdef{radiansangles} and \csdef{degreesangles}
declarations determine if angles are
interpreted as radians or degrees.
%
These three declarations always apply to the active reference system.
\begin{Exemple}
\newcommand{\mypicture}{%
{\thicklines
\xVECTOR(-1,-1)(1,1)
\pictcolor{red}\Circle{1}
\pictcolor{blue}\regularPolygon{1}{4}
\polarreference\degreesangles
\pictcolor{green}\Polygon(1,90)(0,0)(1,-30)}}
\centering
\setlength{\unitlength}{1cm}
\fbox{\begin{Picture}[black!5!white](-1.5,-6.5)(14.5,1.5)
\cartesiangrid(-1,-1)(14,1)
\mypicture
{\referencesystem(3,0)(1,1)(1,0)
\mypicture
\changereferencesystem(0,4)(-1,1)(1,-2)
\mypicture}
\degreesangles
\translateorigin(10,0)
{\rotateaxes{45}
\mypicture}
\translateorigin(3,0)
\symmetrize{45}
\mypicture
\referencesystem(6.5,-4)(7,0)(0,-2)\mypicture
\end{Picture}}
\end{Exemple}

The \csdef{standardreferencesystem} declaration restores the standard 
reference.

\medskip

Changes of reference system can
be used inside or outside the \environ{Picture} environment.
In the next sections we will see what are the effects produced in each case.

\subsubsection{Polar coordinates}
Instead of Cartesian coordinates, we can refer to a point $P$ using the 
polar coordinates $(r,\phi)$ of this point: 
$r$ is the distance from the origin $O$ and $\phi$ is the angle between 
the first coordinate vector and the $OP$ segment.
The \csdef{cartesianreference} and \csdef{polarreference} declarations
establish the coordinates of one or the other type.
By default, the Cartesian coordinates are used, but in some cases
is much easier determine polar coordinates.
Additionally, the \csdef{radiansangles} and \csdef{degreesangles}
declarations
sets angle measuring in radians or in degrees, respectively
(by default, angles are measured in radians).

The following example shows a typical situation in which it is more
appropriate to use polar coordinates:
the \emph{natural} way to enter coordinates on a circle is using 
polar coordinates.

\begin{exemple}
\setlength{\unitlength}{3cm}
\fbox{\begin{Picture}(-1.3,-1.3)(1.3,1.3)
\polarreference
\degreesangles

\renewcommand{\Pictlabelsep}{0.1}

\multiPut(1,0)(0,30){12}{\circle*{0.05}}
    % Put twelve dots, one unit apart, 
    % at 0, 30, 60, ..., 330 degrees

\cPut{90}(1,90){\textsc{xii}}
\cPut{0}(1,0){\textsc{iii}}
\cPut{270}(1,270){\textsc{vi}}
\cPut{180}(1,180){\textsc{ix}}

\pictcolor{blue}\thicklines

\arrowsize{8}{2}
\xtrivVECTOR(0,0)(0.5,37.5)
\xtrivVECTOR(0,0)(0.9,180)

\Put(0,0){\circle*{0.1}}
\linethickness{4pt}
\Circle{1.3}
\end{Picture}}
\end{exemple}

The new commands defined in the \package{xpicture} package and requiring
some kind of coordinates support polar coordinates,
except the \environ{Picture} and \environ{xpicture} environments
and the \cs{cartesianaxes} and \cs{cartesiangrid} environments.
\subsection{The \environ{Picture} (or \environ{xpicture}) environment}
The \package{xpicture} package supports all drawing commands
from standard \LaTeX;
in particular, you can use the \environ{picture} environment.
However, in the expression
\begin{Verbatim}[commandchars=\|\[\]]
\begin{picture}(|TIT[x],|TIT[y])(|TIT[x0],|TIT[y0])
\end{Verbatim}
the pairs of numbers \TTT{(\TIT x,\TIT y)} and
\TTT{(\TIT{x0},\TIT{y0})} always denote standard coordinates,
namely,
the \environ{picture} environment only uses the standard reference,
thus it defines, as drawing area, the rectangle
\TTT{[\TIT{x0},\TIT{x-x0}]}$\times$\TTT{[\TIT{y0},\TIT{y-y0}]},
regardless of whether this is the  active reference.
If we want draw a picture referring coordinates to an alternative reference
system, to determine the appropriate drawing area in absolute coordinates
is not obvious (and often is difficult).
However, the \environdef{Picture} environment 
defines a working area on the active reference system: the
\begin{Verbatim}[commandchars=\|\{\}]
\begin|{Picture|}[|TIT{color}](|TIT{x0},|TIT{y0})(|TIT{x1},|TIT{y1})
\end{Verbatim}
instruction fixes the drawing area
\TTT{[\TIT{x0},\TIT{x1}]}$\times$\TTT{[\TIT{y0},\TIT{y1}]},
refered to the active reference system.
Here, the \TTT{(\TIT{x0},\TIT{y0})} i \TTT{(\TIT{x1},\TIT{y1})}
coordinates are always rectangular
(even when reference in polar coordinates is active).
More precisely, this environment defines a \environ{picture} box
that circumscribes our drawing area.
If the optional argument is used, background is colored in the given 
\textit{color}. 

\emph{Very important: note that the syntax of the 
\environ{picture} environment is not analogous
to the new environment \environ{Picture}}:
Here two pairs of coordinates are required,
\TTT{(\TIT{x0},\TIT{y0})} and \TTT{(\TIT{x1},\TIT{y1})},
representing two opposite corners of the drawing area.\footnote{%
Although it may seem more \emph{logical}
preserve the syntax of \environ{picture} environment, 
it is more natural to define the drawing area in that way.}
Obviously, if the reference sustem is the standard, expression
\begin{Verbatim}[commandchars=\|\[\]]
\begin{Picture}(0,0)(|TIT[x],|TIT[y])
\end{Verbatim}
is equivalent to
\begin{Verbatim}[commandchars=\|\[\]]
\begin{picture}(|TIT[x],|TIT[y])
\end{Verbatim}

The following example shows the boxes produced by the 
\environ{picture} and \environ{Picture} environments.

\medskip

\begin{Exemple}
 \begin{center}
  \setlength{\unitlength}{0.5cm}
  \referencesystem(0,0)(1,-1)(1,1)

  \fbox{\begin{picture}(6,6)(-3,-3)
   \cartesiangrid(-3,-3)(3,3)
  \end{picture}}\qquad
  \fbox{\begin{Picture}(-3,-3)(3,3)
   \cartesiangrid(-3,-3)(3,3)
  \end{Picture}}
 \end{center}
\end{Exemple}

The left picture does not fit the box.
 In fact, some elementary geometric considerations
shown that a square box of $ 12\times12$ units of length must be reserved,
\begin{Verbatim}[commandchars=\|\[\]]
\begin{picture}(12,12)(-6,-6)
\end{Verbatim}
The use of the \environ{Picture} environment frees us to determine the
actual dimensions of the drawing.

The new environment \environdef{xpicture} is an alias to the
\environ{Picture} environment.
Its sintax and its behavior are identical.

On the other hand, the \csdef{draftPictures} declaration 
disables all the instructions defined in this package, 
replacing each picture set in a \environ{Picture} environment
by a parallelogram circumscribed by a white rectangle (the box that shows
the area reserved for the drawing).\footnote{If you use an instruction 
not directly defined by \package{xpicture} (inside of a \environ{Picture}
environment), this instruction may take effect.}

\begin{center}
\setlength{\unitlength}{1cm}
\draftPictures

\begin{minipage}{5cm}\centering
\begin{Picture}(0,0)(5,5)
\end{Picture}

\verb+\standardreferencesystem+
\end{minipage}\quad
\begin{minipage}{7.5cm}\centering
\referencesystem(0,0)(1,0)(0.5,1)
\begin{Picture}(0,0)(5,5)
\end{Picture}

\verb+\referencesystem(0,0)(1,0)(0.5,1)+
\end{minipage}
\end{center}

\subsection{Coordinate axes}
Instruction\ttslashindex{cartesianaxes}
\begin{Verbatim}[commandchars=\|\[\]]
\cartesianaxes(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])
\end{Verbatim}
draws the coordinate axes corresponding to the
\TTT{[\TIT{x0},\TIT{x1}]}$\times$\TTT{[\TIT{y0},\TIT{y1}]} rectangle.
Arguments \TTT{\TIT{x0}}, \TTT{\TIT{y0}},
\TTT{\TIT{x1}} and \TTT{\TIT{y1}} must satisfy the conditions
\TTT{\TIT{x0}}$&lt;$\TTT{\TIT{x1}} and \TTT{\TIT{y0}}$&lt;$\TTT{\TIT{y1}}.
Here, coordinates \TTT{(\TIT{x0},\TIT{y0})} and \TTT{(\TIT{x1},\TIT{y1})}
are always rectangular (even when reference in polar coordinates is active).
\begin{exemple}
\begin{center}
\setlength{\unitlength}{0.75cm}%
\begin{Picture}[black!10!white](-4,-3)(4,3)
\renewcommand{\Pictlabelsep}{0.2}
\cartesianaxes(-3.5,-2.5)(3.5,2.5)
\Put[r](3.5,0){$x$}
\Put[t](0,2.5){$y$}
\end{Picture}
\end{center}
\end{exemple}
\begin{exemple}
\begin{center}
\referencesystem(0,0)(1,0)(0.5,1)
\setlength{\unitlength}{0.75cm}%
\begin{Picture}[black!10!white](-4,-3)(4,3)
\renewcommand{\Pictlabelsep}{0.2}
\cartesianaxes(-3.5,-2.5)(3.5,2.5)
\Put[r](3.5,0){$x$}
\Put[t](0,2.5){$y$}
\end{Picture}
\end{center}
\end{exemple}
The following parameters control the style of the axes, the cut marks
and labels on the axes:

\subsubsection{The style of the axes}
\begin{description}
\item[\csdef{axescolor}] By default, the axes color is \TTT{black}, but
we can change it by redefining the \cs{axescolor} declaration. For example,
\begin{Verbatim}[commandchars=\|\[\]]
\renewcommand{\axescolor}{orange}
\end{Verbatim}

We must use a color name predefined in the package \textsf{xcolor}
or defined by the user (for example, using the \cs{definecolor} command).
\item[\csdef{axesthickness}] Length determining the thickness of axes 
(default \verb+1 pt+).
You can modify it using any command that fixes a length (as \cs{setlength}
or \cs{settowidth}).
\item[\csdef{xunitdivisions}, \csdef{yunitdivisions}] Number of subdivisions of
the unit (in each axis). 
By default, 1. These arguments can also be redefined using
the \cs{renewcommand} command (they must be positive integers).
\end{description}
\begin{exemple}
\renewcommand{\xunitdivisions}{2}
\renewcommand{\yunitdivisions}{3}

\begin{center}
\setlength{\unitlength}{1cm}%
\begin{Picture}(-4,-4)(4,4)
\cartesianaxes(-3.5,-3.5)(3.5,3.5)
\end{Picture}
\end{center}
\end{exemple}
\subsubsection{Axes position}
The coordinate axes (and also tags and cut marks)
are placed by default in the traditional way, on the $y = 0$ (the $x$ axis)
and $x = 0$ (the $y$ axis) lines.
However, sometimes the fact that labels are inside the graphic can be
annoying.\footnote{And produces strange effects when the origin $(0.0)$
is not in the drawing area.}
Alternatively, we can place axes and tags at the 
lower and left sides of the coordinate rectangle.
To choose between these two options we should use the following
declarations:
\begin{description}
\item[\csdef{internalaxes}, \csdef{externalaxes}]
If the \cs{internalaxes} declaration is active, then axes lies
on $y=0$ and $x=0$.

However, if we activate the \cs{externalaxes} declaration, the axes
produced by the instruction
\begin{Verbatim}[commandchars=\|\[\]]
\cartesianaxes(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])
\end{Verbatim}
lies on $y=\TTT{\TIT{y0}}$ and $x=\TTT{\TIT{x0}}$.

By default, the \cs{internalaxes} declaration is active.
\end{description}
\begin{exemple}
\renewcommand{\xunitdivisions}{2}
\renewcommand{\yunitdivisions}{2}

\begin{center}
\externalaxes
\setlength{\unitlength}{1cm}%
\begin{Picture}(-4,-4)(4,4)
\cartesianaxes(-3.5,-3.5)(3.5,3.5)
\end{Picture}
\end{center}
\end{exemple}

\subsubsection{Tags style}
The numerical tags on the axes are made in math mode.
If you need textual labels, put them in a \cs{mbox} or,
using \package{amsmath}, a \cs{text} box.
We can control the color, attributes and distance to the axes of these tags,
redefining
(with \cs{renewcommand}) the following marks:
\begin{description}
\item[\csdef{axeslabelcolor}] The color of the numerical tags on the axes.
By default, this color is identical to the axes color.
\item[\csdef{axeslabelsize}] Size of numerical tags.
By default, \cs{small}.
\item[\csdef{axeslabelmathversion}]
 Mathversion of numerical tags.
By default, \TTT{normal}.\footnote{Standard \emph{math versions}
are \TTT{normal} and \TTT{bold}, but some packages
define other math versions.}
\item[\csdef{axeslabelmathalphabet}] Mathalphabet of numerical tags.
By default, \cs{mathrm}.
\item[\csdef{axislabelsep}] Distance between tags and cut marks,
measured in \cs{unitlength} units;\footnote{The distance between axes and
tags equals \cs{ticssize}$+$\cs{axislabelsep}.}
by default, \verb+0.1+ (see later the description of \cs{makenotics}).
\end{description}

\subsubsection{Tags position}
Position of tags is controlled by two declarations:
\begin{description}
\item[\cs{xlabelpos\{\TIT{position}\}}]\ttslashindex{xlabelpos}
change the relative position of labels in $x$ axis. 
Admissible values are those allowed in the \TTT{\TIT{position}} 
argument of command \cs{Put} (see subsection~\ref{subsec:put}).
Default is \verb+-90+.
\item[\cs{ylabelpos\{\TIT{position}\}}]\ttslashindex{ylabelpos}
change the relative position of labels in $y$ axis. 
Default is \verb+180+.
\end{description}

\subsubsection{Style of cut marks}
Units (and, optionally, unit fractions) are marked over axes with small
segments,
the style of which is controlled by the following parameters:
\begin{description}
\item[\csdef{ticssize}, \csdef{secundaryticssize}]
These lengths control the size of the tics:
\cs{ticssize} is half the length of main cuts
(by default, \verb+4pt+)
and \cs{secundaryticssize} is half the length of secundary cuts
(by default, \verb+2pt+).
\item[\csdef{ticsthickness}] Thickness of the marks on axes 
(by default, \verb+1pt+).
\item[\csdef{ticscolor}] Color of the marks on axes (by default, \verb+black+).
\end{description}
\begin{exemple}
\renewcommand{\axescolor}{blue}
\setlength{\axesthickness}{3pt}
\renewcommand{\xunitdivisions}{2}
\renewcommand{\yunitdivisions}{3}

\renewcommand{\axeslabelcolor}{teal}
\renewcommand{\axeslabelsize}{\footnotesize}
\renewcommand{\axeslabelmathversion}{bold}
\renewcommand{\axeslabelmathalphabet}{\mathsf}
\renewcommand{\axislabelsep}{0.05}
\xlabelpos{ttl}
\ylabelpos{r}

\setlength{\ticssize}{0.2cm}
\setlength{\secundaryticssize}{0.1cm}
\setlength{\ticsthickness}{2pt}
\renewcommand{\ticscolor}{blue!50}

\begin{center}
\degreesangles
\rotateaxes{-30}
\setlength{\unitlength}{0.75cm}%
\begin{Picture}(-5,-4)(5,4)
\cartesianaxes(-4.5,-3.5)(4.5,3.5)
\end{Picture}
\end{center}
\end{exemple}

\subsubsection{Removing and directly printing cut marks and labels}
\begin{description}
 \item [\csdef{maketics}, \csdef{makenotics}]
 These two declarations determine if
 divisions on the axes should be marked or not.
 By default the \cs{maketics} declaration is active.

If divisions are not marked, the \csdef{axislabelsep}
declaration determines the distance between axes and labels.
\end{description}
\begin{exemple}
\begin{center}
\setlength{\unitlength}{0.75cm}%
\begin{Picture}(-4.5,-2.5)(4.5,2.5)
\makenotics
\cartesianaxes(-4,-2)(4,2)
\end{Picture}
\end{center}
\end{exemple}
\begin{description}
 \item [\csdef{makelabels}, \csdef{makenolabels}] Two declarations
determining whether numerical labels on the axes must appear or not.
By default, the \cs{makelabels} declaration is active.
\end{description}
\begin{exemple}
\begin{center}
\setlength{\unitlength}{0.75cm}%
\begin{Picture}(-4.5,-2.5)(4.5,2.5)
\makenolabels
\cartesianaxes(-4,-2)(4,2)
\end{Picture}
\end{center}
\end{exemple}

Declarations \cs{makenotics} and \cs{makenolabels}
can be useful when you want to show only some specific coordinates, 
when the points to be highlighted on the axes are not integers
and when you need to print labels in some special format. In  this cases 
you can plot tics and/or print labels using the following commands.
\begin{description}
\item [\cs{plotxtic\{\TIT{x-coor}\}}, \cs{plotytic\{\TIT{y-coor}\}}] 
\ttslashindex{plotxtic}\ttslashindex{plotytic}
plot a tic for the given \TIT{x} or \TIT{y} coordinate.
\item [\cs{printxlabel\{\TIT{x-coor}\}\{\TIT{label}\}}, 
       \cs{printylabel\{\TIT{y-coor}\}\{\TIT{label}\}}] 
\ttslashindex{printxlabel}\ttslashindex{printylabel}
print \TIT{label}
for the given \TIT{x} or \TIT{y} coordinate. Labels are printed in math mode.
\item [\cs{printxticlabel\{\TIT{x-coor}\}\{\TIT{label}\}}, 
       \cs{printyticlabel\{\TIT{y-coor}\}\{\TIT{label}\}}] 
plot a tic and print \TIT{label} for the given \TIT{x} or \TIT{y} coordinate.
\end{description}
\begin{exemple}
\begin{center}
\setlength{\unitlength}{1cm}%
\begin{Picture}(-4.5,-0.5)(4.5,3.5)
\makenolabels
\makenotics
\cartesianaxes(-4,0)(4,3)

\plotytic{0.5}
\printylabel{0.5}{1/2}
\printxticlabel{2}{2}

\Polyline(2,0)(2,0.5)(0,0,5)
\thicklines
\SCALEfunction{0.125}{\SQUAREfunction}{\F}
\PlotFunction[3]{\F}{-4}{4}
\end{Picture}
\end{center}
\end{exemple}

Multiple equally spaced tics and/or labels can be drawn simultaneously:
\begin{description}
\item [\cs{plotxtics\{\TIT{firstcoor}\}\{\TIT{incr}\}\{\TIT{bound}\}},
       \cs{plotytics\{\TIT{firstcoor}\}\{\TIT{incr}\}\{\TIT{bound}\}}]
\ttslashindex{plotxtics}\ttslashindex{plotytics}       
plot several (\TIT{x} or \TIT{y}) tics,
from the initial coordinate \TIT{firstcoor}; \TIT{incr} is the distance 
between consecutive tics, and the last tic is not in a position 
greater than \TIT{bound}.
\item [\cs{printxlabels[\TIT{digits}]\{\TIT{firstcoor}\}\{\TIT{incr}\}%
       \{\TIT{bound}\}}, 
       \cs{printylabels[\TIT{digits}]\{\TIT{firstcoor}\}\{\TIT{incr}\}%
       \{\TIT{bound}\}}] 
\ttslashindex{printxlabels}\ttslashindex{printylabels} print several labels,
from the initial coordinate \TIT{firstcoor}; \TIT{incr} is the distance 
between consecutive label positions, 
and the last position is not greater than \TIT{bound}. 
The optional argument \TIT{digits} is the number of decimal digits to be 
printed (by default, numbers are printed with its natural number of decimals).
\item [\cs{printxticslabels[\TIT{digits}]\{\TIT{firstcoor}\}\{\TIT{incr}\}%
          \{\TIT{bound}\}}]\ttslashindex{printxticslabels} 
 plot \TIT{x} tics and labels simultaneously.
\item [\cs{printyticslabels[\TIT{digits}]\{\TIT{firstcoor}\}\{\TIT{incr}\}%
          \{\TIT{bound}\}}]\ttslashindex{printyticslabels}
 plot \TIT{y} tics and labels simultaneously.      
\end{description}
\begin{exemple}
\externalaxes
\setlength{\unitlength}{1cm}
\renewcommand{\axeslabelsize}{\tiny}
\referencesystem(0,0)(1.5,0)(0,2)
\begin{center}
\begin{Picture}(-2.5,-1.5)(2.5,1.5)
\makenotics
\makenolabels
\cartesianaxes(-2.25,-1.25)(2.25,1.25)
\printxticslabels[1]{-2}{0.5}{2.25}
\printyticslabels[4]{-1}{0.25}{1}
\end{Picture}
\end{center}
\end{exemple}
\begin{Exemple}
\setlength{\unitlength}{1cm}
\begin{center}
\begin{Picture}(-7,-2.5)(7,2.5)
{\referencesystem(0,0)(\numberHALFPI,0)(0,1)
\renewcommand{\xunitdivisions}{2}
\renewcommand{\yunitdivisions}{2}
\makenolabels
\renewcommand{\Pictlabelsep}{0.25}
\cartesianaxes(-4.2,-2.2)(4.2,2.2)

\printylabels{-2}{0.5}{2}

\highestlabel{$-3\pi/2$}
\printxlabel{-4}{-2\pi}
\printxlabel{-3}{-3\pi/2}
\printxlabel{-2}{-\pi}
\printxlabel{-1}{-\pi/2}
\printxlabel{1}{\pi/2}
\printxlabel{2}{\pi}
\printxlabel{3}{3\pi/2}
\printxlabel{4}{2\pi}
}
\end{Picture}
\end{center}
\end{Exemple}

\subsection{Cartesian grids}
As an alternative to the \cs{cartesianaxes} command,
we can use \csdef{cartesiangrid},
to better visualize the coordinates:
\begin{Verbatim}[commandchars=\|\[\]]
\cartesiangrid(|begin[math]x0,y0|end[math])(|begin[math]x1,y1|end[math])
\end{Verbatim}
\begin{exemple}
\definecolor{myblue}{cmyk}{1,1,0,0.5}
\renewcommand{\gridcolor}{myblue}
\renewcommand{\secundarygridcolor}{cyan}
\setlength{\gridthickness}{0.5pt}
\setlength{\secundarygridthickness}{0.1pt}
\renewcommand{\xunitdivisions}{5}
\renewcommand{\yunitdivisions}{5}
\renewcommand{\axeslabelsize}{\footnotesize}
\begin{center}
\setlength{\unitlength}{1cm}
\begin{Picture}(-3.5,-2.5)(3.5,2.5)
\cartesiangrid(-3.4,-2.4)(3.4,2.4)
\end{Picture}
\end{center}
\end{exemple}
\begin{exemple}
\definecolor{myblue}{cmyk}{1,1,0,0.5}
\renewcommand{\gridcolor}{myblue}
\renewcommand{\secundarygridcolor}{cyan}
\setlength{\gridthickness}{0.5pt}
\setlength{\secundarygridthickness}{0.1pt}
\renewcommand{\xunitdivisions}{5}
\renewcommand{\yunitdivisions}{5}
\renewcommand{\axeslabelsize}{\footnotesize}
\begin{center}
\setlength{\unitlength}{1cm}
\referencesystem(0,0)(1,0)(0.25,1)
\externalaxes
\begin{Picture}(-4,-3)(4,3)
\cartesiangrid(-3.4,-2.4)(3.4,2.4)
\end{Picture}
\end{center}
\end{exemple}

\subsubsection{Grid style}
Note that, in addition to the parameters outlined above, there are the 
following ones, which control the style of the grid 
(as in previous cases, these parameters are changed
by redefining them with the \cs{renewcommand} declaration,
or using the usual instructions when they are lengths).

\begin{description}
 \item [\csdef{gridcolor}] determines the color of main divisions in the grid
(regardless of the axes color). By default, this color is \verb+gray+.
 \item [\csdef{secundarygridcolor}] determines the color of secundary 
divisions in the grid.
By default, \verb+lightgray+).
\item[\csdef{gridthickness}] thickness of main divisions 
(by default, \verb+0.4pt+).
\item[\csdef{secundarygridthickness}] thickness of secundary divisions
(by default, \verb+0.2pt+).
\end{description}
\subsection{Polar grids}
Finally, instead of Cartesian axes, we can construct a polar grid
(obviously, this option will be interesting when we use polar coordinates).
\ttslashindex{polargrid}
\begin{Verbatim}[commandchars=\|\[\]]
\polargrid{|TIT[radius]}{|TIT[circledivs]}
\end{Verbatim}
(\TTT{\TIT{radius}} and \TTT{\TIT{circledivs}} are, respectively,
the radius and the number of divisions of the circle
(\TTT{\TIT{circledivs}}must be a positive integer).

This command supports the same parameters that \cs{cartesianaxes} and
\cs{cartesiangrid} (when they makes sense), and also the following:
\begin{description}
\item[\csdef{runitdivisions}] Number of radial subdivisions of the unit.
By default, $1$ (it must be a positive integer).
\end{description}
\begin{exemple}
\renewcommand{\runitdivisions}{2}
\setlength{\unitlength}{0.75cm}
\renewcommand{\gridcolor}{magenta}
\begin{center}
\begin{Picture}(-4,-4)(4,4)
\polargrid{3.5}{12}
\end{Picture}
\end{center}
\end{exemple}
\begin{exemple}
\renewcommand{\runitdivisions}{2}
\setlength{\unitlength}{0.75cm}
\renewcommand{\gridcolor}{magenta}
\referencesystem(0,0)(1,-1)(0.5,0.5)
\begin{center}
\begin{Picture}(-3.5,-3.5)(3.5,3.5)
\polargrid{3.5}{12}
\end{Picture}
\end{center}
\end{exemple}
\begin{description}
\item[\csdef{degreespolarlabels}, \csdef{radianspolarlabels}]
Arcs are printed, by default, in  radians. 
If you want angular units mesured in degrees,
use the  \csdef{degreespolarlabels} declaration (obviously,
\csdef{radianspolarlabels} recovers tags in radians).
\end{description}
\begin{exemple}
\begin{center}
\degreespolarlabels
\setlength{\unitlength}{1cm}
\begin{Picture}(-4,-4)(4,4)
\polargrid{3}{24}
\end{Picture}
\end{center}
\end{exemple}
\begin{description}
\item[\csdef{rlabelpos}] Relative position of labels in polar axis.
Admissible values are those allowed in the \TTT{\TIT{position}} 
argument of command \cs{Put} (see subsection~\ref{subsec:put}).
Default is \verb+bbr+.
\end{description}
\begin{exemple}
\begin{center}
\setlength{\unitlength}{1cm}
\begin{Picture}(-4,-4)(4,4)
\rlabelpos{b}
\polargrid{3.5}{10}
\end{Picture}
\end{center}
\end{exemple}


To remove tags on the polar axis and angles you can use the 
\csdef{makenolabels} declaration.

\section[Alternatives to some standard commands]{%
             Alternatives to standard commands
             \cs{put},\cs{multiput}, \cs{line}, and \cs{vector}}
Standard commands used inside the \environ{picture} environment
are not modified by this package
(although if we include these commands in the body of a \environ{Picture}
environment).
In particular, there does not affect the \cs{referencesystem} declaration.
This package introduces similar commands to those which are sensitive to the
active reference system and give us a greater control over their behavior.
These are the instructions described below.

\subsection{Extensions of the \cs{put} command}\label{subsec:put}
 \begin{description}
 \item[\csdef{Put}, \csdef{cPut}, \csdef{rPut}]
\mbox{}

\begin{Verbatim}[commandchars=\|\{\}]
\Put[|TIT{position}](|TIT{x},|TIT{y})|{|TIT{object}|}
\Put*[|TIT{position}](|TIT{x},|TIT{y})|{|TIT{object}|}
\cPut|{|TIT{position}|}(|TIT{x},|TIT{y})|{|TIT{object}|}
\rPut|{|TIT{position}|}(|TIT{x},|TIT{y})|{|TIT{object}|}
\rPut*|{|TIT{position}|}(|TIT{x},|TIT{y})|{|TIT{object}|}
\end{Verbatim}
place the drawing pointer in the point
of  coordinates \verb+(+\TTT{\TIT{x}}\verb+,+\TTT{\TIT{y}}\verb+)+
with respect to the active reference system (which may coincide or not with
the standard system).
These commands differ in the criteria used to determine the precise position
of the object.

Involved parameters are (see below)
\ttslashindex{Pictlabelsep}
\ttslashindex{defaultPut}
\ttslashindex{highestlabel}
\begin{Verbatim}[commandchars=\|\[\]]
\Pictlabelsep|{|TIT[distance]|}
\defaultPut|{c|}/\defaultPut|{r|}
\highestlabel|{|TIT[text]|}
\end{Verbatim}
\medskip

In the following example, the red circle (included as an argument in the
\cs{put} command) is at the point
of standard coordinates $(1,-1)$; however, in the case of the
blue circle, coordinates $(1,-1)$ refer to the active reference system.
\begin{exemple}
\begin{center}
\setlength{\unitlength}{0.75cm}
\referencesystem(0,0)(1,-1)(1,1)
\begin{Picture}(-2.5,-2.5)(2.5,2.5)
\cartesiangrid(-2,-2)(2,2)
\pictcolor{red}
\put(1,-1){\circle*{0.25}}
\pictcolor{blue}
\Put(1,-1){\circle*{0.25}}
\end{Picture}
\end{center}
\end{exemple}

Recall that coordinates can be rectangular or polar, and angles may
be measured in radians or in degrees.
\begin{exemple}
\begin{center}
\setlength{\unitlength}{1cm}
\begin{Picture}(-2.5,-2.5)(2.5,2.5)
\cartesiangrid(-2,-2)(2,2)
\polarreference
\pictcolor{blue}
\Put(1,\numberHALFPI){\circle*{0.25}}
\degreesangles
\pictcolor{red}
\Put(1,180){\circle*{0.25}}
\end{Picture}
\end{center}
\end{exemple}
\subsubsection{Accurate positioning of the graphical object}
The \TTT{\TIT{position}} argument allows us to fix the relative position of
\TTT{\TIT{object}} respect to point \TTT{(\TIT{x},\TIT{y})}.
Note that this argument is optional in \cs{Put} and \cs{Put*},
but mandatory in the other commands we are describing.
The purpose of this parameter is to rationalize the disposition of
objects, especially when they are not strictly graphical objects
(but labels, text boxes or mathematical formulas). In these cases, 
the appropriate choice of coordinates seems a problem that is not well
solved with standard instructions, despite the special syntax of the 
\cs{makebox} command in the \environ{picture} environment. 
For example, in this picture (which we made using only the standard 
\LaTeX{} commands)
\begin{center}
\setlength{\unitlength}{2cm}

\begin{picture}(7,3)(-0.5,-1.5)

\put(0,0){\line(1,0){7}}
\put(0,-1.5){\line(0,1){3}}
\put(0,-1.5){\line(0,-1){0}}
\multiput(1.570796,-0.1)(1.570796,0){4}{\line(0,1){0.2}}
\multiput(-0.1,-1)(0,1){3}{\line(1,0){0.2}}

\qbezier(0,0)(1,1)(1.570796,1)
\qbezier(1.570796,1)(2.141593,1)(3.141593,0)
\qbezier(3.141593,0)(4.141593,-1)(4.712389,-1)
\qbezier(4.712389,-1)(5.283185,-1)(6.283185,0)

\put(-1.570796,0){%
   \qbezier(1.570796,1)(2.141593,1)(3.141593,0)
   \qbezier(3.141593,0)(4.141593,-1)(4.712389,-1)
   \qbezier(4.712389,-1)(5.283185,-1)(6.283185,0)}
\put(4.712389,0){\qbezier(0,0)(1,1)(1.570796,1)}

\put(2.356194,0.707107){$\sin x$}
\put(6.283185,1){$\cos x$}
\put(-0.15,-1){\makebox(0,0)[r]{$-1$}}
\put(-0.15,0){\makebox(0,0)[r]{$0$}}
\put(-0.15,1){\makebox(0,0)[r]{$1$}}
\put(1.570796,-0.15){\makebox(0,0)[t]{$\pi/2$}}
\put(3.141593,-0.15){\makebox(0,0)[t]{$\pi$}}
\put(4.712389,-0.15){\makebox(0,0)[t]{$3\pi/2$}}
\put(6.283185,-0.15){\makebox(0,0)[t]{$2\pi$}}
\end{picture}
\end{center}
we have located numerical labels ($0$, $1$, $\pi$\ldots) at 
\TTT{0.15\cs{unitlength}} of its \emph{natural} position over the axes,
while the reference points of tags
``$\sin x$'' and ``$\cos x$'' are just in points $(3\pi/4,\sin(3\pi/4))$ and 
$(2\pi,1)$, using these instructions:
\begin{Verbatim}
\put(2.356194,0.707107){$\sin x$}
\put(6.283185,1){$\cos x$}
\put(-0.15,-1){\makebox(0,0)[r]{$-1$}}
\put(-0.15,0){\makebox(0,0)[r]{$0$}}
\put(-0.15,1){\makebox(0,0)[r]{$1$}}
\put(1.570796,-0.15){\makebox(0,0)[t]{$\pi/2$}}
\put(3.141593,-0.15){\makebox(0,0)[t]{$\pi$}}
\put(4.712389,-0.15){\makebox(0,0)[t]{$3\pi/2$}}
\put(6.283185,-0.15){\makebox(0,0)[t]{$2\pi$}}
\end{Verbatim}

If we change the value of \cs{unitlength}, then these values become
inappropriate and we need to change several lines of code.
\begin{center}
\setlength{\unitlength}{1cm}

\begin{picture}(7,3)(-0.5,-1.5)

\put(0,0){\line(1,0){7}}
\put(0,-1.5){\line(0,1){3}}
\put(0,-1.5){\line(0,-1){0}}
\multiput(1.570796,-0.1)(1.570796,0){4}{\line(0,1){0.2}}
\multiput(-0.1,-1)(0,1){3}{\line(1,0){0.2}}

\qbezier(0,0)(1,1)(1.570796,1)
\qbezier(1.570796,1)(2.141593,1)(3.141593,0)
\qbezier(3.141593,0)(4.141593,-1)(4.712389,-1)
\qbezier(4.712389,-1)(5.283185,-1)(6.283185,0)

\put(-1.570796,0){%
   \qbezier(1.570796,1)(2.141593,1)(3.141593,0)
   \qbezier(3.141593,0)(4.141593,-1)(4.712389,-1)
   \qbezier(4.712389,-1)(5.283185,-1)(6.283185,0)}
\put(4.712389,0){\qbezier(0,0)(1,1)(1.570796,1)}

\put(2.356194,0.707107){$\sin x$}
\put(6.283185,1){$\cos x$}
\put(-0.15,-1){\makebox(0,0)[r]{$-1$}}
\put(-0.15,0){\makebox(0,0)[r]{$0$}}
\put(-0.15,1){\makebox(0,0)[r]{$1$}}
\put(1.570796,-0.15){\makebox(0,0)[t]{$\pi/2$}}
\put(3.141593,-0.15){\makebox(0,0)[t]{$\pi$}}
\put(4.712389,-0.15){\makebox(0,0)[t]{$3\pi/2$}}
\put(6.283185,-0.15){\makebox(0,0)[t]{$2\pi$}}
\end{picture}
\end{center}

Note that, regarding labels along the $x$ axis, instead of aligning them to a 
fixed distance of this axis, there would be better to align the baselines
($\pi$ and $2\pi$ should go down);
some of these labels should
move slightly to the right or to the left to avoid that it cut the graph.
Finally, the tag ``$\cos x$'' should be vertically centered
(with respect to the curve) and slightly moved to the right.
\medskip

Using the \package{xpicture} package we construct this picture 
in the following way:
\begin{Exemple}
\MULTIPLY{3}{\numberQUARTERPI}{\numberTQPI}
\SIN{\numberTQPI}{\sinTQPI}

\begin{center}
\setlength{\unitlength}{2cm}
\begin{Picture}(-0.5,-1.5)(6.5,1.5)
{\referencesystem(0,0)(\numberHALFPI,0)(0,1)
\makenolabels
\renewcommand{\Pictlabelsep}{0.1}
\highestlabel{$-3\pi/2$}
\cartesianaxes(0,-1.5)(4.25,1.5)

\rPut{l}(0,-1){$-1$}                           % put the y-axis labels at left
\rPut{l}(0,0){$0$}
\rPut{l}(0,1){$1$}
\rPut*{bbl}(1,0){$\pi/2$}                      % put "\pi/2" at bbl
\rPut*{b}(2,0){$\pi$}                          % put "\pi" at bottom
\rPut*{bbr}(3,0){$3\pi/2$}                     % put "3\pi/2" at bbr
\rPut*{b}(4,0){$2\pi$}                         % put "2\pi" at bottom

\rPut*{b}(0,0){\pictcolor{gray}\xLINE(0.75,0)(4.25,0)}} % \baseline of x-labels

\PlotFunction[8]{\COSfunction}{0}{\numberTWOPI}
\PlotFunction[8]{\SINfunction}{0}{\numberTWOPI}

\Put[NE](\numberTQPI,\sinTQPI){$\sin x$}       % put "\sin x" at NorthEast
\Put[E](\numberTWOPI,1){$\cos x$}              % put "\cos x" at East
\end{Picture}
\end{center}
\end{Exemple}
Here we used several tools to draw the graphs of the functions.
But aside from this, commands \cs{Put}, \cs{rPut} and \cs{rPut*} have allowed 
we to determine the logical position of objects in a much more 
reasonable way.\footnote{Regarding to labels on coordinated axes 
a better choice would be to use other specific commands, 
as \cs{printxlabels}. Here we have chosen \cs{rPut} because we are
illustrating this instruction.}

Argument \TTT{\TIT{position}} supports multiple values:
\begin{description}
\item[An integer or decimal number,] determining the angle (in degrees)
where \TTT{\TIT{object}} is placed,
 with respect to the reference point \TTT{(\TIT{x},\TIT{y})}.
\end{description}

\begin{Exemple}
\begin{center}
\setlength{\unitlength}{1cm}
\begin{Picture}(0,-1)(9,1)
\makenolabels
\renewcommand{\axescolor}{lightgray}\renewcommand{\ticscolor}{lightgray}
\cartesiangrid(0,-1)(8,1)
\pictcolor{blue}
\Put[0](0,0){0}
\Put[45](1,0){45}
\Put[90](2,0){90}
\Put[135](3,0){135}
\Put[180](4,0){180}
\Put[225](5,0){225}
\Put[270](6,0){270}
\Put[315](7,0){315}
\Put[360](8,0){360}
\end{Picture}
\end{center}
\end{Exemple}
\begin{description}
\item[Letter \TTT{c}] (from \emph{center}),
which places the center of \TTT{\TIT{object}} at point
\verb+(+\TTT{\TIT{x}}\verb+,+\TTT{\TIT{y}}\verb+)+.
\end{description}
\begin{exemple}
\begin{center}
\setlength{\unitlength}{2cm}
\begin{Picture}(-1,-1)(1,1)
\cartesianaxes(-1,-1)(1,1)
\pictcolor{blue}
\Put[c](0,0){A CENTERED BOX}
\end{Picture}
\end{center}
\end{exemple}
Note that this option is not equivalent to the suppression of the optional 
argument, because in that case
the reference point of \TTT{\TIT{object}} is located
in \verb+(+\TTT{\TIT{x}}\verb+,+\TTT{\TIT{y}}\verb+)+.
\begin{exemple}
\begin{center}
\setlength{\unitlength}{2cm}
\begin{Picture}(-1,-1)(1,1)
\cartesianaxes(-1,-1)(1,1)
\pictcolor{blue}
\Put(0,0){A NONCENTERED BOX}
\end{Picture}
\end{center}
\end{exemple}
\begin{description}
\item[Letters or letter combinations \TTT N, \TTT E, \TTT S, \TTT W,
\TTT{NE}, \TTT{SE}, \TTT{SW}, \TTT{NW},
\TTT{NNE}, \TTT{ENE}, \TTT{ESE}, \TTT{SSE}, \TTT{SSW}, \TTT{WSW}, \TTT{WNW},
\TTT{NNW}]\mbox{}

Abbreviation of \emph{North}, \emph{East}\ldots,  \emph{North-East}\ldots,  
\emph{North-North-East}\ldots

For example, the
\begin{Verbatim}
\Put[NE](0,0){A}
\end{Verbatim}
instruction writes ``\verb+A+''  \emph{at north-east} of point \verb+(0,0)+.
\item[Letters o letter combinations  \TTT t, \TTT  r, \TTT  b, \TTT  l,
 \TTT{tr}, \TTT{br}, \TTT{bl}, \TTT{tl},
 \TTT{ttr}, \TTT{rtr}, \TTT{rbr}, \TTT{bbr}, \TTT{bbl}, \TTT{lbl}, \TTT{ltl},
 \TTT{ttl}]\mbox{}
 
Abbreviation of \emph{top}, \emph{right}\ldots,  \emph{top-right}\ldots,  
\emph{top-top-right}\ldots

For example,
\begin{Verbatim}
\Put[tr](0,0){A}
\end{Verbatim}
writes ``\verb+A+''  \emph{at top and right} of point \verb+(0,0)+.

Parameter \cs{Pictlabelsep} determines the distance between the graphical 
object and the given point.
In the following examples we have made this argument very big to clearly 
appreciate the positioning of objects.
\end{description}
\begin{exemple}
\renewcommand{\Pictlabelsep}{1}
\begin{center}
\setlength{\unitlength}{2.5cm}%

\begin{Picture}(-1.5,-1.5)(1.5,1.5)
\Put[N](0,0){N}
\Put[S](0,0){S}
\Put[E](0,0){E}
\Put[W](0,0){W}
\Put[NE](0,0){NE}
\Put[SE](0,0){SE}
\Put[SW](0,0){SW}
\Put[NW](0,0){NW}
%
\Put[NNE](0,0){NNE}
\Put[ENE](0,0){ENE}
\Put[ESE](0,0){ESE}
\Put[SSE](0,0){SSE}
\Put[SSW](0,0){SSW}
\Put[WSW](0,0){WSW}
\Put[WNW](0,0){WNW}
\Put[NNW](0,0){NNW}
\Put(0,0){\Circle{1}}
\xLINE(-1,0)(1,0)
\xLINE(0,-1)(0,1)
\end{Picture}
\end{center}
\end{exemple}
\begin{exemple}
\renewcommand{\Pictlabelsep}{1}
\begin{center}
\setlength{\unitlength}{2.5cm}%

\begin{Picture}(-1.5,-1.5)(1.5,1.5)
\Put[t](0,0){t}
\Put[r](0,0){r}
\Put[b](0,0){b}
\Put[l](0,0){l}
\Put[tr](0,0){tr}
\Put[br](0,0){br}
\Put[bl](0,0){bl}
\Put[tl](0,0){tl}
\Put[ttr](0,0){ttr}
\Put[rtr](0,0){rtr}
\Put[rbr](0,0){rbr}
\Put[bbr](0,0){bbr}
\Put[bbl](0,0){bbl}
\Put[lbl](0,0){lbl}
\Put[ltl](0,0){ltl}
\Put[ttl](0,0){ttl}
\Put(0,0){%
   \regularPolygon[45]{\numberSQRTTWO}{4}}
\xLINE(-1,0)(1,0)
\xLINE(0,-1)(0,1)
\end{Picture}
\end{center}
\end{exemple}
\end{description}
\paragraph{Rectangular o circular distance?}
Commands \cs{rPut} and \cs{cPut} differ only in the criterion they use
to determine the distance between the reference point and the graphical object.
Command \cs{rPut} places the object (outside of) 
the square centered at the reference point and side \verb+2\Pictlabelsep+, 
while \cs{cPut} places it in the cercle of radius \verb+\Pictlabelsep+
(letters \verb+r+ and \verb+c+ mean, respectively,
a \emph{rectangular} and \emph{circular} layout).%
\footnote{For the mathematicians: command \cs{cPut} uses the euclidean norm 
(or 2-norm), while \cs{rPut} uses the infinite norm.}
Although, for small values of the \cs{Pictlabelsep} parameter,
the difference is subtle and usually not very significant, it is generally best
to use the circular version (because it corresponds to the natural concept of 
distance) and reserve the rectangular version
to objects that are placed on horizontal or vertical lines.
\begin{Exemple}
\begin{center}
\setlength{\unitlength}{1.5cm}
\renewcommand{\Pictlabelsep}{1}

\begin{Picture}(-1.5,-1.5)(2,1.5)
\regularPolygon[45]{\numberSQRTTWO}{4}
\Put(0,0){\circle*{0.1}}
\rPut{45}(0,0){r}
\xLINE(0,0)(0,-1)
\thicklines
\renewcommand{\Pictlabelsep}{0.1}
\xLINE(0,0)(1,1)
\xLINE(0,0)(1,0)
\xtrivVECTOR(0,-1)(1,-1)
\xtrivVECTOR(1,-1)(0,-1)
\rPut{b}(0.5,-1){\footnotesize\textbackslash Pictlabelsep}
\xtrivVECTOR(1,-1)(1,0)
\xtrivVECTOR(1,0)(1,-1)
\rPut{r}(1,-0.5){\footnotesize\textbackslash Pictlabelsep}
\polarreference\degreesangles
\xArc{0.3}{0}{45}
\degreesangles
\Put[22.5](0.3,22.5){$45^{\mathrm o}$}
\end{Picture}
\begin{Picture}(-1.5,-1.5)(2,1.5)
\Put(0,0){\circle*{0.1}}
\cPut{45}(0,0){c}
\Circle{1}
\thicklines
\xLINE(0,0)(\numberCOSXLV,\numberCOSXLV)
\xLINE(0,0)(1,0)
\xtrivVECTOR(0,0)(0,-1)
\xtrivVECTOR(0,-1)(0,0)
\renewcommand{\Pictlabelsep}{0.1}
\rPut{r}(0,-0.5){\footnotesize\textbackslash Pictlabelsep}
\polarreference\degreesangles
\xArc{0.3}{0}{45}
\degreesangles
\Put[22.5](0.3,22.5){$45^{\mathrm o}$}
\end{Picture}
\end{center}
\end{Exemple}

Note that if the commands we use are \cs{rPut} or \cs{cPut}, then the 
positioners
\verb+t, r, tr+\ldots are equivalent to the corresponding \verb+N, E, NE+\ldots
However, the \cs{Put} command choose between rectangular or circular layout
following this criteria:
\begin{itemize}
 \item Positioners of \emph{compass} type (like \verb+NE+) use the circular
layout.
 \item Positioners \verb+t, tr+, et cetera use the rectangular layout.
 \item If the positioner is an angle (a number), it uses a default position
which is set using the \cs{defaultPut} declaration:
\verb+\defaultPut{c}+
determines a circular distance, while
\verb+\defaultPut{r}+
determines the rectangular alternative.
\end{itemize}
\begin{exemple}
\renewcommand{\Pictlabelsep}{1}
\begin{center}
\setlength{\unitlength}{2.5cm}%

\begin{Picture}(-1.5,-1.5)(1.5,1.5)
\defaultPut{c}
\Put[45](0,0){c}
\defaultPut{r}
\Put[45](0,0){r}
\regularPolygon[45]{\numberSQRTTWO}{4}
\Put(0,0){\Circle{1}}
\xLINE(-1,0)(1,0)
\xLINE(0,-1)(0,1)
\end{Picture}
\end{center}
\end{exemple}
\paragraph{Alignment by the baseline}
Starred versions \cs{Put*} and \cs{rPut*} allow us to align by the baseline
objects positioned below the reference point.
To use these commands, user must decide which is the higher object to be 
positioned, and introduce it as an argument of
the \csdef{highestlabel} declaration. For example, typing
\begin{Verbatim}
\highestlabel{\Huge A}
\end{Verbatim}
we reserve a sufficient vertical space to write the character {\Huge A}.

It should be noted that starred versions behave differently
only when the position of the object stands
under the reference point, with positioners
\verb+bbl+, \verb+b+ or \verb+bbr+, or with an appropiate angle
(as \verb+-90+ or \verb+300+); otherwise (including
\verb$S$, \verb$SSW$, et cetera),
the \cs{Put*} and \cs{rPut*} commands are equivalent
to the non-starred commands
 \cs{Put} and \cs{rPut}.
\begin{exemple}
\begin{center}
\setlength{\unitlength}{1cm}

\begin{Picture}(-3.5,-1.5)(3.5,1.5)
\xLINE(-3.5,0)(3.5,0)
\multiPut(-3,-0.1)(1,0){7}{\xLINE(0,0)(0,0.2)}
\highestlabel{\Huge A}
\renewcommand{\Pictlabelsep}{0.2}
\Put*[bbl](-3,0){\small A}
\Put*[b](-2,0){\normalsize A}
\Put*[-100](-1,0){\large A}
\Put*[-90](0,0){\Large A}
\Put*[270](1,0){\LARGE A}
\Put*[300](2,0){\huge A}
\Put*[bbr](3,0){\Huge A}
\Put*[bbl](-3.5,0){%
   \pictcolor{gray}\xLINE(0,0)(7,0)}
\end{Picture}
\end{center}
\end{exemple}
 

When a \environ{Picture} environment starts,
highest label is set to \verb+\normalfont\normalsize$1$+
(i.e., the high of a \emph{normal} $1$).
\subsection{Alternatives to the \cs{multiput} command}
The \package{xpicture} package introduces two families of commands
to generalize the \cs{multiput} command:
\begin{enumerate}
 \item The natural generalization, with all versions,
 \ttslashindex{multiPut}\ttslashindex{multicPut}\ttslashindex{multirPut}
\begin{Verbatim}[commandchars=\|\{\},commentchar=\%]
\multiPut[|TIT{position}](|TIT{x0},|TIT{y0})(|TIT{|(|Delta|)x},%
|TIT{|(|Delta|)y})|{|TIT{n}|}|{|TIT{object}|}
\multiPut*[|TIT{position}](|TIT{x0},|TIT{y0})(|TIT{|(|Delta|)x}%
,|TIT{|(|Delta|)y})|{|TIT{n}|}|{|TIT{object}|}
\multicPut|{|TIT{position}|}(|TIT{x0},|TIT{y0})(|TIT{|(|Delta|)x}%
,|TIT{|(|Delta|)y})|{|TIT{n}|}|{|TIT{object}|}
\multirPut|{|TIT{position}|}(|TIT{x0},|TIT{y0})(|TIT{|(|Delta|)x}%
,|TIT{|(|Delta|)y})|{|TIT{n}|}|{|TIT{object}|}
\multirPut*|{|TIT{position}|}(|TIT{x0},|TIT{y0})(|TIT{|(|Delta|)x}%
,|TIT{|(|Delta|)y})|{|TIT{n}|}|{|TIT{object}|}
\end{Verbatim}
These commands compose \TIT{n} copies of \TTT{\TIT{object}}
in $(\TIT{x0},\TIT{y0})$, $(\TIT{x0}+\Delta x,\TIT{y0}+\Delta y)$,
 $(\TIT{x0}+2\Delta x,\TIT{y0}+2\Delta y)$,\ldots, 
 $(\TIT{x0}+(\TIT n-1)\Delta x,\TIT{y0}+(\TIT n-1)\Delta y)$.
\item A new command group,
\ttslashindex{multiPlot}\ttslashindex{multicPlot}\ttslashindex{multirPlot}
\begin{Verbatim}[commandchars=\|\{\},commentchar=\%]
\multiPlot[|TIT{position}]|{|TIT{object}|}(|TIT{x0},|TIT{y0})(|TIT{x1},%
|TIT{y1})...(|TIT{xn},|TIT{yn})
\multiPlot*[|TIT{position}]|{|TIT{object}|}(|TIT{x0},|TIT{y0})(|TIT{x1},%
|TIT{y1})...(|TIT{xn},|TIT{yn})
\multicPlot|{|TIT{position}|}|{|TIT{object}|}(|TIT{x0},|TIT{y0})(|TIT{x1},%
|TIT{y1})...(|TIT{xn},|TIT{yn})
\multirPlot|{|TIT{position}|}|{|TIT{object}|}(|TIT{x0},|TIT{y0})(|TIT{x1},%
|TIT{y1})...(|TIT{xn},|TIT{yn})
\multirPlot*|{|TIT{position}|}|{|TIT{object}|}(|TIT{x0},|TIT{y0})(|TIT{x1},%
|TIT{y1})...(|TIT{xn},|TIT{yn})
\end{Verbatim}
These commands compose the done object in several positions, that are freely
entered as a list of coordinate pairs.
\end{enumerate}
\begin{exemple}
\begin{center}
\setlength{\unitlength}{1cm}
\referencesystem(0,0)(1,-1)(1,1)
\begin{Picture}(-2.5,-2.5)(2.5,2.5)
\cartesiangrid(-2,-2)(2,2)
\pictcolor{blue}
\multiPut(-2,-2)(1,1){5}{\circle*{0.25}}
\pictcolor{red}
\multiPlot{\circle*{0.25}}(-1,-2)(2,1)(-2,2)
\end{Picture}
\end{center}
\end{exemple}
\begin{exemple}
\begin{center}
\setlength{\unitlength}{1cm}
\referencesystem(0,0)(1,-1)(1,1)
\begin{Picture}(-2.5,-2.5)(2.5,2.5)
\cartesiangrid(-2,-2)(2,2)
\pictcolor{blue}
\multiPut[b](-2,-2)(1,1){5}{\circle*{0.25}}
\pictcolor{red}
\multiPlot[NE]{\circle*{0.25}}(-1,-2)(2,1)(-2,2)
\end{Picture}
\end{center}
\end{exemple}
\subsection{Alternatives to \cs{line} and \cs{vector}}
\begin{description}
\item[\csdef{xLINE}] This command draws line segments:
\begin{Verbatim}[commandchars=\|\[\]]
\xLINE(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])
\end{Verbatim}
draws the line segment between the two points
\verb+(+\TIT{x0}\verb+,+\TIT{y0}\verb+)+ and
\verb+(+\TIT{x1}\verb+,+\TIT{y1}\verb+)+
(Cartesian or polar coordinates, in the active reference system).
This allows us to draw any segment in any direction.
\item[\csdef{xVECTOR}, \csdef{xtrivVECTOR}] plot arrows:
\begin{Verbatim}[commandchars=\|\[\]]
\xVECTOR(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])
\xtrivVECTOR(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])
\end{Verbatim}
draw an arrow between points
\verb+(+\TIT{x0}\verb+,+\TIT{y0}\verb+)+ and
\verb+(+\TIT{x1}\verb+,+\TIT{y1}\verb+)+.
The \cs{xtrivVECTOR} command draw an arrow
the end of which simply consists of a pair of segments
(\setlength{\unitlength}{1cm}%
\begin{Picture}(0,-0.1)(0.5,0.1)\xtrivVECTOR(0,0)(0.5,0)\end{Picture}).
length and aperture of the end of arrow are controled by the instruction
\ttslashindex{arrowsize}
\begin{Verbatim}[commandchars=\|\[\]]
\arrowsize{|TIT[xlen]}{|TIT[ylen]}
\end{Verbatim}
where the two parameters are non-negative numbers:
the first one for the length (in points); second
for the half of the aperture. Default is
\begin{Verbatim}
\arrowsize{5}{2}
\end{Verbatim}
\begin{exemple}
\setlength{\unitlength}{0.75cm}
\referencesystem(0,0)(1,0)(0.25,0.75)
\begin{Picture}(-4.5,-4.5)(4.5,4.5)
\cartesiangrid(-4,-4)(4,4)
\thicklines
\pictcolor{blue}
\xLINE(-4,0)(1,4)
\Put(1,-3){\xLINE(0,0)(3,2)}
\pictcolor{red}
\xtrivVECTOR(0,0)(2,3)
\xtrivVECTOR(0,0)(2,0)
\arrowsize{10}{4}
\xtrivVECTOR(0,0)(-2,-1)

\pictcolor{magenta}
\xVECTOR(-3,-3)(-3,3)
\xVECTOR(-3,-3)(-2,-2)
\end{Picture}
\end{exemple}
\item[\csdef{xline}, \csdef{xvector}, \csdef{xtrivvector}]
draw lines and vectors using the standard \LaTeX{} syntax
(but without any restriction in allowed parameters,
that can be integer or decimal numbers, positive, negative or zero).
\begin{Verbatim}[commandchars=\|\[\]]
\xline(|TIT[x],|TIT[y]){|TIT[size]}
\xvector(|TIT[x],|TIT[y]){|TIT[size]}
\xtrivvector(|TIT[x],|TIT[y]){|TIT[size]}
\end{Verbatim}
\begin{exemple}
\setlength{\unitlength}{0.75cm}
\referencesystem(0,0)(1,0)(0.25,0.75)
\begin{Picture}(-4.5,-4.5)(4.5,4.5)
\cartesiangrid(-4,-4)(4,4)
\thicklines
\pictcolor{blue}
\Put(-4,0){\xline(5,4){5}}
\Put(1,-3){\xline(3,2){3}}
\pictcolor{red}
\Put(0,0){\xtrivvector(2,3){2}}
\xtrivvector(1,0){2}
\arrowsize{10}{4}
\Put(0,0){\xtrivvector(2,1){-2}}

\pictcolor{magenta}
\Put(-3,-3){\xvector(0,1){6}}
\Put(-3,-3){\xvector(1,1){1}}
\end{Picture}
\end{exemple}

If you want to draw only an arrowhead (without any line)
you can use either the
\csdef{zerovector}/\csdef{zerotrivvector}
or \cs{xvector}/\cs{xtrivvector} commands:
\begin{Verbatim}[commandchars=\|\[\]]
\zerovector(|TIT[x],|TIT[y])
\zerotrivvector(|TIT[x],|TIT[y])
\xvector(|TIT[x],|TIT[y]){0}
\xtrivvector(|TIT[x],|TIT[y]){0}
\end{Verbatim}
\end{description}
\subsection{Polygons anf polygonal lines}
The \package{pict2e} and \package{curve2e} packages include
specific instructions for drawing polygonal lines and polygons.
We introduce new versions of these
commands in order to refer to the active reference system.
\begin{description}
\item[\csdef{Polyline}] draws polygonal lines.
Logically, we must pass the list of vertices:
\begin{Verbatim}[commandchars=\|\[\]]
\Polyline(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])...(|TIT[xn],|TIT[yn])
\end{Verbatim}
\item[\csdef{Polygon}] plots polygons, ie, closed polygonal lines:
\begin{Verbatim}[commandchars=\|\[\]]
\Polygon(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])...(|TIT[xn],|TIT[yn])
\end{Verbatim}
is equivalent to
\begin{Verbatim}[commandchars=\|\[\],commentchar=\%]
\Polyline(|TIT[x0],|TIT[y0])(|TIT[x1],|TIT[y1])...(|TIT[xn],|TIT[yn])%
(|TIT[x0],|TIT[y0])
\end{Verbatim}

\begin{exemple}
\setlength{\unitlength}{0.75cm}
\referencesystem(0,0)(1,0)(0.25,0.75)
\begin{Picture}(-4.5,-4.5)(4.5,4.5)
\externalaxes
\cartesiangrid(-4,-4)(4,4)
\linethickness{1pt}
\pictcolor{blue}
\Polyline(-2,2)(-3,-1)(0,0)(2,3)(2,2)
\pictcolor{red}
\Polygon(0,0)(1,1)(3,1)(1,-1)
\end{Picture}
\end{exemple}
\item[\csdef{regularPolygon}] draws regular polygons:
\begin{Verbatim}[commandchars=\|\(\)]
\regularPolygon[|TIT(initial angle)]{|TIT(radius)}{|TIT(sides)}
\end{Verbatim}
makes the regular polygon with the given radius and sides.
The optional argument (zero, by default) determines
the slope of the first vertex, always measured in degrees.
\begin{exemple}
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{Picture}(-7.5,-7.5)(7.5,7.5)
\externalaxes
\cartesiangrid(-7,-7)(7,7)
\pictcolor{blue}
\regularPolygon{1}{5}
\Put(-4,0){\regularPolygon{2}{6}}
\Put(3,3){\regularPolygon{2}{4}}
\Put(-4,-4){\regularPolygon[45]{2}{4}}
\Put(4,-4){\regularPolygon[90]{2.5}{11}}
\Put(-4,4){\regularPolygon[90]{3}{3}}
\end{Picture}
\end{center}
\end{exemple}

\end{description}
\section{Drawing curves}
This section highlights the true potentiality of the \package{xpicture} 
package.
We will describe the instructions that can be used to easily (and effectively) 
represent
several interesting curves: Firstly, conic sections and arcs.
Then, any piecewise regular curve
(including graphs of real variable functions, in rectangular or polar 
coordinates,
and ---in a more general way--- curves defined by parametric equations).
\subsection{Conic sections}
The \package{xpicture} package defines new commands to draw conic sections:
 ellipses, circles, hyperbolas and parabolas.
\subsubsection{Circles}
We can draw the circle of implicit equation $x^2+y^2=r^2$ typing
\ttslashindex{Circle}
\begin{Verbatim}[commandchars=\|\[\]]
\Circle{|TIT[r]}
\end{Verbatim}
Note than the standard command \cs{circle}
requeres the diameter as mandatory argument, while here we must insert the 
radius.
\subsubsection{Ellipses}
To draw the ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ enter the
following instruction:\ttslashindex{Ellipse}
\begin{Verbatim}[commandchars=\|\[\]]
\Ellipse{|TIT[a]}{|TIT[b]}
\end{Verbatim}
\begin{exemple}
\setlength{\unitlength}{0.5cm}
\renewcommand{\axeslabelsize}{\footnotesize}
\begin{Picture}(-5.5,-4.5)(5.5,4.5)
\cartesiangrid(-5,-4)(5,4)
\pictcolor{blue}
\Ellipse{4}{3}
\Circle{2}
\end{Picture}

\referencesystem(0,0)(1,0)(0.5,0.5)
\begin{Picture}(-5.5,-4.5)(5.5,4.5)
\cartesiangrid(-5,-4)(5,4)
\pictcolor{blue}
\Ellipse{4}{3}
\Circle{2}
\end{Picture}
\end{exemple}
\subsubsection{Hyperbolas}
Since the hyperbolas and parabolas are not bounded curves, to define the
portion of the curve that we want to draw  we need to specify the
maximum values for the $x$ and $y$ variables.\ttslashindex{Hyperbola}
\begin{Verbatim}[commandchars=\|\[\]]
\Hyperbola{|TIT[a]}{|TIT[b]}{|TIT[xmax]}{|TIT[ymax]}
\end{Verbatim}
draws the hyperbola
$\displaystyle\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,
where variables $x$ and $y$ are limited, respectively,
to the $\TTT{[-\TIT{xmax}}, \TTT{\TIT{xmax}]}$ and 
$\TTT{[-\TIT{ymax}}, \TTT{\TIT{ymax}]}$ intervals.
This curve is well defined if the parameter \TTT{\TIT{xmax}}
is greater than \TTT{\TIT{a}}. Otherwise, \package{xpicture} returns an error 
message and does not draw any curve.

In the following example, we show the hyperbola 
$\displaystyle\frac{x^2}{5^2}-\frac{y^2}{2^2}=1$
and its asymptotes,
using the \cs{xLINE} command (these asymptotes are lines $2x=\pm5y$,
passing through $(\pm16,\pm6.4)$).
\begin{Exemple}
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{Picture}(-17,-9)(17,9)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesiangrid(-16,-8)(16,8)
\pictcolor{blue}
\Hyperbola{5}{2}{16}{8}
\pictcolor{orange}
\xLINE(16,6.4)(-16,-6.4)
\xLINE(-16,6.4)(16,-6.4)
\end{Picture}
\end{center}
\end{Exemple}

Instructions \csdef{lHyperbola} and \csdef{rHyperbola} draw, respectively,
only the \emph{left} or only the \emph{right} branch of the given hyperbola
(here, is interpreted as \emph{right} branch this one that belongs to positive
values of variable $x$).
\begin{exemple}
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{Picture}(-5.5,-5.5)(5.5,5.5)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesiangrid(-5,-5)(5,5)
\pictcolor{red}
\lHyperbola{2}{3}{5}{5}
\pictcolor{blue}
\rHyperbola{2}{3}{5}{5}
\end{Picture}
\end{center}
\end{exemple}
\subsubsection{Parabolas}
Instruction\ttslashindex{Parabola}
\begin{Verbatim}[commandchars=\|\[\]]
\Parabola{|TIT[a]}{|TIT[xmax]}{|TIT[ymax]}
\end{Verbatim}
draw the parabola $x=ay^2$, varying $x$, at most, in the interval
$[0,\TTT{\TIT{xmax}}]$
(if \TTT{\TIT{a}} is positive) or in $[-\TTT{\TIT{xmax}},0]$
(for negative values of \TTT{\TIT{a}}),
and $y$ in $[-\TTT{\TIT{ymax}},\TTT{\TIT{ymax}}]$.
Parameters \TTT{\TIT{xmax}} and \TTT{\TIT{ymax}}
must be positive.
\begin{exemple}
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{Picture}(-5.5,-5.5)(5.5,5.5)
\cartesiangrid(-5,-5)(5,5)
\pictcolor{blue}
\Parabola{2}{5}{5}
\Parabola{0.2}{5}{5}
\pictcolor{orange}
\Parabola{-2}{5}{5}
\Parabola{-0.2}{5}{5}
\end{Picture}
\end{center}
\end{exemple}
\medskip

All commands drawing conic sections or arcs divide the curve in 
\csdef{defaultplotdivs} pieces (8, by default). To obtain a greather
accuracy, you can redefine this parameter.

\medskip

Note that all these commands draw conic sections centered
at the coordinate origin, so that their
principal axes coincide with the coordinate axes. If we
want to move his
center to any other point, we can do it moving in advance
the origin of coordinates or simply
including the command as an argument of the \cs{Put} command.
\begin{Exemple}
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{Picture}(-11,-8)(11,8)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesiangrid(-10,-7)(11,7)
\pictcolor{blue}
\Put(2,3){\Ellipse{4}{3}}
\Put(2,3){\Circle{0.25}}
\pictcolor{orange}
\Put(2,-3){\Hyperbola{5}{2}{9}{3}}
\Put(2,-3){\Circle{0.25}}
\pictcolor{green}
\translateorigin(-10,2)
\Parabola{0.5}{21}{5}
\Circle{0.25}
\end{Picture}
\end{center}
\end{Exemple}
But, if the symmetry axes of our curve are not parallel to the coordinate
axes,\footnote{That is, in mathematical terms,
if the eigenvectors of the underlying quadratic form are not the canonical 
vectors.}
then we will need a rotation of axes.
\begin{Exemple}
\setlength{\unitlength}{0.5cm}
\begin{center}
\begin{Picture}(-10.5,-7.5)(10.5,7.5)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesiangrid(-10,-7)(10,7)
{%
\pictcolor{blue}
\translateorigin(5,3)
\rotateaxes{\numberSIXTHPI}
\Ellipse{4}{3}
\xLINE(-4,0)(4,0)
\xLINE(0,-3)(0,3)
}
\degreesangles
{%
\pictcolor{orange}
\translateorigin(-3,0)
\rotateaxes{110}
\Hyperbola{3}{2}{6}{4}
\xLINE(-6,-4)(6,4)
\xLINE(6,-4)(-6,4)
}
\pictcolor{green}
\translateorigin(5,-6)
\rotateaxes{72}
\Parabola{1}{4}{3}
\xLINE(0,-2)(0,2)
\xLINE(0,0)(4,0)
\end{Picture}
\end{center}
\end{Exemple}
Note that we made a couple of changes of local reference system (one for each
curve) within the drawing.
We can use the recourse to the change of coordinates also to
draw the hyperbola $\displaystyle\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ and the
parabola $y=ax^2$.
Note than \verb+\referencesystem(0,0)(0,1)(1,0)+
(or \verb+\symmetrize{\numberQUARTERPI}+)
makes vertical the $x$ axis and horizontal the
$y$ axis.\footnote{We will use this trick later
to plot inverse functions.}
\begin{exemple}
\setlength{\unitlength}{0.5cm}
\begin{center}
\begin{Picture}(-5.5,-5.5)(5.5,5,5)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesiangrid(-5,-5)(5,5)
\referencesystem(0,0)(0,1)(1,0)
\pictcolor{blue}
\Parabola{0.22}{5}{5}
\pictcolor{red}
\Hyperbola{2}{3}{5}{5}
\end{Picture}
\end{center}
\end{exemple}

\subsection{Arcs (of conic sections)}
The instructions described above allow us to draw whole circles, ellipses
hyperbolas and parabolas. More generally, we can represent any portion of 
these curves, ie, circular, elliptic, hyperbolic and parabolic arcs.
\ttslashindex{xArc}\ttslashindex{circularArc}
\begin{Verbatim}[commandchars=\|\[\]]
\xArc{|TIT[r]}{|TIT[angle1]}{|TIT[angle2]}
\circularArc{|TIT[r]}{|TIT[angle1]}{|TIT[angle2]}
\end{Verbatim}
These two instructions are equivalent.
They draw the arc of the circle centered at $(0,0)$
with radius $\TIT{r}$
and limited by the $\TTT{\TIT{angle1}}$ and $\TTT{\TIT{angle2}}$
angles.
\begin{exemple}
\setlength{\unitlength}{0.5cm}
\begin{center}
\begin{Picture}(-5.5,-5.5)(5.5,5,5)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesianaxes(-5,-5)(5,5)
\pictcolor{gray}
\circularArc{3}{\numberPI}{\numberTWOPI}
\pictcolor{red}
\xLINE(-2,2)(-2,5)
\xLINE(-2,2)(-5,2)
\degreesangles
\Put(-2,2){\circularArc{1}{90}{180}}
\pictcolor{blue}
\polarreference
\Put(1,30){\xLINE(0,0)(4,30)}
\Put(1,30){\xLINE(0,0)(4,60)}
\Put(1,30){\circularArc{2}{30}{60}}
\end{Picture}
\end{center}
\end{exemple}
\begin{exemple}
\SUBTRACT{\numberGOLD}{1}{\midaB}
\COPY{1}{\midaA}
\ADD{\midaA}{\midaB}{\Mida}
\setlength{\unitlength}{5cm}
\newcommand{\espiral}{%
    \Put(0,0){\begin{Picture}(0,0)(0,0)
      \translateorigin(\midaA,0)
      \pictcolor{red}
      \circularArc{\midaA}{\numberHALFPI}{\numberPI}
      \pictcolor{blue}
      \xLINE(0,0)(0,\midaA)
   \end{Picture}
   }
   \COPY{\midaA}{\Mida}
   \COPY{\midaB}{\midaA}
   \SUBTRACT{\Mida}{\midaA}{\midaB}
   \translateorigin(\Mida,\midaB)
   \changereferencesystem(0,\midaA)(0,-1)(1,0)
}
\renewcommand{\defaultplotdivs}{2}

\begin{center}
\begin{Picture}(0,0)(\numberGOLD,1)
   \Polygon(0,0)(\Mida,0)(\Mida,1)(0,1)
   % Plot 8 circular arcs
   \espiral\espiral\espiral\espiral
   \espiral\espiral\espiral\espiral
\end{Picture}

Golden rectangles and spiral
\end{center}
\end{exemple}
\ttslashindex{ellipticArc}
\begin{Verbatim}[commandchars=\|\[\]]
\ellipticArc{|TIT[a]}{|TIT[b]}{|TIT[angle1]}{|TIT[angle2]}
\end{Verbatim}
This instruction draws the arc of the ellipse centered at
 $(0,0)$ with semiaxes $\TIT{a}$
and $\TIT{b}$, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$,
limited by angles $\TTT{\TIT{angle1}}$ and $\TTT{\TIT{angle2}}$.
\begin{exemple}
\setlength{\unitlength}{0.5cm}
\begin{center}
\begin{Picture}(-0.5,-3.5)(5.5,3.5)
\degreesangles
\ellipticArc{2}{3}{-90}{90}
\ellipticArc{5}{3}{-90}{90}
\end{Picture}
\end{center}
\end{exemple}
\ttslashindex{lhyperbolicArc}\ttslashindex{rhyperbolicArc}
\begin{Verbatim}[commandchars=\|\[\]]
\rhyperbolicArc{|TIT[a]}{|TIT[b]}{|TIT[y1]}{|TIT[y2]}
\lhyperbolicArc{|TIT[a]}{|TIT[b]}{|TIT[y1]}{|TIT[y2]}
\end{Verbatim}
Draw the arc (of the right or left branch, respectively)
of the hyperbola
 $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ included between $y=\TTT{\TIT{y1}}$ and 
 $y=\TTT{\TIT{y2}}$.
\begin{exemple}
\setlength{\unitlength}{0.5cm}
\begin{center}
\begin{Picture}(-5.5,-5.5)(5.5,5,5)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesianaxes(-5,-5)(5,5)
\pictcolor{red}
\lhyperbolicArc{2}{3}{-4}{0}
\pictcolor{blue}
\rhyperbolicArc{2}{3}{-2}{5}
\end{Picture}
\end{center}
\end{exemple}
\ttslashindex{parabolicArc}
\begin{Verbatim}[commandchars=\|\[\]]
\parabolicArc{|TIT[a]}{|TIT[y1]}{|TIT[y2]}
\end{Verbatim}
Draw the arc of the parabola
 $x=ay^2$ included between $y=\TTT{\TIT{y1}}$ and $y=\TTT{\TIT{y2}}$.
\begin{exemple}
\setlength{\unitlength}{1cm}
\begin{center}
\begin{Picture}(-2.5,-2.5)(2.5,2,5)
\renewcommand{\axeslabelsize}{\footnotesize}
\cartesianaxes(-2,-2)(2,2)
\pictcolor{red}
\parabolicArc{-2}{-1}{0}
\pictcolor{blue}
\parabolicArc{0.5}{0}{2}
\end{Picture}
\end{center}
\end{exemple}
\subsection{Real variable functions}\label{subsec:real}
The \package{xpicture} package provides us two commands
to draw the graph of a function:
\csdef{PlotFunction} and \csdef{PlotPointsOfFunction}.
\begin{Verbatim}[commandchars=\|\(\)]
\PlotFunction[|TIT(n)]{\|TIT(functionname)}{\|TIT(tzero)}{\|TIT(tone)}
\PlotPointsOfFunction{|TIT(n)}{\|TIT(functionname)}{\|TIT(tzero)}{\|TIT(tone)}
\end{Verbatim}
Note that the parameter $\TTT{\TIT{n}}$ is optional in one of these 
instructions and mandatory in the other one.
In the case of \csdef{PlotFunction},
if we do not use this optional parameter,
a quadratic approximation of the function
\cs{\TIT{functionname}}
in the $[\cs{\TIT{tzero}},\cs{\TIT{tone}}]$ interval is drawn.
\begin{exemple}
\setlength{\unitlength}{1cm}
\begin{Picture}(-2.5,-0.5)(3.5,4.5)
\cartesianaxes(-2,0)(2,4)
\pictcolor{blue}
\PlotFunction{\SQUAREfunction}{-2}{2}
\Put[E](2,4){$f(t)=t^2$}
\end{Picture}
\end{exemple}
Now, this almost never provides a right graphic.
To draw curves with a greater accuracy we should use the parameter,
\TTT{\TIT{n}},
dividing the interval in \TTT{\TIT{n}} subintervals.
\begin{exemple}
\setlength{\unitlength}{1cm}
\CUBE{1.5}{\mymax}
\begin{Picture}(-2,-4)(2,4)
\cartesianaxes(-1.5,-\mymax)(1.5,\mymax)
\pictcolor{blue}
\PlotFunction[8]{\CUBEfunction}{-1.5}{1.5}
\Put[E](1.5,\mymax){$f(t)=t^3$}
\end{Picture}
\end{exemple}

On the other hand, the \csdef{PlotPointsOfFunction} command
plots $\TTT{\TIT{n}}+1$ \emph{points}, uniformly distributed
about the $x$-axis.
\begin{exemple}
\setlength{\unitlength}{1cm}
\CUBE{1.5}{\mymax}
\begin{Picture}(-2,-4)(2,4)
\cartesianaxes(-1.5,-\mymax)(1.5,\mymax)
\pictcolor{blue}
\PlotPointsOfFunction{24}{\CUBEfunction}{-1.5}{1.5}
\Put[E](1.5,\mymax){$f(t)=t^3$}
\end{Picture}
\end{exemple}

By default, \cs{PlotPointsOfFunction} plot \emph{points} as a filled circle
of diameter \verb+0.1\unitlength+. But you can modifie this diameter, by 
redefining the \csdef{pointmarkdiam} parameter.
\begin{exemple}
\setlength{\unitlength}{1cm}
\CUBE{1.5}{\mymax}
\renewcommand{\pointmarkdiam}{0.3}
\begin{Picture}(-2,-4)(2,4)
\cartesianaxes(-1.5,-\mymax)(1.5,\mymax)
\pictcolor{blue}
\PlotPointsOfFunction{24}{\CUBEfunction}{-1.5}{1.5}
\Put[E](1.5,\mymax){$f(t)=t^3$}
\end{Picture}
\end{exemple}

Moreover, you can select another symbol for these points, redefining 
\csdef{pointmark}.
\begin{exemple}
\setlength{\unitlength}{1cm}
\CUBE{1.5}{\mymax}
\renewcommand{\pointmark}{$\diamond$}
\begin{Picture}(-2,-4)(2,4)
\cartesianaxes(-1.5,-\mymax)(1.5,\mymax)
\pictcolor{blue}
\PlotPointsOfFunction{24}{\CUBEfunction}{-1.5}{1.5}
\Put[E](1.5,\mymax){$f(t)=t^3$}
\end{Picture}
\end{exemple}

Naturally, in order to apply these commands the function must be defined.
\package{xpicture}
loads the \packagedef{calculus} package, which
predefines some of the most common elementary functions
and includes several tools to build new ones.
The predefined functions are the following:
\begin{center}
\begin{tabular}{l&gt;{$}l&lt;{$}&gt;{\qquad}l&gt;{$}l&lt;{$}}
 \csdef{ZEROfunction} &amp; f(t)=0 &amp;
 \csdef{ONEfunction}  &amp; f(t)=1    \\
 \csdef{IDENTITYfunction}  &amp; f(t)=t &amp;
 \csdef{RECIPROCALfunction}  &amp; f(t)=1/t \\
 \csdef{SQUAREfunction}  &amp; f(t)=t^2 &amp;
 \csdef{CUBEfunction}  &amp; f(t)=t^3 \\
 \csdef{SQRTfunction}  &amp; f(t)=\sqrt t \\
 \csdef{EXPfunction}  &amp; f(t)=\exp t &amp;
 \csdef{LOGfunction} &amp; f(t)=\log t \\
 \csdef{COSfunction}  &amp; f(t)=\cos t &amp;
 \csdef{SINfunction}  &amp; f(t)=\sin t \\
 \csdef{TANfunction}  &amp; f(t)=\tan t &amp;
 \csdef{COTfunction}  &amp; f(t)=\cot t \\
 \csdef{COSHfunction}  &amp; f(t)=\cosh t &amp;
 \csdef{SINHfunction}  &amp; f(t)=\sinh t \\
 \csdef{TANHfunction}  &amp; f(t)=\tanh t &amp;
 \csdef{COTHfunction}  &amp; f(t)=\coth t \\
 \csdef{HEAVISIDEfunction} &amp; f(t)=\begin{cases}
                              0 &amp; \text{si $t&lt;0$} \\
                              1 &amp; \text{si $t\geq0$}
                          \end{cases}
\end{tabular}
\end{center}

\begin{Exemple}
\setlength{\unitlength}{1cm}
\linethickness{1.5pt}
\centering
\begin{Picture}(-5,-5)(6,5)
\externalaxes\makenotics
\cartesiangrid(-4.5,-4.5)(4.5,4.5)
\pictcolor{red}
\PlotFunction{\IDENTITYfunction}{-4.5}{4.5}
\Put[tr](4.5,4.5){$y=x$}

\DIVIDE{1}{4.5}{\minx}
\pictcolor{magenta}
\PlotFunction[10]{\RECIPROCALfunction}{\minx}{4.5}
\PlotFunction[10]{\RECIPROCALfunction}{-\minx}{-4.5}
\Put[r](4.5,\minx){$y=1/x$}

\SQRT{4.5}{\maxx}
\pictcolor{cyan}
\PlotFunction[10]{\SQUAREfunction}{-\maxx}{\maxx}
\Put[tr](\maxx,4.5){$y=x^2$}

\pictcolor{blue}
\PlotFunction[10]{\CUBEfunction}{-1.6509}{1.6509}
\Put[t](1.6509,4.5){$y=x^3$}
\end{Picture}
\end{Exemple}

\begin{Exemple}
\setlength{\unitlength}{1cm}
\linethickness{1.5pt}
\centering
\begin{Picture}(-7,-4.5)(7,4.5)
{\makenolabels
\changereferencesystem(0,0)(\numberHALFPI,0)(0,1)
\cartesiangrid(-4,-4)(4,4)
\highestlabel{$2\pi$}
\printylabels{-4}{1}{4}
\printxlabel{-4}{-2\pi}
\printxlabel{-2}{-\pi}
\printxlabel{2}{\pi}
\printxlabel{4}{2\pi}}
\pictcolor{red}
\PlotFunction[16]{\COSfunction}{-\numberTWOPI}{\numberTWOPI}
\pictcolor{blue}
\PlotFunction[16]{\SINfunction}{-\numberTWOPI}{\numberTWOPI}
\pictcolor{magenta}
\PlotFunction[6]{\TANfunction}{-1.3258}{1.3258}
\end{Picture}
\end{Exemple}

From these basic functions we can define many others,
using the following \emph{operations}:
\newcommand{\functoper}{%
   \{\cs{\TIT{function1}}\}\{\cs{\TIT{function2}}\}\{\cs{\TIT{newfunction}}\}}
\begin{description}
\item[Constant function:]\mbox{}

\csdef{CONSTANTfunction}\{\TIT{k}\}\{\cs{\TIT{newfunction}}\}

 Example: defining the  $F(t)=5$ function:

\cs{CONSTANTfunction}\{5\}\{\cs{F}\}

\item[Sum function:]\mbox{}

\csdef{SUMfunction}\functoper


Example: defining the $F(t)=t^2+t^3$ function:

\cs{SUMfunction}\{\cs{SQUAREfunction}\}\{\cs{CUBEfunction}\}\{\cs{F}\}

\item[Difference function:]\mbox{}

\csdef{SUBTRACTfunction}\functoper

Example: defining the $F(t)=t^2-t^3$ function:

\cs{SUBTRACTfunction}\cs{SQUAREfunction}\cs{CUBEfunction}\{\cs{F}\}

\item[Product function:]\mbox{}

\csdef{PRODUCTfunction}\functoper

Example: defining the $F(t)=\mathrm e^t\cos t$ function:

\cs{PRODUCTfunction}\cs{EXPfunction}\cs{COSfunction}\{\cs{F}\}

\item[Quotient function:]\mbox{}

\csdef{QUOTIENTfunction}\functoper

Example: defining the $F(t)=\mathrm e^t/\cos t$ function:

\cs{QUOTIENTfunction}\cs{EXPfunction}\cs{COSfunction}\{\cs{F}\}

\item[Composition of two functions:]\mbox{}

\csdef{COMPOSITIONfunction}\functoper

Example: defining the $F(t)=\mathrm e^{\cos t}$ function:

\cs{COMPOSITIONfunction}\cs{EXPfunction}\cs{COSfunction}\{\cs{F}\}

\item[Scaled function:]\mbox{}

\csdef{SCALEfunction}\{\TIT{k}\}\{\cs{\TIT{function}}\}%
   \{\cs{\TIT{newfunction}}\}

Example: defining the $F(t)=3{\cos t}$ function:

\cs{SCALEfunction}\{3\}\cs{COSfunction}\{\cs{F}\}

\item[Scaled variable:]\mbox{}

\csdef{SCALEVARIABLEfunction}\{\TIT{k}\}\{\cs{\TIT{function}}\}%
   \{\cs{\TIT{newfunction}}\}

Example: defining the $F(t)=\cos 3t$ function:

\cs{SCALEVARIABLEfunction}\{3\}\cs{COSfunction}\{\cs{F}\}

\item[Power function:] (exponent enter positiu)\mbox{}

\csdef{POWERfunction}\{\cs{\TIT{function}}\}\{\TIT{n}\}%
   \{\cs{\TIT{newfunction}}\}

Example: defining the $F(t)=t^5$ function:

\cs{POWERfunction}\cs{IDENTITYfunction}\{5\}\{\cs{F}\}

\item[Linear combination:]\mbox{}

\csdef{LINEARCOMBINATIONfunction}\{\TIT{a}\}\{\cs{\TIT{function1}}\}%
  \{\TIT{b}\}\{\cs{\TIT{function2}}\}\{\cs{\TIT{newfunction}}\}

Example: defining the $F(t)=2t-3\cos t$ function:

\cs{LINEARCOMBINATIONfunction}\{2\}\cs{IDENTITYfunction}\{-3\}%
   \cs{COSfunction}\{\cs{F}\}
\end{description}

By combining properly these operations, we can draw graphs of many functions.
Some examples are shown in next pages.
\newpage

First, we will draw the function $f(t)=t^3-2t$,
dividing the interval $[-2,2]$ in ten subintervals.
The simplest way to construct this function is as a linear combination of
$f_1(t)=t^3$ and $f_2(t)=t$.

\begin{exemple}
\LINEARCOMBINATIONfunction
               {1}{\CUBEfunction}
               {-2}{\IDENTITYfunction}
               {\Ffunction}
\begin{center}
\setlength{\unitlength}{1cm}
\begin{Picture}(-2.5,-4.5)(2.5,4.5)
\cartesianaxes(-2,-4)(2,4)
\pictcolor{blue}
\PlotFunction[10]{\Ffunction}{-2}{2}
\Put[rbr](2,4){$f(t)=t^3-2t$}
\end{Picture}
\end{center}
\end{exemple}
\newpage

Graph of $g(t)=t\cos t$. We multiply the identity and the cosine functions:

\begin{Exemple}
\setlength{\unitlength}{0.5cm}
\begin{center}
\begin{Picture}(-11,-11)(11,11)
\cartesianaxes(-10,-10)(10,10)
\PRODUCTfunction{\IDENTITYfunction}{\COSfunction}{\Gfunction}
\pictcolor{red}
\PlotFunction[30]{\Gfunction}{-10}{10}
\end{Picture}
\end{center}
\end{Exemple}
\newpage

Graph of $f(t)=(\cos t)^3$.

\begin{Exemple}
\setlength{\unitlength}{1cm}
\begin{center}
\begin{Picture}(-7,-3)(7,3)
\cartesianaxes(-\numberTWOPI,-2)(\numberTWOPI,2)
\POWERfunction{\COSfunction}{3}{\Ffunction}
\pictcolor{blue}
\PlotFunction[50]{\Ffunction}{-\numberTWOPI}{\numberTWOPI}
\end{Picture}
\end{center}
\end{Exemple}
\newpage

Graph of $g(t)=t\cos t\sin t$.
Note that in this case we have two operations:
First, we define the $f(t)=t \cos t$, multiplying the identity and cosine 
functions; then, we multiply by the sine function.
\begin{Exemple}
\begin{center}
\setlength{\unitlength}{0.75cm}
 \begin{Picture}(-11,-6)(11,6)
\cartesianaxes(-10,-5)(10,5)
\PRODUCTfunction{\IDENTITYfunction}{\COSfunction}{%
                            \Ffunction}
\PRODUCTfunction{\Ffunction}{\SINfunction}{\Gfunction}
\pictcolor{red}
\PlotFunction[40]{\Gfunction}{-10}{10}
\end{Picture}
\end{center}
\end{Exemple}

Graph of $g(t)=\arcsin t$. The \package{calculus} package
not support, for now, the inverse trigonometric functions; but we can plot
these functions (or any other inverse function)
swapping coordinated axes.

\begin{exemple}
\begin{center}
\setlength{\unitlength}{2cm}
\begin{Picture}(-1.5,-2)(1.5,2)
\makenolabels\makenotics
\cartesianaxes
     (-1,-\numberHALFPI)(1,\numberHALFPI)
\printxticslabels{-1}{0.5}{1}
\printyticlabel{-\numberHALFPI}{-\pi/2}
\printyticlabel{-\numberQUARTERPI}{-\pi/4}
\printyticlabel{\numberQUARTERPI}{\pi/4}
\printyticlabel{\numberHALFPI}{\pi/2}
\pictcolor{red}
\symmetrize{\numberQUARTERPI}
\PlotFunction[4]{\SINfunction}
     {-\numberHALFPI}{\numberHALFPI}
\end{Picture}
\end{center}
\end{exemple}
\subsubsection{Polynomial functions}
Although polynomial functions can be easily defined as
linear combinations of power functions,
to facilitate our work, the \package{calculus} package predefines
polynomials of
1, 2, and 3 degrees by these commands:
\cs{newlpoly} (new \emph{linear} polynomial), \cs{newqpoly}
(new \emph{quadratic} polynomial),
and \cs{newcpoly} (new \emph{cubic} polynomial):
\begin{description}
\item[\csdef{newlpoly}\{\cs{\TIT{newfunction}}\}\{\TIT a\}\{\TIT b\}]
stores the
$p(t)=\TTT{\TIT{a}}+\TTT{\TIT{b}}t$
function in the
 \cs{\TIT{newfunction}} command.
\item[\csdef{newqpoly}%
   \{\cs{\TIT{newfunction}}\}\{\TIT a\}\{\TIT b\}\{\TIT c\}]
stores the
$p(t)=\TTT{\TIT{a}}+\TTT{\TIT{b}}t+\TTT{\TIT{c}}t^2$
function in the
\cs{\TIT{newfunction}} command.
\item[\csdef{newcpoly}%
\{\cs{\TIT{newfunction}}\}\{\TIT a\}\{\TIT b\}\{\TIT c\}\{\TIT d\}]
stores the
$p(t)=\TTT{\TIT{a}}+\TTT{\TIT{b}}t+\TTT{\TIT{c}}t^2+\TTT{\TIT{d}}t^3$
function in the
\cs{\TIT{newfunction}} command.
\end{description}
\begin{exemple}
% F(t)=-1+2t
   \newlpoly{\poliF}{-1}{2}
% G(t)=-1+2t+t^2
   \newqpoly{\poliG}{-1}{2}{1}
% H(t)=-1+2t+t^2-0,5t^3
   \newcpoly{\poliH}{-1}{2}{1}{-0.5}

\setlength{\unitlength}{1cm}
\begin{Picture}(-4.5,-5.5)(4.5,5.5)
\cartesianaxes(-4,-5)(4,5)
\pictcolor{blue}
\PlotFunction{\poliF}{-2}{3}
\pictcolor{red}
\PlotFunction{\poliG}{-3.5}{1.5}
\pictcolor{orange}
\PlotFunction[10]{\poliH}{-2}{3.5}
\end{Picture}
\end{exemple}

\subsubsection{Possible errors}
In many cases you get a fairly accurate graph dividing the domain into several 
subintervals.
But an indiscriminate use of this method can produce erroneous results.
For example, if inside a subinterval there is
a discontinuity or a point where the function is not differentiable.
Look at the following example.
\medskip

\begin{exemple}
\SUBTRACTfunction{\SQUAREfunction}{\ONEfunction}
                            {\Ffunction}
\QUOTIENTfunction{\IDENTITYfunction}{\Ffunction}
                            {\Gfunction}

\setlength{\unitlength}{0.5cm}

\begin{Picture}(-8,-6)(8,6)
\def\xunitdivisions{2}
\def\yunitdivisions{2}
\renewcommand{\axeslabelsize}{\scriptsize}
\cartesianaxes(-7,-5)(7,5)
\Put(3,3){%
   $\boxed{\displaystyle g(t)=\frac{t}{t^ 2-1}}$}
\pictcolor{red}
\PlotFunction[10]{\Gfunction}{-7}{7}
\end{Picture}
\end{exemple}

Where is the problem?
Our function is $g(t)=t/(t^2-1)$;
this function has a pair of vertical asymptotes
at $t=\pm1$ (the two zeros of denominator).

We made 10 subdivisions of the $[-7,7]$ interval.
Do, we compute the function in points $-7+(14/10)k=-7+(7/5)k$,
$0\leq k\leq10$, ie,
\[
   -7\quad -\frac{28}{5}\quad -\frac{21}{5}\quad -\frac{14}{5}\quad
    -\frac{7}{5}\quad  0\quad  \frac{7}{5}\quad  \frac{14}{5}\quad
     \frac{21}{5}\quad  \frac{28}{5}\quad 7
\]

Singularities are between $-7/5$ and $0$, and between $0$ and $7/5$,
So, the graph is not correct in these intervals.
\medskip

To avoid this problem, we will
draw the function in three intervals, excluding the points where it is
undefined:
\medskip
\begin{exemple}
\SUBTRACTfunction{\SQUAREfunction}{\ONEfunction}
                            {\Ffunction}
\QUOTIENTfunction{\IDENTITYfunction}{\Ffunction}
                            {\Gfunction}
\renewcommand{\axeslabelsize}{\scriptsize}
\setlength{\unitlength}{0.5cm}
\begin{Picture}(-8,-6)(8,6)
\def\xunitdivisions{2}
\def\yunitdivisions{2}
\cartesianaxes(-7,-5)(7,5)
\pictcolor{red}
\PlotFunction[5]{\Gfunction}{-7}{-1.105}
\PlotFunction[5]{\Gfunction}{-0.905}{0}
\PlotFunction[5]{\Gfunction}{0}{0.905}
\PlotFunction[5]{\Gfunction}{1.105}{7}
\end{Picture}
\end{exemple}

(To determine the ends of the ranges of variation
$\pm1.105$ and $\pm0.905$, we solved the equation
$g(t)=5$, to ensure that asymtotic branches are interrupted
at the border of the drawing area).

\subsubsection{Accurate graphs}
In general, to obtain fairly reliable results we must make
 a careful analysis of the behavior of the function,
determining the points where it is undefined or not differentiable,
 the intervals where it is increasing, its extreme values,
points where graph cuts the coordinate axes and, in general,
 all points where the behavior of
function is significant.
From this information, we can chose the appropriate
drawing intervals.
A careful choice of the partition
subintervals in the domain ensures us
that the graph accurately reflects the behavior of the function.

We will see a couple of examples.
First, we draw the sine function in $[-\pi,\pi]$.
This function ant its derivative have no discontinuities,
but it is convenient to choose a number of partitions
being multiple of $4$, to carefully draw
function at the
$k\pi/2$ points.
In fact, a good choice are 24 subdivisions,
to ensure also the well known values of this function
for angles
multiple of $\pi/6$ and $\pi/4$.
\begin{Exemple}
\setlength{\unitlength}{2cm}%

\highestlabel{\normalfont\normalsize$3\pi/2$}
\begin{center}
\begin{Picture}(-3.5,-1.5)(3.5,1.5)
{\referencesystem(0,0)(\numberHALFPI,0)(0,1)
\makenolabels
\cartesianaxes(-2.2,-1.2)(2.2,1.2)}
\printylabels{-1}{1}{1}
\printxlabel{-\numberPI}{-\pi}
\printxlabel{-\numberHALFPI}{-\pi/2}
\printxlabel{\numberHALFPI}{\pi/2}
\printxlabel{\numberPI}{\pi}
\pictcolor{red}
   \PlotFunction[24]{\SINfunction}{-\numberPI}{\numberPI}
\renewcommand{\axeslabelcolor}{red}
\printxlabel{\numberSIXTHPI}{\pi/6}
\printylabel{0.5}{1/2}
\Polyline(\numberSIXTHPI,0)(\numberSIXTHPI,0.5)(0,0.5)
\end{Picture}
\end{center}
\end{Exemple}

Our second example is more complex. Let's graph the function
\[
f(t)=(t^3/3-t^2/2-2t+3)/3
\]

This function has three roots, at
$t=3/2$ and $t=\pm\sqrt{6}$.
Its derivative, $f'(t)=(t^2-t-2)/3$, equals zero at
$t=-1$ and $t=2$, where the function has, respectively,
a relative maximum and a relative minimum.
 The second derivative, $f''(t)=(2t-1)/3$,
 is zero at $t=1/2$, which is an inflexion point.
Interesting points are, then, the following:
\[
     -\sqrt{6},-1,0,1/2,3/2,2,\sqrt{6}
\]

We will plot this function in the
 $[-3,4]$ interval (because it includes all these points),
 but we divide it as
\[
      [-3,-\sqrt{6}]\cup
      [-\sqrt{6},-1]\cup
      [-1,0]\cup
      [0,1/2]\cup
      [1/2,3/2]\cup
      [3/2,2]\cup
      [2,\sqrt{6}]\cup
      [\sqrt{6},4]
\]
\begin{Exemple}
\SQRT{6}{\SQRTSIX}
\newcpoly{\functionf}{1}{-0.66667}{-0.16667}{0.11111}
\setlength{\unitlength}{2cm}
\begin{center}
 \begin{Picture}(-3.5,-2.5)(4.5,3.5)
\renewcommand{\xunitdivisions}{10}
\renewcommand{\yunitdivisions}{10}
\cartesiangrid(-3,-2)(4,3)
\pictcolor{red}
\PlotFunction{\functionf}{-3}{-\SQRTSIX}
\PlotFunction[4]{\functionf}{-\SQRTSIX}{-1}
\PlotFunction[4]{\functionf}{-1}{0}
\PlotFunction[4]{\functionf}{0}{0.5}
\PlotFunction[4]{\functionf}{0.5}{1.5}
\PlotFunction[4]{\functionf}{1.5}{2}
\PlotFunction[4]{\functionf}{2}{\SQRTSIX}
\PlotFunction{\functionf}{\SQRTSIX}{4}
\functionf{-1}{\tempf}{\tempDf}
\xLINE(-1,0)(-1,\tempf)
\functionf{2}{\tempf}{\tempDf}
\xLINE(2,0)(2,\tempf)
\functionf{0.5}{\tempf}{\tempDf}
\xLINE(0.5,0)(0.5,\tempf)
\end{Picture}
\end{center}
\end{Exemple}

\subsection{Polar coordinates curves}
To draw a curve defined in polar form as $\rho =f(t)$, we must
declare it as a polar curve, using the \csdef{POLARfunction}
declaration: writing
\begin{Verbatim}[commandchars=\|\[\]]
\POLARfunction{\|TIT[functionname]}{\|TIT[polarfunction]}
\end{Verbatim}
we declare the new polar curve \cs{\TIT{polarfunction}}
$\rho=\cs{\TIT{functionname}}(t)$.
For example, the \emph{cardioide} curve, $\rho=1+\cos t$,
can be defined in the following way:
\begin{Verbatim}
\SUMfunction{\ONEfunction}{\COSfunction}{\ffunction} % (y=1 + cos t)
\POLARfunction{\ffunction}{\cardioide}
\end{Verbatim}

Curves defined in such a way can be plotted using the
\csdef{PlotParametricFunction} command,
which syntax is analogous to that of \cs{PlotFunction}.

\begin{exemple}
% Cardioide: r = 1+cos t
\SUMfunction{\ONEfunction}{\COSfunction}
            {\ffunction}
\POLARfunction{\ffunction}{\cardioide}
\begin{center}
\def\runitdivisions{2}
\setlength{\unitlength}{1.5cm}
\begin{Picture}(-2.5,-2.5)(2.5,2.5)
\polargrid{2}{24}
\pictcolor{blue}\linethickness{1pt}
 \PlotParametricFunction[20]{%
         \cardioide}{0}{\numberTWOPI}
\end{Picture}
$\rho=1+\cos\phi$
\end{center}
\end{exemple}

\begin{exemple}
% Eight petal rose: r = cos(4t)
\SCALEVARIABLEfunction{4}{\COSfunction}
                      {\ffunction}
\POLARfunction{\ffunction}{\rose}
\begin{center}
\def\runitdivisions{3}
\MULTIPLY{2}{\numberTWOPI}{\numberFOURPI}
\setlength{\unitlength}{2.5cm}

\begin{Picture}(-1.5,-1.5)(1.5,1.5)
\polargrid{1}{16}
\pictcolor{red}\linethickness{1pt}
\PlotParametricFunction[16]\rose{0}{\numberTWOPI}
\end{Picture}
$\rho=\cos 4\phi$
\end{center}
\end{exemple}

\begin{exemple}
% Archimedean spiral: r=0,5t
\SCALEfunction{0.5}{\IDENTITYfunction}{\ffunction}
\POLARfunction{\ffunction}{\archimedes}

\MULTIPLY{2}{\numberTWOPI}{\numberFOURPI}
\setlength{\unitlength}{0.5cm}
\begin{center}
\begin{Picture}(-7,-7)(7,7)
\pictcolor{red}
\PlotParametricFunction[16]{%
                \archimedes}{0}{\numberFOURPI}
\end{Picture}
$2\rho=\phi$
\end{center}
\end{exemple}

\begin{exemple}
\SCALEVARIABLEfunction{3.2}{\SINfunction}{\ffunction}
\SCALEfunction{0.2}{\ffunction}{\gfunction}
\SUMfunction{\ONEfunction}{\gfunction}{\myfunction}
\POLARfunction{\myfunction}{\Rfunction}
\MULTIPLY{10}{\numberPI}{\numberTENPI}
\setlength{\unitlength}{3cm}
\linethickness{2pt}
\begin{center}
\begin{Picture}(-1.2,-1.2)(1.2,1.2)
\pictcolor{orange}
\PlotParametricFunction[120]\Rfunction{0}{\numberTENPI}
\end{Picture}
$\rho=1+2\sin 3.2\phi$
\end{center}
\end{exemple}
\subsection{Parametrically defined curves}\label{subsec:param}
Polar curves are a particular case of parametrically defined curves,
$x=f(t), y=g(t)$. These curves are declared by the
 \csdef{PARAMETRICfunction} command:
\begin{Verbatim}[commandchars=\|\[\],commentchar=\%]
\PARAMETRICfunction{\|TIT[Xfunction]}{\|TIT[Yfunction]}%
{\|TIT[parametricfunction]}
\end{Verbatim}

Once we have defined it,
to draw this curve, we use the \csdef{PlotParametricFunction} as described 
above.
\begin{Exemple}
\POWERfunction{\IDENTITYfunction}{5}{\xfunction}
\PARAMETRICfunction{\xfunction}{\CUBEfunction}{\myparfunction}
\centering
\setlength{\unitlength}{0.75cm}
\begin{Picture}(-11,-6)(11,6)
\cartesiangrid(-10,-5)(10,5)
\pictcolor{blue}
\PlotParametricFunction[10]{\myparfunction}{-1.5849}{0}
\PlotParametricFunction[10]{\myparfunction}{0}{1.5849}
\Put[E](10,4){$\begin{matrix}x=t^5\\y=t^3\end{matrix}$}
\end{Picture}
\end{Exemple}
\begin{exemple}
% A Lissanjous curve: x=sin 3t, y=sin 4t
\SCALEVARIABLEfunction{3}{\SINfunction}{\ffunction}
\SCALEVARIABLEfunction{4}{\SINfunction}{\gfunction}
\PARAMETRICfunction{\ffunction}{\gfunction}{\myfunction}
\MULTIPLY{10}{\numberPI}{\numberTENPI}
\setlength{\unitlength}{3cm}
\linethickness{2pt}
\begin{center}
\begin{Picture}(-1.2,-1.2)(1.2,1.2)
\pictcolor{red}
\PlotParametricFunction[24]\myfunction{0}{\numberTWOPI}
\end{Picture}

$x=\sin 3t,\ y=\sin 4t$
\end{center}
\end{exemple}

Here, we should also take into account the characteristics of the curve
in order to choose appropriate intervals for
the parameter (typically, the points where the function is not defined,
singularities, cuts with axes,
points where some of the derivatives $x',x'',\ldots$ or $y',y''\ldots$) is 
zero\ldots
In the following example, to represent the curve $x=t^2-1$, $y=t^3-t$,
we see that $x$ or $y$ equals zero when $t$ is
$0$, $1$ or $-1$; the first derivatives $x'=2t$, $y'=3t^2-1$,
in $t=0$ and $t=\pm\sqrt3/3$, and second derivative of $y$ in $t=0$.
Thus, we choose an interval containing these values of $t$, such $[-2.2]$,
and this partition of it:
\[
   [-2,2]=[-2,-1]\cup[-1,-\sqrt3/3]\cup[-\sqrt3/3,0]\cup[0,\sqrt3/3]\cup[\sqrt3/3,1]\cup[1,2]
\]

This same curve was depicted with a single instruction
\cs{PlotParametricFunction} dividing
the interval $[-2.2]$ into five subintervals.
Note that the obtained picture is almost identical, but the fact that
partition not includes zero
conceals the fact that the vertical tangent occurs at the point
 $(-1,0)$.
So, one of the most significant features of the curve is not correctly 
displayed.
\begin{Exemple}
\SUBTRACTfunction{\SQUAREfunction}{\ONEfunction}{\Xpart}
\SUBTRACTfunction{\CUBEfunction}{\IDENTITYfunction}{\Ypart}
\PARAMETRICfunction{\Xpart}{\Ypart}{\myparfunction}
\centering
\setlength{\unitlength}{1cm}
\begin{Picture}(-3.5,-6.5)(3.5,6.5)
\cartesiangrid(-3,-6)(3,6)
\pictcolor{blue}
\PlotParametricFunction\myparfunction{-2}{-1}
\PlotParametricFunction\myparfunction{-1}{-0.57735}
\PlotParametricFunction\myparfunction{-0.57735}{0}
\PlotParametricFunction\myparfunction{0}{0.57735}
\PlotParametricFunction\myparfunction{0.57735}{1}
\PlotParametricFunction\myparfunction{1}{2}
\Put[E](3,6){$\begin{matrix}x=t^2-1\\y=t^3-t\end{matrix}$}
\end{Picture}
\qquad
\begin{Picture}(-3.5,-6.5)(3.5,6.5)
\cartesiangrid(-3,-6)(3,6)
\pictcolor{orange}
\PlotParametricFunction[5]\myparfunction{-2}{2}
\Put[E](3,6){$\begin{matrix}x=t^2-1\\y=t^3-t\end{matrix}$}
\end{Picture}
\end{Exemple}
\subsubsection{The curve of the front page}
To conclude this section we will study in detail the example
of the front page of this manual.
This example shows the power,
while the simplicity of the package \package{xpicture}.

It is the transcendent curve named \emph{butterfly},
\begin{gather*}
        x=\sin t \left(\mathrm e^{\cos t} - 2 \cos 4t 
                       + \sin^5\left(\frac t{12}\right)\right) \\
        y=\cos t \left(\mathrm e^{\cos t} - 2 \cos 4t 
                       + \sin^5\left(\frac t{12}\right)\right)
\end{gather*}

We analyze step by step the code we used:
\begin{itemize}
\item First, we calculated some numbers we'll use later:
\begin{inparaenum}[(a)]
\item $1/12$, that appears in the definition of functions $x$ and $y$;
\item $12\times2\pi$, to plot the curve in $[0,24\pi]$ (twelve laps); and
\item$12\times64$, the number of subdivisions we will use
(64 subintervals for each lap).
\VerbatimInput[numbers=left,firstline=3,lastline=5]{xpicture1.tex}
\end{inparaenum}
\item In the next block we do the important work:
the curve is defined step by step.

\begin{compactitem}
\item Define the function $A(t)=\mathrm e^{\cos t}$
\VerbatimInput[numbers=left,firstline=7,lastline=7]{xpicture1.tex}
\item Define  $B(t)=\cos 4t$
\VerbatimInput[numbers=left,firstline=8,lastline=8]{xpicture1.tex}
\item Define $c(t)=\sin t/12$
\VerbatimInput[numbers=left,firstline=9,lastline=9]{xpicture1.tex}
\item Define $C(t)=\sin^5 t/12$
\VerbatimInput[numbers=left,firstline=10,lastline=10]{xpicture1.tex}
\item Define  $AB(t)=\mathrm e^{\cos t}-2\cos 4t$
\VerbatimInput[numbers=left,firstline=11,lastline=11]{xpicture1.tex}
\item Define $ABC(t)=\mathrm e^{\cos t}-2\cos 4t+\sin^5 t/12$
\VerbatimInput[numbers=left,firstline=12,lastline=12]{xpicture1.tex}
\item Define the $x$ and $y$ functions
\VerbatimInput[numbers=left,firstline=13,lastline=16]{xpicture1.tex}
\item And, finally, we declare the parametric curve:
\VerbatimInput[numbers=left,firstline=17,lastline=17]{xpicture1.tex}
\end{compactitem}

\item Now, the picture composition is trivial
(note the use of constants
\cs{divisions} and \cs{phione} we previously calculated):
\VerbatimInput[numbers=left,firstline=19,lastline=21]{xpicture1.tex}
\end{itemize}

\subsection{Drawing curves from a table of values}
All instructions to draw curves described here use the
\csdef{qCurve} command, which draws quadratic B\'ezier curves:
\begin{Verbatim}[commandchars=\|\[\],commentchar=\%]
\qCurve(|TIT[x0],|TIT[y0])(|TIT[u0],|TIT[v0])(|TIT[x1],|TIT[y1])(|TIT[u1],%
|TIT[v1])
\end{Verbatim}
draw a smooth curve between the points $(\TTT{\TIT{x0}},\TTT{\TIT{y0}})$
and $(\TTT{\TIT{x1}},\TTT{\TIT{y1}})$, with tangent vectors
$(\TTT{\TIT{u0}},\TTT{\TIT{v0}})$ and $(\TTT{\TIT{u1}},\TTT{\TIT{v1}})$,
respectively.
\begin{exemple}
\setlength{\unitlength}{1cm}
\begin{Picture}(-0.5,-0.5)(5.5,5.5)
\cartesianaxes(0,0)(5,5)
\pictcolor{blue}
\qCurve(1,2)(1,2)(4,3)(-1,1)
\pictcolor{gray}
\Put(1,2){\xtrivVECTOR(0,0)(1,2)}
\Put(4,3){\xtrivVECTOR(0,0)(-1,1)}
\Polyline(1,0)(1,2)(0,2)
\Polyline(4,0)(4,3)(0,3)
\end{Picture}
\end{exemple}

The \csdef{PlotQuadraticCurve} command generalizes \cs{qCurve}
to an arbitrary number of points. 
\begin{exemple}
\setlength{\unitlength}{1cm}
\begin{Picture}(-0.5,-0.5)(5.5,3.5)
\cartesianaxes(0,0)(5,3)
\pictcolor{blue}
\PlotQuadraticCurve(0,0)(1,0)%
                   (1,1)(1,2)%
                   (3,2)(-1,1)%
                   (5,2)(0,-1)
\end{Picture}
\end{exemple}
This command supports two alternative syntaxes:
\begin{enumerate}[(a)]
\item
\begin{Verbatim}[commandchars=\|\[\],commentchar=\%]
\PlotQuadraticCurve(|TIT[x0],|TIT[y0])(|TIT[u0],|TIT[v0])(|TIT[x1],|TIT[y1])%
(|TIT[u1],|TIT[v1])...(|TIT[xn],|TIT[yn])(|TIT[un],|TIT[vn])
\end{Verbatim}
draws a curve through the points
$(\TTT{\TIT{x0}},\TTT{\TIT{y0}})$,
 $(\TTT{\TIT{x1}},\TTT{\TIT{y1}})$\ldots
 $(\TTT{\TIT{xn}},\TTT{\TIT{yn}})$
with tangent vectors
$(\TTT{\TIT{u0}},\TTT{\TIT{v0}})$, $(\TTT{\TIT{u1}},\TTT{\TIT{v1}})$\dots
$(\TTT{\TIT{un}},\TTT{\TIT{vn}})$.%
\footnote{This command draws a
quadratic curve between each pair of adjacent points.

The \cs{Curve} command, introduced by the
\package{curve2e} package, does a similar job,
but using cubic approximations, instead of quadratic.}

\begin{exemple}
\setlength{\unitlength}{2cm}
\begin{center}
\begin{Picture}(1,1)(-1,-1)
\pictcolor{red}
\PlotQuadraticCurve(1,0)(1,0)(0,1)(0,1)%
                   (-1,0)(-1,0)(0,-1)(0,-1)%
                   (1,0)(1,0)
\pictcolor{blue}
\referencesystem(0,0)%
                (\numberCOSXLV,\numberCOSXLV)%
                (-\numberCOSXLV,\numberCOSXLV)
\PlotQuadraticCurve(1,0)(1,0)(0,1)(0,1)%
                   (-1,0)(-1,0)(0,-1)(0,-1)%
                   (1,0)(1,0)
\end{Picture}
\end{center}
\end{exemple}
\item
\begin{Verbatim}[commandchars=\|\[\],commentchar=\%]
\PlotQuadraticCurve(|TIT[x0],|TIT[y0]){|TIT[angle0]}(|TIT[x1],|TIT[y1])%
{|TIT[angle1]}...(|TIT[xn],|TIT[yn]){|TIT[anglen]}
\end{Verbatim}
draws a curve through the points
$(\TTT{\TIT{x0}},\TTT{\TIT{y0}})$,
 $(\TTT{\TIT{x1}},\TTT{\TIT{y1}})$\ldots
 $(\TTT{\TIT{xn}},\TTT{\TIT{yn}})$
the inclination angles of which, with respect to the $x$ axis,
are \TTT{\TIT{angle0}}, \TTT{\TIT{angle1}}\dots,
  \TTT{\TIT{angle0}} (always measured in degrees).
\begin{exemple}
\setlength{\unitlength}{2cm}
\begin{center}
\begin{Picture}(1,1)(-1,-1)
\pictcolor{red}
\PlotQuadraticCurve(1,0){0}(0,1){90}
                   (-1,0){180}(0,-1){270}
                   (1,0){360}
\pictcolor{blue}
\referencesystem(0,0)%
                (\numberCOSXLV,\numberCOSXLV)%
                (-\numberCOSXLV,\numberCOSXLV)
\PlotQuadraticCurve(1,0){0}(0,1){90}
                   (-1,0){180}(0,-1){270}
                   (1,0){360}
\end{Picture}
\end{center}
\end{exemple}
\end{enumerate}
With the \cs{PlotQuadraticCurve} command you can approximate any smooth curve
passing through a list of points when you know the tangent vectors.
A particular case, particularly interesting (at least in a calculus course)
is the drawing of the graph a function of real variable knowing a table of 
values of the function and its derivative.
To facilitate this work \package{xpicture}
includes the \csdef{PlotxyDyData} command:
\begin{Verbatim}[commandchars=\|\[\],commentchar=\%]
\PlotxyDyData(|TIT[x0],|TIT[y0],|TIT[Dy0])(|TIT[x1],|TIT[y1],|TIT[Dy1])...%
(|TIT[xn],|TIT[yn],|TIT[Dyn])
\end{Verbatim}
plots the graph of a function $y=f(x)$  passing through points
$(\TTT{\TIT{x0}},\TTT{\TIT{y0}})$,
$(\TTT{\TIT{x1}},\TTT{\TIT{y1}})$\ldots
$(\TTT{\TIT{xn}},\TTT{\TIT{yn}})$
with derivatives $\TTT{\TIT{Dy0}}$, $\TTT{\TIT{Dy1}}$\ldots
$\TTT{\TIT{Dyn}}$.
\begin{exemple}
\setlength{\unitlength}{1cm}
\begin{Picture}(-1,-1)(5.5,5.5)
\cartesianaxes(0,0)(5,5)
\pictcolor{blue}
\PlotxyDyData(0,0,2)(1,1,0)(2,2,3)
                          (3,4,0)(5,1,-2)
\pictcolor{gray}
\Put(0,0){\xtrivVECTOR(0,0)(1,2)}
\Put(1,1){\xtrivVECTOR(0,0)(1,0)}
\Put(2,2){\xtrivVECTOR(0,0)(1,3)}
\Put(3,4){\xtrivVECTOR(0,0)(1,0)}
\Put(5,1){\xtrivVECTOR(0,0)(1,-2)}
\end{Picture}
\end{exemple}
\section{Package options and configuration file}
This package is loaded as usual, using the instruction
\cs{usepackage\{\TIT{list of options}\}\{xpicture\}}.
Then, packages \packagedef{pict2e}, \packagedef{curve2e}, \packagedef{xcolor},
\packagedef{calculator}, and \packagedef{calculus} are automatically loaded.
This package is compatible with any system that supports 
\packagedef{xcolor} and \packagedef{pict2e} packages.

The only specific option for this package is \optiondef{draft},
which disables all the instructions defined in this package, 
replacing each picture set in a \environ{Picture} environment
by a parallelogram circumscribed by a white rectangle (the box that shows
the area reserved for the picture).\footnote{This option is equivalent to
a global use of
the \texttt{\textbackslash draftPictures} declaration.}
This option is very useful throughout the production 
of the document, 
since the composition of the drawings slows considerably
the compilation time. 

All other options are passed directly to packages 
\packagedef{pict2e}, \packagedef{curve2e}, and \packagedef{xcolor}.
The most interesting option (from package \package{pict2e})
is \optiondef{pstarrows}; 
if used, arrowheads in vectors are drawn in PSTricks style (instead of the
standard \LaTeX{} style). 
Do not use the \optiondef{hide} or \optiondef{original}
options (from package \package{pict2e}).

You can include your preferred values for configurable \package{xpicture}
parameters 
(like axes or labels style, radians or degrees measure for angles,
radians or degrees labels in polar grids, et cetera)
using the file \texttt{xpicture.cfg}\ttindex{xpicture.cfg}, because,
if exists, this local configuration file is loaded.
If you want to use it, copy the file 
\texttt{xpicture.cfgxmpl}\ttindex{xpicture.cfgxmpl}
(which is distributed along with package \package{xpicture}),
call your copy as \texttt{xpicture.cfg} and put it in your local
\texttt{texmf} tree.
Initially, this file contains the default values for all parameters, but
you edit it to modify everything agreed.
\section{Compatibility with related packages}
As mentioned earlier, this package loads packages 
\packagedef{pict2e}, \packagedef{curve2e}, \packagedef{xcolor},
\packagedef{calculator}, and \packagedef{calculus}. Every command defined in
these packages works fine within a \environ{Picture} environment. The only
restriction to take in account is that colors must be selected with the
\cs{pictcolor} command, because commands \cs{color} and \cs{textcolor}
may cause the appearance of unwanted spaces. Picture commands defined 
in \packagedef{pict2e} and \packagedef{curve2e} can be freely used 
(had in mind, however, that in this case coordinates 
are interpreted as standard), 
and you can use all the techniques for defining and manipulating colors
from \packagedef{color} and \packagedef{xcolor} packages.

Although guidelines for defining and operating with functions
explained in subsections~\ref{subsec:real}--\ref{subsec:param} 
may be enough to compose a lot of graphics, 
in order to take full advantage of this package you must known
packages \packagedef{calculator} and \packagedef{calculus}
with certain depth. Package \package{calculator} 
will set you free of many tedious calculations.
\medskip

On the other hand, \package{xpicture} is widely compatible with other packages
related to the graphics inclusion, composition or modification.
This fact gives us a lot of flexibility when using them together. 

For example, a picture drawn by \package{xpicture} can include external images 
loaded with packages \packagedef{graphics}/\packagedef{graphicx}, 
and you can also manipulate the whole picture with the aid of these packages.
In a similar way, \texttt{pgf/tikz}\ttindex{pgf}\ttindex{tikz} 
pictures can be included inside a 
\package{xpicture} draw. If you use \LaTeX{} and \TTT{dvips} to compile your 
document, you can combine \package{xpicture} with \packagedef{pstricks}.

\printindex
\end{document}
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