[pstricks] drawing implicit (polar) plot

[Wim Neimeijer]

I have a question concerning the implicit polar plot algorithm proposed
by Manfred Braun
and which is used as an example of http:/www.pstricks.de (polar2.tex)

In the postscript code where the streamlines and equipotential functions
are plotted
via a call to parametricplot, I am trying to figure out the mapping
between the complex function
$\Psi$ and the parameter $t$, excerpt of the code attached below.

Question : To be more specific, in the notes from polar2.tex it says,
solve the equation $f(z) = z + a^2/z$
using $z = x + i y$, which would give a quadratic equation ?

My maths gives me : $f(z) = \phi(x,y) + i \Psi(x,y) = $ where $Re f(z) =
( r + \frac{a^2}{r} ) \cos \phi $
and $Im f(z) = ( r - \frac{a^2}{r} ) \sin \phi$ 

Question : How does this lead to a quadratic equation under which
assumption ? 
Question : How is the quadratic equation mapped to the variable $t$ in
the parametricplot ?


I want to understand the lines of code which I indented with HOW 
to plot an other function to make the code a bit more generic.

If someone could help me, I greatly appreciated it.

Thanks in advance

Example taken from www.pstricks.de

=============== EXAMPLE ===============================
\listfiles
\documentclass[a4paper]{article}%  Herbert Voss 2004-10-06
\usepackage{pstricks}
\usepackage{pst-plot}
\usepackage{pstricks-add}
\usepackage{multido}
\usepackage{url}
%
% ----------- first example --------------------
%
\def\func{%
    const0 x sin dup mul 3 mul div neg /p exch def
    p abs sqrt neg /r exch def
    const1 2.0 div neg /q exch def
    p 0 lt {
      p dup mul p mul q dup mul add 0 le {
        q r 3 exp div arccos 
        3 div cos r dup add neg mul
      }{
        q r 3 exp div arch
        3 div cosh r dup add neg mul
      } ifelse
    }{
      q r 3 exp div arsh
      3 div sinh r dup add neg mul      
    } ifelse
}
%
% ----------- second example --------------------
\def\compl{%
  \pstVerb{
    /cadd { 3 -1 roll add 3 1 roll add exch } def
    /csub { exch 4 -1 roll sub neg 3 1 roll sub } def
    /cmul { 3 index 2 index mul 3 index 2 index mul sub 5 1 roll
      4 -1 roll mul 3 1 roll mul add } def
    /crec { 2 copy dup mul exch dup mul add 3 -1 roll 1 index div
      3 1 roll div neg } def
    /creal { pop } def
    /cimag { exch pop } def
    /cabs { dup mul exch dup mul add sqrt } def
    /carg { exch atan } def
    /csqrt { 2 copy cabs sqrt 3 1 roll carg 2 div 2 copy cos mul 3 1
      roll sin mul } def
  }%
}
%
\newcommand\streamline[2]{%
% A streamline is drawn for #1 as the value of the stream function
% (divided by $2u_\infty$).  The potential function, also normalized,
% takes values in the range (-1.5,+1.5).  The (x, y)-coordinates of
% the points may be modified by providing PostScript commands in #2.
% The streamline through the stagnation points (#1 = 0) is excluded.
  \parametricplot{-1.5}{1.5}{
HOW		    t #1 2 copy 2 copy cmul 1 0 csub csqrt cadd #2
  }%
}
\newcommand\equipotline[2]{%
% An equipotential line is drawn in the same way.  The potential must
% not assume the value 1 exactly, because of the singularity at the
% stagnation points.
%  \parametricplot{0}{1.25}{
%    #1 t 2 copy 2 copy cmul 1 0 csub csqrt cadd #2
%  }%

  \parametricplot{0}{1.25}{
HOW  		   #1 t 2 copy 2 copy cmul 1 0 csub csqrt cadd #2
  }%

}

\begin{document}

% This command provides PostScript commands for complex addition,
% subtraction, multiplication etc.

\psset{unit=2cm}
% \compl invokes the predefined complex arithmetics extensions
\begin{pspicture}(-3,-3)(3,3)%
  \compl
    \psset{linecolor=blue}
    %\psline(-3,0)(-1,0) \pscircle(0,0){1} %\psline(1,0)(3,0)
    %\multido{\n=0.125+0.125}{10}{
    %  \streamline{\n}{ }
    %  \streamline{\n}{ neg }
    %}
    \psset{linecolor=red}
    \multido{\n=0+0.125}{8}{
      \equipotline{\n}{ }
      %\equipotline{\n}{ neg }
      %\equipotline{\n}{ exch neg exch }
      %\equipotline{\n}{ neg exch neg exch }
    }
    %\equipotline{0.9999}{ }
    %\equipotline{0.9999}{ neg }
    %\equipotline{0.9999}{ exch neg exch }
    %\equipotline{0.9999}{ neg exch neg exch }
    %\multido{\n=1.125+0.125}{4}{
    %  \equipotline{\n}{ }
    %  \equipotline{\n}{ neg }
    %  \equipotline{\n}{ exch neg exch }
    %  \equipotline{\n}{ neg exch neg exch }
    %}
\end{pspicture}

\end{document}

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