% Copyright 2011 by Alain Matthes
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Public License.


\def\fileversion{1.16 d}
\def\filedate{2018/09/19}  



% Objet :  outils mathématiques pour la géométrie euclideienne avec pgf/tikz
% utilisable de préférence avec un repère orthonormé et le cm comme unité 
% utile pour la compatibilité avec pgf 2
%<--------------------------------------------------------------------------–>
%<--------------------------------------------------------------------------–>
%             Duplicate Length      à revoir pas de pt pas de global
% ||v(CN)||= ||v(AB)|| et v(CN) colineaire à v(CD) 
% A-->#1 B-->#2 C-->#3 D-->#4 N-->#5   ?????
%<--------------------------------------------------------------------------–>
\def\tkzDuplicateLen(#1,#2)(#3,#4){%
\begingroup 
    \tkzCalcLength(#1,#2)\tkzGetLength{tkz@firstlen}% 
    \tkzCalcLength(#3,#4)\tkzGetLength{tkz@secondlen}%
    \FPdiv\tkz@ratio{\tkz@firstlen}{\tkz@secondlen}%
    \tkz@VecKCoLinear[\tkz@ratio](#3,#4,#3){tkzPointResult}%
\endgroup 
}
\let\tkzDuplicateSegment\tkzDuplicateLen     %<--------------------------------------------------------------------------–>
%                    Coordonnées d'un vecteur  (couple de points)  
% Deux points A et B donc un vecteur on récupère les coordonnées de v(AB)
% en cm 
% tkzGetVecCoord en cm ou en pt ???
%<--------------------------------------------------------------------------–>
%result in #3x et #3y #1 et #2 sont les points  
% passage en cm avec fp ? 
\def\tkzGetVectxy(#1,#2)#3{%
\begingroup 
\pgfpointdiff{\pgfpointanchor{#1}{center}}%
             {\pgfpointanchor{#2}{center}}%
\pgfmathparse{\pgf@sys@tonumber{\pgf@x}/28.45274}%
\global\let\tkzresultx\pgfmathresult
\pgfmathparse{\pgf@sys@tonumber{\pgf@y}/28.45274}%
\global\let\tkzresulty\pgfmathresult
\global\expandafter\edef\csname #3x\endcsname{\tkzresultx}% 
\global\expandafter\edef\csname #3y\endcsname{\tkzresulty}% 
\endgroup
}
\let\tkzGetVecCoord\tkzGetVectxy
%<--------------------------------------------------------------------------–>
%<--------------------------------------------------------------------------–>
\def\tkz@numv{0}
\pgfkeys{/tkzdefv/.cd,
K/.code      = {\pgfmathparse{#1}\global\def\tkz@ratio{\pgfmathresult}},
colinear/.code     args = {at #1}{\global\def\tkz@numv{0}%
                                  \global\def\tkz@frompoint{#1}},
orthogonal/.code        = {\global\def\tkz@numv{1}},
linear/.code            = {\global\def\tkz@numv{2}}\pgfmathparse{#1},
normed orthogonal/.code = {\global\def\tkz@numv{3}},
normed linear/.code     = {\global\def\tkz@numv{4}},
} 
\def\tkzDefVector[#1](#2,#3)#4{%
\begingroup 
\pgfkeys{/tkzdefv/.cd,K=1}
\pgfqkeys{/tkzdefv}{#1}
\ifcase\tkz@numv%
 % first case 0
 \tkzDefVectorfrom[\tkz@ratio](#2,#3){#4}
  \or% 1
  \tkz@VecKOrth[\tkz@ratio](#2,#3){#4} 
  \or% 2
  \tkz@VecK[\tkz@ratio](#2,#3){#4}
  \or% 3
  \tkz@VecKOrthNorm[\tkz@ratio](#2,#3){#4}
  \or% 4
  \tkz@VecKCoLinear[#1](#2,#3)#4
  \fi    
\endgroup
} 

\def\tkzDefVectorfrom[#1](#2,#3)#4{%    
\begingroup    
    \pgfpointdiff{\pgfpointanchor{#2}{center}}%
                 {\pgfpointanchor{#3}{center}}%
    \pgf@xa=\pgf@x%
    \pgf@ya=\pgf@y%
     \path[coordinate](\tkz@frompoint)--+(\tkz@ratio\pgf@xa,%
                                          \tkz@ratio\pgf@ya) coordinate (#4);
\endgroup
} 
%<--------------------------------------------------------------------------–>
%         VecKCoLinear  CN = K x AB  #1 pt #2 pt #3 pt #4 nb #5 pt result   
% il faut modifier cette macro : on supprime #3 pour la colinéarité
% Il suffit d'utiliser Replicate ou Duplicate     coeff dans #1
% v(CD)=#1 x v(AB) #1 le coeff; #2-->A #3-->B #4-->C  #5-->N
%<--------------------------------------------------------------------------–>
\def\tkzVecKCoLinear{\pgfutil@ifnextchar[{\tkz@VecKCoLinear}{\tkz@VecKCoLinear[1]}} 
\def\tkz@VecKCoLinear[#1](#2,#3,#4)#5{% 
\begingroup
   \pgfpointdiff{\pgfpointanchor{#2}{center}}%
                {\pgfpointanchor{#3}{center}}%
   \pgf@xa=\pgf@x%
   \pgf@ya=\pgf@y% 
   \pgfmathparse{#1}\edef\tkz@coeff{\pgfmathresult}
   \path[coordinate](#4)--+(\tkz@coeff\pgf@xa,\tkz@coeff\pgf@ya)%
          coordinate (#5);%  
\endgroup
}%  
%<--------------------------------------------------------------------------–>
%         v(AN)=#1 x v(AB)     
%   #1 le coeff; #2--> A #3--> B   #4-->N  tq    #4-#2 = #1*(#3-#2)                                 
%<--------------------------------------------------------------------------–>
\pgfkeys{
  /tkzscalev/.cd,
  ratio/.code      = {\pgfmathparse{#1}\global\edef\tkz@ratio{\pgfmathresult}}
  }  
\def\tkzScaleVector{\pgfutil@ifnextchar[{\tkz@ScaleVector}{%
                                         \tkz@ScaleVector[]}} 
\def\tkz@ScaleVector[#1](#2,#3)#4{% 
\begingroup
\pgfkeys{/tkzscalev/.cd,ratio=-1}
\pgfqkeys{/tkzscalev}{#1}  
   \pgfpointdiff{\pgfpointanchor{#2}{center}}%
                {\pgfpointanchor{#3}{center}}%
   \pgf@xa=\pgf@x%
   \pgf@ya=\pgf@y% 
    \path[coordinate](#2)--++(\pgf@xa *\tkz@ratio,\pgf@ya *\tkz@ratio)%
          coordinate (#4);%   
\endgroup
}%          
%<--------------------------------------------------------------------------–>
%                 Outils pour les vecteurs
%<--------------------------------------------------------------------------–>
%  ce sont des outils élémentaires qui à partir de deux points en définissent 
% un troisième 
% #1 si c'est une option alors c'est un nombre réel
% #2 et #3 sont deux points
% #4 est le nom du point qui résulte de la transformation
% exemple : \tkzVecKNorm (A,B){C} définit un point C tel que AC = 1 et C est %  %  un point de la droite (AC). #1 peut être négatif

%<--------------------------------------------------------------------------–>
%              VectorNormalised ou K*VectorNormalised
% A-->#2 B-->#3 N-->#4      v(AB)  devient v(AN) tq ||v(AN)||=1 si #1=1  
%  sinon ||v(AN)||=#1 
%<--------------------------------------------------------------------------–>
\def\tkzVecKNorm{\pgfutil@ifnextchar[{\tkz@VecKNorm}{\tkz@VecKNorm[1]}} 
\def\tkz@VecKNorm[#1](#2,#3)#4{%
\begingroup
    \tkzpointnormalised{%
    \pgfpointdiff{\pgfpointanchor{#2}{center}}
                 {\pgfpointanchor{#3}{center}}}
    \pgf@xa=\pgf@x\relax%
    \pgf@ya=\pgf@y\relax% 
    \pgfmathparse{#1}\edef\tkz@coeff{\pgfmathresult}
    \FPmul\tkz@coeff{28.45274}{\tkz@coeff}
    \FPmul\tkz@x{\tkz@coeff}{\pgf@sys@tonumber{\pgf@xa}}
    \FPmul\tkz@y{\tkz@coeff}{\pgf@sys@tonumber{\pgf@ya}}
    \path[coordinate](#2)--++(\tkz@x pt,\tkz@y pt)%
          coordinate (#4);%
\endgroup
}%
%<--------------------------------------------------------------------------–>
%         v(AN)=#1 x v(AB)     
%   #1 le coeff; #2--> A #3--> B   #4-->N  tq    #4-#2 = #1*(#3-#2)                                 
%<--------------------------------------------------------------------------–>
\def\tkzVecK{\pgfutil@ifnextchar[{\tkz@VecK}{\tkz@VecK[1]}} 
\def\tkz@VecK[#1](#2,#3)#4{% 
\begingroup 
   \pgfpointdiff{\pgfpointanchor{#2}{center}}%
                {\pgfpointanchor{#3}{center}}%
   \pgf@xa=\pgf@x%
   \pgf@ya=\pgf@y% 
   \pgfmathparse{#1}\edef\tkz@coeff{\pgfmathresult}
    \path[coordinate](#2)--++(\pgf@xa *\tkz@coeff,%
                              \pgf@ya *\tkz@coeff)%
          coordinate (#4);%   
\endgroup
}% 
%<--------------------------------------------------------------------------–>
%                   tkzVector K Orth   coeff dans #1
%    v(AN) perp v(AB)  v(AB) v(AN) sens direct cercle trigo 
% ||v(AN)||=||v(AB)||
%<--------------------------------------------------------------------------–>
\def\tkzVecKOrth{\pgfutil@ifnextchar[{\tkz@VecKOrth}{\tkz@VecKOrth[1]}}  
\def\tkz@VecKOrth[#1](#2,#3)#4{%
\begingroup
    \pgfpointdiff{\pgfpointanchor{#2}{center}}%
                 {\pgfpointanchor{#3}{center}}%
    \pgf@xa=-\pgf@y%
    \pgf@ya=\pgf@x%
    \pgfmathparse{#1}\edef\tkz@coeff{\pgfmathresult}
    \path[coordinate](#2)--++(\tkz@coeff\pgf@xa,\tkz@coeff\pgf@ya)%
          coordinate (#4);%
\endgroup
}% 
%<--------------------------------------------------------------------------–>
%              tkzVecKOrthNorm   coeff dans #1
%    v(AN) perp v(AB)  v(AB) v(AN) sens direct cercle trigo 
% ||v(AN||=1 si #1 est vide ou =1 sinon ||v(AN||=K
%<--------------------------------------------------------------------------–>
\def\tkzVecKOrthNorm{\pgfutil@ifnextchar[{\tkz@VecKOrthNorm}%
                                         {\tkz@VecKOrthNorm[1]}}
\def\tkz@VecKOrthNorm[#1](#2,#3)#4{%    
\begingroup
 \tkzpointnormalised{\pgfpointdiff{\pgfpointanchor{#2}{center}}%
                                  {\pgfpointanchor{#3}{center}}} 
  \pgf@xa=-\pgf@y%
  \pgf@ya=\pgf@x%
  \FPmul\tkz@coeff{28.45274}{#1}
  \FPmul\tkz@x{\tkz@coeff}{\pgf@sys@tonumber{\pgf@xa}}
  \FPmul\tkz@y{\tkz@coeff}{\pgf@sys@tonumber{\pgf@ya}}
  \path[coordinate](#2)--++(\tkz@x pt,\tkz@y pt)%
          coordinate (#4);%    
\endgroup
}%      
%<--------------------------------------------------------------------------–>
%  \tkzpointnormalised    normalise un point A-->A' tq ||v(OA')=1||
% équivalent de \pgfpointnormalised avec fp
% example
% \tkzpointnormalised{%
% \pgfpointdiff{\pgfpointanchor{A}{center}}
%              {\pgfpointanchor{B}{center}}}

% or
% \pgf@x=1 cm
% \pgf@y=12 cm 
% \tkzpointnormalised{} %<--------------------------------------------------------------------------–> 
\def\tkzpointnormalised#1{%
\pgf@process{#1}%
\FPmul{\tkz@sx}{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@x}}
\FPmul{\tkz@sy}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@y}}
\FPadd{\tkz@sxy}{\tkz@sx}{\tkz@sy}
\FProot{\tkz@den}{\tkz@sxy}{2}
\FPdiv{\tkz@coordx}{\pgf@sys@tonumber{\pgf@x}}{\tkz@den}
\FPround{\tkz@coordx}{\tkz@coordx}{5}
\FPdiv{\tkz@coordy}{\pgf@sys@tonumber{\pgf@y}}{\tkz@den}
\FPround{\tkz@coordy}{\tkz@coordy}{5}
\pgf@x = \tkz@coordx pt
\pgf@y = \tkz@coordy pt
}
%<--------------------------------------------------------------------------–>
% restaure and save length
\def\tkz@save@length{%
\global\let\tkz@temp@length\tkzLengthResult}%
\def\tkz@restore@length{%
 \global\let\tkzLengthResult\tkz@temp@length }% 
%<--------------------------------------------------------------------------–>
%<--------------------------------------------------------------------------–>
%    \tkzCalcLength      Distance entre deux points en pt ou en cm  avec FP 
% \veclen mais avec fp 
%  option cm le résultat est en cm sinon en pt
%<--------------------------------------------------------------------------–>

\newif\iftkzLengthIncm
\pgfkeys{
DefVecLen/.cd,
       cm/.is if         = tkzLengthIncm,
       cm/.default       = true}   

\def\tkzCalcLength{\pgfutil@ifnextchar[{\tkz@CalcLength}{\tkz@CalcLength[]}}  
\def\tkz@CalcLength[#1](#2,#3){%
\pgfkeys{DefVecLen/.cd, cm = false}
\pgfqkeys{/DefVecLen}{#1}%   
\begingroup
\tkz@@CalcLength(#2,#3){tkzLengthResult}
\iftkzLengthIncm  
    \FPdiv\tkzFPMathLen{\tkzFPMathLen}{28.45274}
   \FPround\tkzFPMathLen\tkzFPMathLen5\relax% 
   \global\let\tkzLengthResult\tkzFPMathLen  
\fi 
\endgroup
}%
\def\tkz@@CalcLength(#1,#2)#3{%
\pgfpointdiff{\pgfpointanchor{#1}{center}}%
             {\pgfpointanchor{#2}{center}}%
\pgf@xa=\pgf@x%
\pgf@ya=\pgf@y%
\FPeval\tkz@temp@a{\pgfmath@tonumber{\pgf@xa}}%
\FPeval\tkz@temp@b{\pgfmath@tonumber{\pgf@ya}}%
\FPeval\tkz@temp@sum{(\tkz@temp@a*\tkz@temp@a+\tkz@temp@b*\tkz@temp@b)}%
\FProot{\tkzFPMathLen}{\tkz@temp@sum}{2}%
\FPround\tkzFPMathLen\tkzFPMathLen5\relax
\global\expandafter\edef\csname #3\endcsname{\tkzFPMathLen}
}
%<--------------------------------------------------------------------------–>
\def\tkzGetLength#1{%
\global\expandafter\edef\csname #1\endcsname{\tkzLengthResult}}  
%<--------------------------------------------------------------------------–>
%     \tkzpttocm  passage de pt à cm div par 28.45274
%<--------------------------------------------------------------------------–>
\def\tkzpttocm(#1)#2{%
\begingroup  
    \FPdiv\tkz@mathresult{#1}{28.45274}
    \FPround\tkz@mathresult\tkz@mathresult5\relax%  
     \global\let\tkz@mathresult\tkz@mathresult
     \global\expandafter\edef\csname #2\endcsname{\tkz@mathresult}% 
\endgroup
}%
%<--------------------------------------------------------------------------–>
%     \tkzcmtopt  passage de cm à pt mul par 28.45274  %<--------------------------------------------------------------------------–
\def\tkzcmtopt(#1)#2{%
\begingroup 
    \FPmul\tkz@mathresult{#1}{28.45274}
    \FPround\tkz@mathresult\tkz@mathresult5\relax%  
     \global\let\tkz@mathresult\tkz@mathresult
\global\expandafter\edef\csname #2\endcsname{\tkz@mathresult}% 
\endgroup  
}% 
%<--------------------------------------------------------------------------–>
%                          Slope
%<--------------------------------------------------------------------------–>
\def\tkzFindSlope{\tkz@FindSlope}
\def\tkz@FindSlope(#1,#2)#3{%
    \begingroup
 \tkzpointnormalised{\pgfpointdiff{\pgfpointanchor{#1}{center}}%
                                  {\pgfpointanchor{#2}{center}}}    
    \tkz@ax=\pgf@x\relax%
    \tkz@ay=\pgf@y\relax%
    \FPdiv{\tkz@Slope}{\pgfmath@tonumber{\tkz@ay}}{\pgfmath@tonumber{\tkz@ax}}
    \FPround{\tkz@Slope}{\tkz@Slope}{5}
    \global\expandafter\edef\csname #3\endcsname{\tkz@Slope}%  
\endgroup
}
%<--------------------------------------------------------------------------–>
%<----------------– for compatibility --------------------------------------–>
%<--------------------------------------------------------------------------–>
\def\tkzmathanglebetweenpoints#1#2{%
\begingroup
  \pgf@process{\pgfpointdiff{#1}{#2}}%
  %
  % First approximate the angle of the external point...
  %
  \pgf@xa\pgf@x%
  \pgf@ya\pgf@y%
  \pgf@xb\pgf@x%
  \pgf@yb\pgf@y%
  \ifdim\pgf@xa<0pt\relax%
    \pgf@xa-\pgf@xa%
  \fi
  \ifdim\pgf@ya<0pt\relax%
    \pgf@ya-\pgf@ya%
  \fi
  \ifdim\pgf@ya>\pgf@xa%
    \pgf@x\pgf@xa%
    \pgf@y\pgf@ya%
  \else
    \pgf@x\pgf@ya%
    \pgf@y\pgf@xa%
  \fi
  \ifdim\pgf@y=0pt\relax%
    \pgf@x0pt%
  \else 
  \FPdiv\pgfmathresult{1}{\pgfmath@tonumber{\pgf@y}}
  \FPround\pgfmathresult\pgfmathresult5\relax% 
    \pgf@x\pgfmathresult\pgf@x%
  \fi
  \multiply\pgf@x1000\relax%
  \afterassignment\pgfmath@gobbletilpgfmath@%
  \expandafter\c@pgf@counta\the\pgf@x\relax\pgfmath@%
\expandafter\pgf@x\csname pgfmath@atan@\the\c@pgf@counta\endcsname pt\relax%
    \ifdim\pgfmath@ya>\pgfmath@xa\relax%
      \pgf@x-\pgf@x%
      \advance\pgf@x90pt%
    \fi
    \ifdim\pgf@xb<0pt%
      \ifdim\pgf@yb>0pt%
        \pgf@x-\pgf@x%
      \fi
      \advance\pgf@x180pt\relax%
    \else
      \ifdim\pgf@yb<0pt%
      \pgf@x-\pgf@x%
      \advance\pgf@x360pt\relax%
    \fi 
    \fi
    \ifdim\pgf@x>180pt%
    \advance\pgf@x-360pt\relax%
    \fi
    \pgfmath@returnone\pgf@x%
  \endgroup
}

% \tkzmathrotatepointaround
%
% Rotate point #1 about point #2 by #3 degrees.
%
\def\tkzmathrotatepointaround#1#2#3{%
  \pgf@process{%
    \pgf@process{#1}%
    \pgf@xc=\pgf@x%
    \pgf@yc=\pgf@y%
    \pgf@process{#2}%
    \pgf@xa\pgf@x%
    \pgf@ya\pgf@y%
    \pgf@xb\pgf@x%
    \pgf@yb\pgf@y%
    \pgf@x=\pgf@xc%
    \pgf@y=\pgf@yc%
    \advance\pgf@x-\pgf@xa%
    \advance\pgf@y-\pgf@ya%
    \pgfmathsetmacro\tkz@angle{#3}%
    \pgfmathsin@{\tkz@angle}%
    \let\sineangle\pgfmathresult%
    \pgfmathcos@{\tkz@angle}%
    \let\cosineangle\pgfmathresult%
    \pgf@xa\cosineangle\pgf@x%
    \advance\pgf@xa-\sineangle\pgf@y%
    \pgf@ya\sineangle\pgf@x%
    \advance\pgf@ya\cosineangle\pgf@y%
    \pgf@x\pgf@xb%
    \pgf@y\pgf@yb%
    \advance\pgf@x\pgf@xa%
    \advance\pgf@y\pgf@ya%
  }%
}


% \tkzmathanglebetweenlines
%
% Calculate the clockwise angle between a line from point #1
% to point #2 and a line from #3 to point #4.
%
\def\tkzmathanglebetweenlines#1#2#3#4{%
  \begingroup
    \tkzmathanglebetweenpoints{#1}{#2}%
    \let\firstangle\pgfmathresult%
    \tkzmathanglebetweenpoints{#3}{#4}%
    \let\secondangle\pgfmathresult%
    \ifdim\firstangle pt>\secondangle pt\relax%
      \pgfmathadd@{\secondangle}{360}%
      \let\secondangle\pgfmathresult%
    \fi
    \pgfmathsubtract@{\secondangle}{\firstangle}%
    \pgfmath@smuggleone\pgfmathresult%
  \endgroup
} 
% \pgfmathpointreflectalongaxis
%
% Reflects point #2 around an axis centered on #2 at an angle #3.
%
\def\tkzmathreflectpointalongaxis#1#2#3{%
  \pgf@process{%
    \pgfmathanglebetweenpoints{#2}{#1}%
    \pgfmath@tempdima\pgfmathresult pt\relax%
    \pgfmathparse{#3}%
    \advance\pgfmath@tempdima-\pgfmathresult pt\relax%
    \pgfmath@tempdima-2.0\pgfmath@tempdima%
    \pgfmathrotatepointaround{#1}{#2}{\pgfmath@tonumber{\pgfmath@tempdima}}%
  }%
}


% \pgfmathpointintersectionoflineandarc 
%
% A bit experimental at the moment:
%
% Locates the point where a line crosses an eliptical arc. If the line
% does not cross the arc, a meaningless point will result.
%
% #1 the point of the line on the "convex" side of the arc.
% #2 the point of the line on the "concave" side of the arc.
% #3 the center of the eliptical arc.
% #4 start angle of the arc.
% #5 end angle of the arc.
% #6 radii of the arc.
%
\def\tkzmathpointintersectionoflineandarc#1#2#3#4#5#6{%
  \pgf@process{%
    %
    % Get the required angle.
    %
    \pgfmathanglebetweenpoints{#2}{#1}%
    \let\x\pgfmathresult%
    %
    % Get the radii of the arc.
    %
    \pgfmath@in@{and }{#6}%
    \ifpgfmath@in@%
      \pgf@polar@#6\@@%
    \else
      \pgf@polar@#6 and #6\@@%
    \fi
    \edef\xarcradius{\the\pgf@x}%
    \edef\yarcradius{\the\pgf@y}% 
     %
    % Get the start and end angles of the arc...
    % 
    \pgfmathsetmacro\s{#4}%
    \pgfmathsetmacro\e{#5}%
    %
    % ...and also with rounding.
    %
    \pgfmathmod@{\s}{360}%
    \ifdim\pgfmathresult pt<0pt\relax%
      \pgfmathadd@{\pgfmathresult}{360}%
    \fi
    \let\ss\pgfmathresult%
    \pgfmathmod@{\e}{360}%
    \ifdim\pgfmathresult pt<0pt\relax%
      \pgfmathadd@{\pgfmathresult}{360}%
    \fi
    \let\ee\pgfmathresult%
    %
    % Hackery for when arc straddles zero.
    %
    \ifdim\ee pt<\ss pt\relax%
      \pgfmathadd@{\x}{180}%
      \pgfmathmod@{\pgfmathresult}{360}%
      \let\x\pgfmathresult%
    \fi
    \def\m{360}% Measure of nearness.
    \pgfmathadd@{\s}{\e}%
    \pgfmathdivide@{\pgfmathresult}{2}%
    \let\n\pgfmathresult% The best estimate (default to middle of arc).
    \pgfmathloop%
      \pgfmathadd@{\s}{\e}%
      \pgfmathdivide@{\pgfmathresult}{2}%
      \let\p\pgfmathresult%
      \ifdim\p pt=\s pt\relax% 
      \else
        \tkzmathanglebetweenpoints{#2}{%
          \pgfpointadd{#3}{%
            \pgf@x\xarcradius\relax%
            \pgfmathcos@{\p}%
            \pgf@x\pgfmathresult\pgf@x%
            \pgf@y\yarcradius\relax%
            \pgfmathsin@{\p}%
            \pgf@y\pgfmathresult\pgf@y%
          }%
        }%
        %
        % Hackery for when arc straddles zero.
        %
        \ifdim\ee pt<\ss pt\relax%
          \pgfmathadd@{\pgfmathresult}{180}%
          \pgfmathmod@{\pgfmathresult}{360}%
        \fi
        \let\q\pgfmathresult%
        %
        % More hackery...
        %
        \ifdim\x pt>335pt\relax%
          \ifdim\q pt<45pt\relax%
            \pgfmathadd@{\q}{360}%
            \let\q\pgfmathresult%
          \fi
        \fi
        \ifdim\x pt=\q pt% Found it!
            \pgfmathbreakloop% Breaks after current iteration is complete.
          \else
            \ifdim\x pt<\q pt\relax%
              \let\e\p%
            \else
              \let\s\p%
            \fi
          \fi
          \pgfmathsubtract@{\x}{\q}%
          \pgfmathabs@{\pgfmathresult}%
          %
          % Save the estimate if it is better than any previous estimate.
          %
          \ifdim\pgfmathresult pt<\m pt\relax%
            \let\m\pgfmathresult%
            \let\n\p%
          \fi        
    \repeatpgfmathloop%
    \pgfpointadd{#3}{\pgfpointpolar{\n}{\xarcradius and \yarcradius}}%
  }%
}

% \tkzmathangleonellipse
% 
% Find the angle corresponding to a point on the border of an ellispe.
%
% #1 - the point on the border.
% #2 - the radii of the ellipse.
%
\def\tkzmathangleonellipse#1#2{%
  \begingroup
    \pgfmath@in@{and }{#2}%
    \ifpgfmath@in@%
      \pgf@polar@#2\@@%
    \else
      \pgf@polar@#2 and #2\@@%
    \fi
    \pgf@xa\pgf@x%
    \pgf@ya\pgf@y%
    \pgf@process{#1}% 
    \ifdim\pgf@x=0pt\relax%
      \pgfutil@tempdima1pt\relax%
    \else
      \pgfutil@tempdima\pgf@x%
%\pgfmathdivide@{\pgfmath@tonumber{\pgf@xa}}{\pgfmath@tonumber{\pgfutil@tempdima}}% 
\FPdiv\pgfmathresult{\pgfmath@tonumber{\pgf@xa}}{\pgfmath@tonumber{\pgfutil@tempdima}} 
\FPround\pgfmathresult\pgfmathresult5\relax%
      \pgfutil@tempdima\pgfmathresult pt\relax%
    \fi
    \ifdim\pgf@y=0pt\relax%
      \pgfutil@tempdima1pt\relax%
    \else 
     % \pgfmathdivide@{\pgfmath@tonumber{\pgf@y}}{\pgfmath@tonumber{\pgf@ya}}%
      \FPdiv\pgfmathresult{\pgfmath@tonumber{\pgf@y}}{%
                           \pgfmath@tonumber{\pgf@ya}}%
      \FPround\pgfmathresult\pgfmathresult5\relax%
      \pgfutil@tempdima\pgfmathresult\pgfutil@tempdima%
      \pgfmathatan@{\pgfmath@tonumber{\pgfutil@tempdima}}%
    \fi
    %
    \pgfutil@tempdima\pgfmathresult pt\relax%
    \ifdim\pgfutil@tempdima<0pt\relax%
      \advance\pgfutil@tempdima360pt\relax%
    \fi
    \ifdim\pgf@x<0pt\relax%
      \ifdim\pgf@y=0pt\relax%
        \pgfutil@tempdima180pt\relax%
      \else
        \ifdim\pgf@y<0pt\relax%
          \advance\pgfutil@tempdima180pt\relax%
        \else
          \advance\pgfutil@tempdima-180pt\relax%
        \fi
      \fi
    \else
      \ifdim\pgf@x=0pt\relax%
        \ifdim\pgf@y<0pt\relax%
          \pgfutil@tempdima270pt\relax%
        \else
          \pgfutil@tempdima90pt\relax%
        \fi
      \else
        \ifdim\pgf@y=0pt\relax%
          \pgfutil@tempdima0pt\relax%
        \fi
      \fi        
    \fi
    \pgfmath@returnone\pgfutil@tempdima%
  \endgroup
} 

\def\tkzpointborderellipse#1#2{%
  \pgf@process{#2}%
  \pgf@xa=\pgf@x%
  \pgf@ya=\pgf@y%
  \ifdim\pgf@xa=\pgf@ya% circle. that's easy!
    \pgf@process{\pgfpointnormalised{#1}}%
    \pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x%
    \pgf@y=\pgf@sys@tonumber{\pgf@xa}\pgf@y%
  \else
    \ifdim\pgf@xa<\pgf@ya%
      % Ok, first, let's compute x/y:
      \c@pgf@countb=\pgf@ya%
      \divide\c@pgf@countb by65536\relax%
      \divide\pgf@x by\c@pgf@countb%
      \divide\pgf@y by\c@pgf@countb%
      \pgf@xc=\pgf@x%
      \pgf@yc=8192pt%
      \pgf@y=.125\pgf@y%
      \c@pgf@countb=\pgf@y%
      \divide\pgf@yc by\c@pgf@countb%
      \pgf@process{#1}%
      \pgf@y=\pgf@sys@tonumber{\pgf@yc}\pgf@y%
      \pgf@y=\pgf@sys@tonumber{\pgf@xc}\pgf@y%
      \pgf@process{\pgfpointnormalised{}}%
      \pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x%
      \pgf@y=\pgf@sys@tonumber{\pgf@ya}\pgf@y%
    \else
      % Ok, now let's compute y/x:
      \c@pgf@countb=\pgf@xa%
      \divide\c@pgf@countb by65536\relax%
      \divide\pgf@x by\c@pgf@countb%
      \divide\pgf@y by\c@pgf@countb%
      \pgf@yc=\pgf@y%
      \pgf@xc=8192pt%
      \pgf@x=.125\pgf@x%
      \c@pgf@countb=\pgf@x%
      \divide\pgf@xc by\c@pgf@countb%
      \pgf@process{#1}%
      \pgf@x=\pgf@sys@tonumber{\pgf@yc}\pgf@x%
      \pgf@x=\pgf@sys@tonumber{\pgf@xc}\pgf@x%
      \pgf@process{\pgfpointnormalised{}}%
      \pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x%
      \pgf@y=\pgf@sys@tonumber{\pgf@ya}\pgf@y%
    \fi  
  \fi
}    
\endinput   