%% $Id: pstricks-add-doc.tex 39 2008-08-27 21:17:28Z herbert $
\documentclass[10pt]{article}
\listfiles
\usepackage{filecontents}
\begin{filecontents*}{demo0.dat}
    0.1414    0.0052
    0.2828    0.0217
    0.4243    0.0480
    0.5657    0.0890
    0.7071    0.1375
    0.8485    0.1906
    0.9899    0.2663
    1.1314    0.3580
    1.2728    0.4644
    1.4142    0.5801
    1.5556    0.7033
    1.6971    0.8899
    1.8385    1.1143
    1.9799    1.2593
    2.1213    1.5692
    2.2627    3.2331
    2.4042    4.4097
    2.5456    5.8186
    2.6870    7.4441
    2.8284    8.2287
\end{filecontents*}

\begin{filecontents*}{demo1.dat}
1 99447169
2 110351058
3 123557238
4 138346129
5 145050826
6 160363212
7 174000394
8 183856559
9 189128691
10 197634845
11 213257357
12 216899512
13 230152738
14 224144907
15 247410024
16 261168438
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18 326153799
19 319442110
20 310351522
21 381919943
22 438043888
23 357527766
24 603304997
\end{filecontents*}

\begin{filecontents*}{demo2.dat}
  1989 3.08
  1990 3.84
  1991 4.08
  1992 3.21
  1993 5.23
  1994 2.97
  1995 2.53
  1996 2.8
  1997 2.78
  1998 3.15
  1999 2.32
  2000 2.4
  2001 2.46
\end{filecontents*}
\begin{filecontents*}{demo3.dat}
  1989 3.08
  1990 3.1
  1991 3.08
  1992 3.21
  1993 5.0
  1994 2.27
  1995 3.53
  1996 3.8
  1997 2.8
  1998 4.15
  1999 3.32
  2000 1.4
  2001 2.46
\end{filecontents*}

\begin{filecontents*}{data3.dat}
 0.000  -2.50   0.005  -1.99  0.010  -0.30
 0.015  4.13  0.020  13.38  0.025  28.98  0.030  50.90  0.035  77.54  0.040  107.38
 0.045  140.64  0.050  178.40  0.055  219.83  0.060  261.04  0.065  296.91  0.070  324.44
 0.075  343.62  0.080  355.50  0.085  361.69  0.090  365.03  0.095  368.62  0.100  373.63
 0.105  378.46  0.110  380.61  0.115  379.02  0.120  373.82  0.125  365.64  0.130  356.65
 0.135  350.11  0.140  346.68  0.145  342.68  0.150  333.46  0.155  318.08  0.160  300.16
 0.165  284.50  0.170  274.03  0.175  269.79  0.180  271.15  0.185  275.29  0.190  278.41
 0.195  278.36  0.200  276.25  0.205  274.57  0.210  273.76  0.215  271.64  0.220  266.67
 0.225  259.84  0.230  253.63  0.235  249.81  0.240  248.42  0.245  248.77  0.250  250.60
 0.255  254.21  0.260  259.40  0.265  264.37  0.270  266.65  0.275  265.49  0.280  262.59
 0.285  260.82  0.290  261.87  0.295  264.55  0.300  265.84  0.305  263.56  0.310  257.47
 0.315  248.74  0.320  238.67  0.325  227.49  0.330  214.46  0.335  198.64  0.340  179.05
 0.345  154.73  0.350  125.23  0.355  91.14  0.360  53.95  0.365  15.45  0.370  -21.37
 0.375  -49.99  0.380  -62.54  0.385  -56.58  0.390  -37.83  0.395  -15.76  0.400  1.86
 0.405  11.54  0.410  13.66  0.415  10.78  0.420  5.87  0.425  1.24  0.430  -1.95
 0.435  -3.54  0.440  -3.98  0.445  -3.80  0.450  -3.43  0.455  -3.16  0.460  -3.09
 0.465  -3.21  0.470  -3.44  0.475  -3.76  0.480  -4.13  0.485  -4.46  0.490  -4.74
 0.495  -4.99  0.500  -5.30  0.505  -5.66  0.510  -6.05  0.515  -6.44  0.520  -6.73
 0.525  -6.81  0.530  -6.67  0.535  -6.38  0.540  -6.08  0.545  -5.83  0.550  -5.66
 0.555  -5.56  0.560  -5.49  0.565  -5.45  0.570  -5.46  0.575  -5.51  0.580  -5.61
 0.585  -5.73  0.590  -5.81  0.595  -5.84  0.600  -5.83  0.605  -5.82  0.610  -5.84
 0.615  -5.87  0.620  -5.91  0.625  -5.98  0.630  -6.07  0.635  -6.17  0.640  -6.30
 0.645  -6.41  0.650  -6.43  0.655  -6.30  0.660  -6.01  0.665  -5.55  0.670  -4.96
 0.675  -4.37  0.680  -3.94  0.685  -3.79  0.690  -3.94  0.695  -4.33  0.700  -4.83
 0.705  -5.31  0.710  -5.65  0.715  -5.85  0.720  -5.95  0.725  -6.02  0.730  -6.10
 0.735  -6.19  0.740  -6.29  0.745  -6.37  0.750  -6.40  0.755  -6.32  0.760  -6.07
 0.765  -5.64  0.770  -5.08  0.775  -4.47  0.780  -3.84  0.785  -3.19  0.790  -2.58
 0.795  -2.11  0.800  -1.84  0.805  -1.78  0.810  -1.90  0.815  -2.15  0.820  -2.45
 0.825  -2.71  0.830  -2.86  0.835  -2.91  0.840  -2.96  0.845  -3.02  0.850  -3.10
 0.855  -3.20  0.860  -3.36  0.865  -3.52  0.870  -3.63  0.875  -3.64  0.880  -3.52
 0.885  -3.30  0.890  -3.04  0.895  -2.80  0.900  -2.59  0.905  -2.42  0.910  -2.27
 0.915  -0.53 0.920  -0.16 0.925  0.08  0.930  0.13  0.935  -0.05  0.940  -0.41
 0.945  -0.90  0.950  -1.44  0.955  -1.99  0.960  -2.46  0.965  -2.79  0.970  -2.98
 0.975  -3.08  0.980  -3.13  0.985  -3.13  0.990  -3.05  0.995  -2.86  1.000  -2.57
\end{filecontents*}

\begin{filecontents*}{data.dat}
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801 164947
802 165162 803 165373 804 165703 805 165906 806 166207 807 166815
808 167412 809 168041 810 168758 811 168994 812 169548 813 170116
814 170134 815 170275 816 170725 817 171110 818 171652 819 172034
820 172524 821 172996 822 173240 823 174005 824 174380 825 175092
826 175597 827 176176 828 176920 829 177290 830 177982 831 178062
832 178824 833 179596 834 179693 835 180165 836 180650 837 181475
838 182033 839 182121 840 182814 841 183388 842 183971 843 184270
844 184786 845 185560 846 186048 847 186777 848 187228 849 187467
850 187788 851 187831 852 187835 853 187979 854 188220 855 188853
856 189642 857 190249 858 190994 859 191590 860 192304 861 192904
862 193122 863 193575 864 194069 865 194874 866 195488 867 196335
868 197090 869 197859 870 198241 871 198948  872 199153 873 199264
874 199531 875 200155 876 200335 877 201046 878 201848 879 201885
880 201979 881 202107 882 202892 883 203113 884 203406 885 203888
886 204508 887 204880 888 205419 889 206285 890 206301 891 206455
892 206902 893 207157 894 207246 895 207758 896 208566 897 208977
898 209635 899 209773 900 210085 901 210951 902 211525 903 211557
904 212143 905 212342 906 212833 907 213309 908 213515 909 214052
910 214773 911 215134 912 215895 913 216268 914 216808 915 217435
916 217828 917 217888 918 218804 919 219013 920 219563 921 219629
922 220022 923 220829 924 221335 925 221385 926 221772 927 222538
928 223369 929 223421 930 224214 931 224950 932 225376 933 225441
934 225647 935 225874 936 226647 937 227340 938 227981 939 228754
940 229465 941 230385 942 230550 943 231021 944 231829 945 232030
946 232657 947 232855 948 233463 949 234013 950 234622 951 235472
952 235775 953 235914 954 236551 955 236951 956 237532 957 237668
958 238018 959 238404 960 239162 961 239195 962 240002 963 240893
964 241209 965 242172 966 242201 967 242263 968 242465 969 243094
970 243710 971 243799 972 244333 973 244861 974 245416 975 246031
976 246139 977 246486 978 247242 979 247666 980 247869 981 248654
982 249249 983 249324 984 249402 985 250116 986 250382 987 250517
988 251198 989 251929 990 252212 991 252213 992 252493 993 253441
994 253948 995 254830 996 254984 997 255322 998 256249 999 256500
\end{filecontents*}

\begin{filecontents*}{dataMul.dat}
[% file dataMul.dat
0    0    3.375    0.0625 10    5.375    7.1875    4.5
20    7.1875    8.375    6.25 30    5.75    7.75    6.6875
40    2.1875    5.75    5.9375 50    -1.9375    2.1875    4.3125
60    -5.125    -1.8125    0.875 70    -6.4375    -5.3125    -2.6875
80    -4.875    -7.1875    -4.875 90    0    -7.625    -5.625
100    5.5    -6.3125    -5.8125 110    6.8125    -2.75    -4.75
120    5.25    2.875    -0.75
]%
\end{filecontents*}

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\define@key[psset]{}{fontscale}{\pst@checknum{#1}\psk@fontscale }
\define@key[psset]{}{decimals}{\pst@getint{#1}\psk@decimals }
\psset{PSfont=Times-Roman,fontscale=10,valuewidth=10,decimals=-1}
\def\psPrintValue{\pst@object{psPrintValue}}
\def\psPrintValue@i#1{%
  \begin@SpecialObj
  \addto@pscode{
     gsave \psk@PSfont findfont \psk@fontscale scalefont setfont
     #1 \psk@decimals -1 gt { 10 \psk@decimals exp dup 3 1 roll mul cvi exch div } if
     \psk@valuewidth string cvs 0 0 moveto show grestore
  }%
  \end@SpecialObj%
}
\makeatother
\usepackage{pstricks-add}
\let\pstricksaddFV\fileversion
\def\PST{\texttt{PSTricks}}
\newcommand*\PostScript{\textsf{PostScript}\xspace}
%
%
\usepackage{longtable}
\usepackage{pifont}
\def\textat{\protect\makeatletter\texttt{@}\protect\makeatother}
\makeatletter
\renewcommand*\l@section{\@dottedtocline{1}{1.5em}{2.5em}}
\renewcommand*\l@subsection{\@dottedtocline{2}{3.8em}{3.2em}}
\renewcommand*\l@subsubsection{\@dottedtocline{3}{7.0em}{4.1em}}
\renewcommand*\l@paragraph{\@dottedtocline{4}{10em}{5em}}
\renewcommand*\l@subparagraph{\@dottedtocline{5}{12em}{6em}}
\makeatother
\let\psEllipticArc\psellipticarc
\let\psEllipticArcN\psellipticarcn
\let\psWedgeEllipse\psellipticwedge
%\parindent=0pt
\newcommand\verbI[1]{{\small\texttt{#1}}}
\newcommand\CMD[1]{{\texttt{\textbackslash#1}}}
\let\param\texttt
%
\newcommand{\pstEllipse}[5][]{%
  \psset{#1}
  \parametricplot{#4}{#5}{#2\space t cos mul #3\space t sin mul}%
}
%
\newcommand{\pstEllipseWedge}[5][]{%
  \psset{#1}
  \pscustom{%
    \parametricplot{#4}{#5}{#2\space t cos mul #3\space t sin mul}%
    \psline(! #2\space #5\space cos mul #3\space #5\space sin mul)%
       (0,0)%
       (! #2\space #4\space cos mul #3\space #4\space sin mul)%
  }%
}
%
\psset{subgriddiv=0,griddots=5,gridlabels=7pt}
%
\DeclareRobustCommand\cs[1]{\texttt{\char`\\#1}}
\def\PS{PostScript}
%
%\renewcommand{\ttdefault}{ul9}% Luxi Mono

\parindent=0pt
\parskip=1ex plus 5pt

\usepackage[colorlinks,linktocpage]{hyperref}
\makeatletter
\def\verbatim@font{\small\normalfont\ttfamily}
\makeatother
\usepackage{showexpl}
\lstset{preset=\RaggedRight}
\lstdefinestyle{syntax}{backgroundcolor=\color{blue!20},numbers=none,xleftmargin=0pt,xrightmargin=0pt,
    frame=single}
\usepackage{amsmath}

\newdimen\fullWidth
\makeatletter
\renewcommand\ON{%
  \gdef\lst@alloverstyle##1{%
    \fboxrule=0pt
    \fboxsep=0pt
    \fcolorbox{DarkBlue}{DarkBlue}{\textcolor{white}{\bfseries\strut##1}}%
}}
\renewcommand\OFF{%
  \xdef\lst@alloverstyle##1{##1}%
}

\makeatother
\lstset{escapechar=§}


\begin{document}
\fullWidth=\linewidth
\advance\fullWidth by \marginparsep
\advance\fullWidth by \marginparwidth

\title{\texttt{pstricks-add}\\additionals Macros for \texttt{pstricks}%
%\thanks{%
%    This document was written with \texttt{Kile: 1.7 (Qt: 3.1.1; KDE: 3.3;}
%    \url{http://sourceforge.net/projects/kile/}) and the PDF output
%    was build with VTeX/Free (\url{http://www.micropress-inc.com/linux})}
\\
    \small v.\pstricksaddFV}
\author{Dominique Rodriguez and Herbert Vo\ss}
\date{\today}

\maketitle

\begin{abstract}
This version of \verb+pstricks-add+ needs \verb+pstricks.tex+
version >1.04 from June 2004, otherwise the additional macros may
not work as expected. The ellipsis material and the option
\verb+asolid+ (renamed to \verb+eofill+) are
\index{fillstyle!eofill@\texttt{eofill}} now part of the new
\verb+pstricks.tex+ package, available at CTAN or at
\url{http://perce.de/LaTeX/}. \verb+pstricks-add+ will for ever be
an experimental and dynamical package, try it at your own risk.

\begin{itemize}
\item It is important to load \verb+pstricks-add+ as the \textbf{last} PSTricks related package, otherwise
a lot of the macros won't work in the expected way.
\item \verb+pstricks-add+ uses the extended version of the keyval package. So be sure that
you have installed \verb+pst-xkey+ which is part of the
\verb+xkeyval+-package, and that all packages that use the old
keyval interface are loaded \textbf{before} the
\verb+xkeyval+.\cite{xkeyval}
\item the option \verb+tickstyle+ from \verb+pst-plot+ is no longer supported; use \verb+ticksize+ instead.
\item the option \verb+xyLabel+ is no longer supported; use the option \verb+labelFontSize+ instead.
\item if \verb+pstricks-add+ is loaded together with the package  \texttt{pst-func} then  \texttt{InsideArrow}
    of the \CMD{psbezier} macro doesn't work!
\end{itemize}

\end{abstract}

\clearpage
\tableofcontents

\clearpage
%--------------------------------------------------------------------------------------
\part{\texttt{pstricks}}
%--------------------------------------------------------------------------------------

%--------------------------------------------------------------------------------------
\section{Numeric functions}
%--------------------------------------------------------------------------------------

All macros contain a \textat{} in their name, because they are
only for internal use, but it is no problem to use them like other
macros. One can define another name without a \textat{}:
\begin{lstlisting}[style=syntax]
\makeatletter
\let\pstdivide\pst@divide
\makeatother
\end{lstlisting}

or put the macro inside the \verb+\makeatletter+ --
\verb+\makeatother+ sequence.

%--------------------------------------------------------------------------------------
\subsection{\CMD{pst\textat{}divide}}
%--------------------------------------------------------------------------------------

\verb+pstricks+ itself has its own divide macro, called
\verb+\pst@divide+, which can divide two lengths and save the
quotient as a floating point number: \index{Division}
%
\begin{lstlisting}[style=syntax]
\pst@divide{<dividend>}{<divisor>}{<result as a macro>}
\end{lstlisting}

\begin{LTXexample}[width=2cm]
\makeatletter
\pst@divide{34pt}{6pt}\quotient \quotient\\
\pst@divide{-6pt}{34pt}\quotient \quotient
\makeatother
\end{LTXexample}

\noindent this gives the output $5.66666$. The result is not a length!

%--------------------------------------------------------------------------------------
\subsection{\CMD{pst\textat{}mod}}
%--------------------------------------------------------------------------------------
\verb+pstricks-add+ defines an additional numeric function for the modulus:
\index{Modulus}

\begin{lstlisting}[style=syntax]
\pst@mod{<integer>}{<integer>}{<result as a macro>}
\end{lstlisting}

\begin{LTXexample}[width=2cm]
\makeatletter
\pst@mod{34}{6}\modulo \modulo\\
\pst@mod{25}{-6}\modulo \modulo
\makeatother
\end{LTXexample}

\noindent this gives the output $4$. Using this internal numeric
function in documents requires a setting inside the
\verb+makeatletter+ and \verb+makeatother+ environment. It makes
some sense to define a new macroname in the preamble and use it
throughout, e.g. \verb+\let\modulo\pst@mod+.

%--------------------------------------------------------------------------------------
\subsection{\CMD{pst\textat{}max}}
%--------------------------------------------------------------------------------------

\begin{lstlisting}[style=syntax]
\pst@max{<integer>}{<integer>}{<result as count register>}
\end{lstlisting}

\begin{LTXexample}[width=2cm]
\newcount\maxNo
\makeatletter
\pst@max{-34}{-6}\maxNo \the\maxNo\\
\pst@max{0}{11}\maxNo \the\maxNo
\makeatother
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{\CMD{pst\textat{}maxdim}}
%--------------------------------------------------------------------------------------

\begin{lstlisting}[style=syntax]
\pst@maxdim{<dimension>}{<dimension>}{<result as dimension register>}
\end{lstlisting}

\begin{LTXexample}[width=2cm]
\newdimen\maxDim
\makeatletter
\pst@maxdim{34cm}{1234pt}\maxDim \the\maxDim\\
\pst@maxdim{34cm}{123pt}\maxDim \the\maxDim
\makeatother
\end{LTXexample}

%--------------------------------------------------------------------------------------
\subsection{\CMD{pst\textat{}mindim}}
%--------------------------------------------------------------------------------------

\begin{lstlisting}[style=syntax]
\pst@mindim{<dimension>}{<dimension>}{<result as dimension register>}
\end{lstlisting}

\begin{LTXexample}[width=2cm]
\newdimen\minDim
\makeatletter
\pst@mindim{34cm}{1234pt}\minDim \the\minDim\\
\pst@mindim{34cm}{123pt}\minDim \the\minDim
\makeatother
\end{LTXexample}

%--------------------------------------------------------------------------------------
\subsection{\CMD{pst\textat{}abs}}
%--------------------------------------------------------------------------------------

\begin{lstlisting}[style=syntax]
\pst@abs{<integer>}{<result as a count register>}
\end{lstlisting}

\begin{LTXexample}[width=2cm]
\newcount\absNo
\makeatletter
\pst@abs{-34}\absNo \the\absNo\\
\pst@abs{4}\absNo \the\absNo
\makeatother
\end{LTXexample}

%--------------------------------------------------------------------------------------
\subsection{\CMD{pst\textat{}absdim}}
%--------------------------------------------------------------------------------------

\begin{lstlisting}[style=syntax]
\pst@absdim{<dimension>}{<result as a dimension register>}
\end{lstlisting}

\begin{LTXexample}[width=2cm]
\newdimen\absDim
\makeatletter
\pst@absdim{-34cm}\absDim \the\absDim\\
\pst@absdim{4sp}\absDim \the\absDim
\makeatother
\end{LTXexample}

\clearpage
%--------------------------------------------------------------------------------------
\section{Dashed Lines}
%--------------------------------------------------------------------------------------
Tobias Nähring has implemented an enhanced feature for dashed
lines. The number of arguments is no longer limited.

\begin{lstlisting}[style=syntax]
dash=value1[unit] value2[unit] ...
\end{lstlisting}

\begin{LTXexample}[width=0.4\linewidth]
\psset{linewidth=2.5pt,unit=0.6}
\begin{pspicture}(-5,-4)(5,4)
 \psgrid[subgriddiv=0,griddots=10,gridlabels=0pt]
  \psset{linestyle=dashed}
  \pscurve[dash=5mm 1mm 1mm 1mm,linewidth=0.1](-5,4)(-4,3)(-3,4)(-2,3)
  \psline[dash=5mm 1mm 1mm 1mm 1mm 1mm 1mm 1mm 1mm 1mm](-5,0.9)(5,0.9)
  \psccurve[linestyle=solid](0,0)(1,0)(1,1)(0,1)
  \psccurve[linestyle=dashed,dash=5mm 2mm 0.1 0.2,linetype=0](0,0)(-2.5,0)(-2.5,-2.5)(0,-2.5)
  \pscurve[dash=3mm 3mm 1mm 1mm,linecolor=red,linewidth=2pt](5,-4)(5,2)(4.5,3.5)(3,4)(-5,4)
\end{pspicture}
\end{LTXexample}

%--------------------------------------------------------------------------------------
\section{\CMD{rmultiput}: a multiple \CMD{rput}}
%--------------------------------------------------------------------------------------
\verb+PSTricks+ already has a \verb+multirput+, which puts a box n
times with a difference of $dx$ and $dy$ relative to each other.
It is not possible to put it with a different distance from one
point to the next. This is possible with \verb+rmultiput+:
\begin{lstlisting}[style=syntax]
\rmultiput[<options>]{<any material>}(x1,y1)(x2,y2) ... (xn,yn)
\rmultiput*[<options>]{<any material>}(x1,y1)(x2,y2) ... (xn,yn)
\end{lstlisting}

\begin{LTXexample}[width=6.2cm]
\psset{unit=0.75}
\begin{pspicture}(-4,-4)(4,4)
\rmultiput[rot=45]{\red\psscalebox{3}{\ding{250}}}%
    (-2,-4)(-2,-3)(-3,-3)(-2,-1)(0,0)(1,2)(1.5,3)(3,3)
\rmultiput[rot=90,ref=lC]{\blue\psscalebox{2}{\ding{253}}}%
    (-2,2.5)(-2,2.5)(-3,2.5)(-2,1)(1,-2)(1.5,-3)(3,-3)
\psgrid[subgriddiv=0,gridcolor=lightgray]
\end{pspicture}
\end{LTXexample}

\clearpage

%--------------------------------------------------------------------------------------
\section{\CMD{psrotate}: Rotating objects}
%--------------------------------------------------------------------------------------
\CMD{rput} also has an optional argument for rotating objects, but
it always depends on the \CMD{rput} coordinates. With
\CMD{psrotate} the rotating center can be placed anywhere. The
rotation is done with \verb+\pscustom+, all optional arguments are
only valid if they are part of the \verb+\pscustom+ macro.
\begin{lstlisting}[style=syntax]
\psrotate[options](x,y){rot angle}{<object>}
\end{lstlisting}

\begin{LTXexample}[width=0.4\linewidth]
\psset{unit=0.75}
\begin{pspicture}(-0.5,-3.5)(8.5,4.5)
  \psaxes{->}(0,0)(-0.5,-3)(8.5,4.5)
  \psdots[linecolor=red,dotscale=1.5](2,1)
  \psarc[linecolor=red,linewidth=0.4pt,showpoints=true]
        {->}(2,1){3}{0}{60}
  \pspolygon[linecolor=green,linewidth=1pt](2,1)(5,1.1)(6,-1)(2,-2)
  \psrotate(2,1){60}{%
    \pspolygon[linecolor=blue,linewidth=1pt](2,1)(5,1.1)(6,-1)(2,-2)}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=6cm]
\def\canne{%  Idea by Manuel Luque
  \psgrid[subgriddiv=0](-1,0)(1,5)
  \pscustom[linewidth=2mm]{\psline(0,4)\psarcn(0.3,4){0.3}{180}{360}}%
  \pscircle*(0.6,4){0.1}\pstriangle*(0,0)(0.2,-0.3)}
\def\Object{}
\begin{pspicture}(-1,-1)(3,6)
  \canne
  \psrotate(0.3,4){45}{\psset{linecolor=red!50}\canne}
  \psrotate(0.3,4){90}{\psset{linecolor=blue!50}\canne}
  \psrotate(0.3,4){360}{\psset{linecolor=cyan!50}\canne}
  \psdot[linecolor=red](0.3,4)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[pos=t]
\def\majorette{\psline[linewidth=0.5mm](0,2)%  Idea by Manuel Luque
               \pscircle[fillstyle=solid]{0.1}
               \pscircle[fillstyle=solid](0,2){0.1}}
\begin{pspicture}(0,-6)(15,5)
  \psaxes[linewidth=0.5pt]{->}(0,0)(0,-5)(15,5)
  \pstVerb{/V0 10 def /Alpha 45 def}% vitesse initiale, angle de lancement
  \multido{\nT=0.0+0.05,\iA=0+40}{41}{%
    \pstVerb{/nT \nT\space def}%
    \rput(!V0 Alpha cos mul nT mul -9.81 2 div nT dup mul mul V0 Alpha sin mul nT mul add){%
       \psrotate(0,1){\iA}{\majorette\psdot[linecolor=red](0,1)\psdot[linecolor=green](0,2)}}}
  \parametricplot[linecolor=red]{0}{2}{% trajectoire du milieu
     V0 Alpha cos mul t mul -9.81 2 div t dup mul mul V0 Alpha sin mul t mul add 1 add}
  \parametricplot[linecolor=green,plotpoints=360]{0}{2}{% d'une extrÃ©mitÃ©
     V0 Alpha cos mul t mul 800 t mul sin sub % x(t)
     -9.81 2 div t dup mul mul V0 Alpha sin mul t mul add 1 add 800 t mul cos add }%y(t)
\end{pspicture}
\end{LTXexample}


\clearpage
%--------------------------------------------------------------------------------------
\section{\CMD{psChart}: a pie chart}
%--------------------------------------------------------------------------------------

\begin{lstlisting}[style=syntax]
\psChart[<options>]{comma separated value list}{comma separated value list}{radius}
\end{lstlisting}

The special optional arguments for the \CMD{psChart} macro are as follows:

\begin{tabularx}{\linewidth}{@{}>{\ttfamily}lX>{\ttfamily}l@{}}
\textrm{\emph{name}} & \textrm{\emph{description}} & \textrm{\emph{default}}\\\hline
chartSep  & distance from the pie chart center to an outraged pie piece & 10pt\\
chartColor & gray or colored pie (values are: \texttt{gray} or \texttt{color})& gray\\
userColor & a comma separated list of user defined colors for the pie & \{\}
\end{tabularx}

\bigskip
The first mandatory argument is the list of the values and may not be empty. The second
one is a list of outraged pieces, numbered consecutively from 1 to up the total number
of values. The list of user defined colors must be enclosed in braces!

The macro \CMD{psChart} defines for every value three nodes at the half angle and
in distances from 0.75, 1, and 1.25 times of the radius from the origin. The nodes
are named as \verb+psChartI?+, \verb+psChart?+, and \verb+psChartO?+, where ? is the number of
the pie. The letter I leads to the inner node and the letter O to the outer node.
The distance can be changed with the optional arguments \verb+chartNodeI+ and
\verb+chartNodeO+ in the usual way with \verb+\psset{chartNodeI=...,chartNodeO=...}+.

The other one is the node on the circle line.
The origin is by default \texttt{(0,0)}. Moving the pie to another position can be done as
usual with the \CMD{rput}-macro. The used colors are named internally as \verb+chartFillColor?+
and can be used by the user for coloring lines or text.


\begin{LTXexample}[width=6cm]
\begin{pspicture}(-3,-3)(3,3)
\psChart{ 23, 29, 3, 26, 28, 14 }{}{2}
\multido{\iA=1+1}{6}{%
  \psdot(psChart\iA)\psdot(psChartI\iA)\psdot(psChartO\iA)%
  \psline[linestyle=dashed,linecolor=white](psChart\iA)
  \psline[linestyle=dashed](psChart\iA)(psChartO\iA)}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6cm]
\begin{pspicture}(-3,-3)(3,3)
\psChart[chartColor=color]{ 45, 90 }{ 1 }{2}
\ncline[linecolor=-chartFillColor1,
  nodesepB=-20pt]{psChartO1}{psChart1}
\rput[l](psChartO1){%
  \textcolor{chartFillColor1}{pie no 1}}
\ncline[linecolor=-chartFillColor2,
  nodesepB=-20pt]{psChartO2}{psChart2}
\rput[lt](psChartO2){%
  \textcolor{chartFillColor2}{pie no 2}}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=7.5cm]
\psframebox[fillcolor=black!20,
  fillstyle=solid]{%
\begin{pspicture}(-3.5,-3.5)(4.25,3.5)
\psChart[chartColor=color]%
  {23, 29, 3, 26, 28, 14, 17, 4, 9}{}{2}
\multido{\iA=1+1}{9}{%
  \ncline[linecolor=-chartFillColor\iA,
    nodesepB=-10pt]{psChartO\iA}{psChart\iA}
  \rput[l](psChartO\iA){%
    \textcolor{chartFillColor\iA}{pie no \iA}}}
\end{pspicture}}
\end{LTXexample}

\begin{LTXexample}[width=6cm]
\begin{pspicture}(-3,-3)(3,3)
\psChart[userColor={red!30,green!30,
    blue!40,gray,magenta!60,cyan}]%
      { 23, 29, 3, 26, 28, 14 }{1,4}{2}
\end{pspicture}
\end{LTXexample}



\begin{LTXexample}[width=6cm]
\begin{pspicture}(-3,-3)(3,3)
\psChart{ 23, 29, 3, 26, 28, 14 }{}{2}
\multido{\iA=1+1}{6}{\rput*(psChartI\iA){\iA}}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t]
\psset{unit=1.5}
\begin{pspicture}(-3,-3)(3,3)
\psChart[userColor={red!30,green!30,blue!40,gray,cyan!50,
    magenta!60,cyan},chartSep=30pt,shadow=true,shadowsize=5pt]{ 34.5,17.2,20.7,15.5,5.2,6.9}{6}{2}
\psset{nodesepA=5pt,nodesepB=-10pt}
\ncline{psChartO1}{psChart1}\nput{0}{psChartO1}{1000 (34.5\%)}
\ncline{psChartO2}{psChart2}\nput{150}{psChartO2}{500 (17.2\%)}
\ncline{psChartO3}{psChart3}\nput{-90}{psChartO3}{600 (20.7\%)}
\ncline{psChartO4}{psChart4}\nput{0}{psChartO4}{450 (15.5\%)}
\ncline{psChartO5}{psChart5}\nput{0}{psChartO5}{150 (5.2\%)}
\ncline{psChartO6}{psChart6}\nput{0}{psChartO6}{200 (6.9\%)}
\bfseries%
\rput(psChartI1){Taxes}\rput(psChartI2){Rent}\rput(psChartI3){Bills}
\rput(psChartI4){Car}\rput(psChartI5){Gas}\rput(psChartI6){Food}
\end{pspicture}
\end{LTXexample}



\clearpage
%--------------------------------------------------------------------------------------
\section{\CMD{psHomothetie}: central dilatation}
%--------------------------------------------------------------------------------------

\begin{lstlisting}[style=syntax]
\psHomothetie[<options>](<center>){<factor>}{<object>}
\end{lstlisting}

\begin{LTXexample}[width=9cm]
\begin{pspicture}[showgrid=true](-5,-4)(4,8)
\psBill% needs package pst-fun
\psHomothetie[linecolor=blue](4,-3){2}{\psBill}
\psdots[dotsize=3pt,linecolor=red](4,-3)
\pstVerb{ /m -3 -0.85 sub 4 0.6 sub div def }
\psplot[linestyle=dashed,linecolor=red]{-5}{4}{ m x mul m 4 mul sub 3 sub }
\psHomothetie[linecolor=green](4,-3){-0.2}{\psBill}
\end{pspicture}
\end{LTXexample}



\clearpage

%--------------------------------------------------------------------------------------
\section{\CMD{psbrace}}
%--------------------------------------------------------------------------------------
\subsection{Syntax}
\begin{lstlisting}[style=syntax]
\psbrace[<options>](<A>)(<B>){<text>}
\psbrace*[<options>](<A>)(<B>){<text>}
\end{lstlisting}


\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,4)
\psgrid[subgriddiv=0,griddots=10]
\pnode(0,0){A}
\pnode(4,4){B}
\psbrace[linecolor=red,ref=lC](A)(B){Text I}
\psbrace*[linecolor=blue,ref=lC](3,4)(0,1){Text II}
\psbrace[fillcolor=white](3,0)(3,4){III}
\end{pspicture}
\end{LTXexample}

\bigskip
The option \verb|\specialCoor| is enabled, so that all types of coordinates
are possible, (nodename), ($x,y$), ($nodeA|nodeB$), \ldots
The star version fills the inner of the brace with the current linecolor.
With the fillcolor \verb+white+ or any other background color the brace can
be "`unfilled"'.
%--------------------------------------------------------------------------------------
\subsection{Options}
%--------------------------------------------------------------------------------------

Additional to all other available options from \verb|pstricks| or the other
related packages,  there are two new option, named  \verb|braceWidth| and
\verb|bracePos|. All important ones are shown in the following graphics
and table.

\begin{center}
\begin{pspicture}[showgrid=true](10,5)
  \psbrace[braceWidth=1cm,braceWidthInner=1cm,
    braceWidthOuter=1cm,bracePos=0.6,fillcolor=white,
    nodesepA=10mm,nodesepB=10mm](0,5)(10,5){\fbox{Label}}
\pcline{<->}(3,3)(3,4)\ncput*{\footnotesize\ttfamily braceWidth}
\pcline{<->}(3,4)(3,5)\ncput*{\footnotesize\ttfamily braceWidthInner}
\pcline{<->}(3,2)(3,3)\ncput*{\footnotesize\ttfamily braceWidthOuter}
\pcline{<->}(6,1)(6,2)\ncput{\footnotesize\ttfamily nodesepB}
\pcline{<->}(6,1)(7,1)\ncput*{\footnotesize\ttfamily A}
\pcline{<->}(0,0.5)(6,0.5)\ncput*{\footnotesize\ttfamily bracePos}
\psdot[dotscale=2](0,5)\uput[0](0,5){\textbf{A}}
\psdot[dotscale=2](10,5)\uput[180](10,5){\textbf{B}}
\end{pspicture}
\end{center}

A positive value for \verb+nodesepA+ and \verb+nodesepB+ shifts the label to the upper right
and a negative value to the lower left. This does not depends on
the value for the rotating of the label!

\begin{center}
\begin{tabular}{l|l}
name & meaning\\\hline
\verb|braceWidth| & default is \verb+2\pslinewidth+\\
\verb|braceWidthInner| & default is \verb+10\pslinewidth+\\
\verb|braceWidthOuter| & default is \verb+10\pslinewidth+\\
\verb|bracePos| & relative position (default is $0.5$)\\
\verb|nodesepA| & x-separation (default is $0pt$)\\
\verb|nodesepB| & y-separation (default is $0pt$)\\
\verb|rot| & additional rotating for the text (default is $0$)\\
\verb|ref| & reference point for the text (default is c)\\
\verb|fillcolor| & default is black
\end{tabular}
\end{center}

By default the text is written perpendicular to the brace line and
can be changed with the \verb|pstricks| option \verb|rot=...|. The
text parameter can take any object and may also be empty. The
reference point can be any value of the combination of \verb|l|
(left) or \verb|r| (right) and \verb|b| (bottom) or \verb|B|
(Baseline) or \verb|C| (center) or \verb|t| (top), where the
default is \verb|c|, the center of the object.

%--------------------------------------------------------------------------------------
%\subsection{Examples}
%--------------------------------------------------------------------------------------

\begin{LTXexample}
\begin{pspicture}(8,2.5)
\psbrace(0,0)(0,2){\fbox{Text}}%
\psbrace[nodesepA=10pt](2,0)(2,2){\fbox{Text}}
\psbrace[ref=lC](4,0)(4,2){\fbox{Text}}
\psbrace[ref=lt,rot=90,nodesepB=-15pt](6,0)(6,2){\fbox{Text}}
\psbrace[ref=lt,rot=90,nodesepA=-5pt,nodesepB=15pt](8,2)(8,0){\fbox{Text}}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}
\def\someMath{$\int\limits_1^{\infty}\frac{1}{x^2}\,dx=1$}
\begin{pspicture}(8,2.5)
\psbrace[ref=lC](0,0)(0,2){\someMath}%
\psbrace[rot=90](2,0)(2,2){\someMath}
\psbrace[ref=lC](4,0)(4,2){\someMath}
\psbrace[ref=lt,rot=90,nodesepB=-30pt](6,0)(6,2){\someMath}
\psbrace[ref=lt,rot=90,nodesepB=30pt](8,2)(8,0){\someMath}
\end{pspicture}
\end{LTXexample}

%$

\begin{LTXexample}
\begin{pspicture}(\linewidth,5)
\psbrace(0,0.5)(\linewidth,0.5){\fbox{Text}}%
\psbrace[bracePos=0.25,nodesepB=10pt,rot=90](0,2)(\linewidth,2){\fbox{Text}}
\psbrace[ref=lC,nodesepA=-3.5cm,nodesepB=15pt,rot=90](0,4)(\linewidth,4){%
   \fbox{some very, very long wonderful Text}}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=8cm]
\psset{unit=0.8}
\begin{pspicture}(10,11)
\psgrid[subgriddiv=0,griddots=10]
\pnode(0,0){A}
\pnode(4,6){B}
\psbrace[ref=lC](A)(B){One}
\psbrace[rot=180,nodesepA=-5pt,ref=rb](B)(A){Two}
\psbrace[linecolor=blue,bracePos=0.25,ref=lB](8,1)(1,7){Three}
\psbrace[braceWidth=-1mm,rot=180,ref=rB](8,1)(1,7){Four}
\psbrace*[linearc=0.5,fillstyle=none,linewidth=1pt,braceWidth=1.5pt,
  bracePos=0.25,ref=lC](8,1)(8,9){A}
\psbrace(4,9)(6,9){}
\psbrace(6,9)(6,7){}
\psbrace(6,7)(4,7){}
\psbrace(4,7)(4,9){}
\psset{linecolor=red}
\psbrace*[ref=lb](7,10)(3,10){I}
\psbrace*[ref=lb,bracePos=0.75](3,10)(3,6){II}
\psbrace*[ref=lb](3,6)(7,6){III}
\psbrace*[ref=lb](7,6)(7,10){IV}
\end{pspicture}
\end{LTXexample}

%$

\begin{LTXexample}[width=5cm]
\[
\begin{pmatrix}
    \Rnode[vref=2ex]{A}{~1} \\
    & \ddots \\
    && \Rnode[href=2]{B}{1} \\
    &&& \Rnode[vref=2ex]{C}{0} \\
    &&&& \ddots \\
    &&&&& \Rnode[href=2]{D}{0}~ \\
\end{pmatrix}
\]
\psbrace[rot=-90,nodesepB=-0.5,nodesepA=-0.2](B)(A){\small n times}
\psbrace[rot=-90,nodesepB=-0.5,nodesepA=-0.2](D)(C){\small n times}
\end{LTXexample}


\clearpage It is also possible to put a vertical brace around a
default paragraph. This works by setting two invisible nodes at
the beginning and the end of the paragraph. Indentation is
possible with a minipage.

Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\noindent\rnode{A}{}

\vspace*{-1ex}
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\vspace*{-2ex}\noindent\rnode{B}{}\psbrace*[linecolor=red](A)(B){}

Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\medskip\hfill\begin{minipage}{0.95\linewidth}
\noindent\rnode{A}{}

\vspace*{-1ex}
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\vspace*{-2ex}
\noindent\rnode{B}{}\psbrace[linecolor=red](A)(B){}
\end{minipage}

\begin{lstlisting}
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\noindent\rnode{A}{}

\vspace*{-1ex}
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\vspace*{-2ex}\noindent\rnode{B}{}\psbrace[linecolor=red](A)(B){}

Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\medskip\hfill\begin{minipage}{0.95\linewidth}
\noindent\rnode{A}{}

\vspace*{-1ex}
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.
Some nonsense text, which is nothing more than nonsense.

\vspace*{-2ex}\noindent\rnode{B}{}\psbrace[linecolor=red](A)(B){}
\end{minipage}
\end{lstlisting}

\clearpage

%--------------------------------------------------------------------------------------
\section{Random dots}
%--------------------------------------------------------------------------------------
The syntax of the new macro \verb+\psRandom+ is:

\begin{lstlisting}[style=syntax]
\psRandom[<option>]{}
\psRandom[<option>]{<clip path>}
\psRandom[<option>](<xMax,yMax>){<clip path>}
\psRandom[<option>](<xMin,yMin>)(<xMax,yMax>){<clip path>}
\end{lstlisting}

If there is no area for the dots defined, then \verb+(0,0)(1,1)+ in the actual
scale is used for placing the dots. This area should be greater than the clipping
path to be sure that the dots are placed over the full area. The clipping path can
be everything. If no clipping path is given, then the frame \verb+(0,0)(1,1)+
in user coordinates is used.  The new options are:

\begin{center}
\begin{tabular}{l|l|l}
name & default\\\hline
\verb|randomPoints| &   \verb|1000| & number of random dots\tabularnewline
\verb|color| & \verb+false+ & random color\tabularnewline
\end{tabular}
\end{center}


\begin{LTXexample}[width=0.3\linewidth]
\psset{unit=5cm}
\begin{pspicture}(1,1)
  \psRandom[dotsize=1pt,fillstyle=solid](1,1){\pscircle(0.5,0.5){0.5}}
\end{pspicture}
\begin{pspicture}(1,1)
  \psRandom[dotsize=2pt,randomPoints=5000,color,%
      fillstyle=solid](1,1){\pscircle(0.5,0.5){0.5}}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=0.4\linewidth]
\psset{unit=5cm}
\begin{pspicture}(1,1)
  \psRandom[randomPoints=200,dotsize=8pt,dotstyle=+]{}
\end{pspicture}
\begin{pspicture}(1.5,1)
  \psRandom[dotsize=5pt,color](0,0)(1.5,0.8){\psellipse(0.75,0.4)(0.75,0.4)}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}
\psset{unit=2.5cm}
\begin{pspicture}(0,-1)(3,1)
  \psRandom[dotsize=4pt,dotstyle=o,linecolor=blue,fillcolor=red,%
     fillstyle=solid,randomPoints=1000]%
      (0,-1)(3,1){\psplot{0}{3.14}{ x 114 mul sin }}
\end{pspicture}
\end{LTXexample}

\psset{unit=1cm}


\clearpage
 %--------------------------------------------------------------------------------------
\section{Dice}
 %--------------------------------------------------------------------------------------
\CMD{psdice} creates the view of a dice. The number on the dice is the only parameter.
The optional parameters, like the color can be used as usual. The macro is a box of
dimension zero and is placed
at the current point. Use  the \CMD{rput} macro to place it anywhere. The optional
argument \texttt{unit} can be used to scale the dice. the default size of
the dice $1\mathrm{cm}\times1\mathrm{cm}$.

\begin{center}
\begin{pspicture}(-1,-1)(8,9)
\multido{\iA=1+1}{6}{%
  \rput(\iA,7.5){\Huge\psdice[unit=0.75,linecolor=red!80]{\iA}}
  \rput(! -0.5 7 \iA\space sub){\Huge\psdice[unit=0.75,linecolor=blue!70]{\iA}}%
  \multido{\iB=1+1}{6}{%
    \rput(! \iA\space 7 \iB\space sub){%
      \rnode[c]{p\iA\iB}{\makebox[1em][l]{\strut\psPrintValue[fontscale=12]{\iA\space \iB\space add}}}%
    }%
  }%
}
%
\ncbox[linearc=0.35,nodesep=0.2,linestyle=dotted]{p11}{p66}
\ncbox[linearc=0.35,nodesep=0.2,linestyle=dashed]{p15}{p51}
\rput{90}(-1.5,3.5){1. dice}
\rput{0}(3.5,8.5){2. dice}
\psline[linewidth=1.5pt](0.25,0.5)(0.25,8)
\psline[linewidth=1.5pt](-1,6.75)(6.5,6.75)
%
\end{pspicture}
\end{center}

\begin{lstlisting}
\begin{pspicture}(-1,-1)(8,8)
\multido{\iA=1+1}{6}{%
  \rput(\iA,7.5){\Huge\psdice[unit=0.75,linecolor=red!80]{\iA}}
  \rput(! -0.5 7 \iA\space sub){\Huge\psdice[unit=0.75,linecolor=blue!70]{\iA}}%
  \multido{\iB=1+1}{6}{%
    \rput(! \iA\space 7 \iB\space sub){%
      \rnode[c]{p\iA\iB}{\makebox[1em][l]{\strut\psPrintValue[fontscale=12]{\iA\space \iB\space add}}}%
    }%
  }%
}
%
\ncbox[linearc=0.35,nodesep=0.2,linestyle=dotted]{p11}{p66}
\ncbox[linearc=0.35,nodesep=0.2,linestyle=dashed]{p15}{p51}
\rput{90}(-1.5,3.5){1. dice}
\rput{0}(3.5,8.5){2. dice}
\psline[linewidth=1.5pt](0.25,0.5)(0.25,8)
\psline[linewidth=1.5pt](-1,6.75)(6.5,6.75)
%
\end{pspicture}
\end{lstlisting}

\clearpage
%--------------------------------------------------------------------------------------
\section{Arrows}
%--------------------------------------------------------------------------------------
\subsection{Definition}
%--------------------------------------------------------------------------------------
\verb|pstricks-add| defines the following ''arrows``:

\begin{center}
  \bgroup
  \def\myline#1{\psline[linecolor=red,linewidth=0.5pt,arrowscale=1.5]{#1}(0,1ex)(1.3,1ex)}%
  \psset{arrowscale=1.5}
  \begin{tabular}{c@{\qquad}p{3cm}l}%
    Value & Example & Name \\[2pt]\hline
    \verb/-/      & \myline{-}      & None\\
    \verb/<->/    & \myline{<->}    & Arrowheads.\\
    \verb/>-</    & \myline{>-<}    & Reverse arrowheads.\\
    \verb/<<->>/  & \myline{<<->>}  & Double arrowheads.\\
    \verb/>>-<</  & \myline{>>-<<}  & Double reverse arrowheads.\\
    \verb/|-|/    & \myline{|-|}    & T-bars, flush to endpoints.\\
    \verb/|*-|*/  & \myline{|*-|*}  & T-bars, centered on endpoints.\\
    \verb/[-]/    & \myline{[-]}    & Square brackets.\\
    \verb/]-[/    & \myline{]-[}    & Reversed square brackets.\\
    \verb/(-)/    & \myline{(-)}    & Rounded brackets.\\
    \verb/)-(/    & \myline{)-(}    & Reversed rounded brackets.\\
    \verb/o-o/    & \myline{o-o}    & Circles, centered on endpoints.\\
    \verb/*-*/    & \myline{*-*}    & Disks, centered on endpoints.\\
    \verb/oo-oo/  & \myline{oo-oo}  & Circles, flush to endpoints.\\
    \verb/**-**/  & \myline{**-**}  & Disks, flush to endpoints.\\
    \verb/|<->|/  & \myline{|<->|}  & T-bars and arrows.\\
    \verb/|>-<|/  & \myline{|>-<|}  & T-bars and reverse arrows.\\
    \verb/h-h|/   & \myline{h-h}    & left/right hook arrows.\\
    \verb/H-H|/   & \myline{H-H}    & left/right hook arrows.\\
    \verb/v-v|/   & \myline{v-v}    & left/right inside vee arrows.\\
    \verb/V-V|/   & \myline{V-V}    & left/right outside vee arrows.\\
    \verb/f-f|/   & \myline{f-f}    & left/right inside filled arrows.\\
    \verb/F-F|/   & \myline{F-F}    & left/right outside filled arrows.\\
    \verb/t-t|/   & \myline{t-t}    & left/right inside slash arrows.\\[5pt]
    \verb/T-T|/   & \myline{T-T}    & left/right outside slash arrows.\\
  \end{tabular}
  \egroup
\end{center}



You can also mix and match, e.g., \verb/->/, \verb/*-)/ and \verb/[->/ are all valid values
of the \verb|arrows| parameter. The parameter can be set with
\begin{lstlisting}[style=syntax]
\psset{arrows=<type>}
\end{lstlisting}

\noindent or for some macros with a special option, like\\[5pt]
\noindent\verb|\psline[<general options>]{<arrow type>}(A)(B)|\\
\noindent\verb/\psline[linecolor=red,linewidth=2pt]{|->}(0,0)(0,2)/ \ \psline[linecolor=red,linewidth=2pt]{|->}(0,0)(0,2)

\subsection{Multiple arrows}
There are two new options which are only valid for the arrow type \verb+<<+ or \verb+>>+.
\verb+nArrow+ sets both, the \verb+nArrowA+ and the  \verb+nArrowB+ parameter. The meaning
is declared in the following tables. Without setting one of these parameters the behaviour
is like the one described in the old PSTricks manual.

\begin{center}
  \begin{tabular}{lc}%
    Value & Meaning \\[2pt]\hline
    \verb+->>+   & \ -A \\
    \verb+<<->>+ & A-A\\
    \verb+<<-+   & A-\ \\
    \verb+>>-+   & B-\ \\
    \verb+-<<+   & \ -B\\
    \verb+>>-<<+ & B-B\\
    \verb+>>->>+ & B-A\\
    \verb+<<-<<+ & A-B
  \end{tabular}
\end{center}




\begin{center}
  \bgroup
  \psset{linecolor=red,linewidth=1pt,arrowscale=2}%
  \begin{tabular}{lp{2.8cm}}%
    Value & Example \\[2pt]\hline
    \verb+\psline{->>}(0,1ex)(2.3,1ex)+  & \psline{->>}(0,1ex)(2.3,1ex) \\
    \verb+\psline[nArrowsA=3]{->>}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=3]{->>}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=5]{->>}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=5]{->>}(0,1ex)(2.3,1ex)\\
    \verb+\psline{<<-}(0,1ex)(2.3,1ex)+  & \psline{<<-}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=3]{<<-}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=3]{<<-}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=5]{<<-}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=5]{<<-}(0,1ex)(2.3,1ex)\\
    \verb+\psline{<<->>}(0,1ex)(2.3,1ex)+  & \psline{<<->>}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=3]{<<->>}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=3]{<<->>}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=5]{<<->>}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=5]{<<->>}(0,1ex)(2.3,1ex)\\
    \verb+\psline{<<-|}(0,1ex)(2.3,1ex)+  & \psline{<<-|}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=3]{<<-<<}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=3]{<<-<<}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=5]{<<-o}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=5]{<<-o}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=3,nArrowsB=4]{<<-<<}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=3,nArrowsB=4]{<<-<<}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=3,nArrowsB=4]{>>->>}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=3,nArrowsB=4]{>>->>}(0,1ex)(2.3,1ex)\\
    \verb+\psline[nArrowsA=1,nArrowsB=4]{>>->>}(0,1ex)(2.3,1ex)+  & \psline[nArrowsA=1,nArrowsB=4]{>>->>}(0,1ex)(2.3,1ex)\\
  \end{tabular}
  \egroup
\end{center}



\subsection{\texttt{hookarrow}}
\begin{LTXexample}
\psset{arrowsize=8pt,arrowlength=1,linewidth=1pt,nodesep=2pt,shortput=tablr}
\large
\begin{psmatrix}[colsep=12mm,rowsep=10mm]
        &   & $R_2$            \\
        &   &   0   &   & $R_3$\\
$e_b:S$ & 1 &       & 1 & 0    \\
        &   &   0              \\
        &   &   $R_1$          \\
\end{psmatrix}
\ncline{h-}{1,3}{2,3}<{$e_{r2}$}>{$f_{r2}$}
\ncline{-h}{2,3}{3,2}<{$e_1$}
\ncline{-h}{3,1}{3,2}^{$e_s$}_{$f_{s}$}
\ncline{-h}{3,2}{4,3}>{$e_3$}<{$f_3$}
\ncline{-h}{4,3}{3,4}>{$e_4$}<{$f_4$}
\ncline{-h}{3,4}{2,3}>{$e_2$}<{$f_2$}
\ncline{-h}{3,4}{3,5}^{$e_5$}
\ncline{-h}{3,5}{2,5}<{$e_{r3}$}>{$f_{r3}$}
\ncline{-h}{4,3}{5,3}<{$e_{r1}$}>{$f_{r1}$}
\end{LTXexample}



\subsection{\texttt{hookrightarrow} and \texttt{hookleftarrow}}
This is another type of arrow and is abbreviated with \verb+H+.
The length and width of the hook is set by the new options
\verb+hooklength+ and \verb+hookwidth+, which are by default set
to
%
\begin{lstlisting}[style=syntax]
\psset{hooklength=3mm,hookwidth=1mm}
\end{lstlisting}
%
If the line begins with a right hook then the line ends with a left hook and vice versa:

\begin{LTXexample}[width=3cm]
\begin{pspicture}(3,4)
\psline[linewidth=5pt,linecolor=blue,hooklength=5mm,hookwidth=-3mm]{H->}(0,3.5)(3,3.5)
\psline[linewidth=5pt,linecolor=red,hooklength=5mm,hookwidth=3mm]{H->}(0,2.5)(3,2.5)
\psline[linewidth=5pt,hooklength=5mm,hookwidth=3mm]{H-H}(0,1.5)(3,1.5)
\psline[linewidth=1pt]{H-H}(0,0.5)(3,0.5)
\end{pspicture}
\end{LTXexample}



\begin{LTXexample}[width=7.25cm]
$\begin{psmatrix}
E&W_i(X)&&Y\\
&&W_j(X)
\psset{arrows=->,nodesep=3pt,linewidth=2pt}
\everypsbox{\scriptstyle}
\ncline[linecolor=red,arrows=H->,%
  hooklength=4mm,hookwidth=2mm]{1,1}{1,2}
\ncline{1,2}{1,4}^{\tilde{t}}
\ncline{1,2}{2,3}<{W_{ij}}
\ncline{2,3}{1,4}>{\tilde{s}}
\end{psmatrix}$
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{\texttt{ArrowInside} Option}
%--------------------------------------------------------------------------------------

It is now possible to have arrows inside lines and not only at the
beginning or the end. The new defined options

\psset{arrowscale=2,linecolor=red,unit=1cm,linewidth=1.5pt}
\begin{longtable}{l|p{9cm}|p{2.2cm}}
Name & Example & Output\\\hline
\endfirsthead
Name & Example & Output\\\hline
\endhead
\texttt{ArrowInside} &
  \texttt{\textbackslash psline[ArrowInside=->](0,0)(2,0)} &
  \psline[ArrowInside=->](0,0.1)(2,0.1) \\
\texttt{ArrowInsidePos} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsidePos=0.25](0,0)(2,0)}
& \psline[ArrowInside=->, ArrowInsidePos=0.25](0,0.1)(2,0.1) \\
\texttt{ArrowInsidePos} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsidePos=10](0,0)(2,0)}
& \psline[ArrowInside=->, ArrowInsidePos=10](0,0.1)(2,0.1) \\
\texttt{ArrowInsideNo} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsideNo=2](0,0)(2,0)}
& \psline[ArrowInside=->, ArrowInsideNo=2](0,0.1)(2,0.1) \\
\texttt{ArrowInsideOffset} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsideNo=2,\%}
  \hspace*{20pt}\texttt{ArrowInsideOffset=0.1](0,0)(2,0)}
& \psline[ArrowInside=->, ArrowInsideNo=2,ArrowInsideOffset=0.1](0,0.1)(2,0.1) \\
%
\texttt{ArrowInside} & \texttt{\textbackslash psline[ArrowInside=->]\{->\}(0,0)(2,0)} &
  \psline[ArrowInside=->]{->}(0,0)(2,0)\\
\texttt{ArrowInsidePos} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsidePos=0.25]\{->\}(0,0)(2,0)}
  & \psline[ArrowInside=->, ArrowInsidePos=0.25]{->}(0,0)(2,0) \\
\texttt{ArrowInsidePos} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsidePos=10]\{->\}(0,0)(2,0)}
  & \psline[ArrowInside=->, ArrowInsidePos=10]{->}(0,0)(2,0) \\
\texttt{ArrowInsideNo} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsideNo=2]\{->\}(0,0)(2,0)}
  & \psline[ArrowInside=->, ArrowInsideNo=2]{->}(0,0)(2,0) \\
\texttt{ArrowInsideOffset} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{ArrowInsideNo=2,\%}
  \hspace*{20pt}\texttt{ArrowInsideOffset=0.1]\{->\}(0,0)(2,0)}
  & \psline[ArrowInside=->, ArrowInsideNo=2,ArrowInsideOffset=0.1]{->}(0,0)(2,0) \\
%
\texttt{ArrowFill} & \texttt{\textbackslash psline[ArrowFill=false,\%}
  \hspace*{20pt}\texttt{arrowinset=0]\{->\}(0,0)(2,0)} &
  \psline[ArrowFill=false,arrowinset=0]{->}(0,0)(2,0)\\
\texttt{ArrowFill} & \texttt{\textbackslash psline[ArrowFill=false,\%}
  \hspace*{20pt}\texttt{arrowinset=0]\{<<->>\}(0,0)(2,0)} &
  \psline[ArrowFill=false,arrowinset=0]{<<->>}(0,0)(2,0)\\
\texttt{ArrowFill} & \texttt{\textbackslash psline[ArrowInside=->,\%}
  \hspace*{20pt}\texttt{arrowinset=0,\%}\hspace{30pt}
  \hspace*{20pt}\texttt{ArrowFill=false,\%}\hspace{30pt}
  \hspace*{20pt}\texttt{ArrowInsideNo=2,\%}
  \hspace*{20pt}\texttt{ArrowInsideOffset=0.1]\{->\}(0,0)(2,0)}
  & \psline[ArrowInside=->, ArrowFill=false,ArrowInsideNo=2,ArrowInsideOffset=0.1]{->}(0,0)(2,0) \\
\end{longtable}

\medskip
Without the default arrow definition there is only the one inside
the line, defined by the type and the position. The position is
relative to the length of the whole line. $0.25$ means at $25\%$
of the line length. The peak of the arrow gets the coordinates
which are calculated by the macro. If you want arrows with an
absolute position difference, then choose a value greater than
\verb|1|, e.g. \verb|10| which places an arrow every 10 pt. The
default unit \verb|pt| cannot be changed.

\noindent
\begin{tabularx}{\linewidth}{@{\color{red}\vrule width 2pt}lX@{}}
& The \verb+ArrowInside+ takes only arrow definitions like \verb+->+ into account.
Arrows from right to left (\verb+<-+) are not possible and ignored. If you need
such arrows, change the order of the pairs of coordinates for the line or curve macro.
\end{tabularx}

%--------------------------------------------------------------------------------------
\subsection{\texttt{ArrowFill} Option}
%--------------------------------------------------------------------------------------

By default all arrows are filled polygons. With the option
\verb|ArrowFill=false| there are ''white`` arrows. Only for the
beginning/end arrows are they empty, the inside arrows are
overpainted by the line.


\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=3}
\psline[linecolor=red,arrowinset=0]{<->}(0,0)(3,0)
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=3}
\psline[linecolor=red,arrowinset=0,ArrowFill=false]{<->}(0,0)(3,0)
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=3}
\psline[linecolor=red,arrowinset=0,arrowsize=0.2,ArrowFill=false]{<->}(0,0)(3,0)
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=3}
\psline[linecolor=blue,arrowscale=6,ArrowFill=true]{>>->>}(0,0)(3,0)
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=3}
\psline[linecolor=blue,arrowscale=6,ArrowFill=false]{>>->>}(0,0)(3,0)
\rule{3cm}{0pt}\\[30pt]
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=3}
\psline[linecolor=blue,arrowscale=6,ArrowFill=true]{>|->|}(0,0)(3,0)
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=3}
\psline[linecolor=blue,arrowscale=6,ArrowFill=false]{>|->|}(0,0)(3,0)%
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{Examples}
%--------------------------------------------------------------------------------------

All examples are printed with \verb|\psset{arrowscale=2,linecolor=red}|.
\subsubsection{\CMD{psline}}

\bigskip
\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=->]{|<->|}(2,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=-|]{|-|}(2,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=->,ArrowInsideNo=2]{->}(2,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=->,ArrowInsideNo=2,ArrowInsideOffset=0.1]{->}(2,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=-*]{->}(0,0)(2,1)(3,0)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=-*,ArrowInsidePos=0.25]{->}(0,0)(2,1)(3,0)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=-*,ArrowInsidePos=0.25,ArrowInsideNo=2]{->}%
   (0,0)(2,1)(3,0)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=->, ArrowInsidePos=0.25]{->}%
        (0,0)(2,1)(3,0)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[linestyle=none,ArrowInside=->,ArrowInsidePos=0.25]{->}%
        (0,0)(2,1)(3,0)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=-<, ArrowInsidePos=0.75]{->}%
     (0,0)(2,1)(3,0)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true,ArrowInside=-*}
\psline(0,0)(2,1)(3,0)(4,0)(6,2)
\psset{linestyle=none}
\psline[ArrowInsidePos=0](0,0)(2,1)(3,0)(4,0)(6,2)
\psline[ArrowInsidePos=1](0,0)(2,1)(3,0)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,5)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=->,ArrowInsidePos=20](0,0)(3,0)%
       (3,3)(1,3)(1,5)(5,5)(5,0)(7,0)(6,3)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,2)
\psset{arrowscale=2,ArrowFill=true}
\psline[ArrowInside=-|]{<->}(0,2)(2,0)(3,2)(4,0)(6,2)
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsubsection{\CMD{pspolygon}}
%--------------------------------------------------------------------------------------
% Polygons (\pspolygon macro)

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,3)
\psset{arrowscale=2}
\pspolygon[ArrowInside=-|](0,0)(3,3)(6,3)(6,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,3)
\psset{arrowscale=2}
\pspolygon[ArrowInside=->,ArrowInsidePos=0.25]%
     (0,0)(3,3)(6,3)(6,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,3)
\psset{arrowscale=2}
\pspolygon[ArrowInside=->,ArrowInsideNo=4]%
       (0,0)(3,3)(6,3)(6,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,3)
\psset{arrowscale=2}
\pspolygon[ArrowInside=->,ArrowInsideNo=4,%
   ArrowInsideOffset=0.1](0,0)(3,3)(6,3)(6,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,3)
\psset{arrowscale=2}
 \pspolygon[ArrowInside=-|](0,0)(3,3)(6,3)(6,1)
 \psset{linestyle=none,ArrowInside=-*}
 \pspolygon[ArrowInsidePos=0](0,0)(3,3)(6,3)(6,1)
 \pspolygon[ArrowInsidePos=1](0,0)(3,3)(6,3)(6,1)
 \psset{ArrowInside=-o}
 \pspolygon[ArrowInsidePos=0.25](0,0)(3,3)(6,3)(6,1)
 \pspolygon[ArrowInsidePos=0.75](0,0)(3,3)(6,3)(6,1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\begin{pspicture}(6,5)
\psset{arrowscale=2}
  \pspolygon[ArrowInside=->,ArrowInsidePos=20]%
    (0,0)(3,0)(3,3)(1,3)(1,5)(5,5)(5,0)(7,0)(6,3)
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsubsection{\CMD{psbezier}}
%--------------------------------------------------------------------------------------
% Bezier curves (\psbezier macro)

\resetOptions
\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(3,3)
\psset{arrowscale=2}
  \psbezier[ArrowInside=-|](1,1)(2,2)(3,3)
  \psset{linestyle=none,ArrowInside=-o}
  \psbezier[ArrowInsidePos=0.25](1,1)(2,2)(3,3)
  \psbezier[ArrowInsidePos=0.75](1,1)(2,2)(3,3)
  \psset{linestyle=none,ArrowInside=-*}
  \psbezier[ArrowInsidePos=0](1,1)(2,2)(3,3)
  \psbezier[ArrowInsidePos=1](1,1)(2,2)(3,3)
\end{pspicture}
\end{LTXexample}

\resetOptions
\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,3)
\psset{arrowscale=2}
  \psbezier[ArrowInside=->,showpoints=true]%
     {*-*}(2,3)(3,0)(4,2)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,3)
\psset{arrowscale=2}
  \psbezier[ArrowInside=->,showpoints=true,%
      ArrowInsideNo=2](2,3)(3,0)(4,2)
\end{pspicture}
\end{LTXexample}

\resetOptions
\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,3)
\psset{arrowscale=2}
  \psbezier[ArrowInside=->,showpoints=true,%
      ArrowInsideNo=2,ArrowInsideOffset=-0.2]{->}(2,3)(3,0)(4,2)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=5.5cm]
\begin{pspicture}(5,3)
\psset{arrowscale=2}
  \psbezier[ArrowInsideNo=9,ArrowInside=-|,%
    showpoints=true]{*-*}(1,3)(3,0)(5,3)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,3)
\psset{arrowscale=2}
  \psset{ArrowInside=-|}
  \psbezier[ArrowInsidePos=0.25,showpoints=true]{*-*}(2,3)(3,0)(4,2)
  \psset{linestyle=none}
  \psbezier[ArrowInsidePos=0.75](2,3)(3,0)(4,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=5.5cm]
\begin{pspicture}(5,6)
\psset{arrowscale=2}
  \pnode(3,4){A}\pnode(5,6){B}\pnode(5,0){C}
  \psbezier[ArrowInside=->,%
     showpoints=true](A)(B)(C)
  \psset{linestyle=none,ArrowInside=-<}
  \psbezier[ArrowInsideNo=4](A)(B)(C)
  \psset{ArrowInside=-o}
  \psbezier[ArrowInsidePos=0.1](A)(B)(C)
  \psbezier[ArrowInsidePos=0.9](A)(B)(C)
  \psset{ArrowInside=-*}
  \psbezier[ArrowInsidePos=0.3](A)(B)(C)
  \psbezier[ArrowInsidePos=0.7](A)(B)(C)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[pos=t]
\begin{pspicture}(-3,-5)(15,5)
  \psbezier[ArrowInsideNo=19,%
      ArrowInside=->,ArrowFill=false,%
      showpoints=true]{->}(-3,0)(5,-5)(8,5)(15,-5)
\end{pspicture}
\end{LTXexample}



%--------------------------------------------------------------------------------------
\subsubsection{\CMD{pcline}}
%--------------------------------------------------------------------------------------
These examples need the package \verb|pst-node|.

% Lines (\pcline macro)
\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,1)
\psset{arrowscale=2}
\pcline[ArrowInside=->](0,0)(2,1)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,1)
\psset{arrowscale=2}
\pcline[ArrowInside=->]{<->}(0,0)(2,1)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,1)
\psset{arrowscale=2}
\pcline[ArrowInside=-|,ArrowInsidePos=0.75]{|-|}(0,0)(2,1)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=2.5cm]
\psset{arrowscale=2}
\pcline[ArrowInside=->,ArrowInsidePos=0.65]{*-*}(0,0)(2,0)
\naput[labelsep=0.3]{\large$g$}
\end{LTXexample}


\begin{LTXexample}[width=2.5cm]
\psset{arrowscale=2}
\pcline[ArrowInside=->,ArrowInsidePos=10]{|-|}(0,0)(2,0)
\naput[labelsep=0.3]{\large$l$}
\end{LTXexample}



%--------------------------------------------------------------------------------------
\subsubsection{\CMD{pccurve}}
%--------------------------------------------------------------------------------------
These examples also need the package \verb|pst-node|.

\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,2)
\psset{arrowscale=2}
\pccurve[ArrowInside=->,ArrowInsidePos=0.65,showpoints=true]{*-*}(0,0)(2,2)
\naput[labelsep=0.3]{\large$h$}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=2.5cm]
\begin{pspicture}(2,2)
\psset{arrowscale=2}
\pccurve[ArrowInside=->,ArrowInsideNo=3,showpoints=true]{|->}(0,0)(2,2)
\naput[labelsep=0.3]{\large$i$}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,4)
\psset{arrowscale=2}
\pccurve[ArrowInside=->,ArrowInsidePos=20]{|-|}(0,0)(4,4)
\naput[labelsep=0.3]{\large$k$}
\end{pspicture}
\end{LTXexample}


\subsection{Special arrows \texttt{v--V},\texttt{t--T}, and \texttt{f--F}}

Possible optional arguments are

\psset{linecolor=black}

\begin{center}
\begin{tabular}{l|l}
name & meaning\\\hline
\verb|veearrowlength| & default is 3mm\\
\verb|veearrowangle| & default is 30\\
\verb|veearrowlinewidth| & default is 0.35mm\\
\verb|filledveearrowlength| & default is 3mm\\
\verb|filledveearrowangle| & default is 15\\
\verb|filledveearrowlinewidth| & default is 0.35mm\\
\verb|tickarrowlength| & default is 1.5mm\\
\verb|tickarrowlinewidth| & default is 0.35mm\\
\end{tabular}
\end{center}


\begin{LTXexample}[width=4cm]
\psset{unit=5mm}
\begin{pspicture}(4,6)
  \psset{dimen=middle,arrows=c-c,
    arrowscale=2,linewidth=.25mm}
  \psline[linecolor=red,linewidth=.05mm](0,0)(0,6)
  \psline[linecolor=red,linewidth=.05mm](4,0)(4,6)
  \psline{v-v}(0,6)(4,6)
  \psline{v-V}(0,4)(4,4)
  \psline{V-v}(0,2)(4,2)
  \psline{V-V}(0,0)(4,0)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=4cm]
\psset{unit=5mm}
\begin{pspicture}(4,6)
  \psset{dimen=middle,arrows=c-c,
    arrowscale=2,linewidth=.25mm}
  \psline[linecolor=red,linewidth=.05mm](0,0)(0,6)
  \psline[linecolor=red,linewidth=.05mm](4,0)(4,6)
  \psline{f-f}(0,6)(4,6)
  \psline{f-F}(0,4)(4,4)
  \psline{F-f}(0,2)(4,2)
  \psline{F-F}(0,0)(4,0)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=4cm]
\psset{unit=5mm}
\begin{pspicture}(4,6)
  \psset{dimen=middle,arrows=c-c,linewidth=.25mm}
  \psline[linecolor=red,linewidth=.05mm](0,0)(0,6)
  \psline[linecolor=red,linewidth=.05mm](4,0)(4,6)
  \psline{t-t}(0,6)(4,6)
  \psline{t-T}(0,4)(4,4)
  \psline{T-t}(0,2)(4,2)
  \psline{T-T}(0,0)(4,0)
\end{pspicture}
\end{LTXexample}

\subsection{Special arrow option \texttt{arrowLW}}

Only for the arrowtype \texttt{o} and \texttt{*} it is possible to
set the arrowlinewidth with the optional keyword \texttt{arrowLW}.
When scaling an arrow by the keyword \verb+\arrowscale+ the width
of the borderline is also scaled. With the optional argument
\texttt{arrowLW} the line width can be set separately and is not
taken into account by the scaling value.

\begin{LTXexample}[width=4cm]
\begin{pspicture}(4,6)
\psline[arrowscale=3,arrows=*-o](0,5)(4,5)
\psline[arrowscale=3,arrows=*-o,
  arrowLW=0.5pt](0,3)(4,3)
\psline[arrowscale=3,arrows=*-o,
  arrowLW=0.3333\pslinewidth](0,1)(4,1)
\end{pspicture}
\end{LTXexample}


\clearpage
%--------------------------------------------------------------------------------------
\section{\CMD{psFormatInt}}
%--------------------------------------------------------------------------------------
There exist some packages and a lot of code to format an integer like $1\,000\,000$
or $1,234,567$ (in Europe $1.234.567$). But all packages expect a real number as
argument and cannot handle macros as an argument. For this case \verb|pstricks-add|
has a macro \verb|psFormatInt| which can handle both:

\begin{LTXexample}[width=3cm]
\psFormatInt{1234567}\\
\psFormatInt[intSeparator={,}]{1234567}\\
\psFormatInt[intSeparator=.]{1234567}\\
\psFormatInt[intSeparator=$\cdot$]{1234567}\\
\def\temp{965432}
\psFormatInt{\temp}
\end{LTXexample}

With the option \verb|intSeparator| the symbol can be changed to any any non-number character.


%--------------------------------------------------------------------------------------
\section{Color}
%--------------------------------------------------------------------------------------
%--------------------------------------------------------------------------------------
\subsection{Transparent colors}
%--------------------------------------------------------------------------------------

Transparency is now part of the main \texttt{pstricks} package.
But pay attention, the names and syntax have changed and you need
to run \verb+ps2pdf+ with the option
\verb+-dCompatibilityLevel=1.4+.


%--------------------------------------------------------------------------------------
\subsection{,,Manipulating transparent colors''}
%--------------------------------------------------------------------------------------

\verb+pstricks-add+ supports real transparency and a simulated one with hatch lines:
\begin{lstlisting}
\def\defineTColor{\@ifnextchar[{\defineTColor@i}{\defineTColor@i[]}}
\def\defineTColor@i[#1]#2#3{%     transparency "Colors"
  \newpsstyle{#2}{%
     fillstyle=vlines,hatchwidth=0.1\pslinewidth,
     hatchsep=1\pslinewidth,hatchcolor=#3,#1%
  }%
}
\defineTColor{TRed}{red}
\defineTColor{TGreen}{green}
\defineTColor{TBlue}{blue}
\end{lstlisting}

There are three predefined "'transparent"` colors \verb+TRed+,
\verb+TGreen+, \verb+TBlue+. They are used as \PST{} styles and
not as colors:

\resetOptions
\bgroup
\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(-3,-5)(5,5)
\psframe(-1,-3)(5,5) % objet de base
\psrotate(2,-2){15}{%
  \psframe[style=TRed](-1,-3)(5,5)}
\psrotate(2,-2){30}{%
  \psframe[style=TGreen](-1,-3)(5,5)}
\psrotate(2,-2){45}{%
  \psframe[style=TBlue](-1,-3)(5,5)}
\psframe[linewidth=3pt](-1,-3)(5,5)
\psdots[dotstyle=+,dotangle=45,dotscale=3](2,-2) % centre de la rotation
\end{pspicture}
\end{LTXexample}
\egroup

%--------------------------------------------------------------------------------------
\subsection{Calculated colors}
%--------------------------------------------------------------------------------------
The \verb+xcolor+ package (version 2.6) has a new feature for defining colors:
\begin{lstlisting}[style=syntax]
  \definecolor[ps]{<name>}{<model>}{< PS code >}
\end{lstlisting}

\verb+model+ can be one of the color models, which PostScript will
understand, e.g. \verb+rgb+. With this definition the color is
calculated on the PostScript side.
\begin{LTXexample}[pos=t,preset=\centering]
\definecolor[ps]{bl}{rgb}{tx@addDict begin  Red Green Blue end}%
\psset{unit=1bp}
\begin{pspicture}(0,-30)(400,100)
\multido{\iLAMBDA=0+1}{400}{%
  \pstVerb{
    \iLAMBDA\space 379 add dup /lambda exch def
    tx@addDict begin  wavelengthToRGB end
  }%
  \psline[linecolor=bl](\iLAMBDA,0)(\iLAMBDA,100)%
}
\psaxes[yAxis=false,Ox=350,dx=50bp,Dx=50]{->}(-29,-10)(420,100)
\uput[-90](420,-10){$\lambda$[\textsf{nm}]}
\end{pspicture}
\end{LTXexample}


\begin{center}
\newcommand{\Touch}{%
\psframe[linestyle=none,fillstyle=solid,fillcolor=bl,dimen=middle](0.1,0.75)}
\definecolor[ps]{bl}{rgb}{tx@addDict begin Red Green Blue end}%
% Echelle 1cm <-> 40 nm
%         1 nm <-> 0.025 cm
\psframebox[fillstyle=solid,fillcolor=black]{%
\begin{pspicture}(-1,-0.5)(12,1.5)
\multido{\iLAMBDA=380+2}{200}{%
  \pstVerb{
    /lambda \iLAMBDA\space def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! lambda 0.025 mul 9.5 sub 0){\Touch}
}
\multido{\n=0+1,\iDiv=380+40}{11}{%
    \psline[linecolor=white](\n,0.1)(\n,-0.1)
    \uput[270](\n,0){\textbf{\white\iDiv}}}
    \psline[linecolor=white]{->}(11,0)
    \uput[270](11,0){\textbf{\white$\lambda$(nm)}}
\end{pspicture}}

\psframebox[fillstyle=solid,fillcolor=black]{%
\begin{pspicture}(-1,-0.5)(12,1)
  \pstVerb{
    /lambda 656 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 656 0.025 mul 9.5 sub 0){\Touch}
  \pstVerb{
    /lambda 486 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 486 0.025 mul 9.5 sub 0){\Touch}
   \pstVerb{
    /lambda 434 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 434 0.025 mul 9.5 sub 0){\Touch}
  \pstVerb{
    /lambda 410 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 410 0.025 mul 9.5 sub 0){\Touch}
\multido{\n=0+1,\iDiv=380+40}{11}{%
    \psline[linecolor=white](\n,0.1)(\n,-0.1)
    \uput[270](\n,0){\textbf{\white\iDiv}}}
    \psline[linecolor=white]{->}(11,0)
    \uput[270](11,0){\textbf{\white$\lambda$(nm)}}
\end{pspicture}}

Spectrum of hydrogen emission (Manuel Luque)
\end{center}

\begin{lstlisting}
\newcommand{\Touch}{%
\psframe[linestyle=none,fillstyle=solid,fillcolor=bl,dimen=middle](0.1,0.75)}
\definecolor[ps]{bl}{rgb}{tx@addDict begin Red Green Blue end}%
% Echelle 1cm <-> 40 nm
%         1 nm <-> 0.025 cm
\psframebox[fillstyle=solid,fillcolor=black]{%
\begin{pspicture}(-1,-0.5)(12,1.5)
\multido{\iLAMBDA=380+2}{200}{%
  \pstVerb{
    /lambda \iLAMBDA\space def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! lambda 0.025 mul 9.5 sub 0){\Touch}
}
\multido{\n=0+1,\iDiv=380+40}{11}{%
    \psline[linecolor=white](\n,0.1)(\n,-0.1)
    \uput[270](\n,0){\textbf{\white\iDiv}}}
    \psline[linecolor=white]{->}(11,0)
    \uput[270](11,0){\textbf{\white$\lambda$(nm)}}
\end{pspicture}}

\psframebox[fillstyle=solid,fillcolor=black]{%
\begin{pspicture}(-1,-0.5)(12,1)
  \pstVerb{
    /lambda 656 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 656 0.025 mul 9.5 sub 0){\Touch}
  \pstVerb{
    /lambda 486 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 486 0.025 mul 9.5 sub 0){\Touch}
   \pstVerb{
    /lambda 434 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 434 0.025 mul 9.5 sub 0){\Touch}
  \pstVerb{
    /lambda 410 def
    lambda
    tx@addDict begin  wavelengthToRGB end
  }%
 \rput(! 410 0.025 mul 9.5 sub 0){\Touch}
\multido{\n=0+1,\iDiv=380+40}{11}{%
    \psline[linecolor=white](\n,0.1)(\n,-0.1)
    \uput[270](\n,0){\textbf{\white\iDiv}}}
    \psline[linecolor=white]{->}(11,0)
    \uput[270](11,0){\textbf{\white$\lambda$(nm)}}
\end{pspicture}}

Spectrum of hydrogen emission (Manuel Luque)
\end{lstlisting}



%--------------------------------------------------------------------------------------
\subsection{Gouraud shading}
%--------------------------------------------------------------------------------------
\begin{quotation}
Gouraud shading is a method used in computer graphics to simulate the differing effects of
light and colour across the surface of an object. In practice, Gouraud shading is used to
achieve smooth lighting on low-polygon surfaces without the heavy computational requirements
of calculating lighting for each pixel. The technique was first presented by Henri Gouraud in 1971.\\
~\hfill{\small \url{http://www.wikipedia.org}}
\end{quotation}

PostScript level 3 supports this kind of shading and it can only
be seen with Acroread 7 or later. The syntax is easy:

\begin{lstlisting}[style=syntax]
  \psGTriangle(x1,y1)(x2,y2)(x3,y3){color1}{color2}{color3}
\end{lstlisting}

\psset{unit=0.75cm}

\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(0,-.25)(10,10)
  \psGTriangle(0,0)(5,10)(10,0){red}{green}{blue}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(0,-.25)(10,10)
  \psGTriangle*(0,0)(9,10)(10,3){black}{white!50}{red!50!green!95}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(0,-.25)(10,10)
  \psGTriangle*(0,0)(5,10)(10,0){-red!100!green!84!blue!86}
                               {-red!80!green!100!blue!40}
                               {-red!60!green!30!blue!100}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t,preset=\centering]
\definecolor{rose}{rgb}{1.00, 0.84, 0.88}
\definecolor{vertpommepasmure}{rgb}{0.80, 1.0, 0.40}
\definecolor{fushia}{rgb}{0.60, 0.30, 1.0}
\begin{pspicture}(0,-.25)(10,10)
  \psGTriangle(0,0)(5,10)(10,0){rose}{vertpommepasmure}{fushia}
\end{pspicture}
\end{LTXexample}

\newpage
%--------------------------------------------------------------------------------------
\part{\texttt{pst-node}}
%--------------------------------------------------------------------------------------

%--------------------------------------------------------------------------------------
\section{Relative nodes with \CMD{psGetNodeCenter}}
%--------------------------------------------------------------------------------------
The command \CMD{psGetNodeCenter}\verb+{node}+ makes sense only at
the PostScript level. It defines the two variables \verb+node.x+
and \verb+node.y+ which can be used to define relative nodes. The
following example defines the node \verb+MyNode+ and a second one
relative to the first one, with 4 units left and 4 units up.
\verb+node+ must be an existing node name.

\begin{LTXexample}[width=5cm]
\begin{pspicture}[showgrid=true,arrowscale=2](5,5)
\pnode(4.5,0.5){MyNode}
\psdot(MyNode)
\pnode(! \psGetNodeCenter{MyNode}
   MyNode.x 4 sub MyNode.y 4 add){MySecondNode}
\psdot(MySecondNode)
\ncline[linecolor=red]{<->}{MyNode}{MySecondNode}
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\section{\CMD{ncdiag} and \CMD{pcdiag}}
%--------------------------------------------------------------------------------------
With the new option \verb|lineAngle| the lines drawn by the \verb|ncdiag| macro
can now have a specified gradient. Without this option one has to define the two
arms (which maybe zero) and PSTricks draws the connection between them. Now there
is only a static \verb|armA|, the second one \verb|armB| is calculated when an angle
\verb|lineAngle| is defined. This angle is the gradient of the intermediate line
between the two arms. The syntax of \verb|ncdiag| is

\begin{lstlisting}[style=syntax]
\ncdiag[<options>]{<Node A>}{<Node B>}
\pcdiag[<options>](<Node A>)(<Node B>)
\end{lstlisting}


\begin{tabularx}{\linewidth}{l|X}
name & meaning\\\hline
\verb|lineAngle| & angle of the intermediate line segment. Default is 0, which is the same than using \verb|ncdiag| without the \verb|lineAngle| option.\tabularnewline
\end{tabularx}


\resetOptions
\begin{LTXexample}[width=5.5cm]
\begin{pspicture}(5,6)
  \circlenode{A}{A}\quad\circlenode{C}{C}%
    \quad\circlenode{E}{E}
  \rput(0,4){\circlenode{B}{B}}
  \rput(1,5){\circlenode{D}{D}}
  \rput(2,6){\circlenode{F}{F}}
  \psset{arrowscale=2,linearc=0.2,%
    linecolor=red,armA=0.5, angleA=90,angleB=-90}
  \ncdiag[lineAngle=20]{->}{A}{B}
  \ncput*[nrot=:U]{line I}
  \ncdiag[lineAngle=20]{->}{C}{D}
  \ncput*[nrot=:U]{line II}
  \ncdiag[lineAngle=20]{->}{E}{F}
  \ncput*[nrot=:U]{line III}
\end{pspicture}
\end{LTXexample}


The \verb|ncdiag| macro sets the \verb|armB| dynamically to the calculated value. Any
user setting of \verb|armB| is overwritten by the macro. The \verb|armA| could be set to
a zero length:


\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,3)
  \rput(0.5,0.5){\circlenode{A}{A}}
  \rput(3.5,3){\circlenode{B}{B}}
  {\psset{linecolor=red,arrows=<-,arrowscale=2}
  \ncdiag[lineAngle=60,%
      armA=0,angleA=0,angleB=180]{A}{B}
  \ncdiag[lineAngle=60,%
      armA=0,angleA=90,angleB=180]{A}{B}}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,3)
  \rput(1,0.5){\circlenode{A}{A}}
  \rput(4,3){\circlenode{B}{B}}
  {\psset{linecolor=red,arrows=<-,arrowscale=2}
  \ncdiag[lineAngle=60,%
      armA=0.5,angleA=0,angleB=180]{A}{B}
  \ncdiag[lineAngle=60,%
      armA=0,angleA=70,angleB=180]{A}{B}
  \ncdiag[lineAngle=60,%
      armA=0.5,angleA=180,angleB=180]{A}{B}}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=4.5cm]
\begin{pspicture}(4,5.5)
  \cnode*(0,0){2pt}{A}%
  \cnode*(0.25,0){2pt}{C}%
  \cnode*(0.5,0){2pt}{E}%
  \cnode*(0.75,0){2pt}{G}%
  \cnode*(2,4){2pt}{B}%
  \cnode*(2.5,4.5){2pt}{D}%
  \cnode*(3,5){2pt}{F}%
  \cnode*(3.5,5.5){2pt}{H}%
  {\psset{arrowscale=2,linearc=0.2,%
    linecolor=red,armA=0.5, angleA=90,angleB=-90}
  \pcdiag[lineAngle=20]{->}(A)(B)
  \pcdiag[lineAngle=20]{->}(C)(D)
  \pcdiag[lineAngle=20]{->}(E)(F)
  \pcdiag[lineAngle=20]{->}(G)(H)}
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\section{\CMD{ncdiagg} and \CMD{pcdiagg}}
%--------------------------------------------------------------------------------------
This is nearly the same as \verb+\ncdiag+ except that
\verb+armB=0+ and the \verb+angleB+ value is computed by the
macro, so that the line ends at the node with an angle like a
\verb+\pcdiagg+ line. The syntax of \verb|ncdiagg|/\verb+pcdiagg+
is

\begin{lstlisting}[style=syntax]
\ncdiag[<options>]{<Node A>}{<Node B>}
\pcdiag[<options>](<Node A>)(<Node B>)
\end{lstlisting}

\begin{LTXexample}[width=5cm]
\begin{pspicture}(4,6)
  \psset{linecolor=black}
  \circlenode{A}{A}%
  \quad\circlenode{C}{C}%
  \quad\circlenode{E}{E}
  \rput(0,4){\circlenode{B}{B}}
  \rput(1,5){\circlenode{D}{D}}
  \rput(2,6){\circlenode{F}{F}}
  {\psset{arrowscale=2,linearc=0.2,linecolor=red,armA=0.5, angleA=90}
  \ncdiagg[lineAngle=-160]{->}{A}{B}
  \ncput*[nrot=:U]{line I}
  \ncdiagg[lineAngle=-160]{->}{C}{D}
  \ncput*[nrot=:U]{line II}
  \ncdiagg[lineAngle=-160]{->}{E}{F}
  \ncput*[nrot=:U]{line III}}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=5cm]
\begin{pspicture}(4,6)
  \psset{linecolor=black}
  \cnode*(0,0){2pt}{A}%
  \cnode*(0.25,0){2pt}{C}%
  \cnode*(0.5,0){2pt}{E}%
  \cnode*(0.75,0){2pt}{G}%
  \cnode*(2,4){2pt}{B}%
  \cnode*(2.5,4.5){2pt}{D}%
  \cnode*(3,5){2pt}{F}%
  \cnode*(3.5,5.5){2pt}{H}%
  {\psset{arrowscale=2,linearc=0.2,linecolor=red,armA=0.5, angleA=90}
  \pcdiagg[lineAngle=20]{->}(A)(B)
  \pcdiagg[lineAngle=20]{->}(C)(D)
  \pcdiagg[lineAngle=20]{->}(E)(F)
  \pcdiagg[lineAngle=20]{->}(G)(H)}
\end{pspicture}
\end{LTXexample}

The only catch for \verb+\ncdiagg+ is that you need the right
value for \verb+lineAngle+. If the node connection is on the wrong
side of the second node, then choose the corresponding angle,
e.g.: if $20$ is wrong then take $-160$, which differs by $180$.


\begin{LTXexample}[width=4cm]
\begin{pspicture}(4,1.5)
  \circlenode{a}{A}
  \rput[l](3,1){\rnode{b}{H}}
  \ncdiagg[lineAngle=60,angleA=180,armA=.5,nodesepA=3pt,linecolor=blue]{b}{a}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=4cm]
\begin{pspicture}(4,1.5)
  \circlenode{a}{A}
  \rput[l](3,1){\rnode{b}{H}}
  \ncdiagg[lineAngle=60,armA=.5,nodesepB=3pt,linecolor=blue]{a}{b}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=4cm]
\begin{pspicture}(4,1.5)
  \circlenode{a}{A}
  \rput[l](3,1){\rnode{b}{H}}
  \ncdiagg[lineAngle=-120,armA=.5,nodesepB=3pt,linecolor=blue]{a}{b}
\end{pspicture}
\end{LTXexample}

%--------------------------------------------------------------------------------------
\section{\CMD{ncbarr}}
%--------------------------------------------------------------------------------------
This has the same behaviour as \verb+ncbar+, but has 5 segments
and all are horizontal ones. This is the reason why \verb+angleA+
must be $0$ or alternatively $180$. All other values are set to
$0$ by the macro. The intermediate horizontal line is symmetrical
to the distance of the two nodes.


\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=2}%
\circlenode{X}{X}\\[1cm]
\circlenode{Y}{Y}
\ncbarr[angleA=0,arrows=->,arrowscale=2]{X}{Y}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=2}%
\ovalnode{X}{Xxxxx}\\[1cm]
\circlenode{Y}{Yyyy}
\ncbarr[angleA=180,arrows=->,arrowscale=2,linecolor=red]{X}{Y}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\psset{arrowscale=2}%
\ovalnode{X}{Xxxxx}\\[1cm]
\circlenode{Y}{Yyyy}
\ncbarr[angleA=20,arm=1cm,arrows=->,arrowscale=2]{X}{Y}
\end{LTXexample}

%--------------------------------------------------------------------------------------
\section{\CMD{psRelNode} and \CMD{psDefPSPNodes}}
%--------------------------------------------------------------------------------------
With these macros it is possible to put a node relative to a given line or given
\verb+pspicture+-environment. In the frist case the parameters are
the angle and the length factor:
\begin{lstlisting}[style=syntax]
\psRelNode(<P0>)(<P1>){<length factor>}{<end node name>}
\psRelLine[<options>](<P0>)(<P1>){<length factor>}{<end node name>}
\end{lstlisting}

The length factor relates to the distance $\overline{P_0P_1}$ and
the end node name must be a valid nodename and shouldn't contain
any of the special PostScript characters. There are two valid
options:

\begin{tabularx}{\linewidth}{l|l|X}
name & default & meaning\\\hline \verb|angle| & $0$ & angle
between the given line $\overline{P_0P_1}$ and the new one
$\overline{P_0P_{endNode}}$\tabularnewline \verb+trueAngle+ &
false & defines whether the angle refers to the seen line or to
the mathematical one, which respect the scaling factors
\verb+xunit+ and \verb+yunit+.
\end{tabularx}

\begin{LTXexample}[width=7cm]
\begin{pspicture}(7,6)
  \psgrid[gridwidth=0pt,gridcolor=gray,gridlabels=0pt,subgriddiv=2]
  \pnode(3,3){A}\pnode(4,2){B}
  \psline[nodesep=-3,linewidth=0.5pt](A)(B)
  \multido{\iA=0+30}{12}{%
    \psRelNode[angle=\iA](A)(B){2}{C}%
    \qdisk(C){2pt}
    \uput[0](C){\iA}}
\end{pspicture}
\end{LTXexample}

In the second case the new macro \verb+\psDefPSPNodes+ defines nine nodes that corresponds to
nine particular points (namely bottom left, bottom center,
bottom right, center left, center center, center right, top left,
top center, top right) of the \verb+pspicture+ box.

\begin{LTXexample}[width=6cm,wide=false]
\begin{pspicture}[showgrid=true](-1,-1)(4,4)
  \psDefPSPNodes
  \psdots(PSPbl)(PSPbc)(PSPbr)
      (PSPcl)(PSPcc)(PSPcr)(PSPtl)(PSPtc)(PSPtr)
  \uput[90](PSPbl){PSPbl} \uput[90](PSPbc){PSPbc}
  \uput[90](PSPbr){PSPbr} \uput[90](PSPcl){PSPcl}
  \uput[90](PSPcc){PSPcc} \uput[90](PSPcr){PSPcr}
  \uput[90](PSPtl){PSPtl} \uput[90](PSPtc){PSPtc}
  \uput[90](PSPtr){PSPtr}
\end{pspicture}
\end{LTXexample}

The name of the nodes are predefined as:

\begin{lstlisting}[style=syntax]
\psset[pst-PSPNodes]{blName=PSPbl,bcName=PSPbc,brName=PSPbr,
  clName=PSPcl,ccName=PSPcc,crName=PSPcr,tlName=PSPtl,tcName=PSPtc,trName=PSPtr}
\end{lstlisting}

and can be modified in the same way.
%I guess you modified the family to have the pstricks-add one so the
%\xkvview would have to be adapted.




%--------------------------------------------------------------------------------------
\section{\CMD{psRelLine}}
%--------------------------------------------------------------------------------------
With this macro it is possible to plot lines relative to a given one. Parameter are
the angle and the length factor:

\begin{lstlisting}[style=syntax]
\psRelLine(<P0>)(<P1>){<length factor>}{<end node name>}
\psRelLine{<arrows>}(<P0>)(<P1>){<length factor>}{<end node name>}
\psRelLine[<options>](<P0>)(<P1>){<length factor>}{<end node name>}
\psRelLine[<options>]{<arrows>}(<P0>)(<P1>){<length factor>}{<end node name>}
\end{lstlisting}

The length factor relates to the distance $\overline{P_0P_1}$ and
the end node name must be a valid nodename and shouldn't contain
any of the special PostScript characters. There are two valid
options which are described in the foregoing section for
\verb+\psRelNode+.

The following two figures show the same, the first one with a scaling different to $1:1$,
this is the reason why the end points are on an ellipse and not on a circle like in the
second figure.

\begin{LTXexample}[width=5cm]
\psset{yunit=2,xunit=1}
\begin{pspicture}(-2,-2)(3,2)
\psgrid[subgriddiv=2,subgriddots=10,gridcolor=lightgray]
\pnode(-1,0){A}\pnode(3,2){B}
\psline[linecolor=red](A)(B)
\psRelLine[linecolor=blue,angle=30](-1,0)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=blue,angle=-30](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=magenta,angle=90](-1,0)(3,2){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=magenta,angle=-90](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=5cm]
\begin{pspicture}(-2,-2)(3,2)
\psgrid[subgriddiv=2,subgriddots=10,gridcolor=lightgray]
\pnode(-1,0){A}\pnode(3,2){B}
\psline[linecolor=red](A)(B)
\psarc[linestyle=dashed](A){2.23}{-90}{135}
\psRelLine[linecolor=blue,angle=30](-1,0)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=blue,angle=-30](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=magenta,angle=90](-1,0)(3,2){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=magenta,angle=-90](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\end{pspicture}
\end{LTXexample}

\medskip
The following figure has also a different scaling, but has set the
option \verb+trueAngle+, all angles refer to "what you see".

\begin{LTXexample}[width=6.5cm]
\psset{yunit=2,xunit=1}
\begin{pspicture}(-3,-1)(3,2)\psgrid[subgridcolor=lightgray]
\pnode(-1,0){A}\pnode(3,2){B}
\psline[linecolor=red](A)(B)
\psarc(A){2.83}{-45}{135}
\psRelLine[linecolor=blue,angle=30,trueAngle](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=blue,angle=-30,trueAngle](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=magenta,angle=90,trueAngle](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\psRelLine[linecolor=magenta,angle=-90,trueAngle](A)(B){0.5}{EndNode}
\qdisk(EndNode){2pt}
\end{pspicture}
\end{LTXexample}

\medskip
Two examples using \verb+\multido+ to show the behaviour of the
options \verb+trueAngle+ and \verb+angle+.

\medskip
\begin{LTXexample}[width=8cm]
\psset{yunit=4,xunit=2}
\begin{pspicture}(-1,0)(3,2)\psgrid[subgridcolor=lightgray]
\pnode(-1,0){A}\pnode(1,1){B}
\psline[linecolor=red](A)(3,2)
\multido{\iA=0+10}{36}{%
  \psRelLine[linecolor=blue,angle=\iA](B)(A){-0.5}{EndNode}
  \qdisk(EndNode){2pt}
}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=8cm]
\psset{yunit=4,xunit=2}
\begin{pspicture}(-1,0)(3,2)\psgrid[subgridcolor=lightgray]
\pnode(-1,0){A}\pnode(1,1){B}
\psline[linecolor=red](A)(3,2)
\multido{\iA=0+10}{36}{%
  \psRelLine[linecolor=magenta,angle=\iA,trueAngle]{->}(B)(A){-0.5}{EndNode}
}
\end{pspicture}
\end{LTXexample}

\begin{center}
\bgroup
\psset{xunit=0.75\linewidth,yunit=0.75\linewidth,trueAngle}%
\begin{pspicture}(1,0.6)%\psgrid
  \pnode(.3,.35){Vk} \pnode(.375,.35){D} \pnode(0,.4){DST1} \pnode(1,.18){DST2}
  \pnode(0,.1){A1}   \pnode(1,.31){A1}
  { \psset{linewidth=.02,linestyle=dashed,linecolor=gray}%
    \pcline(DST1)(DST2) % <- Druckseitentangente
    \pcline(A2)(A1) % <- Anstr"omrichtung
    \lput*{:U}{\small Anstr"omrichtung $v_{\infty}$} }%
  \psIntersectionPoint(A1)(A2)(DST1)(DST2){Hk}
  \pscurve(Hk)(.4,.38)(Vk)(.36,.33)(.5,.32)(Hk)
  \psParallelLine[linecolor=red!75!green,arrows=->,arrowscale=2](Vk)(Hk)(D){.1}{FtE}
  \psRelLine[linecolor=red!75!green,arrows=->,arrowscale=2,angle=90](D)(FtE){4}{Fn}% why "4"?
  \psParallelLine[linestyle=dashed](D)(FtE)(Fn){.1}{Fnr1}
  \psRelLine[linestyle=dashed,angle=90](FtE)(D){-4}{Fnr2} % why "-4"?
  \psline[linewidth=1.5pt,arrows=->,arrowscale=2](D)(Fnr2)
  \psIntersectionPoint(D)([nodesep=2]D)(Fnr1)([offset=-4]Fnr1){Fh}
  \psIntersectionPoint(D)([offset=2]D)(Fnr1)([nodesep=4]Fnr1){Fv}
  \psline[linecolor=blue,arrows=->,arrowscale=2](D)(Fh)
  \psline[linecolor=blue,arrows=->,arrowscale=2](D)(Fv)
  \psline[linestyle=dotted](Fh)(Fnr1)  \psline[linestyle=dotted](Fv)(Fnr1)
  \uput{.1}[0](Fh){\blue $F_{H}$}   \uput{.1}[180](Fv){\blue $F_{V}$}
  \uput{.1}[-45](Fnr1){$F_{R}$}     \uput{.1}[90](Fn){\color{red!75!green}$F_{N}$}
  \uput{.25}[-90](FtE){\color{red!75!green}$F_{T}$}
\end{pspicture}
\egroup
\end{center}
\begin{lstlisting}
\psset{xunit=0.75\linewidth,yunit=0.75\linewidth,trueAngle}%
\end{center}
\begin{pspicture}(1,0.6)%\psgrid
  \pnode(.3,.35){Vk} \pnode(.375,.35){D} \pnode(0,.4){DST1} \pnode(1,.18){DST2}
  \pnode(0,.1){A1}   \pnode(1,.31){A1}
  { \psset{linewidth=.02,linestyle=dashed,linecolor=gray}%
    \pcline(DST1)(DST2) % <- Druckseitentangente
    \pcline(A2)(A1) % <- Anstr"omrichtung
    \lput*{:U}{\small Anstr"omrichtung $v_{\infty}$} }%
  \psIntersectionPoint(A1)(A2)(DST1)(DST2){Hk}
  \pscurve(Hk)(.4,.38)(Vk)(.36,.33)(.5,.32)(Hk)
  \psParallelLine[linecolor=red!75!green,arrows=->,arrowscale=2](Vk)(Hk)(D){.1}{FtE}
  \psRelLine[linecolor=red!75!green,arrows=->,arrowscale=2,angle=90](D)(FtE){4}{Fn}% why "4"?
  \psParallelLine[linestyle=dashed](D)(FtE)(Fn){.1}{Fnr1}
  \psRelLine[linestyle=dashed,angle=90](FtE)(D){-4}{Fnr2} % why "-4"?
  \psline[linewidth=1.5pt,arrows=->,arrowscale=2](D)(Fnr2)
  \psIntersectionPoint(D)([nodesep=2]D)(Fnr1)([offset=-4]Fnr1){Fh}
  \psIntersectionPoint(D)([offset=2]D)(Fnr1)([nodesep=4]Fnr1){Fv}
  \psline[linecolor=blue,arrows=->,arrowscale=2](D)(Fh)
  \psline[linecolor=blue,arrows=->,arrowscale=2](D)(Fv)
  \psline[linestyle=dotted](Fh)(Fnr1)  \psline[linestyle=dotted](Fv)(Fnr1)
  \uput{.1}[0](Fh){\blue $F_{H}$}   \uput{.1}[180](Fv){\blue $F_{V}$}
  \uput{.1}[-45](Fnr1){$F_{R}$}     \uput{.1}[90](Fn){\color{red!75!green}$F_{N}$}
  \uput{.25}[-90](FtE){\color{red!75!green}$F_{T}$}
\end{pspicture}
\end{lstlisting}


%--------------------------------------------------------------------------------------
\section{\CMD{psParallelLine}}
%--------------------------------------------------------------------------------------
With this macro it is possible to plot lines relative to a given one, which is parallel.
There is no special parameter here.

\begin{lstlisting}[style=syntax]
\psParallelLine(<P0>)(<P1>)(<P2>){<length>}{<end node name>}
\psParallelLine{<arrows>}(<P0>)(<P1>)(<P2>){<length>}{<end node name>}
\psParallelLine[<options>](<P0>)(<P1>)(<P2>){<length>}{<end node name>}
\psParallelLine[<options>]{<arrows>}(<P0>)(<P1>)(<P2>){<length>}{<end node name>}
\end{lstlisting}

The line starts at $P_2$, is parallel to $\overline{P_0P_1}$ and
the length of this parallel line depends on the length factor. The
end node name must be a valid nodename and shouldn't contain any
of the special PostScript characters.

\begin{LTXexample}
\begin{pspicture*}(-5,-4)(5,3.5)
  \psgrid[subgriddiv=0,griddots=5]
  \pnode(2,-2){FF}\qdisk(FF){1.5pt}
  \pnode(-5,5){A}\pnode(0,0){O}
  \multido{\nCountA=-2.4+0.4}{9}{%
    \psParallelLine[linecolor=red](O)(A)(0,\nCountA){9}{P1}
    \psline[linecolor=red](0,\nCountA)(FF)
    \psRelLine[linecolor=red](0,\nCountA)(FF){9}{P2}
  }
  \psline[linecolor=blue](A)(FF)
  \psRelLine[linecolor=blue](A)(FF){5}{END1}
  \psline[linewidth=2pt,arrows=->](2,0)(FF)
\end{pspicture*}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\section{\CMD{psIntersectionPoint}}
%--------------------------------------------------------------------------------------
This macro calculates the intersection point of two lines, given by the four coordinates.
There is no special parameter here.
\begin{lstlisting}[style=syntax]
\psIntersectionPoint(<P0>)(<P1>)(<P2>)(<P3>){<node name>}
\end{lstlisting}

\begin{LTXexample}[width=5.5cm]
\psset{unit=0.5cm}
\begin{pspicture}(-5,-4)(5,5)
  \psaxes[labelFontSize=\scriptstyle,
    dx=2,Dx=2,dy=2,Dy=2]{->}(0,0)(-5,-4)(5,5)
  \psline[linecolor=red,linewidth=2pt](-5,-1)(5,5)
  \psline[linecolor=blue,linewidth=2pt](-5,3)(5,-4)
  \qdisk(-5,-1){2pt}\uput[-90](-5,-1){A}
  \qdisk(5,5){2pt}\uput[-90](5,5){B}
  \qdisk(-5,3){2pt}\uput[-90](-5,3){C}
  \qdisk(5,-4){2pt}\uput[-90](5,-4){D}
  \psIntersectionPoint(-5,-1)(5,5)(-5,3)(5,-4){IP}
  \qdisk(IP){3pt}\uput{0.3}[90](IP){IP}
  \psline[linestyle=dashed](IP|0,0)(IP)(0,0|IP)
\end{pspicture}
\end{LTXexample}

\clearpage
%--------------------------------------------------------------------------------------
\section{\CMD{psLNode} and \CMD{psLCNode}}
%--------------------------------------------------------------------------------------
\CMD{psLNode} interpolates the Line $\overline{AB}$ by the given value and sets a node at this
point. The syntax is
%
\begin{lstlisting}[style=syntax]
\psLNode(P1)(P2){value}{Node name}
\end{lstlisting}

\begin{LTXexample}[width=5cm]
\begin{pspicture}(5,5)
\psgrid[subgriddiv=0,griddots=10]
\psset{linecolor=red}
\psline{o-o}(1,1)(5,5)
\psLNode(1,1)(5,5){0.75}{PI}
\qdisk(PI){4pt}
\psset{linecolor=blue}
\psline{o-o}(4,3)(2,5)
\psLNode(4,3)(2,5){-0.5}{PII}
\qdisk(PII){4pt}
\end{pspicture}
\end{LTXexample}


\bigskip
The \CMD{psLCNode} macro builds the linear combination of the two given
vectors and stores the end of
the new vector as a node. All vectors start at $(0,0)$, so a \verb+\rput+ maybe appropriate.
The syntax is
%
\begin{lstlisting}[style=syntax]
\psLCNode(P1){value 1}(P2){value 2}{Node name}
\end{lstlisting}

\begin{LTXexample}[width=5cm]
\begin{pspicture}(5,5)
\psgrid[subgriddiv=0,griddots=10]
\psset{linecolor=black}
\psline[linestyle=dashed]{->}(3,1.5)
\psline[linestyle=dashed]{->}(0.375,1.5)
\psset{linecolor=red}
\psline{->}(2,1)\psline{->}(0.5,2)
\psLCNode(2,1){1.5}(0.5,2){0.75}{PI}
\psline[linewidth=2pt]{->}(PI)
\psset{linecolor=black}
\psline[linestyle=dashed](3,1.5)(PI)
\psline[linestyle=dashed](0.375,1.5)(PI)
\end{pspicture}
\end{LTXexample}

\clearpage

%--------------------------------------------------------------------------------------
\section{\CMD{nlput} and \CMD{psLDNode}}
%--------------------------------------------------------------------------------------
\CMD{ncput} allows you to set a label relative to the first node
of the last node connection. With \CMD{nlput} this can be done
absolute to a given node. The syntax is different to the other
node connection macros. It uses internally the macro
\CMD{psLDNode} which places a node absolute to two given points,
starting from the first one.

\begin{lstlisting}[style=syntax]
\nlput[options](A)(B){distance}{text}
\psLDNode[options](A)(B){distance}{node name}
\end{lstlisting}


\begin{LTXexample}[width=5cm]
\begin{pspicture}(5,2)
\pnode(0,0){A}
\pnode(5,2){B}
\ncline{A}{B}
\psLDNode(A)(B){1.5cm}{KN}\qdisk(KN){2pt}
\nlput[nrot=:U](A)(B){1cm}{Test}
\nlput[nrot=:D](A)(B){2cm}{Test}
\nlput[nrot=:U](A)(B){3cm}{Test}
\nlput(A)(B){4cm}{Test}
\end{pspicture}
\end{LTXexample}




\clearpage
%--------------------------------------------------------------------------------------
\part{\texttt{pst-plot}}
%--------------------------------------------------------------------------------------
\section{ New options}
%--------------------------------------------------------------------------------------

The axes macro has now two additional optional arguments for placing labels at
the end of the axes:

\begin{lstlisting}[style=syntax]
\psaxes[settings]{arrows}(x0,y0)(x1,y1)(x2,y2)[Xlabel,Xangle][Ylabel,Yangle]
\end{lstlisting}

It has now four optional arguments, one for the setting, one for
the arrows, one for the x-label and one for the y-label. If you
want only a y-label, then leave the x one empty. A missing y-label
is possible. The following examples show how it can be used.

The option \verb+tickstyle=full|top|bottom+ no longer works in the
usual way. Only the additional value \texttt{inner} is valid for
\verb+pstricks-add+, because everything can be set by the
\verb+ticksize+ option. When using the \verb+comma+ or
\verb+trigLabels+ option, the macros \verb+\pshlabel+ and
\verb+\psvlabel+ shouldn't be redefined, because the package does
it itself internally in these cases. However, if you need a
redefinition, then do it for \CMD{pst@@hlabel} and
\CMD{pst@@vlabel} with

\begin{lstlisting}[style=syntax]
\makeatletter
\def\ps@@hlabel#1{...}
\def\ps@@vlabel#1{...}
\makeatother
\end{lstlisting}


{
\ttfamily
\rowcolors{1}{blue!20}{red!30}
\begin{longtable}{lll}
\caption{All new parameters for \texttt{pst-plot}}\\[-5pt]
\rowcolor{white}\textrm{\bfseries Name}         & \textrm{\bfseries Type}  & \textrm{\bfseries Default}\\\hline
\endfirsthead
\rowcolor{white}\textrm{\bfseries Name}         & \textrm{\bfseries Type}  & \textrm{\bfseries Default}\\\hline
\endhead
labels               & <all|x|y|none>   & all\\%ok
xlabelPos      & <bottom,axis,top>& bottom\\
ylabelPos      & <left,axis,right>& left\\
xlabelFactor         & <anything> & \{\textbackslash\@ empty\}\\
ylabelFactor         & <anything> & \{\textbackslash\@ empty\}\\
labelFontSize       & <fontsize macro>    & \{\} \\
trigLabels        & false|true    & false\\
trigLabelBase        & <number>    & 0\\
algebraic             & false|true & false\\ %ok
comma                 & false|true & false\\ %ok
xAxis                 & false|true & true\\%ok
yAxis                 & false|true & true\\%ok
xyAxes                & false|true & true\\%ok
xDecimals            & <number> or empty  & \{\}\\%ok
yDecimals            & <number> or empty  & \{\}\\%ok
xyDecimals            & <number> or empty   & \{\}\\%ok
%xLabel               & <anything> & \{\}\\%ok
%yLabel               & <anything> & \{\}\\%ok
%xyLabel               & <anything> & \{\}\\%ok
ticks                & <all|x|y|none>   & all\\%ok
tickstyle            & full|top|bottom|inner & full\\%ok
subticks             & <number>         & 0\\
xsubticks            & <number>         & 0\\
ysubticks            & <number>         & 0\\
ticksize             & <length [length]>         & -4pt 4pt\\
subticksize          & <number>         & 0.75\\
tickwidth            & <length>         & 0.5\verb+\pslinewidth+\\
subtickwidth         & <length>         & 0.25\verb+\pslinewidth+\\
tickcolor            & <color>          & black\\
xtickcolor           & <color>          & black\\
ytickcolor           & <color>          & black\\
subtickcolor         & <color>          & darkgray\\
xsubtickcolor        & <color>          & darkgray\\
ysubtickcolor        & <color>          & darkgray\\
ticklinestyle        & solid | dashed | dotted | none & solid\\
subticklinestyle        & solid | dashed | dotted | none & solid\\
xlogBase              & <number> or empty   & \{\}\\
ylogBase              & <number> or empty   & \{\}\\
xylogBase              & <number> or empty   & \{\}\\
logLines             & <none|x|y|all>   & none\\
ignoreLines          & <number>          & 0\\
nStep                & <number>          & 1\\
nStart               & <number>          & 0\\
nEnd                 & <number> or empty          & \{\}\\
xStep                & <number>          & 0\\
yStep               & <number>          & 0\\
xStart               & <number> or empty         & \{\}\\
yStart               & <number> or empty         & \{\}\\
xEnd               & <number> or empty         & \{\}\\
yEnd               & <number> or empty         & \{\}\\
plotNo            & <number>          & 1\\
plotNoMax         & <number>          & 1\\
xAxisLabel        & <anything> & \{\textbackslash\@ empty\}\\
yAxisLabel        & <anything> & \{\textbackslash\@ empty\}\\
xAxisLabelPos     & <(x,y)> or empty & \{\textbackslash\@ empty\}\\
yAxisLabelPos     & <(x,y)> or empty & \{\textbackslash\@ empty\}\\
llx               & <length>  & 0pt\\
lly               & <length>  & 0pt\\
urx               & <length>  & 0pt\\
ury               & <length>  & 0pt\\
polarplot         & false|true    & false\\
ChangeOrder        & false|true    & false\\
\end{longtable}
}


\clearpage

%--------------------------------------------------------------------------------------
\subsection{\texttt{xyAxes}, \texttt{xAxis} and \texttt{yAxis}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
xyAxes=true|false
xAxis=true|false
yAxis=true|false
\end{lstlisting}

Sometimes there is only a need for one axis with ticks. In this
case you can set one of the preceding options to false. The
\verb+xyAxes+ only makes sense when you want to set both x and y
to true with only one command, back to the default, because with
\verb+xyAxes=false+ you get nothing with the \verb+psaxes+ macro.


\resetOptions%
\begin{LTXexample}
\begin{pspicture}(5,1)
\psaxes[yAxis=false,linecolor=blue]{->}(0,0.5)(5,0.5)
\end{pspicture}
\begin{pspicture}(1,5)
\psaxes[xAxis=false,linecolor=red]{->}(0.5,0)(0.5,5)
\end{pspicture}
\begin{pspicture}(1,5)
\psaxes[xAxis=false,linecolor=red,
    ylabelPos=right]{->}(0.5,0)(0.5,5)
\end{pspicture}\\[0.5cm]
\begin{pspicture}(5,1)
\psaxes[yAxis=false,linecolor=blue,
    xlabelPos=top]{->}(0,0.5)(5,0.5)
\end{pspicture}
\end{LTXexample}

As seen in the example, a single y axis gets the labels on the left side. This can be
changed with the option \verb+ylabelPos+ or with \verb+xlabelPos+ for the
x-axis.


%--------------------------------------------------------------------------------------
\subsection{\texttt{labels}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
labels=all|x|y|none
\end{lstlisting}

This option is also already in the \verb+pst-plot+ package and
only mentioned here for completeness.

\begin{LTXexample}[width=3.5cm]
\psset{ticksize=6pt}
\begin{pspicture}(-1,-1)(2,2)
\psaxes[labels=all,subticks=5]{->}(0,0)(-1,-1)(2,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[labels=y,subticks=5]{->}(0,0)(-1,-1)(2,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[labels=x,subticks=5]{->}(0,0)(2,2)(-1,-1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[labels=none,subticks=5]{->}(0,0)(2,2)(-1,-1)
\end{pspicture}
\end{LTXexample}



%--------------------------------------------------------------------------------------
\subsection{\texttt{xlabelPos} and \texttt{ylabelPos}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
xlabelPos=bottom|axis|top
ylabelPos=left|axis|right
\end{lstlisting}

By default the labels for ticks are placed at the bottom (x axis)
and left (y-axis). If both axes are drawn in the negative
direction the default is top (x axis) and right (y axis). It be
changed with the two options \texttt{xlabelPos} and
\texttt{ylabelPos}. With the value \texttt{axis} the user can
place the labels depending on the value of \texttt{labelsep},
which is taken into account for \texttt{axis}.

\resetOptions%
\bigskip
\begin{LTXexample}[width=9cm]
\begin{pspicture}(3,3)
\psaxes{->}(3,3)
\end{pspicture}\hspace{2cm}
\begin{pspicture}(3,-3)
\psaxes[xlabelPos=top]{->}(3,-3)
\end{pspicture}
\end{LTXexample}

\vspace{1cm}
\begin{LTXexample}[width=9cm]
\begin{pspicture}(-3,-3)
\psaxes{->}(-3,-3)
\end{pspicture}\hspace{2cm}
\begin{pspicture}(3,3)
\psaxes[labelsep=0pt,
    ylabelPos=axis,
    xlabelPos=axis]{->}(3,3)
\end{pspicture}
\end{LTXexample}

\vspace{1cm}
\begin{LTXexample}[width=5cm]
\begin{pspicture}(-1,1)(3,-3)
\psaxes[xlabelPos=top,
    xticksize=0 20pt,
    yticksize=-20pt 0]{->}(3,-3)
\end{pspicture}
\end{LTXexample}




%--------------------------------------------------------------------------------------
\subsection{Changing the label font size with \texttt{labelFontSize} and \texttt{mathLabel}}
%--------------------------------------------------------------------------------------

This option sets the horizontal \textbf{and} vertical font size
for the labels depending on the option \verb+mathLabel+ for the
text or the math mode. It will be overwritten when another package
or a user defines
\begin{lstlisting}[style=syntax]
\def\pshlabel#1{\labelFontSize ...}
\def\psvlabel#1{\labelFontSize ...}
\def\pshlabel#1{$\labelFontSize ...$}% for mathLabel=true (default)
\def\psvlabel#1{$\labelFontSize ...$}% for mathLabel=true (default)
\end{lstlisting}
in another way. Note that for \verb+mathLabel=true+ the font size
must be set by one of the mathematical styles \verb+\textstyle+,
\verb+\displaystyle+, \verb+\scriptstyle+, or
\verb+\scriptscriptstyle+.

\begin{LTXexample}[width=6cm]
\psset{mathLabel=false}
\begin{pspicture}(-0.25,-0.25)(5,2.25)
\psaxes{->}(5,2.25)[$x$,0][$y$,90]
\end{pspicture}\\[20pt]
\begin{pspicture}(-0.25,-0.25)(5,2.25)
\psaxes[labelFontSize=\footnotesize]{->}(5,2.25)
\end{pspicture}\\[20pt]
\begin{pspicture}(-0.25,-0.25)(5,2.25)
\psaxes[labelFontSize=\footnotesize]{->}(5,2.25)
\end{pspicture}\\[20pt]
\end{LTXexample}

\begin{LTXexample}[width=6cm]
\begin{pspicture}(-0.25,-0.25)(5,2.25)
\psaxes[labelFontSize=\scriptstyle]{->}(5,2.25)[\textbf{x},-90][\textbf{y},0]
\end{pspicture}\\[20pt]
\psset{mathLabel=true}
\begin{pspicture}(-0.25,-0.25)(5,2.25)
\psaxes[labelFontSize=\scriptscriptstyle]{->}(5,2.25)
\end{pspicture}\\[20pt]
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{\texttt{xlabelFactor} and \texttt{ylabelFactor}}
%--------------------------------------------------------------------------------------
When having big numbers as data records then it makes sense to
write the values as ${<number>\cdot 10^{<exp>}}$. These new
options allow you to define the additional part of the value, but
it must be set in math mode when using math operators!

\resetOptions
\begin{LTXexample}
\readdata{\data}{demo1.dat}
\pstScalePoints(1,0.000001){}{}% (x,y){additional x operator}{y op}
\psset{llx=-1cm,lly=-1cm}
\psgraph[ylabelFactor=\cdot 10^6,Dx=5,Dy=100](0,0)(25,750){8cm}{5cm}
   \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
\endpsgraph
\pstScalePoints(1,1){}{}% reset
\end{LTXexample}


\clearpage
%--------------------------------------------------------------------------------------
\subsection{\texttt{comma}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
comma=false|true
\end{lstlisting}
Setting this option to true gives labels with a comma as a decimal separator instead
of the dot. \verb|comma| and \verb|comma=true| is the same.

\resetOptions
\medskip
\begin{LTXexample}[width=5.5cm]
\begin{pspicture}(-0.5,-0.5)(5,5.5)
\psaxes[Dx=1.5,comma,Dy=0.75,dy=0.75]{->}(5,5)
\psplot[linecolor=red,linewidth=3pt]{0}{4.5}%
   {x RadtoDeg cos 2 mul 2.5 add}
\psline[linestyle=dashed](0,2.5)(4.5,2.5)
\end{pspicture}
\end{LTXexample}

%--------------------------------------------------------------------------------------
\subsection{\texttt{xyDecimals}, \texttt{xDecimals} and \texttt{yDecimals}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
xyDecimals=<number>
xDecimals=<any>
yDecimals=<any>
\end{lstlisting}
By default the labels of the axes get numbers with or without
decimals, depending on the numbers. With these options
\verb|??Decimals| it is possible to determine the decimals, where
the option \verb|xyDecimals| sets this identical for both axes.
The default setting \verb|{}| means, that you'll get the standard
behaviour.


\begin{LTXexample}[width=6cm]
\begin{pspicture}(-1.5,-0.5)(5,3.75)
  \psaxes[xyDecimals=2]{->}(0,0)(4.5,3.5)
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[pos=t]
\psset{xunit=10cm,yunit=0.01cm,labelFontSize=\scriptstyle}
\begin{pspicture}(-0.3,-150)(1.5,550.0)
  \psaxes[Dx=0.25,Dy=100,ticksize=-4pt 0,comma=true,xDecimals=3,yDecimals=1]{->}%
    (0,0)(0,-100)(1.4,520)[\textbf{Amp\`ere},-90][\textbf{Voltage},0]
\end{pspicture}
\end{LTXexample}

\resetOptions

%--------------------------------------------------------------------------------------
\subsection{\texttt{trigLabels} and \texttt{trigLabelBase} -- axis with trigonmetrical units}
%--------------------------------------------------------------------------------------
With the option \verb+trigLabels=true+ the labels on the x axis
are trigonometrical ones. The option \verb+trigLabelBase+ set the
denominator of fraction. The default value of 0 is the same as no
fraction. The following constants are defined in the package:
\begin{lstlisting}[style=syntax]
\def§\ON§\psPiFour§\OFF§{12.566371}
\def§\ON§\psPiTwo§\OFF§{6.283185}
\def§\ON§\psPi§\OFF§{3.14159265}
\def§\ON§\psPiH§\OFF§{1.570796327}
\newdimen\pstRadUnit
\newdimen\pstRadUnitInv
§\ON§\pstRadUnit§\OFF§=1.047198cm % this is pi/3
§\ON§\pstRadUnitInv§\OFF§=0.95493cm % this is 3/pi
\end{lstlisting}



Because it is a bit complicated to set the right values, we show
some more examples here.

For \textbf{all} following examples in this section we did a
global

\lstinline[frame=single]|\psset{trigLabels=true,labelFontSize=\scriptstyle}|.


\psset{trigLabels,labelFontSize=\scriptstyle} Translating the
decimal ticks to trigonometrical ones makes no real sense, because
every 1 xunit (1cm) is a tick and the last one is at 6cm.

\clearpage
\begin{minipage}{0.4\fullWidth}
\begin{pspicture}[trigLabels=true](-0.5,-1.25)(6.5,1.25)%
  \pnode(5,0){A}%
  \psaxes{->}(0,0)(-0.5,-1.25)(\psPiTwo,1.25)%
\end{pspicture}
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)%
  \pnode(5,0){A}%
  \psaxes{->}(0,0)(-.5,-1.25)(\psPiTwo,1.25)
\end{pspicture}
\end{lstlisting}
\end{minipage}

\begin{minipage}{0.4\fullWidth}
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)%
  \psaxes[trigLabelBase=3]{->}(0,0)(-0.5,-1.25)(\psPiTwo,1.25)
\end{pspicture}
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)%
  \psaxes[§\ON§trigLabelBase=3§\OFF§]{->}(0,0)(-0.5,-1.25)(\psPiTwo,1.25)
\end{pspicture}
\end{lstlisting}
\end{minipage}



Modifying the ticks to have the last one exactly at the end is
possible with a different dx value ($\frac{\pi}{3}\approx 1.047$):


\begin{minipage}{0.4\fullWidth}
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)\pnode(\psPiTwo,0){C}%
  \psaxes[dx=\pstRadUnit]{->}(0,0)(-0.5,-1.25)(\psPiTwo,1.25)
\end{pspicture}%
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)\pnode(\psPiTwo,0){C}%
  \psaxes[§\ON§dx=\pstRadUnit§\OFF§]{->}(0,0)(-0.5,-1.25)(\psPiTwo,1.25)
\end{pspicture}%
\end{lstlisting}
\end{minipage}


\begin{minipage}{0.4\fullWidth}
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)\pnode(5,0){B}%
  \psaxes[dx=\pstRadUnit,trigLabelBase=3]{->}(0,0)(-0.5,-1.25)(\psPiTwo,1.25)
\end{pspicture}%
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)\pnode(5,0){B}%
  \psaxes[dx=\pstRadUnit,§\ON§trigLabelBase=3§\OFF§] {->}(0,0)(-0.5,-1.25)(\psPiTwo,1.25)
\end{pspicture}%
\end{lstlisting}
\end{minipage}

\ncline[linestyle=dashed,linewidth=0.4pt]{A}{B}

Set everything globally in radian units. Now 6 units on the
$x$-axis are $6\pi$. Using \verb+trigLabelBase=3+ reduces this
value to $2\pi$, a.s.o.

\bigskip
\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)\pnode(6,0){D}%
  \psaxes{->}(0,0)(-0.5,-1.25)(6.5,1.25)%
\end{pspicture}%
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{§\ON§xunit=\pstRadUnit§\OFF§}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)\pnode(6,0){D}%
  \psaxes{->}(0,0)(-0.5,-1.25)(6.5,1.25)%
\end{pspicture}%
\end{lstlisting}
\end{minipage}
\ncline[linestyle=dashed,linewidth=0.4pt]{C}{D}


\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[trigLabelBase=3]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
\end{pspicture}%
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{§\ON§xunit=\pstRadUnit§\OFF§}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[§\ON§trigLabelBase=3§\OFF§]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
\end{pspicture}%
\end{lstlisting}
\end{minipage}



\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[trigLabelBase=4]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
\end{pspicture}%
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{§\ON§xunit=\pstRadUnit§\OFF§}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[§\ON§trigLabelBase=4§\OFF§]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
\end{pspicture}%
\end{lstlisting}
\end{minipage}

\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[trigLabelBase=6]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
\end{pspicture}%
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{§\ON§xunit=\pstRadUnit§\OFF§}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[§\ON§trigLabelBase=6§\OFF§]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
\end{pspicture}%
\end{lstlisting}
\end{minipage}



The best way seems to be to set the $x$-unit to
\verb+\pstRadUnit+. Plotting a function doesn't consider the value
for \verb+trigLabelBase+, it has to be done by the user. The first
example sets the unit locally for the \verb+\psplot+ back to 1cm,
which is needed, because we use this unit on the PostScript side.

\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[trigLabelBase=3]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[xunit=1cm,linecolor=red,linewidth=1.5pt]{0}{\psPiTwo}{x RadtoDeg sin}
\end{pspicture}
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[trigLabelBase=3]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[§\ON§xunit=1cm§\OFF§,linecolor=red,linewidth=1.5pt]{0}{§\ON§\psPiTwo§\OFF§}{x RadtoDeg sin}
\end{pspicture}
\end{lstlisting}
\end{minipage}


\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[trigLabelBase=3]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[linecolor=red,linewidth=1.5pt]{0}{6}{x Pi 3 div mul RadtoDeg sin}
\end{pspicture}
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[trigLabelBase=3]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[linecolor=red,linewidth=1.5pt]{0}{6}{x §\ON§Pi 3 div mul §\OFF§RadtoDeg sin}
\end{pspicture}
\end{lstlisting}
\end{minipage}


\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[dx=1.5]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[xunit=.5cm,linecolor=red,linewidth=1.5pt]{0}{\psPiFour}{x RadtoDeg sin}
\end{pspicture}
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[§\ON§dx=1.5§\OFF§]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[§\ON§xunit=0.5cm§\OFF§,linecolor=red,linewidth=1.5pt]{0}{§\ON§\psPiFour§\OFF§}{x RadtoDeg sin}
\end{pspicture}
\end{lstlisting}
\end{minipage}


\begin{minipage}{0.4\fullWidth}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[dx=0.75,trigLabelBase=2]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[xunit=.5cm,linecolor=red,linewidth=1.5pt]{0}{\psPiFour}{x RadtoDeg sin}
\end{pspicture}
\end{minipage}%
\begin{minipage}{0.6\fullWidth}
\begin{lstlisting}
\psset{xunit=\pstRadUnit}%
\begin{pspicture}(-0.5,-1.25)(6.5,1.25)
  \psaxes[§\ON§dx=0.75§\OFF§,§\ON§trigLabelBase=2§\OFF§]{->}(0,0)(-0.5,-1.25)(6.5,1.25)
  \psplot[§\ON§xunit=0.5cm§\OFF§,linecolor=red,linewidth=1.5pt]{0}{\psPiFour}{x RadtoDeg sin}
\end{pspicture}
\end{lstlisting}
\end{minipage}


It is also possible to set the $x$ unit and $dx$ value to get the
labels right. But this needs some more understanding as to how it
really works. A \verb+xunit=1.570796327+ sets the unit to $\pi/2$
and a \verb+dx=0.666667+ then puts at every $2/3$ of the unit a
tick mark and a label. The length of the $x$-axis is 6.4 units
which is $6.4\cdot 1.570796327cm\approx 10cm$. The function then
is plotted from $0$ to $3\pi=9.424777961$.




\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-0.5,-1.25)(10,1.25)
  \psaxes[xunit=1.570796327,showorigin=false,trigLabelBase=3,dx=0.666667]{->}(0,0)(-0.5,-1.25)(6.4,1.25)
  \psplot[linecolor=red,linewidth=1.5pt]{0}{9.424777961}{%
  x RadtoDeg dup sin exch 1.1 mul cos add}
\end{pspicture}
\end{center}
\begin{lstlisting}
\begin{pspicture}(-0.5,-1.25)(10,1.25)
  \psaxes[§\ON§xunit=1.570796327§\OFF§,§\ON§trigLabelBase=3§\OFF§,§\ON§dx=0.666667§\OFF§]{->}(0,0)(-0.5,-1.25)(6.4,1.25)
  \psplot[linecolor=red,linewidth=1.5pt]{0}{§\ON§9.424777961§\OFF§}{%
  x RadtoDeg dup sin exch 1.1 mul cos add}
\end{pspicture}
\end{lstlisting}

\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-0.5,-1.25)(10,1.25)
  \psaxes[xunit=\psPi,dx=0.25]{->}(0,0)(-0.25,-1.25)(3.2,1.25)
  \psplot[xunit=0.25,plotpoints=500,linecolor=red,linewidth=1.5pt]{0}{37.70}{%
  x RadtoDeg dup sin exch 1.1 mul cos add}
\end{pspicture}
\end{center}
\begin{lstlisting}
\psset{§\ON§unit=1cm§\OFF§}
  \psplot[§\ON§xunit=0.25§\OFF§,§\ON§plotpoints=500§\OFF§,linecolor=red,linewidth=1.5pt]{0}{37.70}{%
  x RadtoDeg dup sin exch 1.1 mul cos add}
\end{pspicture}
\end{lstlisting}


\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-0.5,-2)(10,2)
  \psplot[xunit=0.0625,linecolor=red,linewidth=1.5pt,plotpoints=5000]{0}{150.80}{%
  x RadtoDeg dup sin exch 1.1 mul cos add}
  \psaxes[xunit=\psPi,dx=0.5,Dx=8,subticks=2]{->}(0,0)(-0.1,-2)(3.2,2)
\end{pspicture}
\end{center}
\begin{lstlisting}
\psset{§\ON§unit=1cm§\OFF§}
\begin{pspicture}(-0.5,-1.25)(10,1.25)
  \psplot[§\ON§xunit=0.0625§\OFF§,linecolor=red,linewidth=1.5pt,%
      §\ON§plotpoints=5000§\OFF§]{0}{150.80}%
  {x RadtoDeg dup sin exch 1.1 mul cos add}
  \psaxes[§\ON§xunit=\psPi§\OFF§,§\ON§dx=0.5§\OFF§,§\ON§Dx=8§\OFF§]{->}(0,0)(-0.25,-1.25)(3.2,1.25)
\end{pspicture}
\end{lstlisting}


\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-7,-1.5)(7,1.5)
  \psaxes[trigLabels=true,xunit=\psPi]{->}(0,0)(-2.2,-1.5)(2.2,1.5)
  \psplot[linecolor=red,linewidth=1.5pt]{-7}{7}{x RadtoDeg sin}
\end{pspicture}
\end{center}
\begin{lstlisting}
\begin{pspicture}(-7,-1.5)(7,1.5)
  \psaxes[trigLabels=true,§\ON§xunit=\psPi§\OFF§]{->}(0,0)(-2.2,-1.5)(2.2,1.5)
  \psplot[linecolor=red,linewidth=1.5pt]{-7}{7}{x RadtoDeg sin}
\end{pspicture}
\end{lstlisting}


\begin{center}
\psset{unit=1cm}
\begin{pspicture}(-7,-1.5)(7,1.5)
  \psaxes[trigLabels=true,
  trigLabelBase=2,dx=\psPiH,xunit=\psPi]{->}(0,0)(-2.2,-1.5)(2.2,1.5)
  \psplot[linecolor=red,linewidth=1.5pt]{-7}{7}{x RadtoDeg sin}
\end{pspicture}
\end{center}
\begin{lstlisting}
\begin{pspicture}(-7,-1.5)(7,1.5)
  \psaxes[trigLabels=true,
  trigLabelBase=2,dx=\psPiH,xunit=\psPi]{->}(0,0)(-2.2,-1.5)(2.2,1.5)
  \psplot[linecolor=red,linewidth=1.5pt]{-7}{7}{x RadtoDeg sin}
\end{pspicture}
\end{lstlisting}


\psset{trigLabels=false}



%--------------------------------------------------------------------------------------
\subsection{\texttt{ticks}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
ticks=all|x|y|none
\end{lstlisting}

This option is also already in the \verb+pst-plot+ package and
only mentioned here for some completeness.

\begin{LTXexample}[width=3.5cm]
\psset{ticksize=6pt}
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=all,subticks=5]{->}(0,0)(-1,-1)(2,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=y,subticks=5]{->}(0,0)(-1,-1)(2,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=x,subticks=5]{->}(0,0)(2,2)(-1,-1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=none,subticks=5]{->}(0,0)(2,2)(-1,-1)
\end{pspicture}
\end{LTXexample}

Single ticks can be set with the two macros
%
\begin{lstlisting}[style=syntax]
\psxTick[options](x value){label}
\psyTick[options](y value){label}
\end{lstlisting}


\begin{LTXexample}[width=.5\linewidth]
\begin{psgraph}[Dx=2,Dy=2](0,0)(-4,-2.2)(4,2.2){.5\textwidth}{!}
  \psxTick[linecolor=red](1.5){$x_0$}
  \psyTick[linecolor=blue](1.7){$y_0$}
\end{psgraph}
\end{LTXexample}


% full= 0, top=1, bottom=-1, inner=2  => -1 0 1 2
%--------------------------------------------------------------------------------------
\subsection{\texttt{tickstyle}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
tickstyle=full|top|bottom|inner
\end{lstlisting}

The value \texttt{inner} (not available with the basic \texttt{pstricks} package) is
only valid for the axes style \texttt{frame}.

\medskip
\begin{LTXexample}[pos=t]
\psset{subticks=10}
\begin{pspicture}(-1,-1)(3,3) \psaxes[tickstyle=full]{->}(3,3) \end{pspicture}
\begin{pspicture}(-1,-1)(3,3) \psaxes[tickstyle=top]{->}(3,3) \end{pspicture}
\begin{pspicture}(-1,-1)(3,3) \psaxes[tickstyle=bottom]{->}(3,3)\end{pspicture}
\begin{pspicture}(-1,-1)(3,3)
  \psaxes[axesstyle=frame, tickstyle=inner, ticksize=0 4pt]{->}(3,3)
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{\texttt{ticksize}, \texttt{xticksize}, \texttt{yticksize}}
%--------------------------------------------------------------------------------------

With this new option the recent \verb+tickstyle+ option of
\verb+pst-plot+ is obsolete and no longer supported by
\verb+pstricks-add+.

Syntax:
\begin{lstlisting}[style=syntax]
ticksize=value[unit]
ticksize=value[unit] value[unit]
xticksize=value[unit]
xticksize=value[unit] value[unit]
yticksize=value[unit]
yticksize=value[unit] value[unit]
\end{lstlisting}

\verb+ticksize+ sets both values. The first one is left/below and the optional second
one is right/above of the coordinate axis. The old setting \verb+tickstyle=bottom+ is
now easy to realize, e.g.: \verb+ticksize=-6pt 0+, or vice versa, if the coordinates
are set from positive to negative values.

\medskip
\begin{LTXexample}[width=6cm]
\psset{arrowscale=2}
\begin{pspicture}(-1.5,-1.5)(4,3.5)
  \psaxes[ticksize=0.5cm]{->}(0,0)(-1.5,-1.5)(4,3.5)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6cm]
\psset{arrowscale=2}
\begin{pspicture}(-1.5,-1.5)(4,3.5)
  \psaxes[xticksize=-10pt 0,yticksize=0 10pt]%
    {->}(0,0)(-1.5,-1.5)(4,3.5)
\end{pspicture}
\end{LTXexample}

A grid is also possible by setting the values to the max/min coordinates.

\begin{LTXexample}[width=6cm]
\psset{arrowscale=2}
\begin{pspicture}(-.5,-.5)(5,4.5)
  \psaxes[ticklinestyle=dashed,
    ticksize=0 4cm]{->}(0,0)(-.5,-.5)(5,4.5)
\end{pspicture}
\end{LTXexample}

\clearpage
%--------------------------------------------------------------------------------------
\subsection{\texttt{subticks}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
subticks=<number>
\end{lstlisting}

By default subticks cannot have labels.

\begin{LTXexample}[width=3.5cm]
\psset{ticksize=6pt}
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=all,subticks=5]{->}(0,0)(-1,-1)(2,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=y,subticks=5]{->}(0,0)(-1,-1)(2,2)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=x,subticks=5]{->}(0,0)(2,2)(-1,-1)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\begin{pspicture}(-1,-1)(2,2)
\psaxes[ticks=none,subticks=5]{->}(0,0)(2,2)(-1,-1)
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{\texttt{subticksize}, \texttt{xsubticksize}, \texttt{ysubticksize}}
%--------------------------------------------------------------------------------------

Syntax:
\begin{lstlisting}[style=syntax]
subticksize=value
xsubticksize=value
ysubticksize=value
\end{lstlisting}

\verb+subticksize+ sets both values, which are relative to the ticksize length and
can have any number. 1 sets it to the same length as the main ticks.

\begin{LTXexample}[preset=\centering,pos=t]
\psset{yunit=1.5cm,xunit=3cm}
\begin{pspicture}(-1.25,-4.75)(3.25,.75)
  \psaxes[xticksize=-4.5 0.5,ticklinestyle=dashed,subticks=5,xsubticksize=1,%
     ysubticksize=0.75,xsubticklinestyle=dotted,xsubtickwidth=1pt,
     subtickcolor=gray]{->}(0,0)(-1,-4)(3.25,0.5)
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{\texttt{tickcolor}, \texttt{subtickcolor}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
tickcolor=<color>
xtickcolor=<color>
ytickcolor=<color>
subtickcolor=<color>
xsubtickcolor=<color>
ysubtickcolor=<color>
\end{lstlisting}

\verb+tickcolor+ and \verb+subtickcolor+ set both for the x- and the y-Axis.

\begin{LTXexample}[preset=\centering,pos=t]
\begin{pspicture}(0,-0.75)(10,1)
\psaxes[yAxis=false,labelFontSize=\scriptstyle,ticksize=0 10mm,subticks=10,subticksize=0.75,
  tickcolor=red,subtickcolor=blue,tickwidth=1pt,subtickwidth=0.5pt](10.01,0)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=5cm]
\begin{pspicture}(5,-0.75)(10,1)
\psaxes[yAxis=false,labelFontSize=\scriptstyle,ticksize=0 -10mm,subticks=10,subticksize=0.75,
  tickcolor=red,subtickcolor=blue,tickwidth=1pt,subtickwidth=0.5pt,Ox=5](5,0)(5,0)(10.01,0)
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{\texttt{ticklinestyle} and \texttt{subticklinestyle}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
ticklinestyle=solid|dashed|dotted|none
xticklinestyle=solid|dashed|dotted|none
yticklinestyle=solid|dashed|dotted|none
subticklinestyle=solid|dashed|dotted|none
xsubticklinestyle=solid|dashed|dotted|none
ysubticklinestyle=solid|dashed|dotted|none
\end{lstlisting}

\verb+ticklinestyle+ and \verb+subticklinestyle+ set both values
for the x and y axis. The value \verb+none+ doesn't really makes
sense, because it is the same as \verb+[sub]ticklines=0+

\begin{LTXexample}[preset=\centering,pos=t]
\psset{unit=4cm}
\pspicture(-0.15,-0.15)(2.5,1)
  \psaxes[axesstyle=frame,logLines=y,xticksize=0 1,xsubticksize=1,ylogBase=10,
    tickcolor=red,subtickcolor=blue,tickwidth=1pt,subticks=20,xsubticks=10,
    xticklinestyle=dashed,xsubticklinestyle=dashed](2.5,1)
\endpspicture
\end{LTXexample}



%--------------------------------------------------------------------------------------
\subsection{\texttt{loglines}}
%--------------------------------------------------------------------------------------
Syntax:
\begin{lstlisting}[style=syntax]
loglines=all|x|y
\end{lstlisting}

By default the option \texttt{logLines} sets the ticksize to the maximal length for x, y, or both.
It can be changed, when \emph{after} the option \texttt{logLines} the ticksize is set.

\begin{LTXexample}[pos=t]
\pspicture(-1,-1)(5,5)
   \psaxes[subticks=5,xylogBase=10,logLines=all](5,5)
\endpspicture\hspace{1cm}
\pspicture(-1,-1)(5,5)
   \psaxes[subticks=10,axesstyle=frame,xylogBase=10,logLines=all,ticksize=0 5pt,tickstyle=inner](5,5)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[preset=\centering,pos=t]
\psset{unit=4cm}
\pspicture(-0.15,-0.15)(2.5,2)
  \psaxes[axesstyle=frame,logLines=y,xticksize=max,xsubticksize=1,ylogBase=10,
    tickcolor=red,subtickcolor=blue,tickwidth=1pt,subticks=20,xsubticks=10](2.5,2)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[preset=\centering,pos=t]
\psset{unit=4}
\pspicture(-0.5,-0.3)(3,1.2)
   \psaxes[axesstyle=frame,tickstyle=inner,logLines=x,xlogBase=10,Dy=0.5,tickcolor=red,
     subtickcolor=blue,tickwidth=1pt,ysubticks=5,xsubticks=10](3,1)
\endpspicture
\end{LTXexample}

%--------------------------------------------------------------------------------------
\subsection{\texttt{xylogBase}, \texttt{xlogBase} and \texttt{ylogBase}}
%--------------------------------------------------------------------------------------
There are additional options \verb|xylogBase | xlogBase |
ylogBase| to get one or both axes with logarithmic labels. For an
interval of [$10^{-3} ... 10^2$] choose a \verb|pstricks| interval
of [-3,2]. \verb|pstricks| takes $0$ as the origin of this axes,
which is wrong if we want to have a logarithmic axes. With the
options \verb|Oy| and \verb|Ox| we can set the origin to $-3$, so
that the first label gets $10^{-3}$. If this is not done by the
user then \verb|pstricks-add| does it by default. An alternative
is to set these parameters to empty values \verb|Ox={},Oy={}|, in
this case \verb|pstricks-add| does nothing.


%------------------------------------------------------------------------------------
\subsubsection{\texttt{xylogBase}}
%------------------------------------------------------------------------------------
This mode in math is also called double logarithmic. It is a
combination of the two foregoing modes and the function is now
$y=\log x$ and is shown in the following example.

\medskip
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
  \psplot[linewidth=2pt,linecolor=red]{0.001}{3}{x log}
  \psaxes[xylogBase=10,Oy=-3,Ox=-3]{->}(-3,-3)(3.5,3.5)
  \uput[-90](3.5,-3){x}
  \uput[180](-3,3.5){y}
  \rput(2.5,1){$y=\log x$}
\end{pspicture}
\end{LTXexample}



%--------------------------------------------------------------------------------------------
\subsubsection{\texttt{ylogBase}}
%--------------------------------------------------------------------------------------------
The values for the \texttt{psaxes} y-coordinate are now the
exponents to the base $10$ and for the right function to the base
$e$: $10^{-3} \ldots 10^1$ which corresponds to the given
y-interval $-3\ldots 1.5$, where only integers as exponents are
possible. These logarithmic labels have no effect on the
internally used units. To draw the logarithm function we have to
use the math function
\[y=\log\{\log x\}\]
\[y=\ln\{\ln x\}\]
with an drawing interval of $1.001\ldots 6$.

\medskip
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-0.5,-3.5)(6.5,1.5)
  \psaxes[ylogBase=10,Oy=-3]{->}(0,-3)(6.5,1.5)
  \uput[-90](6.5,-3){x}
  \uput[0](0,1.4){y}
  \rput(5,1){$y=\log x$}
  \psplot[linewidth=2pt,%
  plotpoints=100,linecolor=red]{1.001}{6}{x log log} % log(log(x))
\end{pspicture}
\end{LTXexample}

\medskip
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-0.5,-3.5)(6.5,1.5)
  \psplot[linewidth=2pt,plotpoints=100,linecolor=red]%
    {1.04}{6}{/ln {log 0.4343 div} def x ln ln} % log(x)
  \psaxes[ylogBase=e,Oy=-3]{->}(0,-3)(6.5,1.5)
  \uput[-90](6.5,-3){x}
  \uput[0](0,1.5){y}
  \rput(5,1){$y=\ln x$}
\end{pspicture}
\end{LTXexample}



\medskip
\begin{LTXexample}[width=7cm]
  \begin{pspicture}(-0.5,1.75)(6.5,4.5)
    \psaxes[ylogBase=10,Oy=2]{->}(0,2)(0,2)(6.5,4.5)
  \end{pspicture}
\end{LTXexample}



\medskip
\begin{LTXexample}[width=7cm]
  \begin{pspicture}(-0.5,-0.25)(6.5,4.5)
    \psplot{0}{6}{x x cos add log}                       % x   + cox(x)
    \psplot[linecolor=red]{0}{6}{x 3 exp x cos add log}  % x^3 + cos(x)
    \psplot[linecolor=cyan]{0}{6}{x 5 exp x cos add log} % x^5 + cos(x)
    \psaxes[ylogBase=10]{->}(6.5,4.5)
  \end{pspicture}
\end{LTXexample}



\medskip
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-0.5,-1.25)(6.5,4.5)
  \psplot{0}{6}{x x cos add log}                       % x   + cox(x)
  \psplot[linecolor=red]{0}{6}{x 3 exp x cos add log}  % x^3 + cos(x)
  \psplot[linecolor=cyan]{0}{6}{x 5 exp x cos add log} % x^5 + cos(x)
  \psaxes[ylogBase=10]{->}(0,-1)(0,-1)(6.5,4.5)
\end{pspicture}
\end{LTXexample}



\medskip
\begin{LTXexample}[width=4cm]
\begin{pspicture}(2.5,1.75)(6.5,4.5)
  \psplot[linecolor=cyan]{3}{6}{x 5 exp x cos add log} % x^5 + cos(x)
  \psaxes[ylogBase=10,Ox=3,Oy=2]{->}(3,2)(3,2)(6.5,4.5)
\end{pspicture}
\end{LTXexample}





%--------------------------------------------------------------------------------------
\subsubsection{\texttt{xlogBase}}
%--------------------------------------------------------------------------------------
Now we have to use the easy math function $y=x$ because the x axis is still $\log x$.

\medskip
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
  \psplot[linewidth=2pt,linecolor=red]{-3}{3}{x} % log(x)
  \psplot[linewidth=2pt,linecolor=blue]{-1.3}{1.5}{x 0.4343 div} % ln(x)
  \psaxes[xlogBase=10,Oy=-3,Ox=-3]{->}(-3,-3)(3.5,3.5)
  \uput[-90](3.5,-3){x}
  \uput[180](-3,3.5){y}
  \rput(2.5,1){$y=\log x$}
  \rput[lb](0,-1){$y=\ln x$}
\end{pspicture}
\end{LTXexample}

\begin{center}
\psset{yunit=3cm,xunit=2cm}
\begin{pspicture}(-1.25,-1.25)(4.25,1.5)
  \uput[-90](4.25,-1){x}
  \uput[0](-1,1.25){y}
  \rput(0,1){$y=\sin x$}
  \psplot[linewidth=2pt,plotpoints=5000,linecolor=red]{-1}{3.5}{10 x exp sin }
  \psaxes[xlogBase=10,Oy=-1,Ox=-1]{->}(-1,-1)(4.25,1.25)
  \psline[linestyle=dashed](!0 1)(!90 log 1)(!90 log 0)
\end{pspicture}
\end{center}

\begin{lstlisting}
\psset{yunit=3cm,xunit=2cm}
\begin{pspicture}(-1.25,-1.25)(4.25,1.5)
  \uput[-90](4.25,-1){x}
  \uput[0](-1,1.25){y}
  \rput(0,1){$y=\sin x$}
  \psplot[linewidth=2pt,plotpoints=5000,linecolor=red]{-1}{3.5}{10 x exp sin }
  \psaxes[xlogBase=10,Ox=-1,Oy=-1]{->}(-1,-1)(4.25,1.25)
   \psline[linestyle=dashed](-1,0)(4,0)
   \psline[linestyle=dashed](!-1 1)(!90 log 1)(!90 log -1)
   \psline[linestyle=dashed](!90 log 1)(!180 log 1)(!180 log -1)
\end{pspicture}
\end{lstlisting}


\begin{LTXexample}[width=7cm]
\begin{pspicture}(-3.5,-2.5)(3.5,2.5)
  \psaxes[xlogBase=10]{->}(0,0)(-3.5,-2.5)(3.5,2.5)
  \psplot{-2.5}{2.5}{10 x exp log}
\end{pspicture}
\end{LTXexample}



\medskip
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-3.5,-2.5)(3.5,2.5)
  \psaxes[xlogBase=10,Ox={},Oy={}]{->}(0,0)(-3.5,-2.5)(3.5,2.5)
  \psplot{-2.5}{2.5}{10 x exp log}
\end{pspicture}
\end{LTXexample}


%------------------------------------------------------------------------------------
\subsubsection{No logstyle (\texttt{xylogBase=\{\}})}
%------------------------------------------------------------------------------------
This is only a demonstration that the default option \verb|logBase={}| still works ... :-)

\medskip
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-3.5,-0.5)(3.5,2.5)
  \psplot[linewidth=2pt,linecolor=red,xylogBase={}]{0.5}{3}{x log} % log(x)
  \psaxes{->}(0,0)(-3.5,0)(3.5,2.5)
  \uput[-90](3.5,0){x}
  \uput[180](0,2.5){y}
  \rput(2.5,1){$y=\log x$}
\end{pspicture}
\end{LTXexample}


\newpage
%--------------------------------------------------------------------------------------
\subsection{\texttt{subticks}, \texttt{tickwidth} and \texttt{subtickwidth}}
%--------------------------------------------------------------------------------------


\begin{center}
{\psset{arrowscale=3,arrows=-D>,yAxis=false}
 \psaxes[subticks=8](0,0)(-5,-1)(5,1)\\[1cm]
 \psaxes[subticks=4,ticksize=-4pt 0,xlabelPos=top](0,0)(5,1)(-5,-1)\\
 \psaxes[subticks=4,ticksize=-10pt 0](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=10,ticksize=0 -10pt](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=4,ticksize=0 10pt,xlabelPos=bottom](0,0)(5,5)(-5,-5)\\[1cm]
 \psaxes[subticks=4,ticksize=0 -10pt,xlabelPos=top](0,0)(5,5)(-5,-5)\\[0.25cm]
 \psaxes[subticks=0](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=0,tickcolor=red,linecolor=blue,xlabelPos=top](0,0)(5,5)(-5,-5)\\
 \psaxes[subticks=5,tickwidth=2pt,subtickwidth=1pt](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=0,tickcolor=red,xlabelPos=top](0,0)(5,5)(-5,-5)}
\end{center}
\begin{lstlisting}[xrightmargin=-1.75cm]
\psset{arrowscale=3,arrows=-D>,yAxis=false}
 \psaxes[subticks=8](0,0)(-5,-1)(5,1)\\[1cm]
 \psaxes[subticks=4,ticksize=-4pt 0,xlabelPos=top](0,0)(5,1)(-5,-1)\\
 \psaxes[subticks=4,ticksize=-10pt 0](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=10,ticksize=0 -10pt](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=4,ticksize=0 10pt,xlabelPos=bottom](0,0)(5,5)(-5,-5)\\[1cm]
 \psaxes[subticks=4,ticksize=0 -10pt,xlabelPos=top](0,0)(5,5)(-5,-5)\\[0.25cm]
 \psaxes[subticks=0](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=0,tickcolor=red,linecolor=blue,xlabelPos=top](0,0)(5,5)(-5,-5)\\
 \psaxes[subticks=5,tickwidth=2pt,subtickwidth=1pt](0,0)(-5,-5)(5,5)\\[1cm]
 \psaxes[subticks=0,tickcolor=red,xlabelPos=top](0,0)(5,5)(-5,-5)}
\end{lstlisting}

\clearpage
\vspace*{4cm}
\begin{center}
\psset{arrowscale=3,xAxis=false}
\psaxes[subticks=8]{->}(0,0)(-5,-5)(5,5)\hspace{2em}
\psaxes[subticks=4,ylabelPos=right,ylabelPos=left]{->}(0,0)(5,5)(-5,-5)\hspace{4em}
\psaxes[subticks=4,ticksize=0 4pt]{->}(0,0)(-5,-5)(5,5)\hspace{3em}
\psaxes[subticks=4,ticksize=-4pt 0]{->}(0,0)(-5,-5)(5,5)\hspace{1em}
\psaxes[subticks=4,ticksize=0 4pt,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)\hspace{3em}
\psaxes[subticks=4,ticksize=-4pt 0,linecolor=red,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)\hspace{5em}
\psaxes[subticks=0]{->}(0,0)(-5,-5)(5,5)\hspace{1em}
\psaxes[subticks=0,tickcolor=red,linecolor=blue,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)\hspace{5em}
\psaxes[subticks=5,tickwidth=2pt,subtickwidth=1pt]{->}(0,0)(-5,-5)(5,5)\hspace{1em}
\psaxes[subticks=5,tickcolor=red,tickwidth=2pt,%
  ticksize=10pt,subtickcolor=blue,subticksize=0.75,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)
\end{center}

\vspace*{5cm}
\begin{lstlisting}[xrightmargin=-1.75cm]
\psset{arrowscale=3,xAxis=false}
\psaxes[subticks=8]{->}(0,0)(-5,-5)(5,5)\hspace{2em}
\psaxes[subticks=4,ylabelPos=right,ylabelPos=left]{->}(0,0)(5,5)(-5,-5)\hspace{4em}
\psaxes[subticks=4,ticksize=0 4pt]{->}(0,0)(-5,-5)(5,5)\hspace{3em}
\psaxes[subticks=4,ticksize=-4pt 0]{->}(0,0)(-5,-5)(5,5)\hspace{1em}
\psaxes[subticks=4,ticksize=0 4pt,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)\hspace{3em}
\psaxes[subticks=4,ticksize=-4pt 0,linecolor=red,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)\hspace{5em}
\psaxes[subticks=0]{->}(0,0)(-5,-5)(5,5)\hspace{1em}
\psaxes[subticks=0,tickcolor=red,linecolor=blue,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)\hspace{5em}
\psaxes[subticks=5,tickwidth=2pt,subtickwidth=1pt]{->}(0,0)(-5,-5)(5,5)\hspace{1em}
\psaxes[subticks=5,tickcolor=red,tickwidth=2pt,%
  ticksize=10pt,subtickcolor=blue,subticksize=0.75,ylabelPos=right]{->}(0,0)(5,5)(-5,-5)
\end{lstlisting}

\begin{LTXexample}[width=5.5cm]
\pspicture(5,5.5)
\psaxes[subticks=4,ticksize=6pt,subticksize=0.5,%
   tickcolor=red,subtickcolor=blue]{->}(5.4,5)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=5.5cm]
\pspicture(5,5.5)
   \psaxes[subticks=5,ticksize=0 6pt,subticksize=0.5]{->}(5.4,5)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=5.5cm]
\pspicture(5,5.5)
   \psaxes[subticks=5,ticksize=-6pt 0,subticksize=0.5]{->}(5.4,5)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\pspicture(-3,-3)(3,3.5)
   \psaxes[subticks=5,ticksize=0 6pt,subticksize=0.5]{->}(0,0)(3,3)(-3,-3)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=6.5cm]
\pspicture(0,0.5)(-3,-3)
   \psaxes[subticks=5,ticksize=-6pt 0,subticksize=0.5,linecolor=red]{->}(-3,-3)
\endpspicture
\end{LTXexample}



\begin{LTXexample}[width=5.5cm]
\psset{axesstyle=frame}
\pspicture(5,5.5)
   \psaxes[subticks=4,tickcolor=red,subtickcolor=blue](5,5)
\endpspicture
\end{LTXexample}

\vspace{1cm}
\begin{LTXexample}[width=5.5cm]
\pspicture(5,5.5)
   \psaxes[subticks=5,subticksize=1,subtickcolor=lightgray](5,5)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=5.5cm]
\pspicture(5,5.5)
   \psaxes[subticks=2,subticksize=1,subtickcolor=lightgray](5,5)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\pspicture(3,4.5)
   \psaxes[subticks=5,ticksize=-7pt 0](3,4)
\endpspicture
\end{LTXexample}


\begin{LTXexample}[width=3.5cm]
\pspicture(0,1)(-3,-4)
   \psaxes[subticks=5](-3,-4)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\pspicture(3,4.5)
   \psaxes[axesstyle=axes,subticks=5](3,4)
\endpspicture
\end{LTXexample}

\begin{LTXexample}[width=3.5cm]
\pspicture(0,1)(-3,-4)
  \psaxes[axesstyle=axes,subticks=5,%
    ticksize=0 10pt](-3,-4)
\endpspicture
\end{LTXexample}

\clearpage

%--------------------------------------------------------------------------------------
\subsection[\texttt{algebraic}]{\texttt{algebraic}%\footnote{This part is adapted
%    from the package \texttt{pst-eqdf}, written by Dominique Rodriguez.}
}
%--------------------------------------------------------------------------------------
By default the function in \verb+\psplot+ has to be described in
Reversed Polish Notation. The option \verb+algebraic+ allows you
to do this in the common algebraic notation. E.g.:

\begin{tabular}{l|l}
RPN & algebraic\\\hline
\verb+x ln+ & \verb+ln(x)+\\
\verb+x cos 2.71 x neg 10 div exp mul+ & \verb+cos(x)*2.71^(-x/10)+\\
\verb+1 x div cos 4 mul+ & \verb+4*cos(1/x)+\\
\verb+t cos t sin+ & \verb+cos(t)|sin(t)+
\end{tabular}

Setting the option \verb$algebraic$ to \verb$true$, allow the user
to describe all expression to be written in the classical
algebraic notation (infix notation). The four arithmetic
operations are obviously defined \verb$+-*/$, and also the
exponential operator \verb$^$. The natural priorities are used :
$3+4\times 5^5=3+(4\times (5^5))$, and by default the computation
is done from left to right. The following functions are defined :

\medskip
\begin{tabular}{ll}
\verb$sin$, \verb$cos$, \verb$tan$, \verb$acos$, \verb$asin$ & in radians\\
\verb$log$, \verb$ln$\\
\verb$ceiling$, \verb$floor$, \verb$truncate$, \verb$round$\\
\verb$sqrt$ & square root\\
\verb$abs$ & absolute value\\
\verb$fact$ & for the factorial\\
\verb$Sum$ & for building sums\\
\verb$IfTE$ & for an easy case structure
\end{tabular}

\medskip
These options can be used with \textbf{all} plot macros.

{\bfseries Using the option \verb+algebraic+ implies that all
angles have to be in radians! }

For the \verb+\parametricplot+ the two parts must be divided by the \verb+|+ character:

\begin{LTXexample}[width=2cm]
\begin{pspicture}(-0.5,-0.5)(0.5,0.5)
\parametricplot[algebraic,linecolor=red]{-3.14}{3.14}{cos(t)|sin(t)}
\end{pspicture}
\end{LTXexample}

\resetOptions
\bigskip
%\begin{LTXexample}[pos=t]
\psset{lly=-0.5cm}
\psgraph[trigLabels,dx=\psPi,dy=0.5,Dy=0.5]{->}(0,0)(-10,-1)(10,1){\linewidth}{6cm}
  \psset{algebraic,plotpoints=1000}
  \psplot[linecolor=yellow,linewidth=2pt]{-10}{10}{0.75*sin(x)*cos(x/2)}%
  \psplot[linecolor=red,showpoints=true,plotpoints=101]{-10}{10}{0.75*sin(x)*cos(x/2)}
\endpsgraph
%\end{LTXexample}

\bigskip
\begin{lstlisting}
\psset{lly=-0.5cm}
\psgraph[trigLabels,dx=\psPi,dy=0.5,Dy=0.5]{->}(0,0)(-10,-1)(10,1){\linewidth}{6cm}
  \psset{algebraic,plotpoints=1000}
  \psplot[linecolor=yellow,linewidth=2pt]{-10}{10}{0.75*sin(x)*cos(x/2)}%
  \psplot[linecolor=red,showpoints=true,plotpoints=101]{-10}{10}{0.75*sin(x)*cos(x/2)}
\endpsgraph
\end{lstlisting}


\bigskip
%\begin{LTXexample}[pos=t]
\bgroup
\psset{lly=-0.5cm}
\psgraph(0,-5)(18,3){15cm}{5cm}
  \psset{algebraic,plotpoints=501}
  \psplot[linecolor=yellow, linewidth=4\pslinewidth]{0.01}{18}{ln(x)}%
  \psplot[linecolor=red]{0.01}{18}{ln(x)}
  \psplot[linecolor=yellow,linewidth=4\pslinewidth]{0}{18}{3*cos(x)*2.71^(-x/10)}
  \psplot[linecolor=blue,showpoints=true,plotpoints=51]{0}{18}{3*cos(x)*2.71^(-x/10)}
\endpsgraph
\egroup
%\end{LTXexample}


\bigskip
\begin{lstlisting}
\psset{lly=-0.5cm}
\psgraph(0,-5)(18,3){15cm}{5cm}
  \psset{algebraic,plotpoints=501}
  \psplot[linecolor=yellow, linewidth=4\pslinewidth]{0.01}{18}{ln(x)}%
  \psplot[linecolor=red]{0.01}{18}{ln(x)}
  \psplot[linecolor=yellow,linewidth=4\pslinewidth]{0}{18}{3*cos(x)*2.71^(-x/10)}
  \psplot[linecolor=blue,showpoints=true,plotpoints=51]{0}{18}{3*cos(x)*2.71^(-x/10)}
\endpsgraph
\end{lstlisting}



\clearpage
%--------------------------------------------------------------------------------------
\subsubsection{Using the \texttt{Sum} function}
%--------------------------------------------------------------------------------------

Syntax: \verb+Sum(<index name>,<start>,<step>,<end>,<function>)+

Let's plot the first development of cosine with polynomials:
$\displaystyle\sum_{n=0}^{+\infty}\frac{(-1)^nx^{2n}}{n!}$.

\begin{center}
\bgroup
\psset{algebraic, plotpoints=501, yunit=3}
\def\getColor#1{\ifcase#1 black\or red\or magenta\or yellow\or green\or Orange\or blue\or
  DarkOrchid\or BrickRed\or Rhodamine\or OliveGreen\fi}
\begin{pspicture}(-7,-1.5)(7,1.5)
  \psclip{\psframe(-7,-1.5)(7,1.5)}
    \psplot{-7}{7}{cos(x)}
    \multido{\n=1+1}{10}{%
      \psplot[linewidth=1pt,linecolor=\getColor{\n}]{-7}{7}{%
         Sum(ijk,0,1,\n,(-1)^ijk*x^(2*ijk)/fact(2*ijk))}}
   \endpsclip
  \psaxes(0,0)(-7,-1.5)(7,1.5)
\end{pspicture}
\egroup
\end{center}
\begin{lstlisting}
\psset{algebraic, plotpoints=501, yunit=3}
\def\getColor#1{\ifcase#1 black\or red\or magenta\or yellow\or green\or Orange\or blue\or
  DarkOrchid\or BrickRed\or Rhodamine\or OliveGreen\fi}
\begin{pspicture}(-7,-1.5)(7,1.5)
  \psclip{\psframe(-7,-1.5)(7,1.5)}
    \psplot{-7}{7}{cos(x)}
    \multido{\n=1+1}{10}{%
      \psplot[linewidth=1pt,linecolor=\getColor{\n}]{-7}{7}{%
         Sum(ijk,0,1,\n,(-1)^ijk*x^(2*ijk)/fact(2*ijk))}}
   \endpsclip
  \psaxes(0,0)(-7,-1.5)(7,1.5)
\end{pspicture}
\end{lstlisting}

\clearpage
%--------------------------------------------------------------------------------------
\subsubsection{Using the \texttt{IfTE} function}
%--------------------------------------------------------------------------------------
Syntax: \verb+IfTE(<condition>,<true part>,<false part>)+

Nesting of several \verb+IfTE+ is possible and seen in the
following examples. A classic example is a piece-wise linear
function.

\begin{center}
\begin{pspicture}(-7.5,-2.5)(7.5,6)
  \psaxes{->}(0,0)(-7,-2)(7.5,6)[x,-90][y,0]
  \psset{algebraic=true, plotpoints=21,linewidth=2pt}
  \psplot[linecolor=blue]{-7.5}{7.5}{IfTE(x<-6,8+x,IfTE(x<0,-x/3,IfTE(x<3,2*x,9-x)))}
  \psplot[linecolor=red, plotpoints=101]{-7.5}{7.5}{%
      IfTE(2*x<-2^2*sqrt(9),7+x,IfTE(x<0,x^2/18-1,IfTE(x<3,2*x^2/3-1,8-x)))}%
\end{pspicture}
\end{center}


\begin{lstlisting}
\begin{pspicture}(-7.5,-2.5)(7.5,6)
  \psaxes{->}(0,0)(-7,-2)(7.5,6)[x,-90][y,0]
  \psset{algebraic=true, plotpoints=21,linewidth=2pt}
  \psplot[linecolor=blue]{-7.5}{7.5}{IfTE(x<-6,8+x,IfTE(x<0,-x/3,IfTE(x<3,2*x,9-x)))}
  \psplot[linecolor=red, plotpoints=101]{-7.5}{7.5}{%
      IfTE(2*x<-2^2*sqrt(9),7+x,IfTE(x<0,x^2/18-1,IfTE(x<3,2*x^2/3-1,8-x)))}%
\end{pspicture}
\end{lstlisting}

When you program a piece-wise defined function you must take care
that a plotting point must be put at each point where the
description changes. Use \verb+showpoints=true+ to see what's
going on when there is a problem. You are on the safe side when
you choose a big number for \verb+plotpoints+.

\clearpage


\begin{center}
\psset{unit=0.75}
\begin{pspicture}(-8,-8)(8,8)
  \psaxes{->}(0,0)(-8,-8)(8,8)[x,-90][y,0]
  \psset{plotpoints=1000,linewidth=1pt}
  \psplot[algebraic]{-8}{8}{ceiling(x)}
  \psplot[algebraic, linecolor=yellow]{-8}{8}{rand/(2^31-1)+x}
  \psplot[algebraic, linecolor=red]{-8}{8}{floor(x)}
  \psplot[algebraic, linecolor=blue]{-8}{8}{round(x)}
  \psplot[algebraic, linecolor=green]{-8}{8}{truncate(x)}
  \psplot[algebraic, linecolor=cyan]{-8}{8}{div(mul(4,x),7)}
  \psplot[algebraic, linecolor=gray]{-8}{8}{abs(x)+abs(x-3)-abs(5-5*x/7)}
  \psplot[algebraic, linecolor=gray]{-8}{8}{abs(3*cos(x)+1)}
  \psplot[algebraic, linecolor=magenta]{-8}{8}{floor(8*cos(x))}
\end{pspicture}
\end{center}

\begin{lstlisting}
\psset{unit=0.75}
\begin{pspicture}(-8,-8)(8,8)
  \psaxes{->}(0,0)(-8,-8)(8,8)[x,-90][y,0]
  \psset{plotpoints=1000,linewidth=1pt}
  \psplot[algebraic, linecolor=yellow]{-8}{8}{rand/(2^31-1)+x}
  \psplot[algebraic]{-8}{8}{ceiling(x)}
  \psplot[algebraic, linecolor=red]{-8}{8}{floor(x)}
  \psplot[algebraic, linecolor=blue]{-8}{8}{round(x)}
  \psplot[algebraic, linecolor=green]{-8}{8}{truncate(x)}
  \psplot[algebraic, linecolor=cyan]{-8}{8}{div(mul(4,x),7)}
  \psplot[algebraic, linecolor=gray]{-8}{8}{abs(x)+abs(x-3)-abs(5-5*x/7)}
  \psplot[algebraic, linecolor=gray]{-8}{8}{abs(3*cos(x)+1)}
  \psplot[algebraic, linecolor=magenta]{-8}{8}{floor(8*cos(x))}
\end{pspicture}
\end{lstlisting}


%--------------------------------------------------------------------------------------
\subsection{Plot style \texttt{bar} and option \texttt{barwidth}}
%--------------------------------------------------------------------------------------
This option allows you to draw bars for the data records. The
width of the bars is controlled by the option \verb+barwidth+,
which is set by default to value of \verb+0.25cm+, which is the
total width.

\def\barData{
0 0.03
1 0.11
2 0.28
3 0.84
4 6.70
5 8.55
6 8.77
7 11.09
8 7.18
9 6.20
10 5.78
11 4.19
12 2.37
13 2.26
14 1.68
15 1.03
16 1.37
17 1.34
18 0.92
19 0.67
20 0.87
21 1.20
22 1.98
23 3.99
24 5.08
25 5.17
26 5.78
27 4.44
28 0.11
}

\begin{LTXexample}[preset=\centering,pos=t]
\psset{xunit=.44cm,yunit=.3cm}
\begin{pspicture}(-2,-3)(29,13)
  \psaxes[axesstyle=axes,Ox=1466,Oy=0,Dx=4,Dy=2,xticksize=-6pt 0,
     ylabelFactor={\,\%}]{-}(29,12)
  \listplot[shadow=true,linecolor=blue,plotstyle=bar,barwidth=0.3cm,
     fillcolor=red,fillstyle=solid]{\barData}
  \rput{90}(-3,6.25){Amount}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[preset=\centering,pos=t]
\psset{xunit=.44cm,yunit=.3cm}
\begin{pspicture}(-2,-3)(29,13)
  \psaxes[axesstyle=axes,Ox=1466,Oy=0,Dx=4,Dy=2,ticksize=-4pt 0,
     ylabelFactor={\,\%}]{-}(29,12)
  \listplot[linecolor=blue,plotstyle=bar,barwidth=0.3cm,
     fillcolor=red,fillstyle=crosshatch]{\barData}
  \rput{90}(-3,6.25){Amount}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[preset=\centering,pos=t]
\psset{xunit=.44cm,yunit=.3cm}
\begin{pspicture}(-2,-3)(29,13)
  \psaxes[axesstyle=axes,Ox=1466,Oy=0,Dx=4,Dy=2,ticksize=-4pt 0,
     ylabelFactor={\,\%}]{-}(29,12)
  \listplot[linecolor=blue,plotstyle=bar,barwidth=0.3cm,
     fillcolor=red,fillstyle=vlines]{\barData}
  \listplot[showpoints=true]{\barData}
  \rput{90}(-3,6.25){Amount}
\end{pspicture}
\end{LTXexample}



%------------------------------------------------------------------------------------
\subsection{New options for \CMD{readdata}}
%------------------------------------------------------------------------------------


By default the macro \verb|\readdata| reads every data record,
which could be annoying when you have some text lines at top of
your data files or when there are more than 10000 records to read.


\verb|pstricks-add| defines two additional keys \verb|ignoreLines|
and \verb|nStep|, which allows you to ignore preceding lines, e.g.
\verb|ignoreLines=2|, or to read only a selected part of the data
records, e.g. \verb|nStep=10|, only every 10\textsuperscript{th}
record is saved.

\begin{lstlisting}
\readdata[ignoreLines=2]{\dataA}{stressrawdata.dat}
\readdata[nStep=10]{\dataA}{stressrawdata.dat}
\end{lstlisting}

The default value for  \verb+ignoreLines+ is $0$ and for \verb+nStep+ is $1$.
the following data file has two text lines which shall be ignored by the \verb+\readdata+ macro:

\begin{LTXexample}[width=4cm]
\begin{filecontents*}{pstricks-add-data9.dat}
some nonsense in this line ï¿½ï¿½ï¿½time forcex forcey
0 0.2
1 1
2 4
\end{filecontents*}
\readdata[ignoreLines=2]{\data}{pstricks-add-data9.dat}
\pspicture(2,4)
  \listplot[showpoints=true]{\data}
  \psaxes{->}(2,4)
\endpspicture
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{New options for \texttt{\textbackslash listplot}}
%--------------------------------------------------------------------------------------
By default the plot macros \verb|\dataplot|, \verb|\fileplot| and \verb|\listplot| plot every
data record. The package \verb|pst-plot-add| defines additional keys \verb|nStep, nStart, nEnd| and \verb|xStep, xStart, xEnd|, which allows
to plot only a selected part of the data records, e.g. \verb|nStep=10|. These "`n"'
options mark the number of the record to be plot ($0,1,2,...$) and the "`x"' ones the x-values of the data records.


\begin{center}
\begin{tabular}{l|l}
Name & Default setting\\\hline
\verb|nStart| &  \verb|1|\\
\verb|nEnd|   &  \verb|{}|\\
\verb|nStep|  &  \verb|1|\\
\verb|xStart| &  \verb|{}|\\
\verb|xEnd|   &  \verb|{}|\\
\verb|yStart| &  \verb|{}|\\
\verb|yEnd|   &  \verb|{}|\\
\verb|xStep|  &  \verb|0|\\
\verb|plotNo|  &  \verb|1|\\
\verb|plotNoMax|  &  \verb|1|\\
\verb|ChangeOrder|  &  \verb|false|\\
(\verb+plotstyle+)& \verb+line+
\end{tabular}
\end{center}

These new options are only available
for the \verb|\listplot| macro, which is not a real limitation, because all data records can be read
from a file with the \verb|\readdata| macro (see example files or \cite{dtk02.2:jackson.voss:plot-funktionen}):
\begin{lstlisting}[style=syntax]
\readdata[nStep=10]{\data}{/home/voss/data/data1.dat}
\end{lstlisting}

The use \verb|nStep| and \verb|xStep| options only make real sense
when also using the option \verb|plotstyle=dots|. Otherwise the
coordinates are connected by a line as usual. Also the
\verb|xStep| option needs increasing x values. Note that
\verb+nStep+ can be used for \verb+\readdata+ and for
\verb+\listplot+. If used in both macros then the effect is
multiplied, e.g. \verb+\readdata+ with \verb+nStep=5+ and
\verb+\listplot+ with \verb+nStep=10+ means, that only every
50\textsuperscript{th} data record is read and plotted.

When both, \verb|x/yStart/End| are defined then the values are also compared with
both values.


%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{nStep/xStep}}
%--------------------------------------------------------------------------------------

The datafile \verb|data.dat| contains $1000$ data records. The thin blue line is the plot
of all records with the plotstyle option \verb|curve|.

\resetOptions
\begin{LTXexample}[preset=\centering,pos=t]
\readdata{\data}{data.dat}
\psset{xunit=12.5cm,yunit=0.2mm}
\begin{pspicture}(-0.080,-30)(1,270)
\pstScalePoints(1,1){1000 div}{1000 div}
\psaxes[Dx=200,dx=2.5cm,Dy=100,ticksize=0 5pt,tickstyle=inner,
    subticks=10,ylabelFactor=\cdot10^3,dy=2cm](0,0)(1,250)
\listplot[nStep=50,linewidth=3pt,linecolor=red,plotstyle=dots]{\data}
\listplot[linewidth=1pt,linecolor=blue]{\data}
\end{pspicture}
\end{LTXexample}


\clearpage

%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{nStart/xStart}}
%--------------------------------------------------------------------------------------

\begin{LTXexample}[preset=\centering,pos=t]
\readdata{\data}{data.dat}
\psset{xunit=12.5cm,yunit=0.2mm}
\begin{pspicture}(-0.080,-30)(1,270)
\pstScalePoints(1,1){1000 div}{1000 div}
\psaxes[Dx=200,dx=2.5cm,Dy=100,ticksize=0 5pt,tickstyle=inner,
    subticks=10,ylabelFactor=\cdot10^3,dy=2cm](0,0)(1,250)
\listplot[nStart=200,linewidth=3pt,
    linecolor=blue,plotstyle=dots]{\data}
\listplot[linewidth=1pt,linecolor=blue]{\data}
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{nEnd/xEnd}}
%--------------------------------------------------------------------------------------

\begin{LTXexample}[preset=\centering,pos=t]
\readdata{\data}{data.dat}
\psset{xunit=12.5cm,yunit=0.2mm}
\begin{pspicture}(-0.080,-30)(1,270)
\pstScalePoints(1,1){1000 div}{1000 div}
\psaxes[axesstyle=frame,Dx=200,dx=2.5cm,Dy=100,ticksize=0 5pt,tickstyle=inner,
    subticks=10,ylabelFactor=\cdot10^3,dy=2cm](0,0)(1,250)
\listplot[nStart=200,linewidth=3pt,
    linecolor=blue]{\data}
\listplot[linewidth=1pt,linecolor=blue]{\data}
\end{pspicture}
\end{LTXexample}


\clearpage

%--------------------------------------------------------------------------------------
\subsubsection{Example for all new options}
%--------------------------------------------------------------------------------------

\begin{LTXexample}[preset=\centering,pos=t]
\readdata{\data}{data.dat}
\psset{xunit=12.5cm,yunit=0.2mm}
\begin{pspicture}(-0.080,-30)(1,270)
\pstScalePoints(1,1){1000 div}{1000 div}
\psaxes[axesstyle=frame,Dx=200,dx=2.5cm,Dy=100,,ticksize=0 5pt,tickstyle=inner,
    ylabelFactor=\cdot10^3,dy=2cm](0,0)(1,250)
\listplot[nStart=200, nEnd=800, nStep=50,
    linewidth=3pt,linecolor=blue,plotstyle=dots]{\data}
\end{pspicture}
\end{LTXexample}



%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{xStart}}
%--------------------------------------------------------------------------------------

This example shows the use of the same plot with different units
and different \verb|xStart| value. The blue curve is the original
plot of the data records. To show the important part of the curve
there is another one plotted with a greater \verb|yunit| and a
start value of \verb|xStart=0.35|. This makes it possible to have
a kind of zoom to the original graphic.

\begin{center}
\psset{xunit=10cm, yunit=0.01cm}
\readdata{\data}{data3.dat}
\begin{pspicture}(-0.1,-100)(1.5,700.0)
  \psaxes[Dx=0.25,Dy=100,dy=100\psyunit,ticksize=-4pt 0,%
    labelFontSize={\scriptstyle}]{->}(0,0)(0,-100)(1.4,520)
  \uput[0](1.4,0){\textsf{t [s]}}
  \rput(-0.125,200){\psrotateleft{\small flow [ml/s]}}
  \listplot[linewidth=2pt, linecolor=blue]{\data}
  \rput(0.4,300){
    \pscustom[yunit=0.04cm, linewidth=1pt]{%
      \listplot[xStart=0.355]{\data}
      \psline(1,-2.57)(1,0)(0.355,0)
      \fill[fillstyle=hlines,fillcolor=gray,hatchwidth=0.4pt,hatchsep=1.5pt,hatchcolor=red]%
      \psline[linewidth=0.5pt]{->}(0.7,0)(1.05,0)
    }%
  }
  \psline[linewidth=.01]{->}(0.75,300)(0.4,20)
  \psline[linewidth=.01]{->}(1,290)(1.1,440)
  \rput(1.1,470){\footnotesize leak volume}
  \psline[linewidth=.01]{->}(0.78,200)(1,100)
  \rput[l](1.02,100){\footnotesize closing volume}
\end{pspicture}
\end{center}


\begin{lstlisting}
\psset{xunit=10cm, yunit=0.01cm}
\readdata{\data}{data3.dat}
\begin{pspicture}(-0.1,-100)(1.5,700.0)
  \psaxes[Dx=0.25,Dy=100,dy=100\psyunit,ticksize=-4pt 0,%
    labelFontSize={\scriptstyle}]{->}(0,0)(0,-100)(1.4,520)
  \uput[0](1.4,0){\textsf{t [s]}}
  \rput(-0.125,200){\psrotateleft{\small flow [ml/s]}}
  \listplot[linewidth=2pt, linecolor=blue]{\data}
  \rput(0.4,300){
    \pscustom[yunit=0.04cm, linewidth=1pt]{%
      \listplot[xStart=0.355]{\data}
      \psline(1,-2.57)(1,0)(0.355,0)
      \fill[fillstyle=hlines,fillcolor=gray,hatchwidth=0.4pt,hatchsep=1.5pt,hatchcolor=red]%
      \psline[linewidth=0.5pt]{->}(0.7,0)(1.05,0)
    }%
  }
  \psline[linewidth=.01]{->}(0.75,300)(0.4,20)
  \psline[linewidth=.01]{->}(1,290)(1.1,440)
  \rput(1.1,470){\footnotesize leak volume}
  \psline[linewidth=.01]{->}(0.78,200)(1,100)
  \rput[l](1.02,100){\footnotesize closing volume}
\end{pspicture}
\end{lstlisting}



\resetOptions
%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{yStart}/\texttt{yEnd}}
%--------------------------------------------------------------------------------------

\begin{LTXexample}[preset=\centering,pos=t]
\readdata{\data}{data.dat}
\psset{xunit=12.5cm,yunit=0.2mm}
\begin{pspicture}(-0.080,-30)(1,270)
\pstScalePoints(1,1){1000 div}{1000 div}
\psaxes[axesstyle=frame,Dx=200,dx=2.5cm,Dy=100,ticksize=0 5pt,tickstyle=inner,
    ylabelFactor=\cdot10^3,dy=2cm](0,0)(1,250)
  \psset{linewidth=0.1pt, linestyle=dashed,linecolor=red}
  \psline(0,40)(1,40)
  \psline(0,175)(1,175)
  \listplot[yStart=40000, yEnd=175000,linewidth=3pt,linecolor=blue,plotstyle=dots]{\data}
\end{pspicture}
\end{LTXexample}



%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{plotNo/plotNoMax}}
%--------------------------------------------------------------------------------------
By default the plot macros expect \verb+x|y+ data records, but
when having data files with multiple values for y, like:
\begin{lstlisting}[style=syntax]
x y1 y2 y3 y4 ... yMax
x y1 y2 y3 y4 ... yMax
...
\end{lstlisting}

you can select the y value which should be plotted. The option \verb+plotNo+ marks the plotted
value (default $1$) and the option \verb+plotNoMax+ tells \verb+pst-plot+ how many $y$ values are
present. There are no real restrictions in the maximum number for \verb+plotNoMax+.

We have the following data file:
\begin{lstlisting}[style=syntax]
[% file data.dat
0    0    3.375    0.0625
10    5.375    7.1875    4.5
20    7.1875    8.375    6.25
30    5.75    7.75    6.6875
40    2.1875    5.75    5.9375
50    -1.9375    2.1875    4.3125
60    -5.125    -1.8125    0.875
70    -6.4375    -5.3125    -2.6875
80    -4.875    -7.1875    -4.875
90    0    -7.625    -5.625
100    5.5    -6.3125    -5.8125
110    6.8125    -2.75    -4.75
120    5.25    2.875    -0.75
]%
\end{lstlisting}

\noindent which holds data records for multiple plots (\verb+x y1 y2 y3+). This can be plotted
without any modification to the data file:

\begin{LTXexample}[preset=\centering,pos=t]
\readdata\Data{dataMul.dat}
\psset{xunit=0.1cm, yunit=0.5cm,lly=-0.5cm}
\begin{pspicture}(0,-7.5)(150,10)
\psaxes[Dx=10,Dy=2.5]{->}(0,0)(0,-7.5)(150,7.5)[$\mathbf{x}$,-90][$\mathbf{y}$,0]
\psset{linewidth=2pt,plotstyle=curve}
\listplot[linecolor=green,plotNo=1,plotNoMax=3]{\Data}
\listplot[linecolor=red,plotNo=2,plotNoMax=3]{\Data}
\listplot[linecolor=blue,plotNo=3,plotNoMax=3]{\Data}
\end{pspicture}
\end{LTXexample}

\clearpage


%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{changeOrder}}
%--------------------------------------------------------------------------------------
It is only possible to fill the region between two listplots with
\verb+\pscustom+ if one of them has the values in reverse order.
Otherwise we do not get a closed path. With the option
\verb+ChangeOrder+ the values are used in reverse order:

\begin{LTXexample}[pos=t,preset=\centering]
\begin{filecontents*}{test.dat}
  0 3 8
  2 4 7
  5 5 5.5
  7 3.5 5
  10 2 9
\end{filecontents*}
\psset{lly=-.5cm}
\begin{psgraph}[axesstyle=frame,ticklinestyle=dotted,ticksize=0 10](0,0)(10,10){4in}{2in}%
   \readdata{\data}{test.dat}%
   \pscustom[fillstyle=solid,fillcolor=blue!40]{%
     \listplot[plotNo=2,plotNoMax=2]{\data}%
     \listplot[plotNo=1,plotNoMax=2,ChangeOrder]{\data}}
\end{psgraph}
\end{LTXexample}


\clearpage
%--------------------------------------------------------------------------------------
\subsubsection{Example for \texttt{plotstyle}}
%--------------------------------------------------------------------------------------
The \verb+plotstyle+ option is defined in the package \verb+pst-plot+, but its value
\verb+LSM+ (\textbf{L}east \textbf{S}quare \textbf{Method}) is only valid for the
\verb+pstricks-add+ package. Instead of plotting the data records as dots or a line,
the \verb+listplot+ macro calculates the values for a line $y=v\cdot x+u$ which fits
best all data records.

\bgroup
\centering
\begin{filecontents*}{LSM.dat}
0 1 1 3 2.8 4 3 2.9 2 5 4 4 5 5.5 6 8.2 8 7
\end{filecontents*}
\psset{lly=-.5cm}
\readdata{\data}{LSM.dat}
\begin{psgraph}[arrows=->](0,0)(0,0)(8,8){.5\textwidth}{!}
  \listplot[plotstyle=dots]{\data}
  \listplot[plotstyle=LSM,linecolor=red]{\data}
\end{psgraph}
\egroup


\begin{lstlisting}
\begin{filecontents*}{LSM.dat}
0 1 1 3 2.8 4 3 2.9 2 5 4 4 5 5.5 6 8.2 8 7
\end{filecontents*}
\psset{lly=-.5cm}
\readdata{\data}{LSM.dat}
\begin{psgraph}[arrows=->](0,0)(0,0)(8,8){.5\textwidth}{!}
  \listplot[plotstyle=dots]{\data}
  \listplot[§\ON§plotstyle§\OFF§=§\ON§LSM§\OFF§,linecolor=red]{\data}
\end{psgraph}
\end{lstlisting}


The macro looks for the lowest and biggest x-value and draws the
line for this interval. It is possible to pass other values to the
macro by setting the \verb+xStart+ and/or \verb+xEnd+ options.
They are preset with an empty value \verb+{}+.

\bgroup
\centering
\begin{filecontents*}{LSM.dat}
0 1 1 3 2.8 4 3 2.9 2 5 4 4 5 5.5 6 8.2 8 7
\end{filecontents*}
\readdata{\data}{LSM.dat}
\psset{lly=-1.75cm}
\begin{psgraph}[arrows=->](0,0)(0,0)(8,8){.5\textwidth}{!}
  \listplot[plotstyle=dots]{\data}
  \listplot[PstDebug=1,plotstyle=LSM,xStart=-0.5,xEnd=8.5,linecolor=red]{\data}
\end{psgraph}
\egroup

\begin{lstlisting}
\begin{filecontents*}{LSM.dat}
0 1 1 3 2.8 4 3 2.9 2 5 4 4 5 5.5 6 8.2 8 7
\end{filecontents*}
\readdata{\data}{LSM.dat}
\psset{lly=-1.75cm}
\begin{psgraph}[arrows=->](0,0)(0,0)(8,8){.5\textwidth}{!}
  \listplot[plotstyle=dots]{\data}
  \listplot[§\ON§PstDebug§\OFF§=1,plotstyle=§\ON§LSM§\OFF§,§\ON§xStart§\OFF§=-0.5,§\ON§xEnd§\OFF§=8.5,linecolor=red]{\data}
\end{psgraph}
\end{lstlisting}


With \verb+PstDebug=1+ one gets the equation $y=v\cdot x+u$
printed, beginning at the position (0|-50pt). This cannot be
changed, because it is only for some kind of debugging. Pay
attention for the correct \verb+xStart+- and \verb+xEnd+-values,
when you use the \verb+\pstScalePoints+-Macro. In the following
example we use an x-interval from 0 to 3 to plot the values; first
we subtract 0.003 from all x-values and then scale them with
10000. This is not taken into account for the \verb+xStart+- and
\verb+xEnd+-values.


\bgroup
\centering
\begin{filecontents*}{LSM.dat}
0.003298697 1.397785583
0.003193358 1.615489564
0.003094538 2.044019006
0.003001651 2.259240127
\end{filecontents*}
\readdata{\data}{LSM.dat}
\pstScalePoints(10000,1){ 0.003 sub }{}
\psset{lly=-1.75cm}
\psgraph[arrows=->,Ox=0.0030,Dx=0.0001,dx=\psxunit](0,0)(3.2,3){10cm}{5cm}
  \listplot[showpoints=true,linewidth=1pt,linecolor=blue]{\data}
  \listplot[PstDebug=1,plotstyle=LSM,linewidth=0.1pt,linestyle=dashed,%
    xStart=-0.25,xEnd=3.3]{\data}
\endpsgraph
\egroup

\begin{lstlisting}
\begin{filecontents*}{LSM.dat}
0.003298697 1.397785583
0.003193358 1.615489564
0.003094538 2.044019006
0.003001651 2.259240127
\end{filecontents*}
\readdata{\data}{LSM.dat}
§\ON§\pstScalePoints§\OFF§(10000,1){ 0.003 sub }{}
\psset{lly=-1.75cm}
\psgraph[arrows=->,Ox=0.0030,Dx=0.0001,dx=\psxunit](0,0)(3.2,3){10cm}{5cm}
  \listplot[showpoints=true,linewidth=1pt,linecolor=blue]{\data}
  \listplot[PstDebug=1,plotstyle=§\ON§LSM§\OFF§,linewidth=0.1pt,linestyle=dashed,%
    xStart=-0.25,xEnd=3.3]{\data}
\endpsgraph
\end{lstlisting}


%--------------------------------------------------------------------------------------
\section{Polar plots}
%--------------------------------------------------------------------------------------

With the option \verb+polarplot=false|true+ it is possible to use \verb+\psplot+
in polar mode:
\begin{lstlisting}[style=syntax]
\psplot[polarplot=true,...]{<start angle>}{<end angle>}{<r(alpha)>}
\end{lstlisting}

The equation in PostScript code is interpreted as a function $r=f(\alpha)$, e.g. for the
circle with radius 1 as $r=\sqrt{\sin^2x+\cos^2x}$:

\begin{lstlisting}[style=syntax]
x sin dup mul x cos dup mul add sqrt
\end{lstlisting}


\medskip
\begin{LTXexample}[width=6cm]
\resetOptions
\psset{plotpoints=200,unit=0.75}
\begin{pspicture}*(-5,-5)(3,3)
  \psaxes[arrowlength=1.75,ticksize=2pt,%
    labelFontSize=\scriptstyle,%
    linewidth=0.17mm]{->}(0,0)(-4.99,-4.99)(3,3)
  \rput[Br](3,-.35){$x$}
  \rput[tr](-.15,3){$y$}
  \rput[Br](-.15,-.35){$0$}
  \psset{linewidth=.35mm,polarplot=true}
  \psplot[linecolor=red]{140}{310}{3 neg x sin mul x cos mul x sin 3 exp x cos 3 exp add div}
  \psplot[linecolor=cyan]{140}{310}{6 neg x sin mul x cos mul x sin 3 exp x cos 3 exp add div}
  \psplot[linecolor=blue]{140}{310}{9 neg x sin mul x cos mul x sin 3 exp x cos 3 exp add div}
\end{pspicture}
\end{LTXexample}



\medskip
\begin{LTXexample}[width=5cm]
\resetOptions
\psset{plotpoints=200,unit=1}
\begin{pspicture}(-2.5,-2.5)(2.5,2.5)% Ulrich Dirr
 \psaxes[arrowlength=1.75,%
    ticksize=2pt,linewidth=0.17mm]{->}(0,0)(-2.5,-2.5)(2.5,2.5)
  \rput[Br](2.5,-.35){$x$}
  \rput[tr](-.15,2.5){$y$}
  \rput[Br](-.15,-.35){$0$}
  \psset{linewidth=.35mm,plotstyle=curve,polarplot=true}
  \psplot[linecolor=red]{0}{360}{x cos 2 mul x sin mul}
  \psplot[linecolor=green]{0}{360}{x cos 3 mul x sin mul}
  \psplot[linecolor=blue]{0}{360}{x cos 4 mul x sin mul}
\end{pspicture}
\end{LTXexample}



\medskip
\begin{LTXexample}[width=8cm]
\psset{plotpoints=200,unit=0.5}
\begin{pspicture}(-8.5,-8.5)(9,9)% Ulrich Dirr
\psaxes[Dx=2,dx=2,Dy=2,dy=2,arrowlength=1.75,
    ticksize=2pt,linewidth=0.17mm]{->}(0,0)(-8.5,-8.5)(9,9)
\rput[Br](9,-.7){$x$}
\rput[tr](-.3,9){$y$}
\rput[Br](-.3,-.7){$0$}
%
\psset{linewidth=.35mm,plotstyle=curve,polarplot=true}
\psplot[linecolor=blue]{0}{720}{8 2.5 x mul sin mul}
\end{pspicture}
\end{LTXexample}


\resetOptions

%--------------------------------------------------------------------------------------
\section{\CMD{pstScalePoints}}
%--------------------------------------------------------------------------------------
The syntax is
\begin{lstlisting}[style=syntax]
\pstScalePoints(xScale,xScale){xPS}{yPS}
\end{lstlisting}

\verb+xScale,yScale+ are decimal values used as scaling factors,
the \verb+xPs+ and \verb+yPS+ are additional PostScript code
applied to the x- and y-values of the data records. This macro is
only valid for the \CMD{listplot} macro!

\resetOptions
\begin{LTXexample}[width=6cm]
\def\data{%
  0 0 1 3 2 4 3 1
  4 2 5 3 6 6 }
\begin{pspicture}(-0.5,-1)(6,6)
  \psaxes{->}(0,0)(6,6)
  \listplot[showpoints=true,%
    linecolor=red]{\data}
  \pstScalePoints(1,0.5){}{3 add}
  \listplot[showpoints=true,%
    linecolor=blue]{\data}
\end{pspicture}
\end{LTXexample}

\bigskip
\verb+\pstScalePoints(1,0.5){}{3 add}+ means that \textbf{first} the value $3$ is added
to the $y$ values and \textbf{second} this value is scaled with the factor $0.5$.
As seen for the blue line for $x=0$ we get $y(0)=(0+3)\cdot 0.5=1.5$.

Changes with \verb+\pstScalePoints+ are always global to all following \verb+\listplot+
macros. This is the reason why it is a good idea to reset the values at the end of the
\verb+pspicture+ environment.

\begin{lstlisting}[style=syntax]
\pstScalePoints(1,1){}{}
\end{lstlisting}


\clearpage
%--------------------------------------------------------------------------------------
\part{New commands and environments}
%--------------------------------------------------------------------------------------

%--------------------------------------------------------------------------------------
\section[\texttt{psCancel} environment]{\texttt{psCancel} environment\footnotemark}
%--------------------------------------------------------------------------------------
\footnotetext{Thanks to by Stefano Baroni} This macro works like
the \CMD{cancel} macro from the package of the same name but it
allows as argument any contents, not only letters but also a
complex graphic.

\begin{lstlisting}[style=syntax]
\psCancel*[line options]{contents}%
\end{lstlisting}

All optional arguments for lines and boxes are valid and can be
used in the usual way. The star option fills the underlying box
rectangle with the linecolor. This can be transparent if
\texttt{opacity} is set to a value less than 1. This can be used
in presentation to strike out words, equations, and graphic
objects. Lines can also be transparent when the option
\texttt{strokeopacity} is used.

\begingroup
\psCancel{A} \psCancel[linecolor=red]{Tikz :-)} \quad
\psCancel[linecolor=blue,doubleline=true]{%
  \readdata{\data}{demo1.dat}
  \psset{shift=*,xAxisLabel=x-Axis,yAxisLabel=y-Axis,llx=-13mm,lly=-7mm,
      xAxisLabelPos={c,-1},yAxisLabelPos={-7,c}}
  \pstScalePoints(1,0.00000001){}{}
  \begin{psgraph}[axesstyle=frame,xticksize=0 7.5,yticksize=0 25,subticksize=1,
     ylabelFactor=\cdot 10^8,Dx=5,Dy=1,xsubticks=2](0,0)(25,7.5){5.5cm}{5cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
  \end{psgraph}} \qquad% end of Cancel
\psCancel[linewidth=3pt,linecolor=red,
    strokeopacity=0.5]{\tabular[b]{c}first line\\second line\endtabular}\quad
\psCancel*[linecolor=red!50,opacity=0.5]{\tabular[b]{c}first line\\second line\endtabular}
\quad
\psCancel*[linecolor=blue!30,opacity=0.5]{%
  \readdata{\data}{demo1.dat}
  \psset{shift=*,xAxisLabel=x-Axis,yAxisLabel=y-Axis,llx=-15mm,lly=-7mm,urx=1mm,
      xAxisLabelPos={c,-1},yAxisLabelPos={-7,c}}
  \pstScalePoints(1,0.00000001){}{}
  \begin{psgraph}[axesstyle=frame,xticksize=0 7.5,yticksize=0 25,subticksize=1,
     ylabelFactor=\cdot 10^8,Dx=5,Dy=1,xsubticks=2](0,0)(25,7.5){5.5cm}{5cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
  \end{psgraph}} \quad% end of Cancel
\psCancel[linewidth=4pt,strokeopacity=0.5]{\parbox{8cm}{\[
  \binom{x_R}{y_R} = \underbrace{r\vphantom{\binom{A}{B}}}_{\text{Scaling}}\cdot
    \underbrace{\begin{pmatrix}
        \sin\gamma & -\cos\gamma \\
      \cos \gamma & \sin \gamma \\
      \end{pmatrix}}_{\text{Rotation}} \binom{x_K}{y_K} +
  \underbrace{\binom{t_x}{t_y}}_{\text{Translation}} \]} }% end of psCancel
\endgroup

\bigskip
\begin{lstlisting}
\psCancel{A} \psCancel[linecolor=red]{Tikz :-)} \quad
\psCancel[linecolor=blue,doubleline=true]{%
  \readdata{\data}{demo1.dat}
  \psset{shift=*,xAxisLabel=x-Axis,yAxisLabel=y-Axis,llx=-13mm,lly=-7mm,
      xAxisLabelPos={c,-1},yAxisLabelPos={-7,c}}
  \pstScalePoints(1,0.00000001){}{}
  \begin{psgraph}[axesstyle=frame,xticksize=0 7.5,yticksize=0 25,subticksize=1,
     ylabelFactor=\cdot 10^8,Dx=5,Dy=1,xsubticks=2](0,0)(25,7.5){5.5cm}{5cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
  \end{psgraph}} \qquad% end of Cancel
\psCancel[linewidth=3pt,linecolor=red,
    strokeopacity=0.5]{\tabular[b]{c}first line\\second line\endtabular}\quad
\psCancel*[linecolor=red!50,opacity=0.5]{\tabular[b]{c}first line\\second line\endtabular}
\quad
\psCancel*[linecolor=blue!30,opacity=0.5]{%
  \readdata{\data}{demo1.dat}
  \psset{shift=*,xAxisLabel=x-Axis,yAxisLabel=y-Axis,llx=-15mm,lly=-7mm,urx=1mm,
      xAxisLabelPos={c,-1},yAxisLabelPos={-7,c}}
  \pstScalePoints(1,0.00000001){}{}
  \begin{psgraph}[axesstyle=frame,xticksize=0 7.5,yticksize=0 25,subticksize=1,
     ylabelFactor=\cdot 10^8,Dx=5,Dy=1,xsubticks=2](0,0)(25,7.5){5.5cm}{5cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
  \end{psgraph}} \quad% end of Cancel
\psCancel[linewidth=4pt,strokeopacity=0.5]{\parbox{8cm}{\[
  \binom{x_R}{y_R} = \underbrace{r\vphantom{\binom{A}{B}}}_{\text{Scaling}}\cdot
    \underbrace{\begin{pmatrix}
        \sin\gamma & -\cos\gamma \\
      \cos \gamma & \sin \gamma \\
      \end{pmatrix}}_{\text{Rotation}} \binom{x_K}{y_K} +
  \underbrace{\binom{t_x}{t_y}}_{\text{Translation}} \]} }% end of psCancel
\end{lstlisting}


%--------------------------------------------------------------------------------------
\section{\texttt{psgraph} environment}
%--------------------------------------------------------------------------------------
This new environment does the scaling, it expects as parameter the values (without units!) for the
coordinate system and the values of the physical width and height (with units!). The syntax is:

\begin{lstlisting}[style=syntax]
\psgraph[<axes options>]{<arrows>}%
    (xOrig,yOrig)(xMin,yMin)(xMax,yMax){xLength}{yLength}
...
\endpsgraph

\begin{psgraph}[<axes options>]{<arrows>}%
    (xOrig,yOrig)(xMin,yMin)(xMax,yMax){xLength}{yLength}
...
\end{psgraph}
\end{lstlisting}

where the options are valid \textbf{only} for the the \verb+\psaxes+ macro. The first
two arguments have the usual \verb+PSTricks+ behaviour.
\begin{itemize}
  \item if \verb+(xOrig,yOrig)+ is missing, it is substituted to  \verb+(xMin,xMax)+;
  \item if \verb+(xOrig,yOrig)+ \textbf{and} \verb+(xMin,yMin)+ are missing, they are both
  substituted to \verb+(0,0)+.
\end{itemize}

The y-length maybe given as !, when the macro uses the same unit
as for the x-axis.

%-----------------------------------------------------------------------------

\begin{center}
\readdata{\data}{demo1.dat}
\pstScalePoints(1,0.000001){}{}% (x,y){additional x operator}{y op}
\psset{llx=-1cm,lly=-1cm}
\begin{psgraph}[axesstyle=frame,xticksize=0 759,yticksize=0 25,%
    subticks=0,ylabelFactor=\cdot 10^6,
    Dx=5,dy=100\psyunit,Dy=100](0,0)(25,750){10cm}{6cm} % parameters
   \listplot[linecolor=red,linewidth=2pt,showpoints=true]{\data}
\end{psgraph}
\end{center}

\begin{lstlisting}
\readdata{\data}{demo1.dat}
\pstScalePoints(1,0.000001){}{}% (x,y){additional x operator}{y op}
\psset{llx=-1cm,lly=-1cm}
§\ON§\begin{psgraph}§\OFF§[axesstyle=frame,xticksize=0 759,yticksize=0 25,%
    subticks=0,ylabelFactor=\cdot 10^6,
    Dx=5,dy=100\psyunit,Dy=100](0,0)(25,750){10cm}{6cm} % parameters
   \listplot[linecolor=red,linewidth=2pt,showpoints=true]{\data}
§\ON§\end{psgraph}§\OFF§
\end{lstlisting}

%-----------------------------------------------------------------------------

In the following example, the y unit gets the same value as the one for the x-axis.
\begin{center}
\psset{llx=-1cm,lly=-0.5cm,ury=0.5cm}
\begin{psgraph}(0,0)(5,3){6cm}{!} % x-y-axis with same unit
   \psplot[linecolor=red,linewidth=1pt]{0}{5}{x dup mul 10 div}
\end{psgraph}
\end{center}

\begin{lstlisting}
\psset{llx=-1cm,lly=-0.5cm,ury=0.5cm}
\begin{psgraph}(0,0)(5,3){6cm}§\ON§{!}§\OFF§ % x-y-axis with same unit
   \psplot[linecolor=red,linewidth=1pt]{0}{5}{x dup mul 10 div}
\end{psgraph}
\end{lstlisting}

%-----------------------------------------------------------------------------

\begin{center}
\readdata{\data}{demo1.dat}
\psset{xAxisLabel=x-Axis,yAxisLabel=y-Axis,llx=-.5cm,lly=-1cm,lly=-1cm,ury=0.5cm,
   xAxisLabelPos={c,-1},yAxisLabelPos={-7,c}}
\pstScalePoints(1,0.00000001){}{}
\begin{psgraph}[axesstyle=frame,xticksize=0 7.5,yticksize=0 25,subticksize=1,
     ylabelFactor=\cdot 10^8,Dx=5,Dy=1,xsubticks=2](0,0)(25,7.5){5.5cm}{5cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
\end{psgraph}
\end{center}

\begin{lstlisting}
\readdata{\data}{demo1.dat}
\psset{§\ON§xAxisLabel§\OFF§=x-Axis,§\ON§yAxisLabel§\OFF§=y-Axis,llx=-.5cm,lly=-1cm,ury=0.5cm,
   §\ON§xAxisLabelPos§\OFF§={c,-1},§\ON§yAxisLabelPos§\OFF§={-7,c}}
\pstScalePoints(1,0.00000001){}{}
\begin{psgraph}[axesstyle=frame,xticksize=0 7.5,yticksize=0 25,subticksize=1,
     §\ON§ylabelFactor§\OFF§=\cdot 10^8,Dx=5,Dy=1,xsubticks=2](0,0)(25,7.5){5.5cm}{5cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
\end{psgraph}
\end{lstlisting}

%-----------------------------------------------------------------------------

\begin{LTXexample}[pos=t,preset=\centering]
\readdata{\data}{demo1.dat}
\psset{llx=-0.5cm,lly=-1cm}
\pstScalePoints(1,0.000001){}{}
\psgraph[arrows=->,Dx=5,dy=200\psyunit,Dy=200,subticks=5,ticksize=-10pt 0,
    tickwidth=0.5pt,subtickwidth=0.1pt](0,0)(25,750){5.5cm}{5cm}
\listplot[linecolor=red,linewidth=2pt,showpoints=true,plotstyle=LineToYAxis]{\data}
\endpsgraph
\end{LTXexample}

%-----------------------------------------------------------------------------

\begin{center}
\readdata{\data}{demo1.dat}
\pstScalePoints(1,0.2){}{log}
\psset{lly=-0.75cm}
\psgraph[ylogBase=10,Dx=5,Dy=1,subticks=5](0,0)(25,2){12cm}{4cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
\endpsgraph
\end{center}

\begin{lstlisting}
\readdata{\data}{demo1.dat}
\pstScalePoints(1,0.2){}{log}
\psset{lly=-0.75cm}
\psgraph[§\ON§ylogBase§\OFF§=10,Dx=5,Dy=1,subticks=5](0,0)(25,2){12cm}{4cm}
  \listplot[linecolor=red, linewidth=2pt, showpoints=true]{\data}
\endpsgraph
\end{lstlisting}

%-----------------------------------------------------------------------------

\begin{LTXexample}[pos=t,preset=\centering]
\readdata{\data}{demo0.dat}
\psset{lly=-0.75cm,ury=0.5cm}
\pstScalePoints(1,1){}{log}
\begin{psgraph}[arrows=->,Dx=0.5,ylogBase=10,Oy=-1,xsubticks=10,%
     ysubticks=2](0,-3)(3,1){12cm}{4cm}
  \listplot[linecolor=red,linewidth=2pt,showpoints=true,plotstyle=LineToXAxis]{\data}
\end{psgraph}
\end{LTXexample}


\begin{LTXexample}[pos=t,preset=\centering]
\psset{lly=-0.75cm,ury=0.5cm}
\readdata{\data}{demo0.dat}
\pstScalePoints(1,1){}{log}
\psgraph[arrows=->,Dx=0.5,ylogBase=10,Oy=-1,subticks=4](0,-3)(3,1){6cm}{3cm}
  \listplot[linecolor=red,linewidth=2pt,showpoints=true,plotstyle=LineToXAxis]{\data}
\endpsgraph
\end{LTXexample}


%-----------------------------------------------------------------------------
\begin{center}
\readdata{\data}{demo2.dat}%
\readdata{\dataII}{demo3.dat}%
\pstScalePoints(1,1){1989 sub}{}
\psset{llx=-0.5cm,lly=-1cm, xAxisLabel=Year,yAxisLabel=Whatever,%
     xAxisLabelPos={c,-0.4in},yAxisLabelPos={-0.4in,c}}
\psgraph[axesstyle=frame,Dx=2,Ox=1989,subticks=2](0,0)(12,6){4in}{2in}%
  \listplot[linecolor=red,linewidth=2pt]{\data}
  \listplot[linecolor=blue,linewidth=2pt]{\dataII}
  \listplot[linecolor=cyan,linewidth=2pt,yunit=0.5]{\dataII}
\endpsgraph
\end{center}

\begin{lstlisting}
\readdata{\data}{demo2.dat}%
\readdata{\dataII}{demo3.dat}%
\pstScalePoints(1,1){1989 sub}{}
\psset{llx=-0.5cm,lly=-1cm, §\ON§xAxisLabel§\OFF§=Year,§\ON§yAxisLabel§\OFF§=Whatever,%
     §\ON§xAxisLabelPos§\OFF§={c,-0.4in},§\ON§yAxisLabelPos§\OFF§={-0.4in,c}}
\psgraph[axesstyle=frame,Dx=2,Ox=1989,subticks=2](0,0)(12,6){4in}{2in}%
  \listplot[linecolor=red,linewidth=2pt]{\data}
  \listplot[linecolor=blue,linewidth=2pt]{\dataII}
  \listplot[linecolor=cyan,linewidth=2pt,yunit=0.5]{\dataII}
\endpsgraph
\end{lstlisting}
%-----------------------------------------------------------------------------

%\begin{LTXexample}[pos=t,preset=\centering]
\begin{center}
\readdata{\data}{demo2.dat}%
\readdata{\dataII}{demo3.dat}%
\psset{llx=-0.5cm,lly=-0.75cm,plotstyle=LineToXAxis}
\pstScalePoints(1,1){1989 sub}{2 sub}
\begin{psgraph}[axesstyle=frame,Dx=2,Ox=1989,Oy=2,subticks=2](0,0)(12,4){6in}{3in}
  \listplot[linecolor=red,linewidth=2pt]{\data}
  \listplot[linecolor=blue,linewidth=2pt]{\dataII}
  \listplot[linecolor=cyan,linewidth=2pt,yunit=0.5]{\dataII}
\end{psgraph}
\end{center}
%\end{LTXexample}

\begin{lstlisting}
\readdata{\data}{demo2.dat}%
\readdata{\dataII}{demo3.dat}%
\psset{llx=-0.5cm,lly=-0.75cm,plotstyle=LineToXAxis}
\pstScalePoints(1,1){1989 sub}{2 sub}
\begin{psgraph}[axesstyle=frame,Dx=2,Ox=1989,Oy=2,subticks=2](0,0)(12,4){6in}{3in}
  \listplot[linecolor=red,linewidth=2pt]{\data}
  \listplot[linecolor=blue,linewidth=2pt]{\dataII}
  \listplot[linecolor=cyan,linewidth=2pt,yunit=0.5]{\dataII}
\end{psgraph}
\end{lstlisting}

%\newpage
An example with ticks on every side of the frame and filled areas:

\begin{center}
\def\data{0 0 1 4 1.5 1.75 2.25 4 2.75 7 3 9}
\psset{lly=-0.5cm}
\begin{psgraph}[axesstyle=none,ticks=none](0,0)(3.0,9.0){12cm}{5cm}
  \pscustom[fillstyle=solid,fillcolor=red!40,linestyle=none]{%
    \listplot{\data}
    \psline(3,9)(3,0)}
  \pscustom[fillstyle=solid,fillcolor=blue!40,linestyle=none]{%
    \listplot{\data}
    \psline(3,9)(0,9)}
  \listplot[linewidth=2pt]{\data}
  \psaxes[axesstyle=frame,ticksize=0 5pt,xsubticks=20,ysubticks=4,
    tickstyle=inner,dy=2,Dy=2,tickwidth=1.5pt,subtickcolor=black](0,0)(3,9)
  \rput*(2.5,3){level 1}\rput*(1,7){level 2}
\end{psgraph}
\end{center}

\begin{lstlisting}
\def\data{0 0 1 4 1.5 1.75 2.25 4 2.75 7 3 9}
\psset{lly=-0.5cm}
\begin{psgraph}[axesstyle=none,ticks=none](0,0)(3.0,9.0){12cm}{5cm}
  \pscustom[fillstyle=solid,fillcolor=red!40,linestyle=none]{%
    \listplot{\data}
    \psline(3,9)(3,0)}
  \pscustom[fillstyle=solid,fillcolor=blue!40,linestyle=none]{%
    \listplot{\data}
    \psline(3,9)(0,9)}
  \listplot[linewidth=2pt]{\data}
  \psaxes[axesstyle=frame,ticksize=0 5pt,xsubticks=20,ysubticks=4,
    tickstyle=inner,dy=2,Dy=2,tickwidth=1.5pt,subtickcolor=black](0,0)(3,9)
  \rput*(2.5,3){level 1}\rput*(1,7){level 2}
\end{psgraph}
\end{lstlisting}


%-------------------------------------------------------------------------------------------
\subsection{The new options}
%-------------------------------------------------------------------------------------------

\begin{center}
\begin{tabular}{>{\tt}l>{\tt}ll}
\textrm{name} & \textrm{default} & meaning\\\hline
xAxisLabel & x & label for the x-axis\\
yAxisLabel & y & label for the y-axis\\
xAxisLabelPos & \{\} & where to put the x-label\\
yAxisLabelPos & \{\} & where to put the y-label\\
llx & 0pt & trim for the lower left x\\
lly & 0pt & trim for the lower left y\\
urx & 0pt & trim for the upper right x\\
ury & 0pt & trim for the upper right y
\end{tabular}
\end{center}

There is one restriction in using the trim parameters, they must
been set \textbf{before} \verb+psgraph+ is called. They are
redundant when used as parameters of \verb+psgraph+ itself. The
\verb+?AxisLabelPos+ options can use the letter \verb+c+ for
centering an x-axis or y-axis label. The c is a replacement for
the x or y value. When using values with units, the position is
always measured from the origin of the coordinate system, which
can be outside the visible \verb+pspicture+ environment

\medskip
\resetOptions

\begin{center}
\readdata{\data}{demo2.dat}%
\readdata{\dataII}{demo3.dat}%
\psset{llx=-1cm,lly=-1.25cm,urx=0.5cm,ury=0.1in,xAxisLabel=Year,%
   yAxisLabel=Whatever,xAxisLabelPos={c,-0.4in},%
   yAxisLabelPos={-0.4in,c}}
\pstScalePoints(1,1){1989 sub}{}
\psframebox[linestyle=dashed,boxsep=0pt]{%
\begin{psgraph}[axesstyle=frame,Ox=1989,subticks=2](0,0)(12,6){0.8\linewidth}{2.5in}%
  \listplot[linecolor=red,linewidth=2pt]{\data}%
  \listplot[linecolor=blue,linewidth=2pt]{\dataII}%
  \listplot[linecolor=cyan,linewidth=2pt,yunit=0.5]{\dataII}%
\end{psgraph}%
}
\end{center}


\begin{lstlisting}
\readdata{\data}{demo2.dat}%
\readdata{\dataII}{demo3.dat}%
\psset{llx=-1cm,lly=-1.25cm,urx=0.5cm,ury=0.1in,xAxisLabel=Year,%
   yAxisLabel=Whatever,xAxisLabelPos={c,-0.4in},%
   yAxisLabelPos={-0.4in,c}}
\pstScalePoints(1,1){1989 sub}{}
\psframebox[linestyle=dashed,boxsep=0pt]{%
\begin{psgraph}[axesstyle=frame,Ox=1989,subticks=2](0,0)(12,6){0.8\linewidth}{2.5in}%
  \listplot[linecolor=red,linewidth=2pt]{\data}%
  \listplot[linecolor=blue,linewidth=2pt]{\dataII}%
  \listplot[linecolor=cyan,linewidth=2pt,yunit=0.5]{\dataII}%
\end{psgraph}%
}
\end{lstlisting}


\pstScalePoints(1,1){}{}% reset


%--------------------------------------------------------------------------------------
\subsection{Problems}
%--------------------------------------------------------------------------------------
Floating point operations in \TeX\ are a real mess, which causes a
lot of problems when there are very small or very big units. With
the options of \verb+\pst-plot+ it is possible to choose normal
units (whatever this may be ...), and plot the data as usual.

\begin{LTXexample}[pos=t]
\begin{filecontents*}{test.dat}
3.2345 34.5
3.2364 65.4
3.2438 50.2
\end{filecontents*}

\psset{lly=-0.5cm,llx=-1cm}
\readdata{\data}{test.dat}
\pstScalePoints(1,1){3.23 sub 100 mul}{}
\begin{psgraph}[Ox=3.23,Dx=0.01,dx=\psxunit,Dy=10](0,0)(3,70){0.8\linewidth}{5cm}%
  \listplot[showpoints=true,plotstyle=curve]{\data}
\end{psgraph}
\end{LTXexample}

This example shows some important facts:
\begin{itemize}
\item \verb+3.23 sub 100 mul+: the x values are now $0.45; 0.64; 1.38$
\item \verb+Ox=3.23+: the origin of the x axis is set to $3.23$
\item \verb+Dx=0.01+: the increment of the labels
\item \verb+dx=\psxunit+: uses the calculated unit value to get every unit a label
\item \verb+Dy=10+: increase the y labels by 10
\end{itemize}

Using the internal \verb+\psxunit+ one can have dynamical x-units,
depending on the linewidth of the document.

\resetOptions

%--------------------------------------------------------------------------------------
\section{\CMD{psStep}}
%--------------------------------------------------------------------------------------
\verb+\psStep+ calculates a step function for the upper or lower
sum or the max/min of the Riemann integral definition of a given
function. The available option is

\verb+StepType=lower|upper|Riemann|infimum|supremum+ or alternative
\verb+StepType=l|u|R|i|s+

with \verb+lower+ as the default setting. The syntax of the function is

\verb+\psStep[options](x1,x2){n}{function}+


(x1,x2) is the given interval for the step wise calculated
function, n is the number of the rectangles and \verb+function+ is
the mathematical function in postfix or algebraic notation (with
\verb+algebraic=true+).

\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(-0.5,-0.5)(10,3)
 \psaxes[labelFontSize=\scriptstyle]{->}(10,3)
 \psplot[plotpoints=100,linewidth=1.5pt,algebraic]{0}{10}{sqrt(x)}
 \psStep[linecolor=magenta,StepType=upper,fillstyle=hlines](0,9){9}{x sqrt}
 \psStep[linecolor=blue,fillstyle=vlines](0,9){9}{x sqrt }
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t,preset=\centering]
\psset{plotpoints=200}
\begin{pspicture}(-0.5,-2.25)(10,3)
  \psaxes[labelFontSize=\scriptstyle]{->}(0,0)(0,-2.25)(10,3)
 \psplot[linewidth=1.5pt,algebraic]{0}{10}{sqrt(x)*sin(x)}
 \psStep[algebraic,linecolor=magenta,StepType=upper](0,9){20}{sqrt(x)*sin(x)}
 \psStep[linecolor=blue,linestyle=dashed](0,9){20}{x sqrt x RadtoDeg sin mul}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t,preset=\centering]
\psset{yunit=1.25cm,plotpoints=200}
\begin{pspicture}(-0.5,-1.5)(10,1.5)
 \psaxes[labelFontSize=\scriptstyle]{->}(0,0)(0,-1.5)(10,1.5)
 \psStep[algebraic,StepType=Riemann,fillstyle=solid,fillcolor=black!10](0,10){50}%
    {sqrt(x)*cos(x)*sin(x)}
 \psplot[linewidth=1.5pt,algebraic]{0}{10}{sqrt(x)*cos(x)*sin(x)}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[pos=t,preset=\centering]
\psset{yunit=1.25cm,plotpoints=200}
\begin{pspicture}(-0.5,-1.5)(10,1.5)
 \psaxes[labelFontSize=\scriptstyle]{->}(0,0)(0,-1.5)(10,1.5)
 \psStep[algebraic,StepType=infimum,fillstyle=solid,fillcolor=black!10](0,10){50}%
    {sqrt(x)*cos(x)*sin(x)}
 \psplot[linewidth=1.5pt,algebraic]{0}{10}{sqrt(x)*cos(x)*sin(x)}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t,preset=\centering]
\psset{yunit=1.25cm,plotpoints=200}
\begin{pspicture}(-0.5,-1.5)(10,1.5)
 \psaxes[labelFontSize=\scriptstyle]{->}(0,0)(0,-1.5)(10,1.5)
 \psStep[algebraic,StepType=supremum,fillstyle=solid,fillcolor=black!10](0,10){50}%
    {sqrt(x)*cos(x)*sin(x)}
 \psplot[linewidth=1.5pt,algebraic]{0}{10}{sqrt(x)*cos(x)*sin(x)}
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[pos=t,preset=\centering]
\psset{unit=1.5cm,plotpoints=200}
\begin{pspicture}[plotpoints=200](-0.5,-3)(10,2.5)
  \psStep[algebraic,fillstyle=solid,fillcolor=yellow](0.001,9.5){40}{2*sqrt(x)*cos(ln(x))*sin(x)}
  \psStep[algebraic,StepType=Riemann,fillstyle=solid,fillcolor=blue](0.001,9.5){40}{2*sqrt(x)*cos(ln(x))*sin(x)}
  \psaxes[labelFontSize=\scriptstyle]{->}(0,0)(0,-2.75)(10,2.5)
  \psplot[algebraic,linecolor=white]{0.001}{9.75}{2*sqrt(x)*cos(ln(x))*sin(x)}
  \uput[90](6,1.2){$f(x)=2\cdot\sqrt{x}\cdot\cos{(\ln{x})}\cdot\sin{x}$}
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\section{\CMD{psplotTangent} and option \texttt{Tnormal}}
%--------------------------------------------------------------------------------------
There is an additional option, named \verb+Derive+ for an
alternative function (see following example) to calculate the
slope of the tangent. This will be in general the first
derivative, but can also be any other function. If this option is
different to to the default value \verb+Derive=default+, then this
function is taken to calculate the slope. For the other cases,
\verb+pstricks-add+ builds a secant with -0.00005<x<0.00005,
calculates the slope and takes this for the tangent. This may be
problematic in some cases of special functions or $x$ values, then
it may be appropriate to use the Derive option.

The macro expects three parameters:

\begin{description}
\item[$x$]: the $x$ value of the function for which the tangent should be calculated
\item[$dx$]: the $dx$ to both sides of the $x$ value
\item[$f(x)$]: the function in infix (with option \verb+algebraic+) or the default
postfix (PostScript) notation
\end{description}

The following examples show the use of the algebraic option together with the Derive option.
Remember that using the \verb+algebraic+ option implies that the angles have to be in the
radian unit!

\begin{center}
\bgroup
\def\F{x RadtoDeg dup dup cos exch 2 mul cos add exch 3 mul cos add}
\def\Fp{x RadtoDeg dup dup sin exch 2 mul sin 2 mul add exch 3 mul sin 3 mul add neg}
\psset{plotpoints=1001,algebraic=false}
\begin{pspicture}(-7.5,-2.5)(7.5,4)%X\psgrid
  \psaxes{->}(0,0)(-7.5,-2)(7.5,3.5)
  \psplot[linewidth=3\pslinewidth]{-7}{7}{\F}
  \psset{linecolor=red, arrows=<->, arrowscale=2}
  \multido{\n=-7+1}{8}{\psplotTangent{\n}{1}{\F}}
  \psset{linecolor=magenta, arrows=<->, arrowscale=2}%
  \multido{\n=0+1}{8}{\psplotTangent[linecolor=blue, Derive=\Fp]{\n}{1}{\F}}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\def\F{x RadtoDeg dup dup cos exch 2 mul cos add exch 3 mul cos add}
\def\Fp{x RadtoDeg dup dup sin exch 2 mul sin 2 mul add exch 3 mul sin 3 mul add neg}
\psset{plotpoints=1001}
\begin{pspicture}(-7.5,-2.5)(7.5,4)%X\psgrid
  \psaxes{->}(0,0)(-7.5,-2)(7.5,3.5)
  \psplot[linewidth=3\pslinewidth]{-7}{7}{\F}
  \psset{linecolor=red, arrows=<->, arrowscale=2}
  \multido{\n=-7+1}{8}{\psplotTangent{\n}{1}{\F}}
  \psset{linecolor=magenta, arrows=<->, arrowscale=2}%
  \multido{\n=0+1}{8}{\psplotTangent[linecolor=blue, §\ON§Derive=\Fp§\OFF§]{\n}{1}{\F}}
\end{pspicture}
\end{lstlisting}


\begin{center}
\bgroup
\def\Falg{cos(x)+cos(2*x)+cos(3*x)}   \def\Fpalg{-sin(x)-2*sin(2*x)-3*sin(3*x)}
\begin{pspicture}(-7.5,-2.5)(7.5,4)%\psgrid
  \psaxes{->}(0,0)(-7.5,-2)(7.5,3.5)
  \psplot[linewidth=1.5pt,algebraic,plotpoints=500]{-7.5}{7.5}{\Falg}
  \multido{\n=-7+1}{8}{\psplotTangent[linecolor=red,arrows=<->,arrowscale=2,algebraic]{\n}{1}{\Falg}}
  \multido{\n=0+1}{8}{\psplotTangent[linecolor=magenta,%
     arrows=<->,arrowscale=2,algebraic,Derive={\Fpalg}]{\n}{1}{\Falg}}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\def\Falg{cos(x)+cos(2*x)+cos(3*x)}   \def\Fpalg{-sin(x)-2*sin(2*x)-3*sin(3*x)}
\begin{pspicture}(-7.5,-2.5)(7.5,4)%\psgrid
  \psaxes{->}(0,0)(-7.5,-2)(7.5,3.5)
  \psplot[linewidth=1.5pt,algebraic,plotpoints=500]{-7.5}{7.5}{\Falg}
  \multido{\n=-7+1}{8}{\psplotTangent[linecolor=red,arrows=<->,arrowscale=2,algebraic]{\n}{1}{\Falg}}
  \multido{\n=0+1}{8}{\psplotTangent[linecolor=magenta,%
     arrows=<->,arrowscale=2,algebraic,§\ON§Derive={\Fpalg}§\OFF§]{\n}{1}{\Falg}}
\end{pspicture}
\end{lstlisting}

The next example shows the use of the \verb+Derive+ option to draw
the perpendicular line to the tangent.

\begin{LTXexample}[width=8cm,wide]
\begin{pspicture}(-0.5,-0.5)(7.25,7.25)
  \def\Func{10 x div}
  \psaxes[arrowscale=1.5]{->}(7,7)
  \psplot[linewidth=2pt,algebraic]{1.5}{5}{10/x}
  \psplotTangent[linewidth=.5\pslinewidth,linecolor=red,algebraic]{3}{2}{10/x}
  \psplotTangent[linewidth=.5\pslinewidth,linecolor=blue,algebraic,Derive=(x*x)/10]{3}{2}{10/x}
  \psline[linestyle=dashed](!0 /x 3 def \Func)(!3 /x 3 def \Func)(3,0)
\end{pspicture}
\end{LTXexample}

By setting the optional argument \texttt{Tnormal} one can plot the
normal of the tangent line. It always starts at the given point.

%\resetOptions
\begin{LTXexample}[width=8cm,wide]
\begin{pspicture}(-0.5,-0.5)(7.25,7.25)
  \def\Func{10 x div}
  \psaxes[arrowscale=1.5]{->}(7,7)
  \psplot[linewidth=2pt]{1.5}{5}{\Func}
  \psplotTangent[linewidth=1.5\pslinewidth,linecolor=red]{3}{2}{\Func}
  \psplotTangent[linewidth=1.5\pslinewidth,linecolor=blue,Tnormal]{3}{2}{\Func}
  \psline[linestyle=dashed](!0 /x 3 def \Func)(!3 /x 3 def \Func)(3,0)
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\subsection{A \texttt{polarplot} example}
%--------------------------------------------------------------------------------------

Let's work with the classical cardioid : $r=2(1+\cos(\theta))$ and
$\displaystyle \frac{d r}{d\theta}=-2\sin(\theta)$. The Derive
option always expects the $\frac{d r}{d\theta}$ value and uses
internally the equation for the derivative of implicitly defined
functions:

\[
\frac{dy}{dx}=\frac{r^\prime\cdot\sin\theta + x}{r^\prime\cdot\cos\theta - y}
\]
where $x=r\cdot\cos\theta$ and $y=r\cdot\sin\theta$


\begin{LTXexample}[width=6cm,wide]
\begin{pspicture}(-1,-3)(5,3)%\psgrid[subgridcolor=lightgray]
  \psaxes{->}(0,0)(-1,-3)(5,3)
  \psplot[polarplot,linewidth=3\pslinewidth,linecolor=blue,%
     plotpoints=500]{0}{360}{1 x cos add 2 mul}
\end{pspicture}
\end{LTXexample}

\psset{algebraic=false}
\begin{LTXexample}[width=6cm,wide]
\begin{pspicture}(-1,-3)(5,3)%\psgrid[subgridcolor=lightgray]
  \psaxes{->}(0,0)(-1,-3)(5,3)
  \psplot[polarplot,linewidth=3\pslinewidth,linecolor=blue,plotpoints=500]{0}{360}{1 x cos add 2 mul}
  \multido{\n=0+36}{10}{%
     \psplotTangent[polarplot,linecolor=red,arrows=<->]{\n}{1.5}{1 x cos add 2 mul} }
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=6cm,wide]
\begin{pspicture}(-1,-3)(5,3)%\psgrid[subgridcolor=lightgray]
  \psaxes{->}(0,0)(-1,-3)(5,3)
  \psplot[polarplot,linewidth=3\pslinewidth,linecolor=blue,algebraic,plotpoints=500]{0}{6.289}{2*(1+cos(x))}
  \multido{\r=0.000+0.314}{21}{%
     \psplotTangent[polarplot,Derive=-2*sin(x),algebraic,linecolor=red,arrows=<->]{\r}{1.5}{2*(1+cos(x))} }
\end{pspicture}
\end{LTXexample}



%--------------------------------------------------------------------------------------
\subsection{A \CMD{parametricplot} example}
%--------------------------------------------------------------------------------------

Let's work with a Lissajou curve :
 $\displaystyle\left\{\begin{array}{l}x=3.5\cos(2t)\\y=3.5\sin(6t)\end{array}\right.$
whose derivative is :
 $\displaystyle\left\{\begin{array}{l}x=-7\sin(2t)\\y=21\cos(6t)\end{array}\right.$

The parameter must be the letter $t$ instead of $x$ and when using
the \verb+algebraic+ option you must separate the two equations by
a | (see example).

\begin{LTXexample}[pos=t,wide]
\def\Lissa{t dup 2 RadtoDeg mul cos 3.5 mul exch 6 mul RadtoDeg sin 3.5 mul}%
\psset{yunit=0.6}
\begin{pspicture}(-4,-4)(4,6)
  \parametricplot[plotpoints=500,linewidth=3\pslinewidth]{0}{3.141592}{\Lissa}
  \multido{\r=0.000+0.314}{11}{%
    \psplotTangent[linecolor=red,arrows=<->]{\r}{1.5}{\Lissa} }
  \multido{\r=0.157+0.314}{11}{%
    \psplotTangent[linecolor=blue,arrows=<->]{\r}{1.5}{\Lissa} }
\end{pspicture}\hfill%
\def\LissaAlg{3.5*cos(2*t)|3.5*sin(6*t)}  \def\LissaAlgDer{-7*sin(2*t)|21*cos(6*t)}%
\begin{pspicture}(-4,-4)(4,6)
  \parametricplot[algebraic,plotpoints=500,linewidth=3\pslinewidth]{0}{3.141592}{\LissaAlg}
  \multido{\r=0.000+0.314}{11}{%
    \psplotTangent[algebraic,linecolor=red,arrows=<->]{\r}{1.5}{\LissaAlg} }
  \multido{\r=0.157+0.314}{11}{%
    \psplotTangent[algebraic,linecolor=blue,arrows=<->,%
       Derive=\LissaAlgDer]{\r}{1.5}{\LissaAlg} }
\end{pspicture}
\end{LTXexample}



\resetOptions

\section{Successive derivatives of a function}

The new PostScript function \verb$Derive$ has been added for
plotting successive derivatives of a function. It must be used
with the \verb|algebraic| option. This function has two arguments:

\begin{enumerate}
\item a positive integer which defines the order of the derivative; obviously $0$ means the
  function itself!
\item a function of variable $x$ which can be any function using common operators,
\end{enumerate}

Do not think that the derivative is approximated, the internal PostScript engine will
compute the real derivative using a formal derivative engine.

The following diagram contains the plot of the polynomial:

\[ f(x)=\sum_{i=0}^{14}\frac{(-1)^{i}x^{2i}}{i!}=1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-
          \frac{x^{10}}{10!}+\frac{x^{12}}{12!}-\frac{x^{14}}{14!}\]

and of its first 15 derivatives. It is the sequence definition of
the cosine.


\begin{LTXexample}[pos=t,wide,preset=\centering]
\psset{unit=2}
\def\getColor#1{\ifcase#1 Tan\or RedOrange\or magenta\or yellow\or green\or Orange\or blue\or
  DarkOrchid\or BrickRed\or Rhodamine\or OliveGreen\or Goldenrod\or Mahogany\or
  OrangeRed\or CarnationPink\or RoyalPurple\or Lavender\fi}
\begin{pspicture}[showgrid=true](0,-1.2)(7,1.5)
  \psclip{\psframe[linestyle=none](0,-1.1)(7,1.1)}
  \multido{\in=0+1}{16}{%
     \psplot[linewidth=1pt,algebraic,linecolor=\getColor{\in}]{0}{7}
      {Derive(\in,1-x^2/2+x^4/24-x^6/720+x^8/40320-x^10/3628800+x^12/479001600-x^14/87178291200)}}
  \endpsclip
\end{pspicture}
\end{LTXexample}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \subsection{Other examples}


\begin{LTXexample}[width=3.5cm]
\begin{pspicture}[shift=-2.5,showgrid=true,linewidth=1pt](0,-2)(3,3)
  \psplot[algebraic]{.001}{3}{x*ln(x)}  % f(x)
  \psplot[algebraic,linecolor=red]{.05}{3}{Derive(1,x*ln(x))} % f'(x)=1+ln(x)
\end{pspicture}
\end{LTXexample}


\section{Variable step for plotting a curve}
\subsection{Theory}

As you know with the \verb$\psplot$ macro, the curve is plotted
using a piece-wise linear curve. The step is given by the
parameter \texttt{plotpoints}. For each step between $x_i$ and
$x_{i+1}$, the area defined between the curve and its
approximation (a segment) is majored by this formula :

\begin{minipage}[m]{.5\linewidth}
\[|\varepsilon|\le\frac{M_2(f)(x_{i+1}-x_i)^3}{12}\]

$M_2(f)$ is a majorant of the second derivative of $f$ in the interval $[x_i;x_{i+1}]$.
\end{minipage}
{\psset{unit=1cm, showpoints=false}
\begin{pspicture}[shift=-2,showgrid=true](0,-1)(6,3)
  \pscurve(0,0)(1,1)(3,2.2)(5,2)(6,1)\psline(1,1)(5,2)
  \psline(.5,0)(5.5,0)\psline(1,0)(1,1)\psline(5,0)(5,2)
  \rput[t](1,-.1){$x_n$}\rput[t](5,-.1){$x_{n+1}$}
  \psclip{\pscustom{\psecurve(0,0)(1,1)(3,2.2)(5,2)(6,1)\psline(5,2)}}
    \psframe[fillstyle=solid, fillcolor=gray](0,0)(5,5)
  \endpsclip
  \rput*(3,1.8){$\varepsilon$}
\end{pspicture}}



The parameter \verb$VarStep$ (\verb$false$ by default) activates
the variable step algorithm. It is set to a tolerance defined by
the parameter \verb$VarStepEpsilon$ (\verb+default+ by default,
accept real value). If this parameter is not set by the user, then
it is automatically computed using the default first step given by
the parameter \verb+plotpoints+. Then, for each step, $f''(x_n)$
and $f''(x_{n+1})$ are computed and the smaller is used as
$M_2(f)$, and then the step is approximated. This means that the
step is constant for second order polynomials.

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{The cosine}

  Different value for the tolerance from $0.01$ to $0.000\,1$, a factor $10$ between
  each of them. In black, there is the classic \verb+psplot+ behavior, and in
  magenta the default variable step behavior.

\begin{center}
\bgroup
\psset{algebraic, VarStep=true, unit=2, showpoints=true, linecolor=red}
\begin{pspicture}(-0,-1)(3.14,2)\psgrid
  \psplot[VarStepEpsilon=.01]{0}{3.14}{cos(x)}
  \psplot[VarStepEpsilon=.001]{0}{3.14}{cos(x)+.15}
  \psplot[VarStepEpsilon=.0001]{0}{3.14}{cos(x)+.3}
  \psplot[linecolor=magenta]{0}{3.14}{cos(x)+.45}
  \psplot[VarStep=false, linewidth=2\pslinewidth, linecolor=black]{-0}{3.14}{cos(x)+.6}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{algebraic, VarStep=true, unit=2, showpoints=true, linecolor=red}
\begin{pspicture}[showgrid=true](-0,-1)(3.14,2)
  \psplot[VarStepEpsilon=.01]{0}{3.14}{cos(x)}
  \psplot[VarStepEpsilon=.001]{0}{3.14}{cos(x)+.15}
  \psplot[VarStepEpsilon=.0001]{0}{3.14}{cos(x)+.3}
  \psplot[linecolor=magenta]{0}{3.14}{cos(x)+.45}
  \psplot[VarStep=false,linewidth=1pt,linecolor=black]{-0}{3.14}{cos(x)+.6}
\end{pspicture}
\end{lstlisting}


\clearpage
\subsection{The Napierian Logarithm}

A really classic example which gives a bad beginning, the
tolerance is set to $0.001$.

\begin{center}
\bgroup
\psset{algebraic, VarStep=true, linecolor=red, showpoints=true}
\begin{pspicture}[showgrid=true](0,-5)(16,4)
  \psplot[VarStep=false, linecolor=black]{.01}{16}{ln(x)+1}
  \psplot[linecolor=magenta]{.51}{16}{ln(x-1/2)+1/2}
  \psplot[VarStepEpsilon=.001]{1.01}{16}{ln(x-1)}
  \psplot[VarStepEpsilon=.01]{1.51}{16}{ln(x-1.5)-100/200}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{algebraic, VarStep=true, linecolor=red, showpoints=true}
\begin{pspicture}[showgrid=true](0,-5)(16,4)
  \psplot[VarStep=false, linecolor=black]{.01}{16}{ln(x)+1}
  \psplot[linecolor=magenta]{.51}{16}{ln(x-1/2)+1/2}
  \psplot[VarStepEpsilon=.001]{1.01}{16}{ln(x-1)}
  \psplot[VarStepEpsilon=.01]{1.51}{16}{ln(x-1.5)-100/200}
\end{pspicture}
\end{lstlisting}


\clearpage
\subsection{Sine of the inverse of $x$}
Impossible to draw, but let's try!

\begin{center}
\bgroup
\psset{xunit=64,algebraic,VarStep,linecolor=red,showpoints=true,linewidth=1pt}
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStepEpsilon=.0001]{.01}{.25}{sin(1/x)}
\end{pspicture}\\
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStepEpsilon=.00001]{.01}{.25}{sin(1/x)}
\end{pspicture}\\
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStepEpsilon=.000001]{.01}{.25}{sin(1/x)}
\end{pspicture}\\
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStep=false, linecolor=black]{.01}{.25}{sin(1/x)}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{xunit=64,algebraic,VarStep,linecolor=red,showpoints=true,linewidth=1pt}
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStepEpsilon=.0001]{.01}{.25}{sin(1/x)}
\end{pspicture}\\
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStepEpsilon=.00001]{.01}{.25}{sin(1/x)}
\end{pspicture}\\
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStepEpsilon=.000001]{.01}{.25}{sin(1/x)}
\end{pspicture}\\
\begin{pspicture}[showgrid=true](0,-1)(.5,1)
  \psplot[VarStep=false, linecolor=black]{.01}{.25}{sin(1/x)}
\end{pspicture}
\end{lstlisting}





\subsection{A really complex function}

Just appreciate the difference between the normal behavior and the plotting with the
\texttt{varStep} option. The function is :

\[f(x)=x-\frac{x^2}{10}+\ln(x)+\cos(2x)+\sin(x^2)-1\]

\begin{center}
\bgroup
\psset{xunit=3, algebraic, VarStep, showpoints=true}
\begin{pspicture}[showgrid=true](0,-2)(5,6)
  \psplot[VarStepEpsilon=.0005, linecolor=red]{.1}{5}{x-x^2/10+ln(x)+cos(2*x)+sin(x^2)}
  \psplot[linecolor=magenta]{.1}{5}{x-x^2/10+ln(x)+cos(2*x)+sin(x^2)+.5}
  \psplot[VarStep=false]{.1}{5}{x-x^2/10+ln(x)+cos(2*x)+sin(x^2)-1}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{xunit=3, algebraic, VarStep, showpoints=true}
\begin{pspicture}[showgrid=true](0,-2)(5,6)
  \psplot[VarStepEpsilon=.0005, linecolor=red]{.1}{5}{x-x^2/10+ln(x)+cos(2*x)+sin(x^2)}
  \psplot[linecolor=magenta]{.1}{5}{x-x^2/10+ln(x)+cos(2*x)+sin(x^2)+.5}
  \psplot[VarStep=false]{.1}{5}{x-x^2/10+ln(x)+cos(2*x)+sin(x^2)-1}
\end{pspicture}
\end{lstlisting}


\subsection{A hyperbola}

\begin{center}
\bgroup
\psset{algebraic, showpoints=true, unit=0.75}
\begin{pspicture}(-5,-4)(9,6)
  \psplot[linecolor=black]{-5}{1.8}{(x-1)/(x-2)}
  \psplot[VarStep=true, VarStepEpsilon=.001, linecolor=red]{2.2}{9}{(x-1)/(x-2)}
  \psaxes{->}(0,0)(-5,-4)(9,6)
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{algebraic, showpoints=true, unit=0.75}
\begin{pspicture}(-5,-4)(9,6)
  \psplot[linecolor=black]{-5}{1.8}{(x-1)/(x-2)}
  \psplot[VarStep=true, VarStepEpsilon=.001, linecolor=red]{2.2}{9}{(x-1)/(x-2)}
  \psaxes{->}(0,0)(-5,-4)(9,6)
\end{pspicture}
\end{lstlisting}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Successive derivatives of a polynomial}

\begin{center}
\bgroup
\psset{unit=2, algebraic=true, VarStep=true, showpoints=true, VarStepEpsilon=.001}
\def\getColor#1{\ifcase#1 Tan\or RedOrange\or magenta\or yellow\or green\or Orange\or blue\or
  DarkOrchid\or BrickRed\or Rhodamine\or OliveGreen\or Goldenrod\or Mahogany\or
  OrangeRed\or CarnationPink\or RoyalPurple\or Lavender\fi}
\begin{pspicture}[showgrid=true](0,-1.2)(7,1.5)
  \psclip{\psframe[linestyle=none](0,-1.1)(7,1.1)}
  \multido{\in=0+1}{16}{%
    \psplot[algebraic=true, linecolor=\getColor{\in}]{0.1}{7}
    {Derive(\in,Sum(i,0,1,7,(-1)^i*x^(2*i)/Fact(2*i)))}}
  \endpsclip
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{unit=2, algebraic=true, VarStep=true, showpoints=true, VarStepEpsilon=.001}
\def\getColor#1{\ifcase#1 Tan\or RedOrange\or magenta\or yellow\or green\or Orange\or blue\or
  DarkOrchid\or BrickRed\or Rhodamine\or OliveGreen\or Goldenrod\or Mahogany\or
  OrangeRed\or CarnationPink\or RoyalPurple\or Lavender\fi}
\begin{pspicture}[showgrid=true](0,-1.2)(7,1.5)
  \psclip{\psframe[linestyle=none](0,-1.1)(7,1.1)}
  \multido{\in=0+1}{16}{%
    \psplot[algebraic=true, linecolor=\getColor{\in}]{0.1}{7}
    {Derive(\in,Sum(i,0,1,7,(-1)^i*x^(2*i)/Fact(2*i)))}}
  \endpsclip
\end{pspicture}
\end{lstlisting}


\subsection{The variable step algorithm together with the \texttt{IfTE} primitive}

\begin{center}
\bgroup
\psset{unit=1.5, algebraic, VarStep, showpoints=true, VarStepEpsilon=.001}
\begin{pspicture}[showgrid=true](-7,-2)(2,4)
  \psplot{-7}{2}{IfTE(x<-5,-(x+5)^3/2,IfTE(x<0,0,x^2))}
  \psplot{-7}{2}{5*x/9+26/9}
  \psplot[linecolor=blue]{-7}{2}{(x+7)^30/9^30*4.5-1/2}
  \psplot[linecolor=red]{-6.9}{2}
      {IfTE(x<-6,ln(x+7),IfTE(x<-3,x+6,IfTE(x<0.1415926,sin(x+3)+3,3.1415926-x)))}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{unit=1.5, algebraic, VarStep, showpoints=true, VarStepEpsilon=.001}
\begin{pspicture}[showgrid=true](-7,-2)(2,4)
  \psplot{-7}{2}{IfTE(x<-5,-(x+5)^3/2,IfTE(x<0,0,x^2))}
  \psplot{-7}{2}{5*x/9+26/9}
  \psplot[linecolor=blue]{-7}{2}{(x+7)^30/9^30*4.5-1/2}
  \psplot[linecolor=red]{-6.9}{2}
      {IfTE(x<-6,ln(x+7),IfTE(x<-3,x+6,IfTE(x<0.1415926,sin(x+3)+3,3.1415926-x)))}
\end{pspicture}
\end{lstlisting}



\subsection{Using \CMD{parametricplot}}

\begin{center}
\bgroup
\psset{unit=2.5}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=red,VarStep=true, showpoints=true,
                VarStepEpsilon=.0001]
                {-3.14}{3.14}{cos(3*t)|sin(2*t)}
\end{pspicture}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=blue,VarStep=true, showpoints=false,
                VarStepEpsilon=.0001]
                {-3.14}{3.14}{cos(3*t)|sin(2*t)}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{unit=3}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=red,VarStep=true, showpoints=true,
                VarStepEpsilon=.0001]
                {-3.14}{3.14}{cos(3*t)|sin(2*t)}
\end{pspicture}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=blue,VarStep=true, showpoints=false,
                VarStepEpsilon=.0001]
                {-3.14}{3.14}{cos(3*t)|sin(2*t)}
\end{pspicture}
\end{lstlisting}


\begin{center}
\bgroup
\psset{unit=2.5}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=red,VarStep=true, showpoints=true,
                VarStepEpsilon=.0001]
                {0}{47.115}{cos(5*t)|sin(3*t)}
\end{pspicture}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=blue,VarStep=true, showpoints=false,
                VarStepEpsilon=.0001]
                {0}{47.115}{cos(5*t)|sin(3*t)}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{unit=2.5}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=red,VarStep=true, showpoints=true,
                VarStepEpsilon=.0001]
                {0}{47.115}{cos(5*t)|sin(3*t)}
\end{pspicture}
\begin{pspicture}[showgrid=true](-1,-1)(1,1)
\parametricplot[algebraic=true,linecolor=blue,VarStep=true, showpoints=false,
                VarStepEpsilon=.0001]
                {0}{47.115}{cos(5*t)|sin(3*t)}
\end{pspicture}
\end{lstlisting}


\begin{center}
\bgroup
\psset{xunit=.5}
\begin{pspicture}[showgrid=true](0,0)(12.566,2)
\parametricplot[algebraic,linecolor=red,VarStep, showpoints=true,
        VarStepEpsilon=.01]{0}{12.566}{t+cos(-t-Pi/2)|1+sin(-t-Pi/2)}
\end{pspicture}
%
\begin{pspicture}[showgrid=true](0,0)(12.566,2)
\parametricplot[algebraic,linecolor=blue,VarStep, showpoints=false,
        VarStepEpsilon=.001]{0}{12.566}{t+cos(-t-Pi/2)|1+sin(-t-Pi/2)}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{xunit=.5}
\begin{pspicture}[showgrid=true](0,0)(12.566,2)
\parametricplot[algebraic,linecolor=red,VarStep, showpoints=true,
        VarStepEpsilon=.01]{0}{12.566}{t+cos(-t-Pi/2)|1+sin(-t-Pi/2)}
\end{pspicture}
%
\begin{pspicture}[showgrid=true](0,0)(12.566,2)
\parametricplot[algebraic,linecolor=blue,VarStep, showpoints=false,
        VarStepEpsilon=.001]{0}{12.566}{t+cos(-t-Pi/2)|1+sin(-t-Pi/2)}
\end{pspicture}
\end{lstlisting}


\resetOptions


\section{New math functions and their derivatives}

\subsection{The inverse sine and its derivative}

\begin{center}
\bgroup
\psset{unit=1.5}
\begin{pspicture}[showgrid=true](-1,-2)(1,2)
  \psplot[linecolor=blue,algebraic]{-1}{1}{asin(x)}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,-2)(1,2)
  \psplot[linecolor=blue]{-.999}{.999}{asin(x)}
\end{pspicture}
\hspace{1em}
\begin{pspicture}[showgrid=true](-1,0)(1,4)
  \psplot[linecolor=blue]{-.97}{.97}{Derive(1,asin(x))}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.0001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,0)(1,4)
  \psplot[linecolor=blue]{-.97}{.97}{Derive(1,asin(x))}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{unit=1.5}
\begin{pspicture}[showgrid=true](-1,-2)(1,2)
  \psplot[linecolor=blue,algebraic]{-1}{1}{asin(x)}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,-2)(1,2)
  \psplot[linecolor=blue]{-.999}{.999}{asin(x)}
\end{pspicture}
\hspace{1em}
\begin{pspicture}[showgrid=true](-1,0)(1,4)
  \psplot[linecolor=red]{-.97}{.97}{Derive(1,asin(x))}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.0001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,0)(1,4)
  \psplot[linecolor=red]{-.97}{.97}{Derive(1,asin(x))}
\end{pspicture}
\end{lstlisting}


\subsection{The inverse cosine and its derivative}

\begin{center}
\bgroup
\psset{unit=1.5}
\begin{pspicture}[showgrid=true](-1,0)(1,3)
  \psplot[linecolor=blue,algebraic]{-1}{1}{acos(x)}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,0)(1,3)
  \psplot[linecolor=blue]{-.999}{.999}{acos(x)}
\end{pspicture}
\hspace{1em}
\begin{pspicture}[showgrid=true](-1,-4)(1,-1)
  \psplot[linecolor=blue]{-.97}{.97}{Derive(1,acos(x))}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.0001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,-4)(1,-1)
  \psplot[linecolor=blue]{-.97}{.97}{Derive(1,acos(x))}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{unit=1.5}
\begin{pspicture}[showgrid=true](-1,0)(1,3)
  \psplot[linecolor=blue,algebraic]{-1}{1}{acos(x)}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,0)(1,3)
  \psplot[linecolor=blue]{-.999}{.999}{acos(x)}
\end{pspicture}
\hspace{1em}
\begin{pspicture}[showgrid=true](-1,-4)(1,-1)
  \psplot[linecolor=red]{-.97}{.97}{Derive(1,acos(x))}
\end{pspicture}
\hspace{1em}
\psset{algebraic, VarStep, VarStepEpsilon=.0001, showpoints=true}
\begin{pspicture}[showgrid=true](-1,-4)(1,-1)
  \psplot[linecolor=red]{-.97}{.97}{Derive(1,acos(x))}
\end{pspicture}
\end{lstlisting}



\subsection{The inverse tangent and its derivative}

\begin{center}
\bgroup
\begin{pspicture}[showgrid=true](-4,-2)(4,2)
\psset{algebraic=true}
  \psplot[linecolor=blue,linewidth=1pt]{-4}{4}{atg(x)}
  \psplot[linecolor=red,VarStep, VarStepEpsilon=.0001, showpoints=true]{-4}{4}{Derive(1,atg(x))}
\end{pspicture}
\hspace{1em}
\begin{pspicture}[showgrid=true](-4,-2)(4,2)
\psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=blue]{-4}{4}{atg(x)}
  \psplot[linecolor=red]{-4}{4}{Derive(1,atg(x))}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\begin{pspicture}[showgrid=true](-4,-2)(4,2)
\psset{algebraic=true}
  \psplot[linecolor=blue,linewidth=1pt]{-4}{4}{atg(x)}
  \psplot[linecolor=red,VarStep, VarStepEpsilon=.0001, showpoints=true]{-4}{4}{Derive(1,atg(x))}
\end{pspicture}
\hspace{1em}
\begin{pspicture}[showgrid=true](-4,-2)(4,2)
\psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=blue]{-4}{4}{atg(x)}
  \psplot[linecolor=red]{-4}{4}{Derive(1,atg(x))}
\end{pspicture}
\end{lstlisting}

\subsection{Hyperbolic functions}

\begin{center}
\bgroup
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{sh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{ch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{th(x)}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\hspace{1em}
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true, VarStep=true, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{sh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{ch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{th(x)}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{sh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{ch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{th(x)}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\hspace{1em}
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true, VarStep=true, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{sh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{ch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{th(x)}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\end{lstlisting}



\begin{center}
\bgroup
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{Derive(1,sh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{Derive(1,ch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{Derive(1,th(x))}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\hspace{1em}
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true, VarStep=true, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{Derive(1,sh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{Derive(1,ch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{Derive(1,th(x))}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true,linewidth=1pt}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{Derive(1,sh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{Derive(1,ch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{Derive(1,th(x))}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\hspace{1em}
\begin{pspicture}(-3,-4)(3,4)
\psset{algebraic=true, VarStep=true, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-2}{2}{Derive(1,sh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{-2}{2}{Derive(1,ch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-3}{3}{Derive(1,th(x))}
  \psaxes{->}(0,0)(-3,-4)(3,4)
\end{pspicture}
\end{lstlisting}



\begin{center}
\bgroup
\begin{pspicture}(-7,-3)(7,3)
\psset{algebraic=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Argsh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{1}{7}{Argch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-.99}{.99}{Argth(x)}
  \psaxes{->}(0,0)(-7,-3)(7,3)
\end{pspicture}\\[\baselineskip]
\begin{pspicture}(-7,-3)(7,3)
  \psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Argsh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{1.001}{7}{Argch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-.99}{.99}{Argth(x)}
  \psaxes{->}(0,0)(-7,-3)(7,3)
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\begin{pspicture}(-7,-3)(7,3)
\psset{algebraic=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Argsh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{1}{7}{Argch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-.99}{.99}{Argth(x)}
  \psaxes{->}(0,0)(-7,-3)(7,3)
\end{pspicture}\\[\baselineskip]
\begin{pspicture}(-7,-3)(7,3)
  \psset{algebraic, VarStep, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Argsh(x)}
  \psplot[linecolor=blue,linewidth=1pt]{1.001}{7}{Argch(x)}
  \psplot[linecolor=green,linewidth=1pt]{-.99}{.99}{Argth(x)}
  \psaxes{->}(0,0)(-7,-3)(7,3)
\end{pspicture}
\end{lstlisting}



\begin{center}
\bgroup
\begin{pspicture}(-7,-0.5)(7,6)
\psset{algebraic=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Derive(1,Argsh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{1.014}{7}{Derive(1,Argch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-.9}{.9}{Derive(1,Argth(x))}
  \psaxes{->}(0,0)(-7,0)(7,6)
\end{pspicture}\\[\baselineskip]
\begin{pspicture}(-7,-0.5)(7,6)
\psset{algebraic=true}
  \psset{algebraic=true, VarStep=true, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Derive(1,Argsh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{1.014}{7}{Derive(1,Argch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-.9}{.9}{Derive(1,Argth(x))}
  \psaxes{->}(0,0)(-7,0)(7,6)
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\begin{pspicture}(-7,-0.5)(7,6)
\psset{algebraic=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Derive(1,Argsh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{1.014}{7}{Derive(1,Argch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-.9}{.9}{Derive(1,Argth(x))}
  \psaxes{->}(0,0)(-7,0)(7,6)
\end{pspicture}\\[\baselineskip]
\begin{pspicture}(-7,-0.5)(7,6)
\psset{algebraic=true}
  \psset{algebraic=true, VarStep=true, VarStepEpsilon=.001, showpoints=true}
  \psplot[linecolor=red,linewidth=1pt]{-7}{7}{Derive(1,Argsh(x))}
  \psplot[linecolor=blue,linewidth=1pt]{1.014}{7}{Derive(1,Argch(x))}
  \psplot[linecolor=green,linewidth=1pt]{-.9}{.9}{Derive(1,Argth(x))}
  \psaxes{->}(0,0)(-7,0)(7,6)
\end{pspicture}
\end{lstlisting}

\clearpage
%--------------------------------------------------------------------------------------
\section[\CMD{psplotDiffEqn} -- solving diffential equations]{\CMD{psplotDiffEqn} -- solving diffential equations}
%--------------------------------------------------------------------------------------


 A differential equation of first order is like

\begin{align} y^\prime=f(x,y,y^\prime) \end{align}


where $y$ is a function of $x$. We define some vectors $Y=[y, y',
\cdots , y^{(n-1)}]$ and $Y^\prime=[y^\prime, y^{\prime\prime},
\cdots , y^{n}]$, depending on the order $n$. The syntax of the
macro is

\begin{lstlisting}[style=syntax]
\psplotDiffEqn[options]{x0}{x1}{y0}{f(x,y,y',...)}
\end{lstlisting}

\begin{itemize}\setlength\itemsep{0pt}\setlength\parsep{0pt}\setlength\parskip{0pt}
\item \verb+options+: the \verb+\psplotDiffEqn+ specific options and all other of PSTricks, which
make sense;
\item $x_0$: the start value;
\item $x_1$: the end value of the definition interval;
\item $y_0$: the initial values for $y(x_0)\ y'(x_0)\ \ldots$;
\item $f(x,y,y',...)$: the differential equation, depending to the number of initial values, e.g.:
    \verb+{0 1}+ for $y_0$ are two initial values, so that we have a differential equation of
    second order $f(x,y,y')$ and the macro leaves $y\ y'$ on the stack.
\end{itemize}

The new options are:


\begin{itemize}\setlength\itemsep{0pt}\setlength\parsep{0pt}\setlength\parskip{0pt}
\item \param{method}: integration method (\verb+euler+ for order 1 euler method, \verb+rk4+ for
  4\textsuperscript{th} order Runge-Kutta method);
\item \param{whichabs}: select the abscissa for plotting the graph, by default it is
  $x$, but you can specify a number which represent a position in the vector $y$;
\item \param{whichord}: same as precedent for the ordinate, by default $y(0)$;
\item \param{plotfuncx}: describe a ps function for the abscissa, parameter
  \param{whichabs} becomes useless;
\item \param{plotfuncy}: idem for the ordinate;
\item \param{buildvector}: boolean parameter for specifying the input-output of the
  $f$ description:
  \begin{description}
  \item[\texttt{true}] (default): $y$ is put on the stack element by element, $y'$
    must be given in the same way;
  \item[\texttt{false}]: $y$ is put on the stack as a vector, $y'$ must be returned
  in the same way;
  \end{description}

\item \param{algebraic}: algebraic description for $f$, \param{buildvector}
  parameter is useless when activating this option.
\end{itemize}



\subsection{Variable step for differential equations}

A new algorithm has been added for adjusting the step according to the variations of
the curve. The parameter \verb+method+ has a new possible value : \verb+varrkiv+ to
activate the \textsc{Runge-Kutta} method with variable step, then the parameter
\verb+varsteptol+ (real value; \verb+.01+ by default) can control the tolerance of
the algortihm.

\begin{center}
\bgroup
\def\Funct{neg}\def\FunctAlg{-y[0]}
\psset{xunit=1.5, yunit=8, showpoints=true}
\begin{pspicture}[showgrid=true](0,0)(10,1.2)
  \psplot[linewidth=6\pslinewidth, linecolor=green, showpoints=false]{0}{10}{2.71828182846 x neg exp}
  \psplotDiffEqn[linecolor=magenta, method=varrkiv, varsteptol=.1, plotpoints=2]{0}{10}{1}{\Funct}
  \rput(0,.0){\psplotDiffEqn[linecolor=blue, method=varrkiv, varsteptol=.01, plotpoints=2]{0}{10}{1}{\Funct}}
  \rput(0,.1){\psplotDiffEqn[linecolor=Orange, method=varrkiv, varsteptol=.001, plotpoints=2]{0}{10}{1}{\Funct}}
  \rput(0,.2){\psplotDiffEqn[linecolor=red, method=varrkiv, varsteptol=.0001, plotpoints=2]{0}{10}{1}{\Funct}}
  \psset{linewidth=4\pslinewidth,showpoints=false}
  \rput*(3.3,.9){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](3.3,.9){\small RK ordre 4 : $\varepsilon<10^{-1}$}
  \rput*(3.3,.8){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](3.3,.8){\small RK ordre 4 : $\varepsilon<10^{-2}$}
  \rput*(3.3,.7){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](3.3,.7){\small RK ordre 4 : $\varepsilon<10^{-3}$}
  \rput*(3.3,.6){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](3.3,.6){\small RK ordre 4 : $\varepsilon<10^{-4}$}
  \rput*(3.3,.5){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](3.3,.5){\small solution exacte}
\end{pspicture}
{\captionof{figure}{Equation $y'=-y$ with $y_0=1$.}\label{fig:minusexpvarstep}}
\egroup
\end{center}


\begin{lstlisting}[wide=true]
\def\Funct{neg}\def\FunctAlg{-y[0]}
\psset{xunit=1.5, yunit=8, showpoints=true}
\begin{pspicture}[showgrid=true](0,0)(10,1.2)
  \psplot[linewidth=6\pslinewidth, linecolor=green, showpoints=false]{0}{10}{2.71828182846 x neg exp}
  \psplotDiffEqn[linecolor=magenta, method=varrkiv, varsteptol=.1, plotpoints=2]{0}{10}{1}{\Funct}
  \rput(0,.0){\psplotDiffEqn[linecolor=blue, method=varrkiv, varsteptol=.01, plotpoints=2]{0}{10}{1}{\Funct}}
  \rput(0,.1){\psplotDiffEqn[linecolor=Orange, method=varrkiv, varsteptol=.001, plotpoints=2]{0}{10}{1}{\Funct}}
  \rput(0,.2){\psplotDiffEqn[linecolor=red, method=varrkiv, varsteptol=.0001, plotpoints=2]{0}{10}{1}{\Funct}}
  \psset{linewidth=4\pslinewidth,showpoints=false}
  \rput*(3.3,.9){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](3.3,.9){\small RK ordre 4 : $\varepsilon<10^{-1}$}
  \rput*(3.3,.8){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](3.3,.8){\small RK ordre 4 : $\varepsilon<10^{-2}$}
  \rput*(3.3,.7){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](3.3,.7){\small RK ordre 4 : $\varepsilon<10^{-3}$}
  \rput*(3.3,.6){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](3.3,.6){\small RK ordre 4 : $\varepsilon<10^{-4}$}
  \rput*(3.3,.5){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](3.3,.5){\small solution exacte}
\end{pspicture}
\end{lstlisting}



\begin{center}
\bgroup
\def\Funct{exch neg}
\psset{xunit=1.5, yunit=5, method=varrkiv, showpoints=true}%%
\def\quatrepi{12.5663706144}
\begin{pspicture}(0,-1)(10,1.3)
  \psaxes{->}(0,0)(0,-1)(10,1.3)
  \psplot[linewidth=4\pslinewidth, linecolor=green, algebraic=true]{0}{10}{cos(x)}
  \rput(0,.0){\psplotDiffEqn[linecolor=magenta, plotpoints=7, varsteptol=.1]{0}{10}{1 0}{\Funct}}
  \rput(0,.0){\psplotDiffEqn[linecolor=blue, plotpoints=201, varsteptol=.01]{0}{10}{1 0}{\Funct}}
  \rput(0,.1){\psplotDiffEqn[linewidth=2\pslinewidth, linecolor=red, varsteptol=.001]{0}{10}{1 0}{\Funct}}
  \rput(0,.2){\psplotDiffEqn[linecolor=black, varsteptol=.0001]{0}{10}{1 0}{\Funct}}
  \rput(0,.3){\psplotDiffEqn[linecolor=Orange, varsteptol=.00001]{0}{10}{1 0}{\Funct}}
  \psset{linewidth=4\pslinewidth,showpoints=false}
  \rput*(2.3,.9){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](2.3,.9){\small $\varepsilon<10^{-1}$}
  \rput*(2.3,.8){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](2.3,.8){\small $\varepsilon<10^{-2}$}
  \rput*(2.3,.7){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](2.3,.7){\small $\varepsilon<10^{-3}$}
  \rput*(2.3,.6){\psline[linecolor=black](-.75cm,0)}
  \rput*[l](2.3,.6){\small $\varepsilon<10^{-4}$}
  \rput*(2.3,.5){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](2.3,.5){\small $\varepsilon<10^{-5}$}
  \rput*(2.3,.4){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](2.3,.4){\small solution exacte}
\end{pspicture}
{\captionof{figure}{Equation $y''=-y$}\label{fig:trigfunc}}
\egroup
\end{center}

\begin{lstlisting}[wide=true]
\def\Funct{exch neg}
\psset{xunit=1.5, yunit=5, method=varrkiv, showpoints=true}%%
\def\quatrepi{12.5663706144}
\begin{pspicture}(0,-1)(10,1.3)
  \psaxes{->}(0,0)(0,-1)(10,1.3)
  \psplot[linewidth=4\pslinewidth, linecolor=green, algebraic=true]{0}{10}{cos(x)}
  \rput(0,.0){\psplotDiffEqn[linecolor=magenta, plotpoints=7, varsteptol=.1]{0}{10}{1 0}{\Funct}}
  \rput(0,.0){\psplotDiffEqn[linecolor=blue, plotpoints=201, varsteptol=.01]{0}{10}{1 0}{\Funct}}
  \rput(0,.1){\psplotDiffEqn[linewidth=2\pslinewidth, linecolor=red, varsteptol=.001]{0}{10}{1 0}{\Funct}}
  \rput(0,.2){\psplotDiffEqn[linecolor=black, varsteptol=.0001]{0}{10}{1 0}{\Funct}}
  \rput(0,.3){\psplotDiffEqn[linecolor=Orange, varsteptol=.00001]{0}{10}{1 0}{\Funct}}
  \psset{linewidth=4\pslinewidth,showpoints=false}
  \rput*(2.3,.9){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](2.3,.9){\small $\varepsilon<10^{-1}$}
  \rput*(2.3,.8){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](2.3,.8){\small $\varepsilon<10^{-2}$}
  \rput*(2.3,.7){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](2.3,.7){\small $\varepsilon<10^{-3}$}
  \rput*(2.3,.6){\psline[linecolor=black](-.75cm,0)}
  \rput*[l](2.3,.6){\small $\varepsilon<10^{-4}$}
  \rput*(2.3,.5){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](2.3,.5){\small $\varepsilon<10^{-5}$}
  \rput*(2.3,.4){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](2.3,.4){\small solution exacte}
\end{pspicture}
\end{lstlisting}




\begin{center}
\bgroup
\def\Funct{exch}
\psset{xunit=4, yunit=1, method=varrkiv, showpoints=true}%%
\def\quatrepi{12.5663706144}
\begin{pspicture}(0,-0.5)(3,11)
  \psaxes{->}(0,0)(3,11)
  \psplot[linewidth=4\pslinewidth, linecolor=green, algebraic=true]{0}{3}{ch(x)}
  \rput(0,.0){\psplotDiffEqn[linecolor=magenta, varsteptol=.1]{0}{3}{1 0}{\Funct}}
  \rput(0,.3){\psplotDiffEqn[linecolor=blue, varsteptol=.01]{0}{3}{1 0}{\Funct}}
  \rput(0,.6){\psplotDiffEqn[linecolor=red, varsteptol=.001]{0}{3}{1 0}{\Funct}}
  \rput(0,.9){\psplotDiffEqn[linecolor=black, varsteptol=.0001]{0}{3}{1 0}{\Funct}}
  \rput(0,1.2){\psplotDiffEqn[linecolor=Orange, varsteptol=.00001]{0}{3}{1 0}{\Funct}}
  \psset{linewidth=4\pslinewidth,showpoints=false}
  \rput*(2.3,.9){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](2.3,.9){\small $\varepsilon<10^{-1}$}
  \rput*(2.3,.8){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](2.3,.8){\small $\varepsilon<10^{-2}$}
  \rput*(2.3,.7){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](2.3,.7){\small $\varepsilon<10^{-3}$}
  \rput*(2.3,.6){\psline[linecolor=black](-.75cm,0)}
  \rput*[l](2.3,.6){\small $\varepsilon<10^{-4}$}
  \rput*(2.3,.5){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](2.3,.5){\small $\varepsilon<10^{-5}$}
  \rput*(2.3,.4){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](2.3,.4){\small solution exacte}
\end{pspicture}
\captionof{figure}{Equation $y''=y$}
\egroup
\end{center}

\begin{lstlisting}[wide=true]
\def\Funct{exch}
\psset{xunit=4, yunit=1, method=varrkiv, showpoints=true}%%
\def\quatrepi{12.5663706144}
\begin{pspicture}(0,-0.5)(3,11)
  \psaxes{->}(0,0)(3,11)
  \psplot[linewidth=4\pslinewidth, linecolor=green, algebraic=true]{0}{3}{ch(x)}
  \rput(0,.0){\psplotDiffEqn[linecolor=magenta, varsteptol=.1]{0}{3}{1 0}{\Funct}}
  \rput(0,.3){\psplotDiffEqn[linecolor=blue, varsteptol=.01]{0}{3}{1 0}{\Funct}}
  \rput(0,.6){\psplotDiffEqn[linecolor=red, varsteptol=.001]{0}{3}{1 0}{\Funct}}
  \rput(0,.9){\psplotDiffEqn[linecolor=black, varsteptol=.0001]{0}{3}{1 0}{\Funct}}
  \rput(0,1.2){\psplotDiffEqn[linecolor=Orange, varsteptol=.00001]{0}{3}{1 0}{\Funct}}
  \psset{linewidth=4\pslinewidth,showpoints=false}
  \rput*(2.3,.9){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](2.3,.9){\small $\varepsilon<10^{-1}$}
  \rput*(2.3,.8){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](2.3,.8){\small $\varepsilon<10^{-2}$}
  \rput*(2.3,.7){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](2.3,.7){\small $\varepsilon<10^{-3}$}
  \rput*(2.3,.6){\psline[linecolor=black](-.75cm,0)}
  \rput*[l](2.3,.6){\small $\varepsilon<10^{-4}$}
  \rput*(2.3,.5){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](2.3,.5){\small $\varepsilon<10^{-5}$}
  \rput*(2.3,.4){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](2.3,.4){\small solution exacte}
\end{pspicture}
\end{lstlisting}





\subsection{Equation of second order}

Here is the traditional simulation of two stars attracting each
other according to the classical gravitation law in
$\displaystyle\frac{1}{r^2}$. In 2-Dimensions, the system to be
solved is composed of four second order differential equations. In
order to be described, each of them gives two first order
equations, then we obtain a 8 sized vectorial equation. In the
following example the masses of the stars are 1 and 20.

\[
\left\{
\begin{array}[m]{l}
  x''_1=\displaystyle\frac{M_2}{r^2}\cos(\theta)\\
  y''_1=\displaystyle\frac{M_2}{r^2}\sin(\theta)\\
  x''_2=\displaystyle\frac{M_1}{r^2}\cos(\theta)\\
  y''_2=\displaystyle\frac{M_1}{r^2}\sin(\theta)\\
\end{array}
\right.
\mbox{ avec }
\left\{
\begin{array}[m]{l}
  r^2=(x_1-x_2)^2+(y_1-y_2)^2\\
  \cos(\theta)=\displaystyle\frac{(x_1-x_2)}{r}\\
  \sin(\theta)=\displaystyle\frac{(y_1-y_2)}{r}\\
\end{array}
\right.
\mbox{%
\begin{pspicture}[shift=-2](5,4)\psset{arrowscale=2}
  \psframe[linewidth=.75\pslinewidth](5,4)
  \pstGeonode[PosAngle={-90,90}](1,1){M_1}(4,3){M_2}
  \pstHomO[HomCoef=.33, PointSymbol=none]{M_1}{M_2}[F_1]
  \psline[arrows=->](M_1)(F_1)
  \pstHomO[HomCoef=.33, PointSymbol=none]{M_2}{M_1}[F_2]
  \psline[arrows=->, arrowscale=2](M_2)(F_2)
  \pstGeonode[PointSymbol=none, PointName=none](M_2|M_1){A}
  \psline[linewidth=.5\pslinewidth](M_1)(A)
  \pstMarkAngle{A}{M_1}{M_2}{$\theta$}
  \ncline[linewidth=.5\pslinewidth, offset=.5, arrows=<->]{M_1}{M_2}
  \ncput*{$r$}
\end{pspicture}}
\]

\begin{table}[!htbp]
  \centering\small
    \begin{tabular}{|l@{}>{\ttfamily}l@{}>{ \ttfamily \%\% }l|}
      \hline
      && x1 y1 x'1 y'1 x2 y2 x'2 y'2\\
      &/yp2 exch def /xp2 exch def /ay2 exch def /ax2 exch def&mise en variables\\
      &/yp1 exch def /xp1 exch def /ay1 exch def /ax1 exch def&mise en variables\\
      &/ro2 ax2 ax1 sub dup mul ay2 ay1 sub dup mul add def&calcul de r*r\\
      &xp1 yp1&\\
      &ax2 ax1 sub ro2 sqrt div ro2 div&calcul de x''1\\
      &ay2 ay1 sub ro2 sqrt div ro2 div&calcul de y''1\\
      &xp2 yp2&\\
      &3 index -20 mul&calcul de x''2=-20x''1\\
      &3 index -20 mul&calcul de y''2=-20y''1\\
      \hline
    \end{tabular}
    \caption{\PostScript source code for the gravitational interaction}\label{intgravcode}
\end{table}

\begin{table}[!htbp]
  \centering
    \small\newcommand{\POW}{\symbol{'136}}
    \begin{tabular}{|l@{}>{\ttfamily}l@{}>{ \ttfamily \%\% }l|}
      \hline
      &y[2]|&y'[0]\\
      &y[3]|&y'[1]\\
      &(y[4]-y[0])/((y[4]-y[0])\POW 2+(y[5]-y[1])\POW 2)\POW 1.5|&y'[2]=y''[0]\\
      &(y[5]-y[1])/((y[4]-y[0])\POW 2+(y[5]-y[1])\POW 2)\POW 1.5|&y'[3]=y''[1]\\
      &y[6]|&y'[4]\\
      &y[7]|&y'[5]\\
      &20*(y[0]-y[4])/((y[4]-y[0])\POW 2+(y[5]-y[1])\POW 2)\POW 1.5|&y'[6]=y''[4]\\
      &20*(y[1]-y[5])/((y[4]-y[0])\POW 2+(y[5]-y[1])\POW 2)\POW 1.5&y'[7]=y''[5]\\
      \hline
    \end{tabular}
    \caption{Algebraic description for the gravitational interaction}\label{intgravalgcode}
\end{table}

\newcommand\Grav{%
  /yp2 exch def /xp2 exch def /ay2 exch def /ax2 exch def
  /yp1 exch def /xp1 exch def /ay1 exch def /ax1 exch def
  /ro2 ax2 ax1 sub dup mul ay2 ay1 sub dup mul add def
  xp1 yp1
  ax2 ax1 sub ro2 sqrt div ro2 div
  ay2 ay1 sub ro2 sqrt div ro2 div
  xp2 yp2
  3 index -20 mul
  3 index -20 mul}
\newcommand\GravAlg{%
  y[2]|y[3]|%
  (y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|%
  (y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|%
  y[6]|y[7]|%
  20*(y[0]-y[4])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|%
  20*(y[1]-y[5])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5}
%%  0  1   2   3  4  5   6   7
%% x1 y1 x'1 y'1 x2 y2 x'2 y'2


\begin{LTXexample}[width=5cm,wide]
\def\InitCond{ 1  1  .1  0 -1 -1  -2   0}
\begin{pspicture}[shift=-2,showgrid=true](-3,-1.75)(2,1.5)
  \psplotDiffEqn[whichabs=0, whichord=1, linecolor=blue, method=rk4, plotpoints=100]{0}{3.95}{\InitCond}{\Grav}
  \psset{showpoints=true,whichabs=4, whichord=5}
  \psplotDiffEqn[linecolor=black, method=varrkiv, varsteptol=.0001, plotpoints=200]{0}{3.9}{\InitCond}{\Grav}
\end{pspicture}
\end{LTXexample}
\vspace{-2ex}
{\captionof{figure}{Gravitational interaction: fixed landmark, trajectory of the stars}\label{fig:InterGravRepFix}}



\bigskip
\begin{LTXexample}[width=5cm,wide]
\def\InitCond{ 1  1  .1  0 -1 -1  -2   0}
\begin{pspicture}[shift=-1.5,showgrid=true](-4,-1.75)(1,1)
  \psplotDiffEqn[linecolor=red, plotpoints=200,method=varrkiv, varsteptol=.0001, showpoints=true,
      plotfuncx=y dup 4 get exch 0 get sub,
      plotfuncy=dup 5 get exch 1 get sub ]{0}{3.9}{\InitCond}{\Grav}
\end{pspicture}
\end{LTXexample}
\vspace{-2ex}
{\captionof{figure}{Gravitational interaction : landmark defined by one star}\label{fig:IGnewrep}}


\begin{center}
\bgroup
\def\InitCond{ 1  1  .1   0 -1 -1  -2   0}
\psset{xunit=2}
\begin{pspicture}[showgrid=true](0,0)(8,9)
  \psset{showpoints=true}
  \psplotDiffEqn[linecolor=red, method=varrkiv, plotpoints=2, varsteptol=.0001,
      plotfuncy=dup 6 get dup mul exch 7 get dup mul add sqrt]{0}{8}{\InitCond}{\Grav}
  \psplotDiffEqn[linecolor=blue, method=varrkiv, plotpoints=2, varsteptol=.0001,
      plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt]{0}{8}{\InitCond}{\Grav}
\end{pspicture}
\captionof{figure}{Gravitational interaction : speeds of the
stars} \egroup
\end{center}

\begin{lstlisting}
\psset{xunit=2}
\begin{pspicture}[showgrid=true](0,0)(8,9)
  \psset{showpoints=true}
  \psplotDiffEqn[linecolor=red, method=varrkiv, plotpoints=2, varsteptol=.0001,
      plotfuncy=dup 6 get dup mul exch 7 get dup mul add sqrt]{0}{8}{\InitCond}{\Grav}
  \psplotDiffEqn[linecolor=blue, method=varrkiv, plotpoints=2, varsteptol=.0001,
      plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt]{0}{8}{\InitCond}{\Grav}
\end{pspicture}
\end{lstlisting}

%--------------------------------------------------------------------------------------
\subsubsection{Simple equation of first order $y'=y$}
%--------------------------------------------------------------------------------------

For the initial value $y(0)=1$ we have the solution $y(x)=e^x$. $y$ is always
on the stack, so we have to do nothing. Using the \verb+algebraic+ option, we write it
as \verb$y[0]$. The following example shows different solutions depending to the number of plotpoints
with $y_0=1$:


\begin{center}
\bgroup
\psset{xunit=4, yunit=.4}
\begin{pspicture}(3,19)\psgrid[subgriddiv=1]
  \psplot[linewidth=6\pslinewidth, linecolor=green]{0}{3}{Euler x exp}
  \psplotDiffEqn[linecolor=magenta,plotpoints=16,algebraic=true]{0}{3}{1}{y[0]}
  \psplotDiffEqn[linecolor=blue,plotpoints=151]{0}{3}{1}{}
  \psplotDiffEqn[linecolor=red,method=rk4,plotpoints=15]{0}{3}{1}{}
  \psplotDiffEqn[linecolor=Orange,method=rk4,plotpoints=4]{0}{3}{1}{}
  \psset{linewidth=4\pslinewidth}
  \rput*(0.35,19){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](0.35,19){\small Euler order 1 $h=0{,}2$}
  \rput*(0.35,17){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](0.35,17){\small Euler order 1 $h=0{,}02$}
  \rput*(0.35,15){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](0.35,15){\small RK ordre 4 $h=1$}
  \rput*(0.35,13){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](0.35,13){\small RK ordre 4 $h=0{,}2$}
  \rput*(0.35,11){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](0.35,11){\small solution exacte}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{xunit=4, yunit=.4}
\begin{pspicture}(3,19)\psgrid[subgriddiv=1]
  \psplot[linewidth=6\pslinewidth, linecolor=green]{0}{3}{Euler x exp}
  \psplotDiffEqn[linecolor=magenta,plotpoints=16,algebraic=true]{0}{3}{1}{y[0]}
  \psplotDiffEqn[linecolor=blue,plotpoints=151]{0}{3}{1}{}
  \psplotDiffEqn[linecolor=red,method=rk4,plotpoints=15]{0}{3}{1}{}
  \psplotDiffEqn[linecolor=Orange,method=rk4,plotpoints=4]{0}{3}{1}{}
  \psset{linewidth=4\pslinewidth}
  \rput*(0.35,19){\psline[linecolor=magenta](-.75cm,0)}
  \rput*[l](0.35,19){\small Euler order 1 $h=0{,}2$}
  \rput*(0.35,17){\psline[linecolor=blue](-.75cm,0)}
  \rput*[l](0.35,17){\small Euler order 1 $h=0{,}02$}
  \rput*(0.35,15){\psline[linecolor=Orange](-.75cm,0)}
  \rput*[l](0.35,15){\small RK ordre 4 $h=1$}
  \rput*(0.35,13){\psline[linecolor=red](-.75cm,0)}
  \rput*[l](0.35,13){\small RK ordre 4 $h=0{,}2$}
  \rput*(0.35,11){\psline[linecolor=green](-.75cm,0)}
  \rput*[l](0.35,11){\small solution exacte}
\end{pspicture}
\end{lstlisting}

%--------------------------------------------------------------------------------------
\subsubsection{$y'=\displaystyle\frac{2-ty}{4-t^2}$}% $
%--------------------------------------------------------------------------------------

For the initial value $y(0)=1$ the exact solution is
$y(x)=\displaystyle\frac{t+\sqrt{4-t^2}}{2}$. The function $f$
described in PostScript code is like (y is still on the stack):
\begin{lstlisting}[style=syntax]
x              %% y x
mul            %% x*y
2 exch sub     %% 2-x*y
4 x dup mul    %% 2-x*y 4 x^2
sub            %% 2-x*y 4-x^2
div            %% (2-x*y)/(4-x^2)
\end{lstlisting}
\noindent
The following example uses $y_0=1$.

\begin{lstlisting}[style=syntax]
\newcommand{\InitCond}{1}
\newcommand{\Func}{x mul 2 exch sub 4 x dup mul sub div}
\newcommand{\FuncAlg}{(2-x*y[0])/(4-x^2)}
\end{lstlisting}

\begin{center}
\bgroup
\psset{xunit=6.4, yunit=9.6, showpoints=false}
\begin{pspicture}(0,1)(2,1.5)  \psgrid[griddots=10](0,1)(2,1.5)
  { \psset{linewidth=4\pslinewidth,linecolor=lightgray}
  \psplot{0}{1.8}{x dup dup mul 4 exch sub sqrt add 2 div}
  \psplot{1.8}{2}{x dup dup mul 4 exch sub sqrt add 2 div} }
  \def\InitCond{1}
  \def\Func{x mul 2 exch sub 4 x dup mul sub div}
  \psplotDiffEqn[linecolor=magenta, plotpoints=20]{0}{1.9}{\InitCond}{\Func}
  \psplotDiffEqn[linecolor=blue, plotpoints=191]{0}{1.9}{\InitCond}{\Func}
  \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11,%
     algebraic=true]{0}{1.9}{\InitCond}{(2-x*y[0])/(4-x^2)}
  \psplotDiffEqn[linecolor=Orange, method=rk4, plotpoints=21,%
     algebraic=true]{0}{1.9}{\InitCond}{(2-x*y[0])/(4-x^2)}
  \psset{linewidth=4\pslinewidth}\small
  \rput*(0,1.4){\psline[linecolor=magenta](-.75cm,0)}\rput*[l](0,1.4){Euler order 1 $h=0{,}1$}
  \rput*(0,1.35){\psline[linecolor=blue](-.75cm,0)}\rput*[l](0,1.35){Euler order 1 $h=0{,}01$}
  \rput*(0,1.3){\psline[linecolor=Orange](-.75cm,0)}\rput*[l](0,1.3){RK order 4 $h=0{,}19$}
  \rput*(0,1.25){\psline[linecolor=red](-.75cm,0)}\rput*[l](0,1.25){RK order 4 $h=0{,}095$}
  \rput*(0,1.2){\psline[linecolor=lightgray](-.75cm,0)}\rput*[l](0,1.2){exactly}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}[xrightmargin=-1cm,xleftmargin=-1cm]
\psset{xunit=6.4, yunit=9.6, showpoints=false}
\begin{pspicture}(0,1)(2,1.7)  \psgrid[subgriddiv=5]
  { \psset{linewidth=4\pslinewidth,linecolor=lightgray}
  \psplot{0}{1.8}{x dup dup mul 4 exch sub sqrt add 2 div}
  \psplot{1.8}{2}{x dup dup mul 4 exch sub sqrt add 2 div} }
  \def\InitCond{1}
  \def\Func{x mul 2 exch sub 4 x dup mul sub div}
  \psplotDiffEqn[linecolor=magenta, plotpoints=20]{0}{1.9}{\InitCond}{\Func}
  \psplotDiffEqn[linecolor=blue, plotpoints=191]{0}{1.9}{\InitCond}{\Func}
  \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11,%
     algebraic=true]{0}{1.9}{\InitCond}{(2-x*y[0])/(4-x^2)}
  \psplotDiffEqn[linecolor=Orange, method=rk4, plotpoints=21,%
     algebraic=true]{0}{1.9}{\InitCond}{(2-x*y[0])/(4-x^2)}
  \psset{linewidth=4\pslinewidth}
  \rput*(0.3,1.6){\psline[linecolor=magenta](-.75cm,0)}\rput*[l](0.3,1.6){\small Euler order 1 $h=0{,}1$}
  \rput*(0.3,1.55){\psline[linecolor=blue](-.75cm,0)}\rput*[l](0.3,1.55){\small Euler order 1 $h=0{,}01$}
  \rput*(0.3,1.5){\psline[linecolor=Orange](-.75cm,0)}\rput*[l](0.3,1.5){\small RK order 4 $h=0{,}19$}
  \rput*(0.3,1.45){\psline[linecolor=red](-.75cm,0)}\rput*[l](0.3,1.45){\small RK order 4 $h=0{,}095$}
  \rput*(0.3,1.4){\psline[linecolor=lightgray](-.75cm,0)}\rput*[l](0.3,1.4){\small exactly}
\end{pspicture}
\end{lstlisting}


%--------------------------------------------------------------------------------------
\subsubsection{$y'=-2xy$}
%--------------------------------------------------------------------------------------

For $y(-1)=\frac{1}{e}$ we get $y(x)=e^{-x^2}$.

\begin{center}
\bgroup
\psset{unit=4}
\begin{pspicture}(-1,0)(3,1.1)\psgrid
  \psplot[linewidth=4\pslinewidth,linecolor=gray]{-1}{3}{Euler x dup mul neg exp}
  \psset{plotpoints=9}
  \psplotDiffEqn[linecolor=cyan]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psplotDiffEqn[linecolor=yellow, method=rk4]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psset{plotpoints=21}
  \psplotDiffEqn[linecolor=blue]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psplotDiffEqn[linecolor=Orange, method=rk4]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psset{linewidth=2\pslinewidth}
  \rput*(2,1){\psline[linecolor=Orange](-0.25,0)}
  \rput*[l](2,1){RK}
  \rput*(2,.9){\psline[linecolor=blue](-0.25,0)}
  \rput*[l](2,.9){\textsc{Euler}-1}
  \rput*(2,.8){\psline[linecolor=gray](-0.25,0)}
  \rput*[l](2,.8){solution}
\end{pspicture}
\egroup
\end{center}


\begin{lstlisting}
\psset{unit=4}
\begin{pspicture}(-1,0)(3,1.1)\psgrid
  \psplot[linewidth=4\pslinewidth,linecolor=gray]{-1}{3}{Euler x dup mul neg exp}
  \psset{plotpoints=9}
  \psplotDiffEqn[linecolor=cyan]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psplotDiffEqn[linecolor=yellow, method=rk4]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psset{plotpoints=21}
  \psplotDiffEqn[linecolor=blue]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psplotDiffEqn[linecolor=Orange, method=rk4]{-1}{3}{1 Euler div}{x -2 mul mul}
  \psset{linewidth=2\pslinewidth}
  \rput*(2,1){\psline[linecolor=Orange](-0.25,0)}
  \rput*[l](2,1){RK}
  \rput*(2,.9){\psline[linecolor=blue](-0.25,0)}
  \rput*[l](2,.9){\textsc{Euler}-1}
  \rput*(2,.8){\psline[linecolor=gray](-0.25,0)}
  \rput*[l](2,.8){solution}
\end{pspicture}
\end{lstlisting}


%--------------------------------------------------------------------------------------
\subsubsection{Spiral of Cornu}
%--------------------------------------------------------------------------------------

The integrals of Fresnel :
\begin{align} x & =\int^t_0\cos\frac{\pi t^2}{2}\mathrm{d}t \\
 y & =\int^t_0\sin\frac{\pi t^2}{2}\mathrm{d}t \\
\intertext{with}
 \dot{x} &= \cos\frac{\pi t^2}{2} \\
 \dot{y} & =\sin\frac{\pi t^2}{2}
 \end{align}

\begin{lstlisting}
\psset{unit=8}
\begin{pspicture}(1,1)\psgrid[subgriddiv=5]
  \psplotDiffEqn[whichabs=0,whichord=1,linecolor=red,method=rk4,algebraic,%
     plotpoints=500,showpoints=true]{0}{10}{0 0}{cos(Pi*x^2/2)|sin(Pi*x^2/2)}
\end{pspicture}
\end{lstlisting}


\begin{center}
\bgroup
\psset{unit=8}
\begin{pspicture}(1,1)\psgrid[subgriddiv=5]
  \psplotDiffEqn[whichabs=0,whichord=1,linecolor=red,method=rk4,algebraic,%
     plotpoints=500,showpoints=true]{0}{10}{0 0}{cos(Pi*x^2/2)|sin(Pi*x^2/2)}
\end{pspicture}
\egroup
\end{center}



%--------------------------------------------------------------------------------------
\subsubsection{Lotka-Volterra}
%--------------------------------------------------------------------------------------

The Lotka-Volterra model describes interactions between two species in an ecosystem, a predator and a prey. This represents our first multi-species model. Since we are considering two species, the model will involve two equations, one which describes how the prey population changes and the second which describes how the predator population changes.

For concreteness let us assume that the prey in our model are rabbits, and that the predators are foxes. If we let $R(t)$ and $F(t)$ represent the number of rabbits and foxes, respectively, that are alive at time t, then the Lotka-Volterra model is:

\begin{align}
\dot R &= a\cdot R - b\cdot R\cdot F\\
\dot F &= e\cdot b\cdot R\cdot F - c\cdot F
\end{align}

where the parameters are defined by:
\begin{description}
\item[a] is the natural growth rate of rabbits in the absence of predation,
\item[c] is the natural death rate of foxes in the absence of food (rabbits),
\item[b] is the death rate per encounter of rabbits due to predation,
\item[e] is the efficiency of turning predated rabbits into foxes.
\end{description}

The Stella model representing the Lotka-Volterra model will be slightly more complex than the single species models we've dealt with before. The main difference is that our model will have two stocks (reservoirs), one for each species. Each species will have its own birth and death rates. In addition, the Lotka-Volterra model involves four parameters rather than two. All told, the Stella representation of the Lotka-Volterra model will use two stocks, four flows, four converters and many connectors.

\bgroup
\def\InitCond{ 0 10 10}%% xa ya xl
\def\Faiglelapin{\Vaigle*(y[2]-y[0])/sqrt(y[1]^2+(y[2]-y[0])^2)|%
                 -\Vaigle*y[1]/sqrt(y[1]^2+(y[2]-y[0])^2)|%
                 -\Vlapin}
\def\Vlapin{1}  \def\Vaigle{1.6}
\psset{unit=.7,subgriddiv=0,gridcolor=lightgray,method=adams,algebraic,%
   plotpoints=20,showpoints=true}
\begin{pspicture}(-3,-8.25)(8,10)\psgrid[griddots=10]
 \psplotDiffEqn[plotfuncy=pop 0,whichabs=2,linecolor=red]{0}{10}{\InitCond}{\Faiglelapin}
 \psplotDiffEqn[whichabs=0,whichord=1,linecolor=black,method=rk4]{0}{10}{\InitCond}{\Faiglelapin}
  \psplotDiffEqn[whichabs=0,whichord=1,linecolor=blue]{0}{10}{\InitCond}{\Faiglelapin}
\end{pspicture}\hfill
\begin{pspicture}(0,-0.25)(10,14)\psgrid
 \psplotDiffEqn[plotfuncy=dup 1 get dup mul exch dup 0 get exch 2 get sub dup
    mul add sqrt,linecolor=red,method=rk4]{0}{10}{\InitCond}{\Faiglelapin}
 \psplotDiffEqn[plotfuncy=dup 1 get dup mul exch dup 0 get exch 2 get sub dup
    mul add sqrt,linecolor=blue]{0}{10}{\InitCond}{\Faiglelapin}
 \psplotDiffEqn[plotfuncy=pop Func aload pop pop dup mul exch dup mul add sqrt,
    linecolor=yellow]{0}{10}{\InitCond}{\Faiglelapin}
\end{pspicture}
\egroup
\begin{lstlisting}[label={fig:aiglelapin}]
\def\InitCond{ 0 10 10}%% xa ya xl
\def\Faiglelapin{\Vaigle*(y[2]-y[0])/sqrt(y[1]^2+(y[2]-y[0])^2)|%
                 -\Vaigle*y[1]/sqrt(y[1]^2+(y[2]-y[0])^2)|%
                 -\Vlapin}
\def\Vlapin{1}  \def\Vaigle{1.6}
\psset{unit=.7,subgriddiv=0,gridcolor=lightgray,method=adams,algebraic,%
   plotpoints=20,showpoints=true}
\begin{pspicture}(-3,-8)(5,10)\psgrid[griddots=10]
 \psplotDiffEqn[plotfuncy=pop 0,whichabs=2,linecolor=red]{0}{10}{\InitCond}{\Faiglelapin}
 \psplotDiffEqn[whichabs=0,whichord=1,linecolor=black,method=rk4]{0}{10}{\InitCond}{\Faiglelapin}
  \psplotDiffEqn[whichabs=0,whichord=1,linecolor=blue]{0}{10}{\InitCond}{\Faiglelapin}
\end{pspicture}\hfill
\begin{pspicture}(10,12)\psgrid
 \psplotDiffEqn[plotfuncy=dup 1 get dup mul exch dup 0 get exch 2 get sub dup
    mul add sqrt,linecolor=red,method=rk4]{0}{10}{\InitCond}{\Faiglelapin}
 \psplotDiffEqn[plotfuncy=dup 1 get dup mul exch dup 0 get exch 2 get sub dup
    mul add sqrt,linecolor=blue]{0}{10}{\InitCond}{\Faiglelapin}
 \psplotDiffEqn[plotfuncy=pop Func aload pop pop dup mul exch dup mul add sqrt,
    linecolor=yellow]{0}{10}{\InitCond}{\Faiglelapin}
\end{pspicture}
\end{lstlisting}


\clearpage
%--------------------------------------------------------------------------------------
\subsubsection{$y''=y$}
%--------------------------------------------------------------------------------------

Beginning with the initial equation $\displaystyle y(x)=Ae^x+Be^{-x}$ we get the hyperbolic
trigonometrical functions.

\begin{center}
\bgroup
\def\Funct{exch}   \psset{xunit=5cm, yunit=0.75cm}
\begin{pspicture}(0,-0.25)(2,7)\psgrid[subgriddiv=1,griddots=10]
 \psplot[linewidth=4\pslinewidth, linecolor=green]{0}{2}{Euler x exp}  %%e^x
 \psplotDiffEqn[linecolor=magenta, plotpoints=11]{0}{2}{1 1}{\Funct}
 \psplotDiffEqn[linecolor=blue, plotpoints=101]{0}{2}{1 1}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11]{0}{2}{1 1}{\Funct}
 \psplot[linewidth=4\pslinewidth, linecolor=green]{0}{2}{Euler dup x exp  %%ch(x)
    exch x neg exp add 2 div}
 \psplotDiffEqn[linecolor=magenta, plotpoints=11]{0}{2}{1 0}{\Funct}
 \psplotDiffEqn[linecolor=blue, plotpoints=101]{0}{2}{1 0}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11]{0}{2}{1 0}{\Funct}
 \psplot[linewidth=4\pslinewidth, linecolor=green]{0}{2}{Euler dup x exp
     exch x neg exp sub 2 div}  %%sh(x)
 \psplotDiffEqn[linecolor=magenta, plotpoints=11]{0}{2}{0 1}{\Funct}
 \psplotDiffEqn[linecolor=blue, plotpoints=101]{0}{2}{0 1}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11]{0}{2}{0 1}{\Funct}
 \rput*(1.3,.9){\psline[linecolor=magenta](-.75cm,0)}\rput*[l](1.3,.9){\small\textsc{Euler} order 1 $h=1$}
 \rput*(1.3,.8){\psline[linecolor=blue](-.75cm,0)}\rput*[l](1.3,.8){\small\textsc{Euler} order 1 $h=0{,}1$}
 \rput*(1.3,.7){\psline[linecolor=red](-.75cm,0)}\rput*[l](1.3,.7){\small RK order 4 $h=1$}
 \rput*(1.3,.6){\psline[linecolor=green](-.75cm,0)}\rput*[l](1.3,.6){\small exact solution}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}[label={fig:minusexp}]
\def\Funct{exch}   \psset{xunit=5cm, yunit=0.75cm}
\begin{pspicture}(0,-0.25)(2,7)\psgrid[subgriddiv=1,griddots=10]
 \psplot[linewidth=4\pslinewidth, linecolor=green]{0}{2}{Euler x exp}  %%e^x
 \psplotDiffEqn[linecolor=magenta, plotpoints=11]{0}{2}{1 1}{\Funct}
 \psplotDiffEqn[linecolor=blue, plotpoints=101]{0}{2}{1 1}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11]{0}{2}{1 1}{\Funct}
 \psplot[linewidth=4\pslinewidth, linecolor=green]{0}{2}{Euler dup x exp  %%ch(x)
    exch x neg exp add 2 div}
 \psplotDiffEqn[linecolor=magenta, plotpoints=11]{0}{2}{1 0}{\Funct}
 \psplotDiffEqn[linecolor=blue, plotpoints=101]{0}{2}{1 0}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11]{0}{2}{1 0}{\Funct}
 \psplot[linewidth=4\pslinewidth, linecolor=green]{0}{2}{Euler dup x exp
     exch x neg exp sub 2 div}  %%sh(x)
 \psplotDiffEqn[linecolor=magenta, plotpoints=11]{0}{2}{0 1}{\Funct}
 \psplotDiffEqn[linecolor=blue, plotpoints=101]{0}{2}{0 1}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=11]{0}{2}{0 1}{\Funct}
 \rput*(1.3,.9){\psline[linecolor=magenta](-.75cm,0)}\rput*[l](1.3,.9){\small\textsc{Euler} order 1 $h=1$}
 \rput*(1.3,.8){\psline[linecolor=blue](-.75cm,0)}\rput*[l](1.3,.8){\small\textsc{Euler} order 1 $h=0{,}1$}
 \rput*(1.3,.7){\psline[linecolor=red](-.75cm,0)}\rput*[l](1.3,.7){\small RK order 4 $h=1$}
 \rput*(1.3,.6){\psline[linecolor=green](-.75cm,0)}\rput*[l](1.3,.6){\small exact solution}
\end{pspicture}
\end{lstlisting}

%--------------------------------------------------------------------------------------
\subsubsection{$y''=-y$}
%--------------------------------------------------------------------------------------
\begin{center}
\bgroup
\def\Funct{exch neg}
\psset{xunit=1, yunit=4}
\def\quatrepi{12.5663706144}%%4pi=12.5663706144
\begin{pspicture}(0,-1.25)(\quatrepi,1.25)\psgrid[subgriddiv=1,griddots=10]
 \psplot[linewidth=4\pslinewidth,linecolor=green]{0}{\quatrepi}{x RadtoDeg cos}%%cos(x)
 \psplotDiffEqn[linecolor=blue, plotpoints=201]{0}{3.1415926}{1 0}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=31]{0}{\quatrepi}{1 0}{\Funct}
 \psplot[linewidth=4\pslinewidth,linecolor=green]{0}{\quatrepi}{x RadtoDeg sin}  %%sin(x)
 \psplotDiffEqn[linecolor=blue,plotpoints=201]{0}{3.1415926}{0 1}{\Funct}
 \psplotDiffEqn[linecolor=red,method=rk4, plotpoints=31]{0}{\quatrepi}{0 1}{\Funct}
 \rput*(3.3,.9){\psline[linecolor=magenta](-.75cm,0)}\rput*[l](3.3,.9){\small Euler order 1 $h=1$}
 \rput*(3.3,.8){\psline[linecolor=blue](-.75cm,0)}\rput*[l](3.3,.8){\small Euler order 1 $h=0{,}1$}
 \rput*(3.3,.7){\psline[linecolor=red](-.75cm,0)}\rput*[l](3.3,.7){\small RK order 4 $h=1$}
 \rput*(3.3,.6){\psline[linecolor=green](-.75cm,0)}\rput*[l](3.3,.6){\small exact solution}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}[label={fig:minusexp2}]
\def\Funct{exch neg}
\psset{xunit=1, yunit=4}
\def\quatrepi{12.5663706144}%%4pi=12.5663706144
\begin{pspicture}(0,-1.25)(\quatrepi,1.25)\psgrid[subgriddiv=1,griddots=10]
 \psplot[linewidth=4\pslinewidth,linecolor=green]{0}{\quatrepi}{x RadtoDeg cos}%%cos(x)
 \psplotDiffEqn[linecolor=blue, plotpoints=201]{0}{3.1415926}{1 0}{\Funct}
 \psplotDiffEqn[linecolor=red, method=rk4, plotpoints=31]{0}{\quatrepi}{1 0}{\Funct}
 \psplot[linewidth=4\pslinewidth,linecolor=green]{0}{\quatrepi}{x RadtoDeg sin}  %%sin(x)
 \psplotDiffEqn[linecolor=blue,plotpoints=201]{0}{3.1415926}{0 1}{\Funct}
 \psplotDiffEqn[linecolor=red,method=rk4, plotpoints=31]{0}{\quatrepi}{0 1}{\Funct}
 \rput*(3.3,.9){\psline[linecolor=magenta](-.75cm,0)}\rput*[l](3.3,.9){\small Euler order 1 $h=1$}
 \rput*(3.3,.8){\psline[linecolor=blue](-.75cm,0)}\rput*[l](3.3,.8){\small Euler order 1 $h=0{,}1$}
 \rput*(3.3,.7){\psline[linecolor=red](-.75cm,0)}\rput*[l](3.3,.7){\small RK order 4 $h=1$}
 \rput*(3.3,.6){\psline[linecolor=green](-.75cm,0)}\rput*[l](3.3,.6){\small exact solution}
\end{pspicture}
\end{lstlisting}

%--------------------------------------------------------------------------------------
\subsubsection{The mechanical pendulum: $y''=-\frac{g}{l}\sin(y)$}% $
%--------------------------------------------------------------------------------------

For small oscillations $\sin(y)\simeq y$:

\[ y(x)=y_0\cos\left(\sqrt{\frac{g}{l}}x\right) \]

The function $f$ is written in PostScript code:

\begin{lstlisting}[style=syntax]
exch RadtoDeg sin -9.8 mul %% y' -gsin(y)
\end{lstlisting}

\begin{center}
\bgroup
\def\Func{y[1]|-9.8*sin(y[0])}
\psset{yunit=2,xunit=4,algebraic=true,linewidth=1.5pt}
\begin{pspicture}(0,-2.25)(3,2.25)
  \psaxes{->}(0,0)(0,-2)(3,2)
  \psplot[linewidth=3\pslinewidth, linecolor=Orange]{0}{3}{.1*cos(sqrt(9.8)*x)}
  \psset{method=rk4,plotpoints=50,linecolor=blue}
  \psplotDiffEqn{0}{3}{.1 0}{\Func}
  \psplot[linewidth=3\pslinewidth,linecolor=Orange]{0}{3}{.25*cos(sqrt(9.8)*x)}
  \psplotDiffEqn{0}{3}{.25 0}{\Func}
  \psplotDiffEqn{0}{3}{.5 0}{\Func}
  \psplotDiffEqn{0}{3}{1 0}{\Func}
  \psplotDiffEqn[plotpoints=100]{0}{3}{Pi 2 div 0}{\Func}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}[label=fig:second]
\def\Func{y[1]|-9.8*sin(y[0])}
\psset{yunit=2,xunit=4,algebraic=true,linewidth=1.5pt}
\begin{pspicture}(0,-2.25)(3,2.25)
  \psaxes{->}(0,0)(0,-2)(3,2)
  \psplot[linewidth=3\pslinewidth, linecolor=Orange]{0}{3}{.1*cos(sqrt(9.8)*x)}
  \psset{method=rk4,plotpoints=50,linecolor=blue}
  \psplotDiffEqn{0}{3}{.1 0}{\Func}
  \psplot[linewidth=3\pslinewidth,linecolor=Orange]{0}{3}{.25*cos(sqrt(9.8)*x)}
  \psplotDiffEqn{0}{3}{.25 0}{\Func}
  \psplotDiffEqn{0}{3}{.5 0}{\Func}
  \psplotDiffEqn{0}{3}{1 0}{\Func}
  \psplotDiffEqn[plotpoints=100]{0}{3}{Pi 2 div 0}{\Func}
\end{pspicture}
\end{lstlisting}

%--------------------------------------------------------------------------------------
\subsubsection{$y''=-\frac{y'}{4}-2y$}% $
%--------------------------------------------------------------------------------------

For $y_0=5$ and $y'_0=0$ the solution is:

\[
5e^{-\frac{x}{8}}\left(\cos\left(\omega x\right)+\frac{\sin(\omega x)}{8\omega}\right)
\mbox{ avec } \omega=\frac{\sqrt{127}}{8}
\]

\iffalse
La fonction $f$ est dï¿½rite par le code PostScript suivant :

\begin{lstlisting}[style=syntax]
dup             %% y y' y'
3 1 roll        %% y' y y'
-4 div          %% y' y y'/-4
exch            %% y' y'/-4 y
2 mul           %% y' y'/-4 2y
sub             %% y' y'/-4-2y
\end{lstlisting}

\fi
\begin{center}
\bgroup
\psset{xunit=.6,yunit=0.8,plotpoints=500}
\begin{pspicture}(0,-4.25)(26,5.25)
  \psaxes{->}(0,0)(0,-4)(26,5)
  \psplot[plotpoints=200,linewidth=4\pslinewidth,linecolor=gray]{0}{26}{%
     Euler x -8 div exp x 127 sqrt 8 div mul RadtoDeg dup cos 5 mul exch sin 127 sqrt div 5 mul add mul}
  \psplotDiffEqn[linecolor=red,linewidth=5\pslinewidth]{0}{26}{5 0}
     {dup 3 1 roll -4 div exch 2 mul sub}
  \psplotDiffEqn[linecolor=black,algebraic]{0}{26}{5 0} {y[1]|-y[1]/4-2*y[0]}
  \psset{method=rk4, plotpoints=50}
  \psplotDiffEqn[linecolor=blue,linewidth=5\pslinewidth]{0}{26}{5 0}{%
      dup 3 1 roll -4 div exch 2 mul sub}
  \psplotDiffEqn[linecolor=black,algebraic=true]{0}{26}{5 0}{y[1]|-y[1]/4-2*y[0]}
\end{pspicture}
\egroup
\end{center}

\begin{lstlisting}
\psset{xunit=.6,yunit=0.8,plotpoints=500}
\begin{pspicture}(0,-4.25)(26,5.25)
  \psaxes{->}(0,0)(0,-4)(26,5)
  \psplot[plotpoints=200,linewidth=4\pslinewidth,linecolor=gray]{0}{26}{%
     Euler x -8 div exp x 127 sqrt 8 div mul RadtoDeg dup cos 5 mul exch sin 127 sqrt div 5 mul add mul}
  \psplotDiffEqn[linecolor=red,linewidth=5\pslinewidth]{0}{26}{5 0}
     {dup 3 1 roll -4 div exch 2 mul sub}
  \psplotDiffEqn[linecolor=black,algebraic]{0}{26}{5 0} {y[1]|-y[1]/4-2*y[0]}
  \psset{method=rk4, plotpoints=50}
  \psplotDiffEqn[linecolor=blue,linewidth=5\pslinewidth]{0}{26}{5 0}{%
      dup 3 1 roll -4 div exch 2 mul sub}
  \psplotDiffEqn[linecolor=black,algebraic=true]{0}{26}{5 0}{y[1]|-y[1]/4-2*y[0]}
\end{pspicture}
\end{lstlisting}

\iffalse

\newpage
%--------------------------------------------------------------------------------------
\subsubsection{Gravitation: two stars and more}
%--------------------------------------------------------------------------------------

\iffalse
L'exemple le plus complexe dans l'ï¿½riture de $f$, mais aussi dans l'ï¿½uilibrage de
la simulation\footnote{n'hï¿½itez pas ï¿½modifier des paramï¿½res, ajouter des astres,
  et vous verrez que le systï¿½e solaire est miraculeux : il est stable !}. $f$ est
gï¿½ï¿½ique : elle peut traiter un nombre d'astres quelconques chacun ayant sa propre
masse. Dans cette section on va en faire interagir quatre. Cela donne pour le moteur
d'affichage un systï¿½e de 16 ï¿½uations du premier ordre ï¿½rï¿½oudre, et le rï¿½ultat
est remarquable.

\fi
                x1 y1 x'1 y'1 x2 y2 x'2 y'2 x3 y3 x'3    y'3  x4   y4  x'4  y'4
  \def\InitCond{ 1  1   0   0 -1 -1  -2   2  3  3 2.86 -2.86 5.5 -3.5 -2.1 -2.1}
  \def\tMAX{10}
  \newcommand{\MunMdeux}{40}
  \newcommand{\MdeuxMtrois}{40}
  \psset{unit=1, showpoints=false, plotpoints=400}%%plotpoints=375}
  \newcommand{\Test}{
    %% masses des astres
    /Weigth [ 40 1 1 2 ] def
    /NbAstor Weigth length def
    /ABVect { 3 -1 roll exch sub 3 1 roll sub exch } def
    %% calculs des distances entre masses
    1 1 NbAstor 1 sub {
      dup /i exch def 1 add 1 NbAstor {
        /j exch def
        y i 1 sub 4 mul get y j 1 sub 4 mul get sub dup mul
        y i 1 sub 4 mul 1 add get y j 1 sub 4 mul 1 add get sub dup mul add
        } for
      } for
    NbAstor dup 1 sub mul 2 div cvi array astore /ro2 exch def
    %% calculs des forces
    1 1 NbAstor {
      /i exch def
      y i 1 sub 4 mul 4 getinterval aload pop 4 2 roll
      /xf 0 def /yf 0 def
      1 1 NbAstor {
        dup i eq
        { pop }
        { /j exch def
        2 copy y j 1 sub 4 mul 2 getinterval aload pop ABVect
        Weigth j 1 sub get
        ro2 i j 2 copy lt { exch } if dup
        dup 1 sub dup NbAstor mul 3 1 roll mul 2 div sub 3 1 roll sub add 1 sub
        cvi get dup sqrt mul div exch 1 index mul 3 1 roll mul exch
        yf add /yf exch def xf add /xf exch def } ifelse
        } for pop pop xf neg yf neg
      } for
      NbAstor 4 mul array astore
    }

\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(-9,-6.25)(6,6)
 \psgrid[subgriddiv=0,gridcolor=lightgray](-9,-6)(6,6)
 \psset{buildvector=true,method=rk4,linewidth=1.5pt}
 \psplotDiffEqn[whichabs=0,whichord=1,linecolor=blue]{0}{\tMAX}{\InitCond}{\Test}
 \psplotDiffEqn[whichabs=4,whichord=5,linecolor=red]{0}{\tMAX}{\InitCond}{\Test}
 \psplotDiffEqn[whichabs=4,whichord=5,linecolor=Orange,method=adams]{0}{\tMAX}{\InitCond}{\Test}
 \psplotDiffEqn[whichabs=8,whichord=9,linecolor=magenta]{0}{\tMAX}{\InitCond}{\Test}
 \psplotDiffEqn[whichabs=12,whichord=13,linecolor=green]{0}{\tMAX}{\InitCond}{\Test}
 \psplotDiffEqn[whichabs=12,whichord=13,linecolor=yellow,method=adams]{0}{\tMAX}{\InitCond}{\Test}
\end{pspicture}
\end{LTXexample}


\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(-10,-7.25)(5,5)
  \psgrid[subgriddiv=0,gridcolor=lightgray](-10,-7)(5,5)
  \psset{buildvector=true,method=rk4,linewidth=1.5pt}
  \psplotDiffEqn[linecolor=red,plotfuncx=y dup 4 get exch 0 get sub ,
     plotfuncy=dup 5 get exch 1 get sub ]{0}{\tMAX}{\InitCond}{\Test}
  \psplotDiffEqn[linecolor=magenta,plotfuncx=y dup 8 get exch 0 get sub ,
     plotfuncy=dup 9 get exch 1 get sub ]{0}{\tMAX}{\InitCond}{\Test}
  \psplotDiffEqn[linecolor=green,plotfuncx=y dup 12 get exch 0 get sub ,
     plotfuncy=dup 13 get exch 1 get sub ]{0}{\tMAX}{\InitCond}{\Test}
\end{pspicture}
\end{LTXexample}

\fi


%--------------------------------------------------------------------------------------
\section{\CMD{psMatrixPlot}}
%--------------------------------------------------------------------------------------
\begin{filecontents}{matrix.dat}
/dotmatrix [ %
0  1  1  0  0  0  0  1  1  1
0  1  1  0  1  1  1  0  1  0
1  0  1  1  0  0  0  1  1  0
0  0  1  0  0  0  0  0  1  1
1  1  1  1  1  0  1  0  0  1
0  0  1  1  0  1  0  1  1  1
1  0  0  0  1  1  0  0  0  1
0  0  0  1  1  1  0  1  1  0
1  1  0  0  0  0  1  0  0  1
1  0  1  0  0  1  1  1  0  0
] def
\end{filecontents}


This macro allows you to visualize a matrix. The datafile must be
defined as a PostScript matrix named \verb+/dotmatrix+:
\begin{lstlisting}[style=syntax]
/dotmatrix [ %  <------------ important line
0  1  1  0  0  0  0  1  1  1
0  1  1  0  1  1  1  0  1  0
1  0  1  1  0  0  0  1  1  0
0  0  1  0  0  0  0  0  1  1
1  1  1  1  1  0  1  0  0  1
0  0  1  1  0  1  0  1  1  1
1  0  0  0  1  1  0  0  0  1
0  0  0  1  1  1  0  1  1  0
1  1  0  0  0  0  1  0  0  1
1  0  1  0  0  1  1  1  0  0
] def        %  <------------ important line
\end{lstlisting}

Only the value 0 is important, in which case nothing happens, and
for all other cases a dot is printed. The syntax of the macro is:
\begin{lstlisting}[style=syntax]
  \psMatrixPlot[options]{rows}{columns}{data file}
\end{lstlisting}

The matrix is scanned line by line from the the first one to the
last. In general it appears as a bottom-to-top version of the
above listed matrix, the first row $0\,1\,1\,0\,0\,0\,0\,1\,1\,1$
is the first plotted line ($y=1$). With the option
\verb+ChangeOrder=true+ it looks exactly like the above view.

\bgroup
\begin{center}
%\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(-0.5,-0.75)(11,11)
  \psaxes{->}(11,11)
  \psMatrixPlot[dotsize=1.1cm,dotstyle=square*,linecolor=magenta]%
    {10}{10}{matrix.dat}
  \psMatrixPlot[dotsize=.5cm,dotstyle=o,ChangeOrder]{10}{10}{matrix.dat}
\end{pspicture}
%\end{LTXexample}

\begin{lstlisting}
\begin{pspicture}(-0.5,-0.75)(11,11)
  \psaxes{->}(11,11)
  \psMatrixPlot[dotsize=1.1cm,dotstyle=square*,linecolor=magenta]%
    {10}{10}{matrix.dat}
  \psMatrixPlot[dotsize=.5cm,dotstyle=o,ChangeOrder]{10}{10}{matrix.dat}
\end{pspicture}
\end{lstlisting}

\begin{LTXexample}[pos=t,preset=\centering]
\begin{pspicture}(-0.5,-0.75)(11,11)
  \psaxes{->}(11,11)
  \psMatrixPlot[dotscale=3,dotstyle=*,linecolor=blue]{10}{8}{matrix.dat}
\end{pspicture}
\end{LTXexample}
\end{center}
\egroup

%--------------------------------------------------------------------------------------
\section{\CMD{psforeach}}
%--------------------------------------------------------------------------------------

The macro \CMD{psforeach} allows a loop with an individual
increment.

\begin{lstlisting}[style=syntax]
\psforeach{variable}{value list}{action}
\end{lstlisting}

\begin{LTXexample}
\begin{pspicture}[showgrid=true](5,5)
  \psforeach{\nA}{0, 1, 1.5, 3, 5}{%
    \psdot[dotscale=3](\nA,\nA)}
\end{pspicture}
\end{LTXexample}


%--------------------------------------------------------------------------------------
\section{\CMD{resetOptions}}
%--------------------------------------------------------------------------------------

Sometimes it is difficult to know what options, which are changed
inside a long document, are different to the default ones. With
this macro all options belonging to \verb+pst-plot+ can be reset.
This refers to all options of the packages \verb+pstricks+,
\verb+pst-plot+ and \verb+pst-node+.


\appendix


%--------------------------------------------------------------------------------------
\section{PostScript}
%--------------------------------------------------------------------------------------

PostScript uses the stack system and the LIFO system, "'Last In, First Out"`.

\newlength{\Li}\settowidth{\Li}{Function}
\begin{table}[htbp]
  \begin{center}{\ttfamily
    \begin{tabular}{|l|r@{ $\rightarrow$ }l|}\hline
    \multirow{2}{\Li}{\normalfont Function}&\multicolumn{2}{c|}{\normalfont Meaning}\\
    &\normalfont on stack before & \normalfont after\\\hline\hline
    add&$x\quad y$&$x+y$\\\hline
    sub&$x\quad y$&$x-y$\\\hline
    mul&$x\quad y$&$x\times y$\\\hline
    div&$x\quad y$&$x\div y$\\\hline
    sqrt&$x$&$\sqrt{x}$\\\hline
    abs&$x$&$|x|$\\\hline
    neg&$x$&$-x$\\\hline
    cos&$x$&$\cos(x)$ ($x$ in degrees)\\\hline
    sin&$x$&$\sin(x)$ ($x$  in degrees)\\\hline
    tan&$x$&$\tan(x)$ ($x$  in degrees)\\\hline
    atan&$y\quad x$&$\angle{(\vec{Ox};\vec{OM})}$ (in degrees of $M(x,y)$)\\\hline
    ln&$x$&$\ln(x)$\\\hline
    log&$x$&$\log(x)$\\\hline
    array&$n$&\normalfont$v$ (of dimension $n$)\\\hline
    aload&$v$&$x_1\quad x_2\quad \cdots\quad x_n\quad v$\\\hline
    astore&$x_1\quad x_2\quad \cdots\quad x_n\quad v$&$v$\\\hline
    pop&$x$&\\\hline
    dup&$x\quad x$&\\\hline
    roll&$x_1\quad x_2\quad \cdots\quad x_n\quad n p$&\\\hline
  \end{tabular}}
    \caption{Some primitive PostScript macros}\label{tab:primpost}
  \end{center}
\end{table}


\clearpage
\section{List of all optional arguments for \texttt{pstricks-add}}

\xkvview{family=pstricks-add,columns={key,type,default}}


%--------------------------------------------------------------------------------------
\section{Credits}
%--------------------------------------------------------------------------------------
{Hendri Adriaens | }
{Stefano Baroni | }
{Martin Chicoine | }
{Gerry Coombes | }
{Ulrich Dirr | }
{Christophe Fourey | }
{Hubert G\"a\ss lein |}
{Denis Girou | }
{Peter Hutnick | }
{Christophe Jorssen | }
{Uwe Kern | }
{Manuel Luque | }
{Jens-Uwe Morawski |}
{Tobias N\"ahring |}
{Rolf Niepraschk |}
{Alan Ristow |}
{Arnaud Schmittbuhl |}
{Timothy Van Zandt}



\nocite{*}
\bgroup
\RaggedRight
\bibliographystyle{plain}
\bibliography{pstricks-add-doc}
\egroup

%--------------------------------------------------------------------------------------
\section{Change log}
%--------------------------------------------------------------------------------------

See file Changes

%\clearpage
%\section{The code}

%\lstinputlisting{pstricks-add.tex}


\end{document}
