texlive[52457] Master: pst-eucl (20oct19)
commits+karl at tug.org
commits+karl at tug.org
Sun Oct 20 22:57:52 CEST 2019
Revision: 52457
http://tug.org/svn/texlive?view=revision&revision=52457
Author: karl
Date: 2019-10-20 22:57:52 +0200 (Sun, 20 Oct 2019)
Log Message:
-----------
pst-eucl (20oct19)
Modified Paths:
--------------
trunk/Master/texmf-dist/doc/generic/pst-eucl/Changes
trunk/Master/texmf-dist/doc/generic/pst-eucl/pst-eucl-doc.pdf
trunk/Master/texmf-dist/doc/generic/pst-eucl/pst-eucl-doc.tex
trunk/Master/texmf-dist/tex/generic/pst-eucl/pst-eucl.tex
trunk/Master/tlpkg/libexec/ctan2tds
Modified: trunk/Master/texmf-dist/doc/generic/pst-eucl/Changes
===================================================================
--- trunk/Master/texmf-dist/doc/generic/pst-eucl/Changes 2019-10-20 17:28:18 UTC (rev 52456)
+++ trunk/Master/texmf-dist/doc/generic/pst-eucl/Changes 2019-10-20 20:57:52 UTC (rev 52457)
@@ -1,10 +1,15 @@
pst-eucl.pro --------
-1.01 2012/09/21 - fix for introduced bug
-1.00 2011/08/05 - fix bug in /InterLines
+1.01 2012/09/21 - fix for introduced bug
+1.00 2011/08/05 - fix bug in /InterLines
pst-eucl.tex --------
-
+1.66 2019/10/20 - add macros to operate the node coordinates, \pstAbscissa, \pstOrdinate, \pstMoveNode etc.
+ - add optional parameters angleA and angleB for \pstCircleOA and \pstCircleAB.
+ - add optional parameters to output the inner circle center and outer circle center for \pstTriangleIC and \pstTriangleOC.
+ - add macros to draw the tangent line and tangent node of circle.
+ - add macros to draw the external and internal common tangent lines of two circles.
+ - add macros to draw conics (ellipse, parabola and hyperbola) and their geometrical elements, such as focus, directrix and intersections.
1.65 2019/08/19 - new type for angle
1.64 2019/01/31 - fix for PointName and pstInterCC
1.63 2019/01/27 - fix for PointSymbol=none for pstTriangle
@@ -43,7 +48,7 @@
1.36 2010/08/23 - fix for \pstMarkAngle (hv)
1.35 2009/01/19 - new option labelColor (hv)
1.34 2006/01/28 - use \psscalebox instead of \scalebox
- - small tweaks
+ - small tweaks
%% 2000-10-16 : creation of the file from a first LaTeX protype sty file
%% 2001-05-7 : distribution of the first beta version
%% 2002-03-21 : distribution of the second beta version
Modified: trunk/Master/texmf-dist/doc/generic/pst-eucl/pst-eucl-doc.pdf
===================================================================
(Binary files differ)
Modified: trunk/Master/texmf-dist/doc/generic/pst-eucl/pst-eucl-doc.tex
===================================================================
--- trunk/Master/texmf-dist/doc/generic/pst-eucl/pst-eucl-doc.tex 2019-10-20 17:28:18 UTC (rev 52456)
+++ trunk/Master/texmf-dist/doc/generic/pst-eucl/pst-eucl-doc.tex 2019-10-20 20:57:52 UTC (rev 52457)
@@ -4,6 +4,8 @@
\usepackage{pst-eucl}
\let\pstEuclideFV\fileversion
\usepackage{multicol}
+\usepackage{ntheorem}
+\newtheorem{theorem}{Theorem}
\usepackage{pst-plot,paralist}
\usepackage[mathscr]{eucal}
\lstset{pos=l,wide=false,language=PSTricks,
@@ -42,7 +44,7 @@
\item Michael Vulis for his fast testing of the documentation using
V\TeX\ which leads to the correction of a bug in the \PS\ code;
\item Manuel Luque and Olivier Reboux for their remarks and their examples.
- \item Alain Delplanque for its modification propositions on automatic
+ \item Alain Delplanque for its modification theorems on automatic
placing of points name and the ability of giving a list of points in
\Lcs{pstGeonode}.
\end{compactitem}
@@ -51,9 +53,10 @@
\vfill
\noindent
-Thanks to:
+Thanks to:
Manuel Luque;
-Thomas Söll.
+Thomas Söll;
+Liao Xiongfei.
@@ -195,19 +198,61 @@
\rnode{ici}{here}.
\nccurve[arrowscale=2]{->}{ici}{B_1}
+After v1.65, we add macros \Lcs{pstAbscissa} and \Lcs{pstOrdinate} to
+get the abscissa and ordinate of the specified node, so it is possible
+to define a new node from an already constructed node with them.
+
+\begin{BDef}
+\Lcs{pstAbscissa}\Largb{A}\\
+\Lcs{pstOrdinate}\Largb{A}
+\end{BDef}
+
+Note that the value of abscissa or ordinate are transformed to the \texttt{User coordinate},
+and then put into the stack of \PS, so they can be used to do some compound arithmetic
+without concerned the \texttt{xunit} and \texttt{yunit} in the \PST{} \texttt{SpecialCoor}
+function. You need the other third package to do float arithmetic operation,
+like \Lcs{pscalculate} \footnote{Provided by package \texttt{pst-calculate},
+sometimes it results the numbers more than 9 fraction digits,
+which are not supported good by \PST\space with '! number too big' issue.} to generate the numerical values,
+or the expandable command \Lcs{fpeval}\footnote{Provided by package \texttt{xfp},
+it can truncate the fraction part digits using the \texttt{trunc} function perfectly,
+e.g. \texttt{\textbackslash{}fpeval\{trunc(18/7,3)\}}.} to get a purely numerical result,
+
+The macro \Lcs{pstMoveNode} use them to move node $A$ by abscissa increment $dx$
+and ordinate increment $dy$ to get the target node $B$.
+
+\begin{BDef}
+\Lcs{pstMoveNode}\OptArgs($dx$,\kern 1pt$dy$)\Largb{$A$}\Largb{$B$}
+\end{BDef}
+
+for example:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\def\ra{3.0}\def\rb{4.0}
+\pstGeonode[PosAngle=-90](1.0,1.5){A}
+\pstGeonode[PosAngle=90](! \pstAbscissa{A} 1 add \pstOrdinate{A} 2 add){B}
+\pstLineAB[linecolor=blue]{A}{B}
+\pstMoveNode[PosAngle=-90,PointSymbol=asterisk](3,2){A}{C}
+\pstLineAB[linecolor=red]{A}{C}
+\pstMoveNode[PosAngle=-90,PointSymbol=diamond](\pscalculate{sqrt(\ra*\ra+\rb*\rb)/2},\pscalculate{\ra*\rb/(2*(\ra+\rb))}){A}{D}
+\pstLineAB[linecolor=cyan]{A}{D}
+\pstMoveNode[PosAngle=-90](\pstAbscissa{B} 3 div,\pstOrdinate{B} neg 3 div){D}{E}
+\pstLineAB[linecolor=green]{A}{E}
+\end{pspicture}
+\end{LTXexample}
+
%\subsubsection{User defined axes}
-\Lcs{pstOIJGeonode} creates a list of points in the landmark $(O;I;J)$. Possible
+\Lcs{pstOIJGeonode} creates a list of points in the landmark $(O;I;J)$. Possible
parameters are \Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PosAngle},
- \Lkeyword{PointSymbol}, and \Lkeyword{PtNameMath}.
+\Lkeyword{PointSymbol}, and \Lkeyword{PtNameMath}.
+
\begin{BDef}
\Lcs{pstOIJGeonode}\OptArgs\coord1\Largb{$A_1$}\Largb{$O$}\Largb{$I$}\Largb{$J$}
\coord2\Largb{$A_2$}\ldots\cAny\Largb{$A_n$}
\end{BDef}
-\clearpage
-
-
\begin{LTXexample}[width=5.6cm,pos=l]
\psset{unit=.7}
\begin{pspicture*}[showgrid=true](-4,-4)(4,4)
@@ -268,7 +313,7 @@
The three commands of the family \Lkeyval{MarkHash} draw a line whose inclination is
controled by the parameter \Lkeyword{MarkAngle} (default is 45). Their width and colour
-depends of the width and color of the line when the drawing is done, ass shown is the
+depends of the width and color of the line when the drawing is done, as shown is the
next example.
@@ -277,12 +322,12 @@
\begin{pspicture}[showgrid=true](-2,-2)(2,2)
\rput{18}{%
\pstGeonode[PosAngle={0,90,180,-90}](2,0){A}(2;72){B}
- (2;144){C}(2;216){D}(2;288){E}}
- \pstSegmentMark{A}{B}
+ (2;144){C}(2;216){D}(2;288){E}}
+ \pstSegmentMark[SegmentSymbol=none]{A}{B}
\pstSegmentMark[linecolor=green]{B}{C}
\psset{linewidth=2\pslinewidth}
\pstSegmentMark[linewidth=2\pslinewidth]{C}{D}
- \pstSegmentMark{D}{E}
+ \pstSegmentMark[MarkAngle=90]{D}{E}
\pstSegmentMark{E}{A}
\end{pspicture}
\end{LTXexample}
@@ -294,18 +339,16 @@
\subsection{Triangles}
-The more classical figure, it has its own macro for a quick definition:
+The more classical figure, it has its own macro \Lcs{pstTriangle} for a quick definition:
\begin{BDef}
-\Lcs{pstTriangle}\OptArgs\coord1\Largb{A}\coord2\Largb{B}\coord3\Largb{C}\\
-\Lcs{pstTriangleIC}\OptArgs\Largb{A}\Largb{B}\Largb{C}\\
-\Lcs{pstTriangleOC}\OptArgs\Largb{A}\Largb{B}\Largb{C}
+\Lcs{pstTriangle}\OptArgs\coord1\Largb{A}\coord2\Largb{B}\coord3\Largb{C}
\end{BDef}
\begin{sloppypar}
Valid optional arguments are \Lkeyword{PointName},
- \Lkeyword{PointNameSep}, %\Lkeyword{PosAngle},
+ \Lkeyword{PointNameSep}, %\Lkeyword{PosAngle},
\Lkeyword{PointSymbol}, \Lkeyword{PointNameA},
\Lkeyword{PosAngleA}, \Lkeyword{PointSymbolA}, \Lkeyword{PointNameB},
\Lkeyword{PosAngleB}, \Lkeyword{PointSymbolB}, \Lkeyword{PointNameC},
@@ -326,6 +369,14 @@
\Lkeyword{PointSymbol}. The management of the default value followed the
same rule.
+The macros \Lcs{pstTriangleIC} and \Lcs{pstTriangleOC} are used to draw the inner circle
+and outer circle of triangle $ABC$.
+
+\begin{BDef}
+\Lcs{pstTriangleIC}\OptArgs\Largb{A}\Largb{B}\Largb{C}\OptArg{I}\OptArg{H}\\
+\Lcs{pstTriangleOC}\OptArgs\Largb{A}\Largb{B}\Largb{C}\OptArg{O}
+\end{BDef}
+
\begin{LTXexample}[width=5cm,pos=l]
\begin{pspicture}[showgrid](-2,-2)(2,2)
\pstTriangle[PointSymbol=square,PointSymbolC=o,
@@ -336,10 +387,20 @@
\end{pspicture}
\end{LTXexample}
-The center of the inner circle is called \verb|IC_O| and the outer circle \verb|OC_O|. They are
-only defined, if the macros \Lcs{pstTriangleIC} and \Lcs{pstTriangleOC} are used.
+The center of the inner circle is called \verb|IC_O| as default and the outer circle \verb|OC_O| as default,
+but you can change the node names by the optional parameters \OptArg{I} \OptArg{H} and \OptArg{O}.
+The optional node name $H$ is a node on the inner circle,
+so you can operate the inner circle by center $I$ and node $H$ later.
+The inner center $I$, node $H$ and outer circle center $O$ are not printed out as default,
+but you can setup \Lkeyword{PointSymbol} and \Lkeyword{PointName} to display them.
+For example:
+\begin{lstlisting}
+\pstTriangleIC[PosAngle={-90,160},PointName={I,none},PointSymbol={*,none}]{A}{B}{C}[I][D]
+\pstTriangleOC[PosAngle=90,PointSymbol=*,PointName=X]{A}{B}{C}[X]
+\end{lstlisting}
+
\subsection{Angles}
Each angle is defined with three points. The vertex is the second
@@ -379,7 +440,7 @@
\begin{sloppypar}
-Valid optional arguments are \Lkeyword{MarkAngleRadius}, \Lkeyword{LabelAngleOffset},
+Valid optional arguments are \Lkeyword{MarkAngleRadius}, \Lkeyword{LabelAngleOffset},
\Lkeyword{MarkAngleType} and
\Lkeyword{Mark}
%
@@ -415,18 +476,18 @@
\psset{PointSymbol=none,PointNameMathSize=\scriptstyle,PointNameSep=6pt,
RightAngleSize=0.15,PosAngle={135,225,-45,45}}
\pstGeonode(1,2){A}(1,1){B}(2,1){C}(2,2){D}%
-\pstRightAngle[fillstyle=solid,fillcolor=blue!40]{C}{B}{A}
-\pstRightAngle{D}{C}{B} \pstRightAngle{A}{D}{C}
+\pstRightAngle[fillstyle=solid,fillcolor=blue!40]{C}{B}{A}
+\pstRightAngle{D}{C}{B} \pstRightAngle{A}{D}{C}
\pstRightAngle{B}{A}{D} \pspolygon(A)(B)(C)(D)
\psset{RightAngleType=suisseromand}
\pstGeonode(3,2){A}(3,1){B}(4,1){C}(4,2){D}%
-\pstRightAngle[fillstyle=solid,fillcolor=blue!40]{C}{B}{A}
-\pstRightAngle{D}{C}{B} \pstRightAngle{A}{D}{C}
+\pstRightAngle[fillstyle=solid,fillcolor=blue!40]{C}{B}{A}
+\pstRightAngle{D}{C}{B} \pstRightAngle{A}{D}{C}
\pstRightAngle{B}{A}{D} \pspolygon(A)(B)(C)(D)
\psset{RightAngleType=german}
\pstGeonode(5,2){A}(5,1){B}(6,1){C}(6,2){D}%
-\pstRightAngle[fillstyle=solid,fillcolor=blue!40]{C}{B}{A}
-\pstRightAngle{D}{C}{B} \pstRightAngle{A}{D}{C}
+\pstRightAngle[fillstyle=solid,fillcolor=blue!40]{C}{B}{A}
+\pstRightAngle{D}{C}{B} \pstRightAngle{A}{D}{C}
\pstRightAngle{B}{A}{D} \pspolygon(A)(B)(C)(D)
\end{pspicture}
\end{LTXexample}
@@ -469,47 +530,104 @@
\begin{LTXexample}[width=5cm,pos=l]
\begin{pspicture}[showgrid](-2,-2)(2,2)
\pstGeonode(1,1){A}(-1,-1){B}
-\pstLineAB[nodesepA=-.4,nodesepB=-1,
+\pstLineAB[nodesepA=-.4,nodesepB=-1,
linecolor=green]{A}{B}
\pstLineAB[nodesep=.4,linecolor=red]{A}{B}
\end{pspicture}
\end{LTXexample}
+The macro \Lcs{pstLine} draws a new line with two nodes, or two coordinates
+or one node and one coordinate. This macro is similar with \Lcs{pstLineAB},
+but more compatible.
+\begin{BDef}
+\Lcs{pstLine}\OptArgs\Largb{A}\Largb{B}\\
+\Lcs{pstLine}\OptArgs\Largb{A}\cAny\\
+\Lcs{pstLine}\OptArgs\cAny\Largb{B}\\
+\Lcs{pstLine}\OptArgs\cAny\cAny
+\end{BDef}
+The macros \Lcs{pstLineAA} and \Lcs{pstLineAS} draw a new line with one node,
+the slope \texttt{angle} between the line and the horizontal axis, or the
+slope \texttt{gradient} of the line, and create a new node $B$ on the line.
+\begin{BDef}
+\Lcs{pstLineAA}\OptArgs\Largb{A}\Largb{angle}\Largb{B}\\
+\Lcs{pstLineAA}\OptArgs\cAny\Largb{angle}\Largb{B}\\
+\Lcs{pstLineAS}\OptArgs\Largb{A}\Largb{gradient}\Largb{B}\\
+\Lcs{pstLineAS}\OptArgs\cAny\Largb{gradient}\Largb{B}
+\end{BDef}
- \subsection{Circles}
+Here are some examples:
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\pstGeonode[PosAngle=90](1.5,1.5){A}
+% draw a line with angle atan(2/1), about 63.43 degree.
+\pstLineAA[linecolor=red,nodesep=-0.5,PosAngle=90]{A}{2 1 atan}{B}
+\pstLineAA[linecolor=yellow,nodesep=-0.5,PosAngle=-120]{A}{-45}{C}
+\pstLineAS[linecolor=green,nodesep=-0.5,PosAngle=30]{A}{-0.5}{D}
+% draw a line with gradient (cos50/sin50).
+\pstLineAS[linecolor=cyan,nodesep=-0.5]{A}{50 cos 50 sin div}{E}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstLineAbsNode} creates a new node $C$ whose abscissa
+is the given value $x_1$ on the line $AB$. The macro \Lcs{pstLineOrdNode} creates a new node $C$ whose ordinate is the given value $y_1$ on the line $AB$.
+You can input $x_1$ or $y_1$ as any number(e.g, 2.0),
+or use \Lcs{pscalculate} or \Lcs{fpeval} to get a purely numerical result,
+or use \Lcs{pstAbscissa} and \Lcs{pstOrdinate} to get the abscissa and ordinate of any other node.
+
+\begin{BDef}
+\Lcs{pstLineAbsNode}\OptArgs\Largb{A}\Largb{B}\Largb{$x_1$}\Largb{C}\\
+\Lcs{pstLineOrdNode}\OptArgs\Largb{A}\Largb{B}\Largb{$y_1$}\Largb{C}
+\end{BDef}
+
+For example,
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\pstGeonode[PosAngle=-40](0.8,0.5){A}
+\pstGeonode[PosAngle=-40](1.2,1.0){B}
+\pstLineAB[linecolor=red,nodesep=-0.5]{A}{B}
+\pstLineAbsNode[PosAngle=-40,PointSymbol=o]{A}{B}{2.5}{C}
+\pstLineOrdNode[PosAngle=-40,PointSymbol=o]{A}{B}{3.0}{D}
+\pstLineAB[linecolor=blue,nodesep=-0.5]{C}{D}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{Circles}
+
A circle can be defined either with its center and a point of its
circumference, or with two diameterly opposed points. There is two
commands :
-
\begin{BDef}
-\Lcs{pstCircleOA}\OptArgs\Largb{O}\Largb{A}\\
-\Lcs{pstCircleAB}\OptArgs\Largb{O}\Largb{A}\\
-\Lcs{pstDistAB}\OptArgs\Largb{A}\Largb{B}\\
-\Lcs{pstDistVal}\OptArgs\Largb{x}
+\Lcs{pstCircleOA}\OptArgs\Largb{O}\Largb{A}\OptArg{angleA}\OptArg{angleB}\\
+\Lcs{pstCircleAB}\OptArgs\Largb{O}\Largb{A}\OptArg{angleA}\OptArg{angleB}
\end{BDef}
-%\Lcs{pstCircleOA} draws the circle of center $O$ crossing $A$. Possible options are \Lkeyword{Radius} and
-% \Lkeyword{Diameter}.
-
-%\Lcs{pstCircleAB} draws the circle of diameter $AB$ with the same options.
+\Lcs{pstCircleOA} draws the circle of center $O$ crossing $A$ from \Lkeyword{angleA} to \Lkeyword{angleB}, going counter clockwise.
+Possible options are \Lkeyword{Radius} and \Lkeyword{Diameter}.
+\Lcs{pstCircleAB} draws the circle of diameter $AB$ with the same options.
+
For the first macro, it is possible to omit the second point and then
to specify a radius or a diameter using the parameters \Lkeyword{Radius}
and \Lkeyword{Diameter}. The values of these parameters must be specified
with one of the two following macros :
+\begin{BDef}
+\Lcs{pstDistAB}\OptArgs\Largb{A}\Largb{B}\\
+\Lcs{pstDistVal}\OptArgs\Largb{x}
+\end{BDef}
+
%\Lcs{pstDistAB} Specifies distance $AB$ for the parameters
% \Lkeyword{Radius}, \Lkeyword{Diameter} and \Lkeyword{DistCoef}.
-
+%
%\Lcs{pstDistVal} Specifies a numerical value for the parameters
% \Lkeyword{Radius}, \Lkeyword{Diameter}, and \Lkeyword{DistCoef}.
-
The first specifies a distance between two points. The parameter
\Lkeyword{DistCoef} can be used to specify a coefficient to reduce or
@@ -529,41 +647,23 @@
\item {\color{RoyalBlue} the circle whose diameter is $BC$.}
\end{compactitem}
-\enlargethispage{3\normalbaselineskip}
-
-\bigskip
-\begin{pspicture}[showgrid](-4,-3.3)(5,3)
-\psset{linewidth=2\pslinewidth}
-\pstGeonode[PosAngle={0,-135,90},PointSymbol={*,*,square}](1,0){A}(-2,-1){B}(0,1){C}
-\pstCircleOA[linecolor=red]{A}{B}
-\pstCircleOA[linecolor=green, DistCoef=2 3 div, Radius=\pstDistAB{A}{C}]{A}{}
-\pstCircleOA[linecolor=blue, Radius=\pstDistAB{B}{C}]{A}{}
-\pstCircleOA[linecolor=Sepia, Radius=\pstDistAB{A}{C}]{B}{}
-\pstCircleOA[linecolor=Aquamarine, Diameter=\pstDistAB{A}{C}]{B}{}
-\pstCircleAB[linecolor=RoyalBlue]{B}{C}
-\end{pspicture}
-
-
\clearpage
-\begin{lstlisting}
-\begin{pspicture}[showgrid](-4,-4)(5,3)
+\begin{LTXexample}[width=\linewidth,pos=t]
+\begin{pspicture}[showgrid](-4,-3.3)(4,3)
\psset{linewidth=2\pslinewidth}
\pstGeonode[PosAngle={0,-135,90},PointSymbol={*,*,square}](1,0){A}(-2,-1){B}(0,1){C}
\pstCircleOA[linecolor=red]{A}{B}
\pstCircleOA[linecolor=green, DistCoef=2 3 div, Radius=\pstDistAB{A}{C}]{A}{}
-\pstCircleOA[linecolor=blue, Radius=\pstDistAB{B}{C}]{A}{}
+\pstCircleOA[linecolor=blue, Radius=\pstDistAB{B}{C}]{A}{}[45][270]
\pstCircleOA[linecolor=Sepia, Radius=\pstDistAB{A}{C}]{B}{}
-\pstCircleOA[linecolor=Aquamarine, Diameter=\pstDistAB{A}{C}]{B}{}
+\pstCircleOA[linecolor=Aquamarine, Diameter=\pstDistAB{A}{C}]{B}{}[80][320]
\pstCircleAB[linecolor=RoyalBlue]{B}{C}
\end{pspicture}
-\end{lstlisting}
+\end{LTXexample}
+\subsection{Circle arcs}
- \subsection{Circle arcs}
-
-
-
\begin{BDef}
\Lcs{pstArcOAB}\OptArgs\Largb{O}\Largb{A}\Largb{B}\\
\Lcs{pstArcnOAB}\OptArgs\Largb{O}\Largb{A}\Largb{B}
@@ -588,14 +688,80 @@
\end{pspicture}
\end{LTXexample}
-\subsection{Curved abscissa}
+\subsection{Circle tangent}
-A point can be positioned on a circle using its curved abscissa.
+The macro \Lcs{pstCircleTangentLine} is used to draw a tangent line $AT$ from a point $A$ on the circle,
+and the macro \Lcs{pstCircleTangentNode} is used to draw the tangent points $T_1$ and $T_2$ from a point $P$ out of the circle.
+\begin{BDef}
+\Lcs{pstCircleTangentLine}\OptArgs\Largb{O}\Largb{A}\Largb{T}\\
+\Lcs{pstCircleTangentNode}\OptArgs\Largb{O}\Largb{A}\Largb{P}\Largb{T1}\Largb{T2}
+\end{BDef}
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\psset{nodesep=-0.8}
+\pstGeonode[PosAngle={90,120,-30}](1,1){O}(-1,0){T}(3,0){S}
+\pstCircleOA[Radius=\pstDistVal{1.5},linecolor=red]{O}{}
+\pstCircleRotNode[Radius=\pstDistVal{1.5},PosAngle=-30,RotAngle=-30]{O}{}{A}
+\pstCircleTangentLine[PosAngle=-10,PointName=A_1]{O}{A}{A1}
+\pstCircleRotNode[Radius=\pstDistVal{1.5},PosAngle=90,RotAngle=90]{O}{}{B}
+\pstCircleTangentLine[PosAngle=90,PointName=B_1]{O}{B}{B1}
+\pstCircleTangentNode[Radius=\pstDistVal{1.5},PosAngle={150,90},PointName={T_1,T_2}]{O}{}{T}{T1}{T2}
+\pstCircleTangentNode[PosAngle={80,200},PointName={S_1,S_2}]{O}{A}{S}{S1}{S2}
+\end{pspicture}
+\end{LTXexample}
+The macro \Lcs{pstCircleExternalCommonTangent} is used to find the external common tangent lines of two circle $A(O_1)$ and $B(O_2)$,
+and the macro \Lcs{pstCircleInternalCommonTangent} is used to find the internal common tangent lines of two circle $A(O_1)$ and $B(O_2)$.
+They both create four tangent point nodes $T_1,T_2,T_3,T_4$, where $T_1,T_2$ lie on circle $A(O_1)$, and $T_3,T_4$ lie on circle $B(O_2)$.
\begin{BDef}
+\Lcs{pstCircleExternalCommonTangent}\OptArgs\Largb{$O_1$}\Largb{A}\Largb{$O_2$}\Largb{B}\Largb{$T_1$}\Largb{$T_2$}\Largb{$T_3$}\Largb{$T_4$}\\
+\Lcs{pstCircleInternalCommonTangent}\OptArgs\Largb{$O_1$}\Largb{A}\Largb{$O_2$}\Largb{B}\Largb{$T_1$}\Largb{$T_2$}\Largb{$T_3$}\Largb{$T_4$}
+\end{BDef}
+
+You can use \Lkeyword{RadiusA} and \Lkeyword{RadiusB} to define the two circles like as following:
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(3,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\pstGeonode[PosAngle=-90](-1,0){O1}
+\pstGeonode[PosAngle=-60](1.5,1.5){O2}
+\pstCircleOA[Radius=\pstDistVal{2},linecolor=red]{O1}{}
+\pstCircleOA[Radius=\pstDistVal{1},linecolor=blue]{O2}{}
+\pstCircleExternalCommonTangent[RadiusA=\pstDistVal{2},RadiusB=\pstDistVal{1},PosAngle={90,-60,90,-60}]{O1}{}{O2}{}{P}{Q}{R}{S}
+\pstLine[nodesep=-1]{P}{R}
+\pstLine[nodesep=-1]{Q}{S}
+\pstCircleInternalCommonTangent[RadiusA=\pstDistVal{2},RadiusB=\pstDistVal{1},PosAngle={120,60,120,60}]{O1}{}{O2}{}{H}{I}{J}{K}
+\pstLine[nodesep=-1]{H}{J}
+\pstLine[nodesep=-1]{I}{K}
+\end{pspicture}
+\end{LTXexample}
+
+You also can use \Lkeyword{DiameterA} and \Lkeyword{DiameterB} to define the two circles like as following:
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(3,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\pstGeonode[PosAngle=-90](-1,0){O1}
+\pstGeonode[PosAngle=-60](1.5,1.5){O2}
+\pstCircleOA[Diameter=\pstDistVal{3},linecolor=red]{O1}{}
+\pstCircleOA[Diameter=\pstDistVal{2},linecolor=blue]{O2}{}
+\pstCircleExternalCommonTangent[DiameterA=\pstDistVal{3},DiameterB=\pstDistVal{2},PosAngle={100,-60,90,-60}]{O1}{}{O2}{}{P}{Q}{R}{S}
+\pstLine[nodesep=-1]{P}{R}
+\pstLine[nodesep=-1]{Q}{S}
+\pstCircleInternalCommonTangent[DiameterA=\pstDistVal{3},DiameterB=\pstDistVal{2},PosAngle={80,-60,-90,140}]{O1}{}{O2}{}{H}{I}{J}{K}
+\pstLine[nodesep=-1]{H}{J}
+\pstLine[nodesep=-1]{I}{K}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{Curved abscissa}
+
+A point can be positioned on a circle using its curved abscissa.
+
+
+\begin{BDef}
\Lcs{pstCurvAbsNode}\OptArgs\Largb{O}\Largb{A}\Largb{B}\Largb{Abs}
\end{BDef}
@@ -623,6 +789,55 @@
\end{pspicture}
\end{LTXexample}
+
+A point can be positioned on a circle using its absolute abscissa or ordinate too.
+You can input $x_1$ or $y_1$ as any number(e.g, 2.0), or use \Lcs{pscalculate} or \Lcs{fpeval} to generate the value,
+or use \Lcs{pstAbscissa} and \Lcs{pstOrdinate} to get the abscissa and ordinate of any other node.
+
+\begin{BDef}
+\Lcs{pstCircleAbsNode}\OptArgs\Largb{O}\Largb{A}\Largb{$x_1$}\Largb{C}\Largb{C}\\
+\Lcs{pstCircleOrdNode}\OptArgs\Largb{O}\Largb{A}\Largb{$y_1$}\Largb{C}\Largb{C}
+\end{BDef}
+
+for example,
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-1)(4,4)
+\pstGeonode[PosAngle=60](1.5,1.5){O}
+\pstGeonode[PosAngle=-30](2.5,0){A}
+\pstCircleOA[linecolor=red]{O}{A}
+\pstCircleAbsNode[PosAngleA=-60,PosAngleB=60,PointSymbol=*]{O}{A}{1.0}{C}{D}
+\pstCircleOrdNode[PosAngleA=150,PosAngleB=30,PointSymbol=*]{O}{A}{1.0}{E}{F}
+\pstLineAB[linestyle=dashed,linecolor=gray!40,nodesep=-0.5]{C}{D}
+\pstLineAB[linestyle=dashed,linecolor=gray!40,nodesep=-0.5]{E}{F}
+\end{pspicture}
+\end{LTXexample}
+
+A point can be positioned on a circle using its rotation angle by macro \Lcs{pstCircleRotNode}.
+The rotation angle should be passed by the \Lkeyword{RotAngle} in the \texttt{Options}.
+The circle is defined by center $O$ and point $A$ on the circle or \Lkeyword{Radius} in parameter.
+If you not set \Lkeyword{RotAngle}, the default value is $60^\circ$.
+
+\begin{BDef}
+\Lcs{pstCircleRotNode}\OptArgs\Largb{O}\Largb{A}\Largb{X}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-1)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\psset{Radius=\pstDistVal{2.0}}
+\pstGeonode[PosAngle=0](1.5,1.5){O}
+\pstCircleOA[linecolor=red]{O}{}
+\pstCircleRotNode[PosAngle=0,RotAngle=0]{O}{}{A}
+\pstCircleRotNode[PosAngle=60]{O}{}{B} % default 60 degree
+\pstCircleRotNode[PosAngle=90,RotAngle=90]{O}{}{C}
+\pstCircleRotNode[PosAngle=150,RotAngle=\pscalculate{3*360/7}]{O}{}{D}
+\pstCircleRotNode[PosAngle=180,RotAngle=180]{O}{}{E}
+\pstCircleRotNode[PosAngle=230,RotAngle=230]{O}{}{F}
+\pstCircleRotNode[PosAngle=270,RotAngle=270]{O}{}{G}
+\pstCircleRotNode[PosAngle=-45,RotAngle=-45]{O}{}{H}
+\end{pspicture}
+\end{LTXexample}
+
\subsection{Generic curve}
It is possible to generate a set of points using a loop, and to give
@@ -658,6 +873,1871 @@
\end{pspicture}
\end{LTXexample}
+\section{Conics}
+\subsection{Standard Ellipse}
+The Standard Ellipse $E$ with coordinate translation is defined by center $O(x_0,y_0)$,
+the half of the major axis $max(abs(a),abs(b))$, the half of the minor axis $min(abs(a),abs(b))$,
+the equation as following:
+\begin{equation}\label{FunctionOfStandardEllipse}
+\dfrac{(x-x_0)^2}{a^2}+\dfrac{(y-y_0)^2}{b^2}=1
+\end{equation}
+Sometimes we use the parametric function of the Standard Ellipse with coordinate translation:
+\begin{equation}\label{ParametricFunctionOfEllipse}
+\left\{\begin{array}{l}
+x=a\cos\alpha+x_0\\
+y=b\sin\alpha+y_0
+\end{array}\right.
+\end{equation}
+
+The Macro \Lcs{pstEllipse} is used to draw a Standard Ellipse with center $O$ from
+\Lkeyword{angleA} to \Lkeyword{angleB}, going counter clockwise.
+It combines the function like \Lcs{psellipse} and \Lcs{psellipticarc} in \PST.
+If \Lkeyword{angleA} and \Lkeyword{angleB} are not specified,
+the macro will draw the whole ellipse.
+
+\begin{BDef}
+\Lcs{pstEllipse}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{angleA}\OptArg{angleB}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.4}\def\rb{0.8}\def\rot{56}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+%\psellipse[linecolor=red!60](O)(\ra,\rb)
+\pstEllipse[linecolor=red!60](O)(\ra,\rb)[0][120]
+\pstEllipse[linecolor=green!60,linestyle=dashed,arrows=->,arrowscale=1.2](O)(\ra,\rb)[120][200]
+\pstEllipse[linecolor=blue!60](O)(\ra,\rb)[200][300]
+\pstEllipse[linecolor=purple!60,linestyle=dashed,arrows=->,arrowscale=1.2](O)(\ra,\rb)[300][360]
+\pstEllipse[linecolor=cyan!60](O)(\rb,\ra)
+\end{pspicture}
+\end{LTXexample}
+
+Now you can draw some points on this Ellipse using macro \Lcs{pstEllipseNode} or \Lcs{pstEllipseRotNode}.
+The macro \Lcs{pstEllipseNode} requires an explicit parameter $t$ as $\alpha$ in equation (\ref{ParametricFunctionOfEllipse})
+to calculate the point; but the macro \Lcs{pstEllipseRotNode} requires an implicit parameter \Lkeyword{RotAngle}
+as $\alpha$ in equation (\ref{ParametricFunctionOfEllipse}) to calculate the point.
+
+\begin{BDef}
+\Lcs{pstEllipseNode}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{$t$}\Largb{P}\\
+\Lcs{pstEllipseRotNode}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{P}
+\end{BDef}
+
+The following is the example, note that the \Lkeyword{RotAngle} is not $\angle{HOX}$ in geometrical,
+but $\angle{HOA}$ or $\angle{HOB}$.
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.4}\def\rb{0.8}\def\rot{56}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+%\psellipse[linecolor=red!60](O)(\ra,\rb)
+\pstEllipse[linecolor=red!60](O)(\ra,\rb)
+\pstEllipseNode[PosAngle=180](O)(\ra,\rb){180}{P}
+\pstEllipseRotNode[PosAngle=0,RotAngle=0](O)(\ra,\rb){Q}
+\pstEllipseRotNode[PosAngle=90,RotAngle=90](O)(\ra,\rb){M}
+\pstEllipseRotNode[PosAngle=-90,RotAngle=-90](O)(\ra,\rb){N}
+\pstCircleOA[linecolor=blue!60,Radius=\pstDistVal{\ra}]{O}{}
+\pstCircleRotNode[PosAngle=\rot,RotAngle=\rot,Radius=\pstDistVal{\ra}]{O}{}{A}
+\pstCircleOA[linecolor=green!60,Radius=\pstDistVal{\rb}]{O}{}
+\pstCircleRotNode[PosAngle=180,RotAngle=\rot,Radius=\pstDistVal{\rb}]{O}{}{B}
+\pstEllipseRotNode[PosAngle=30,RotAngle=\rot](O)(\ra,\rb){X}
+\pstProjection[PosAngle=-90]{P}{Q}{A}[H]
+\pstLineAB[linestyle=dashed]{A}{O}
+\pstLineAB[linestyle=dashed]{A}{H}
+\pstLineAB[linestyle=dashed]{B}{X}
+\pstLineAB[linestyle=dashed]{O}{H}
+\pstMarkAngle[LabelSep=.6,MarkAngleRadius=.3,MarkAngleType=double,fillcolor=red!30,fillstyle=solid]{H}{O}{A}{$\rot^\circ$}
+\end{pspicture}
+\end{LTXexample}
+
+The macros \Lcs{pstEllipseAbsNode} and \Lcs{pstEllipseOrdNode} are used to get the two nodes $A$ and $B$
+whose abscissas or ordinates are the given value $x_1$ or $y_1$ on the Standard Ellipse $E$.
+
+If there is no such point satisfied this condition, then the nodes $A$ and $B$ will be put at the origin.
+
+\begin{BDef}
+\Lcs{pstEllipseAbsNode}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstEllipseOrdNode}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.0}\def\rb{-1.2}
+\pstGeonode[PosAngle=-50,PointNameSep=0.2](2,2){O}
+\pstEllipse[linecolor=red!40](O)(\ra,\rb)
+\pstEllipse[linecolor=blue!40](O)(\rb,\ra)
+\pstEllipseAbsNode[PosAngle={120,200}](O)(\ra,\rb){2.5}{A}{B}
+\pstEllipseAbsNode(O)(\ra,\rb){6}{X}{Y} % not exist
+\pstEllipseOrdNode(O)(\ra,\rb){2.5}{A'}{B'}
+\pstEllipseOrdNode(O)(\ra,\rb){6}{X'}{Y'} % not exist
+\end{pspicture}
+\end{LTXexample}
+
+Here we find the focus node of Standard Ellipse! Please use macro \Lcs{pstEllipseFocusNode} to do this work.
+
+\begin{BDef}
+\Lcs{pstEllipseFocusNode}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{A}\Largb{B}
+\end{BDef}
+
+For example:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.0}\def\rb{-1.2}
+\pstGeonode[PosAngle=-50,PointNameSep=0.2](2,2){O}
+\pstEllipse[linecolor=red!40](O)(\ra,\rb)
+\pstEllipse[linecolor=blue!40](O)(\rb,\ra)
+\pstEllipseFocusNode(O)(\ra,\rb){L}{R}
+\pstEllipseFocusNode(O)(\rb,\ra){D}{U}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstEllipseDirectrixLine} is used to draw the two directrix lines of Standard Ellipse,
+and create two new nodes on each of them. The nodes $L_x$, $L_y$ are on the left/down directrix line,
+and $R_x$, $R_y$ are on the right/up directrix line. They are lie on the tangent line of the vertex
+on the other axis.
+
+\begin{BDef}
+\Lcs{pstEllipseDirectrixLine}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{$L_x$}\Largb{$L_y$}\Largb{$R_x$}\Largb{$R_y$}
+\end{BDef}
+
+For example:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.0}\def\rb{-1.2}
+\pstGeonode[PosAngle=-50,PointNameSep=0.2](2,2){O}
+\pstEllipse[linecolor=red!40](O)(\ra,\rb)
+\pstEllipse[linecolor=blue!40](O)(\rb,\ra)
+\pstEllipseDirectrixLine[PointName={L_x,L_y,R_x,R_y},PosAngle={210,210,-30,-30},nodesep=-1,linecolor=red!40](O)(\ra,\rb){Lx}{Ly}{Rx}{Ry}
+\pstEllipseDirectrixLine[PointName={D_x,D_y,U_x,U_y},PosAngle={-30,-30,30,30},nodesep=-1,linecolor=blue!40](O)(\rb,\ra){Dx}{Dy}{Ux}{Uy}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{Lx}{Rx}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{Ly}{Ry}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{Dx}{Ux}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{Dy}{Uy}
+\end{pspicture}
+\end{LTXexample}
+
+Sometimes we need to find the intersection of Ellipse and line,
+the Macro \Lcs{pstEllipseLineInter} can do this work, and it can handle any type of line,
+i.e, horizontal, vertical or others lines. It get the two intersection $C$ and $D$ of the
+Standard Ellipse $E$ and the given line $AB$. When there is none intersection,
+$C$ and $D$ are both put at the origin; When there is only on intersection, it will be saved
+at node $C$, and $D$ will be put at the origin.
+
+\begin{BDef}
+\Lcs{pstEllipseLineInter}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{$A$}\Largb{$B$}\Largb{$C$}\Largb{$D$}
+\end{BDef}
+
+Here is examples:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.0}\def\rb{-1.2}
+\pstGeonode[PosAngle=-50,PointNameSep=0.2](2,2){O}
+\pstEllipse[linecolor=red!40](O)(\ra,\rb)
+\pstEllipse[linecolor=blue!40](O)(\rb,\ra)
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{0,1}{3,4}
+\pstEllipseLineInter[PosAngle={-90,90}](O)(\ra,\rb){0,1}{3,4}{C}{D}
+\pstEllipseLineInter[PosAngle={-90,90}](O)(\rb,\ra){0,1}{3,4}{C'}{D'}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{1.5,0}{1.5,4}
+\pstEllipseLineInter[PosAngle={40,60}](O)(\ra,\rb){1.5,0}{1.5,4}{E}{F}
+\pstEllipseLineInter[PosAngle={40,130}](O)(\rb,\ra){1.5,1}{1.5,4}{E'}{F'}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{4,2.5}{0,2.5}
+\pstEllipseLineInter[PosAngle={130,50}](O)(\ra,\rb){4,2.5}{0,2.5}{G}{H}
+\pstEllipseLineInter[PosAngle={130,50}](O)(\rb,\ra){4,2.5}{0,2.5}{G'}{H'}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstEllipsePolarNode} is use to draw the tangent line of a point $A$ or $B$
+on the Standard Ellipse. It draws the every tangent line through the point $A$ and $B$ on
+the Standard Ellipse $E$ and get the insection node $T$ of the two tangent lines.
+We call $T$ as the polar point of chord $AB$ as normal.
+
+\begin{BDef}
+\Lcs{pstEllipsePolarNode}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{$A$}\Largb{$B$}\Largb{$T$}
+\end{BDef}
+
+We use the following theorem to find the node $T$:
+\begin{theorem}\label{EllipsePolarPointTheorem}
+Give chord $AB$ on the ellipse, we draw any other two chords $PQ$ and $RS$, $AB$ and $PQ$ intersect at $I$,
+$AQ$ and $BP$ intersect at $X$, $AP$ and $BQ$ intersect at $Y$, we call $XY$ is the polar line of point $I$.
+Also $AB$ and $RS$ intersect at $J$, $AR$ and $BS$ intersect at $M$, $AS$ and $BR$ intersect at $N$,
+we call $MN$ is the polar line of point $J$. Then the intersection $T$ of $XY$ and $MN$ is the polar point of chord $AB$,
+i.e. $TA$ is the tangent line through $A$ and $TB$ is the tangent line through $B$.
+\end{theorem}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\rb{2.0}\def\ra{-1.2}
+\pstGeonode[PosAngle=-50,PointNameSep=0.2](2,2){O}
+\pstEllipse[linecolor=red!40](O)(\ra,\rb)
+\pstLine[nodesep=-0.8,linecolor=black!40,linestyle=dashed]{1,2}{2.5,3.5}
+\pstEllipseLineInter[PosAngle={-100,90}](O)(\ra,\rb){1,2}{2.5,3.5}{A}{B}
+\pstEllipsePolarNode[PosAngle=120](O)(\ra,\rb){A}{B}{T}
+% Here are the auxiliary lines to explain Theorem 1.
+\pstEllipseRotNode[PosAngle=0,RotAngle=5](O)(\ra,\rb){P}
+\pstEllipseRotNode[PosAngle=-10,RotAngle=-61](O)(\ra,\rb){Q}
+\pstEllipseRotNode[PosAngle=-100,RotAngle=-92](O)(\ra,\rb){R}
+\pstEllipseRotNode[PosAngle=0,RotAngle=-30](O)(\ra,\rb){S}
+\pstInterLL[PosAngle=-90]{A}{Q}{B}{P}{X}
+\pstInterLL[PosAngle=-10]{A}{P}{B}{Q}{Y}
+\pstInterLL[PosAngle=-90]{A}{R}{B}{S}{M}
+\pstInterLL[PosAngle=190]{A}{S}{B}{R}{N}
+\psset{linestyle=dashed,linecolor=gray!40}
+\pstLine{A}{Q}\pstLine{B}{P}\pstLine{A}{P}\pstLine{B}{Q}
+\pstLine{A}{R}\pstLine{B}{S}\pstLine{A}{S}\pstLine{B}{R}
+\pstLine{Q}{X}\pstLine{Q}{Y}\pstLine{P}{X}\pstLine{P}{Y}
+\pstLine{R}{M}\pstLine{S}{M}\pstLine{T}{Y}\pstLine{T}{N}
+\pstLine[linestyle=dashed,linecolor=red!40]{X}{Y}
+\pstLine[linestyle=dashed,linecolor=red!40]{M}{N}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstEllipseTangentNode} is use to draw the tangent line of a point $T$
+out of the Standard Ellipse $E$. It draw the two tangent lines through the point $T$
+to the Standard Ellipse $E$ and get the node $A$ and $B$ on the Ellipse.
+
+\begin{BDef}
+\Lcs{pstEllipseTangentNode}\OptArgs\Largr{O}\Largr{$a,b$}\Largb{$T$}\Largb{$A$}\Largb{$B$}
+\end{BDef}
+
+We use the following theorem to find the tangent node of the given $T$:
+\begin{theorem}\label{EllipseTangentPointTheorem}
+Give point $T$ outside of the ellipse, we draw any other two chords $TPQ$ and $TRS$,
+let $PS$ and $QR$ intersect at $I$, $PR$ and $QS$ intersect at $X$, $XI$ and Ellipse intersect at $A$ and $B$,
+then $TA$ is the tangent line through $A$ and $TB$ is the tangent line through $B$.
+\end{theorem}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.0}\def\rb{-1.2}
+\pstGeonode[PosAngle=-50,PointNameSep=0.2](2,2){O}
+\pstEllipse[linecolor=red!40](O)(\ra,\rb)
+\pstGeonode[PosAngle=-50,PointNameSep=0.2](-1,-1){T}
+\pstEllipseTangentNode[PosAngle=120](O)(\ra,\rb){T}{A}{B}
+% Here are the auxiliary lines to explain Theorem 2.
+\pstEllipseRotNode[PointName=none,RotAngle=71](O)(\ra,\rb){P0}
+\pstEllipseRotNode[PointName=none,RotAngle=31](O)(\ra,\rb){R0}
+\pstEllipseLineInter[PosAngle=0](O)(\ra,\rb){T}{P0}{P}{Q}
+\pstEllipseLineInter[PosAngle=0](O)(\ra,\rb){T}{R0}{R}{S}
+\pstInterLL[PosAngle=0]{P}{S}{Q}{R}{I}
+\pstInterLL[PosAngle=0]{P}{R}{Q}{S}{X}
+\psset{linestyle=dashed,linecolor=gray!40}
+\pstLine{T}{P}\pstLine{P}{Q}\pstLine{T}{R}\pstLine{R}{S}
+\pstLine{P}{S}\pstLine{Q}{R}\pstLine{P}{R}\pstLine{Q}{S}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{General Ellipse}
+Now we will introduce some macros for the General Ellipse as same as the Standard Ellipse.
+The General Ellipse $E$ with coordinate translation and rotation is defined by center $O(x_0,y_0)$,
+the half of the major axis $max(abs(a),abs(b))$, the half of the minor axis $min(abs(a),abs(b))$,
+and the rotation angle $\theta$ of the major axis.
+
+The equation can be got from the parametric function of the ellipse equation (\ref{ParametricFunctionOfEllipse}),
+using the rotation transform formula:
+\begin{equation}\label{RotationTransformFormula}
+\left\{\begin{array}{l}
+x'=x\cos\theta-y\sin\theta\\
+y'=x\sin\theta+y\cos\theta
+\end{array}\right.
+\end{equation}
+then we have
+\begin{equation}
+\left\{\begin{array}{l}
+x'=(a\cos\alpha+x_0)\cos\theta-(b\sin\alpha+y_0)\sin\theta=a\cos\alpha\cos\theta-b\sin\alpha\sin\theta+x_0'\\
+y'=(a\cos\alpha+x_0)\sin\theta+(b\sin\alpha+y_0)\cos\theta=a\cos\alpha\sin\theta+b\sin\alpha\cos\theta+y_0'
+\end{array}\right.
+\end{equation}
+where the $x_0'$ and $y_0'$ are the coordinate of the given center $O$ after rotation.
+So we get the parametric function of the General Ellipse with coordinate translation and rotation as following:
+\begin{equation}\label{ParametricFunctionOfGeneralEllipse}
+\left\{\begin{array}{l}
+x=a\cos\alpha\cos\theta-b\sin\alpha\sin\theta+x_0\\
+y=a\cos\alpha\sin\theta+b\sin\alpha\cos\theta+y_0
+\end{array}\right.
+\end{equation}
+
+The Macro \Lcs{pstGeneralEllipse} is used to draw a General Ellipse with center $O$ from
+\Lkeyword{angleA} to \Lkeyword{angleB}, going counter clockwise.
+If \Lkeyword{angleA} and \Lkeyword{angleB} are not specified,
+the macro will draw the whole ellipse.
+If you not input rotation angle $\theta$, the default value is $0^\circ$,
+at this time, the result of this macro is same as \Lcs{pstEllipse}.
+That is, \Lcs{pstGeneralEllipse} is more complex than \Lcs{pstEllipse}!
+
+\begin{BDef}
+\Lcs{pstGeneralEllipse}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\OptArg{angleA}\OptArg{angleB}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.4}\def\rb{-1.5}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+\pstGeneralEllipse[linecolor=red!40](O)(\ra,\rb)[0]
+\pstGeneralEllipse[linecolor=gray!10](O)(\ra,\rb)[10]
+\pstGeneralEllipse[linecolor=gray!20](O)(\ra,\rb)[20]
+\pstGeneralEllipse[linecolor=gray!30](O)(\ra,\rb)[30]
+\pstGeneralEllipse[linecolor=gray!40](O)(\ra,\rb)[40]
+\pstGeneralEllipse[linecolor=magenta!40](O)(\ra,\rb)[50]
+\end{pspicture}
+\end{LTXexample}
+
+Similarly, we can location the points on the General Ellipse using the macros
+\Lcs{pstGeneralEllipseNode}, \Lcs{pstGeneralEllipseRotNode}, \Lcs{pstGeneralEllipseAbsNode}
+and \Lcs{pstGeneralEllipseOrdNode} as following.
+
+\begin{BDef}
+\Lcs{pstGeneralEllipseNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstGeneralEllipseRotNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{A}\\
+\Lcs{pstGeneralEllipseAbsNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstGeneralEllipseOrdNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+Some examples all together:
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.4}\def\rb{-1.5}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+\pstGeneralEllipse[linecolor=magenta!40](O)(\ra,\rb)[50]
+\pstGeneralEllipseNode[PosAngle=30](O)(\ra,\rb)[50]{30}{A}
+\pstGeneralEllipseRotNode[PosAngle=120,RotAngle=120](O)(\ra,\rb)[50]{B}
+\pstGeneralEllipseRotNode[PosAngle=0,RotAngle=0](O)(\ra,\rb)[50]{C}
+\pstGeneralEllipseRotNode[PosAngle=0,RotAngle=90](O)(\ra,\rb)[50]{D}
+\pstGeneralEllipseRotNode[PosAngle=-90,RotAngle=180](O)(\ra,\rb)[50]{E}
+\pstGeneralEllipseRotNode[PosAngle=90,RotAngle=-90](O)(\ra,\rb)[50]{F}
+\pstGeneralEllipseAbsNode[PosAngle={60,240}](O)(\ra,\rb)[50]{2}{I}{J}
+\pstGeneralEllipseOrdNode[PosAngle={-40,210}](O)(\ra,\rb)[50]{1}{M}{N}
+\pstLineAB[nodesep=-1,linecolor=blue!40]{C}{E}
+\pstLineAB[nodesep=-1,linecolor=blue!40]{D}{F}
+\end{pspicture}
+\end{LTXexample}
+
+Using macro \Lcs{pstGeneralEllipseFocusNode} to find the two focus nodes, and macro \\
+\Lcs{pstGeneralEllipseDirectrixLine} to get the two directrix lines.
+
+\begin{BDef}
+\Lcs{pstGeneralEllipseFocusNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstGeneralEllipseDirectrixLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{A}
+\end{BDef}
+
+for example,
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{2.4}\def\rb{-1.5}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+\pstGeneralEllipse[linecolor=magenta!40](O)(\ra,\rb)[50]
+\pstGeneralEllipseFocusNode[PosAngle={-40,-40}](O)(\ra,\rb)[50]{L}{R}
+\pstGeneralEllipseDirectrixLine[PointName={L_x,L_y,R_x,R_y},nodesep=-1,linecolor=magenta](O)(\ra,\rb)[50]{Lx}{Ly}{Rx}{Ry}
+\pstLine[nodesep=-1,linecolor=red!40]{L}{R}
+\pstLine[nodesep=-1,linecolor=red!40,linestyle=dashed]{Lx}{Rx}
+\pstLine[nodesep=-1,linecolor=red!40,linestyle=dashed]{Ly}{Ry}
+\end{pspicture}
+\end{LTXexample}
+
+Using \Lcs{pstGeneralEllipseLineInter} to get the two intersections $C$ and $D$ of the General Ellipse $E$ and the given line $AB$!
+
+\begin{BDef}
+\Lcs{pstGeneralEllipseLineInter}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{A}\Largb{B}\Largb{C}\Largb{D}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{1.5}\def\rb{-2.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+\pstGeneralEllipse[linecolor=blue!40](O)(\ra,\rb)[50]
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{0,1}{1.5,4}
+\pstGeneralEllipseLineInter[PosAngle={-90,90}](O)(\ra,\rb)[50]{0,1}{1.5,4}{A}{B}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{0,3}{3,3}
+\pstGeneralEllipseLineInter[PosAngle={-90,240}](O)(\ra,\rb)[50]{0,3}{3,3}{C}{D}
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{1,0}{1,4}
+\pstGeneralEllipseLineInter[PosAngle={30,10}](O)(\ra,\rb)[50]{1,1}{1,4}{E}{F}
+\end{pspicture}
+\end{LTXexample}
+
+Using \Lcs{pstGeneralEllipsePolarNode} to find the polar point $T$ of chord $AB$,
+please refer to Theorem \ref{EllipsePolarPointTheorem}.
+
+\begin{BDef}
+\Lcs{pstGeneralEllipsePolarNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{A}\Largb{B}\Largb{T}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{1.5}\def\rb{-2.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+\pstGeneralEllipse[linecolor=blue!40](O)(\ra,\rb)[50]
+\pstLine[nodesep=-0.5,linecolor=black!40,linestyle=dashed]{0,1}{1.5,4}
+\pstGeneralEllipseLineInter[PosAngle={-90,90}](O)(\ra,\rb)[50]{0,1}{1.5,4}{A}{B}
+\pstGeneralEllipsePolarNode[PosAngle=90](O)(\ra,\rb)[50]{A}{B}{T}
+\end{pspicture}
+\end{LTXexample}
+
+Using \Lcs{pstGeneralEllipseTangentNode} to find the tangent point $A$ and $B$ of outside point $T$,
+please refer to Theorem \ref{EllipseTangentPointTheorem}.
+
+\begin{BDef}
+\Lcs{pstGeneralEllipseTangentNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{T}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,0)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\ra{1.5}\def\rb{-2.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,2){O}
+\pstGeneralEllipse[linecolor=blue!40](O)(\ra,\rb)[50]
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](-1,-1){P}
+\pstGeneralEllipseTangentNode[PosAngle=90](O)(\ra,\rb)[50]{P}{X}{Y}
+\end{pspicture}
+\end{LTXexample}
+
+\clearpage
+
+\subsection{Standard Parabola}
+The Standard Parabola $P$ with coordinate translation is defined by vertex $O(x_0,y_0)$,
+the half of the focus chord axis $abs(p)$.
+Note that the sign of $p$ indicates the direction of the parabola.
+
+The equation can be written as:
+\begin{equation}\label{FunctionOfStandardParabola}
+(x-x_0)^2=2p(y-y_0)
+\end{equation}
+and the parametric function can be written as:
+\begin{equation}\label{ParametricFunctionOfStandardParabola}
+\left\{\begin{array}{l}
+x=t+x_0\\
+y=\dfrac{t^2}{2p}+y_0
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstParabola} is used to draw a Parabola from $x_1$ to $x_2$ with Vertex $O$,
+the half of the focus chord axis $abs(p)$.
+
+\begin{BDef}
+ \Lcs{pstParabola}\OptArgs\Largr{O}\Largb{$p$}\Largb{$x_1$}\Largb{$x_2$}
+\end{BDef}
+
+The macro \Lcs{pstParabolaNode} is used to draw a node whose parameter is the given value $t$ on parabola,
+please refer to equation (\ref{ParametricFunctionOfStandardParabola}).
+The macro \Lcs{pstParabolaAbsNode} is used to draw a node whose abscissa is the given value $x_1$ on parabola.
+The macro \Lcs{pstParabolaOrdNode} is used to draw a node whose ordinate is the given value $y_1$ on parabola.
+Note that \Lcs{pstParabolaOrdNode} will create two nodes $A$ and $B$ at most time.
+
+\begin{BDef}
+\Lcs{pstParabolaNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstParabolaAbsNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{$x_1$}\Largb{A}\\
+\Lcs{pstParabolaOrdNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstParabolaNode[PosAngle=-90](O){\p}{1.5}{A}
+\pstParabolaAbsNode[PosAngle=-90,PointName=X_1](O){\p}{1.5}{X1}
+\pstParabolaOrdNode[PosAngle=40,PointName={Y_1,Y_2}](O){\p}{1.5}{Y1}{Y2}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstParabolaFocusNode} is used to find the focus of the parabola,
+and the macro \Lcs{pstParabolaDirectrixLine} is used to find the directrix line of the parabola.
+
+\begin{BDef}
+\Lcs{pstParabolaFocusNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{F}\\
+\Lcs{pstParabolaDirectrixLine}\OptArgs\Largr{O}\Largb{$p$}\Largb{$L_x$}\Largb{$L_y$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstParabolaFocusNode[linecolor=red!40,PosAngle=50](O){\p}{F}
+\pstParabolaDirectrixLine[linecolor=red!40,nodesepA=-1.8,nodesepB=-1,PosAngle={-50,-50}](O){\p}{A}{B}
+\pstLine[linecolor=red!40,nodesepA=-0.8,nodesepB=-2.5]{A}{F}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstParabolaLineInter} is used to find the intersections $C$ and $D$ of the parabola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstParabolaLineInter}\OptArgs\Largr{O}\Largb{$p$}\Largb{A}\Largb{B}\Largb{C}\Largb{D}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,0){O}
+\pstParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstLine[linecolor=gray!40,nodesepA=-0.8,nodesepB=-0.8]{0,2}{4,1}
+\pstParabolaLineInter[linecolor=gray!40,PosAngle={120,210}](O){\p}{0,2}{4,1}{P}{Q}
+\pstLine[linecolor=purple!40,nodesepA=-0.8,nodesepB=-0.8]{2.5,0}{2.5,3}
+\pstParabolaLineInter[linecolor=purple!40,PosAngle={0,210}](O){\p}{2.5,0}{2.5,3}{U}{V}
+\pstLine[linecolor=green!40,nodesepA=-2.5,nodesepB=-1.6]{1.5,2.5}{0.5,2.5}
+\pstParabolaLineInter[linecolor=green!40,PosAngle={210,210}](O){\p}{1.5,2.5}{0.5,2.5}{M}{N}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstParabolaPolarNode} is used to find the polar point $T$ of chord $AB$ on Parabola $P$.
+
+\begin{BDef}
+\Lcs{pstParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\Largr{F}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\Largr{F}\OptArg{$L_x$}\OptArg{$L_y$}\Largb{A}\Largb{B}\Largb{T}
+\end{BDef}
+
+We use the following theorem to find the polar point $T$ of chord $AB$:
+\begin{theorem}\label{ParabolaPolarPointTheorem}
+Give any chord $AB$ on parabola, drawing two focal chord $AFC$ and $BFD$, where $F$ is the focus of parabola,
+then drawing $FX$ which is perpendicular to $AFC$ at point $F$, and intersect with the directrix line at $X$;
+also drawing $FY$ which is perpendicular to $BFD$ at point $F$, and intersect with the directrix line at $Y$.
+Then the intersection $T$ of $AX$ and $BY$ is the polar point of chord $AB$.
+\end{theorem}
+
+If you don't know the focus $F$, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstLine[linecolor=gray!40,nodesepA=-0.8,nodesepB=-0.8]{0,2}{4,1}
+\pstParabolaLineInter[linecolor=gray!40,PosAngle={120,210}](O){\p}{0,2}{4,1}{P}{Q}
+% if you don't know focus F or directrix line
+\pstParabolaPolarNode[linecolor=purple!40,PosAngle=-90](O){\p}{P}{Q}{T}
+\end{pspicture}
+\end{LTXexample}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstParabolaFocusNode[linecolor=red!40](O){\p}{F}
+\pstLine[linecolor=gray!40,nodesepA=-0.8,nodesepB=-0.8]{0,2}{4,1}
+\pstParabolaLineInter[linecolor=gray!40,PosAngle={120,210}](O){\p}{0,2}{4,1}{P}{Q}
+% if you know focus F, but don't known directrix line
+\pstParabolaPolarNode[linecolor=purple!40,PosAngle=-90](O){\p}(F){P}{Q}{T}
+\end{pspicture}
+\end{LTXexample}
+
+\vspace{1cm}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstParabolaFocusNode[linecolor=red!40](O){\p}{F}
+\pstParabolaDirectrixLine[linecolor=red!40,nodesepA=-2.8,nodesepB=-2,PosAngle={-50,-50}](O){\p}{A}{B}
+\pstLineAB[linecolor=red!40,nodesepA=-0.8,nodesepB=-2.5]{A}{F}
+\pstLine[linecolor=gray!40,nodesepA=-0.8,nodesepB=-0.8]{0,2}{4,1}
+\pstParabolaLineInter[linecolor=gray!40,PosAngle={120,210}](O){\p}{0,2}{4,1}{P}{Q}
+% if you know focus F and also directrix line
+\pstParabolaPolarNode[linecolor=purple!40,PosAngle=-90](O){\p}(F)[A][B]{P}{Q}{T}
+\end{pspicture}
+\end{LTXexample}
+
+\vspace{10pt}
+
+The macro \Lcs{pstParabolaTangentNode} is used to find the two nodes $A$ and $B$ on the Parabola through the point $T$.
+
+\begin{BDef}
+\Lcs{pstParabolaTangentNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{T}\Largb{A}\Largb{B}
+\end{BDef}
+
+We use the following theorem to find the tangent node $A$ and $B$ of outside point $T$:
+\begin{theorem}\label{ParabolaTangentPointTheorem}
+Give point $T$ outside of the parabola, we draw any other two chords $TPQ$ and $TRS$,
+$PS$ and $QR$ intersect at $I$, $PR$ and $QS$ intersect at $X$, $XI$ and Parabola intersect at $A$ and $B$,
+then $TA$ is the tangent line through $A$ and $TB$ is the tangent line through $B$.
+\end{theorem}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstGeonode[PosAngle=-90](1.5,-1){T}
+\pstParabolaTangentNode[linecolor=red!50,PosAngle={80,140},PointName={T_1,T_2}](O){\p}{T}{T1}{T2}
+\pstGeonode[PosAngle=-90](2,-1){P}
+\pstParabolaTangentNode[linecolor=red!50,PosAngle={80,140},PointName={P_1,P_2}](O){\p}{P}{P1}{P2}
+\pstGeonode[PosAngle=-90](2.3,-1){X}
+\pstParabolaTangentNode[linecolor=red!50,PosAngle={80,140},PointName={X_1,X_2}](O){\p}{X}{X1}{X2}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{Standard Inversion Parabola}
+The Inversion Parabola $P$ with coordinate translation is defined by vertex $O(x_0,y_0)$,
+the half of the focus chord axis $abs(p)$.
+Note that the sign of $p$ indicates the direction of the parabola.
+The equation can be written as:
+\begin{equation}\label{StandardInversionParabola}
+(y-y_0)^2=2p(x-x_0)
+\end{equation}
+and the parametric function can be written as:
+\begin{equation}\label{ParametricFunctionOfStandardInversionParabola}
+\left\{\begin{array}{l}
+x=\dfrac{t^2}{2p}+x_0\\
+y=t+y_0
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstIParabola} is used to draw a Standard Inversion Parabola from $y_1$ to $y_2$ with Vertex $O$,
+the half of the focus chord axis $abs(p)$.
+
+\begin{BDef}
+\Lcs{pstIParabola}\OptArgs\Largr{O}\Largb{$p$}\Largb{$y_1$}\Largb{$y_2$}
+\end{BDef}
+
+The macro \Lcs{pstIParabolaNode} is used to draw a node whose parameter is the given value $t$ on parabola,
+please refer to equation (\ref{ParametricFunctionOfStandardInversionParabola}).
+The macro \Lcs{pstIParabolaAbsNode} is used to draw a node whose abscissa is the given value $x_1$ on parabola.
+The macro \Lcs{pstIParabolaOrdNode} is used to draw a node whose ordinate is the given value $y_1$ on parabola.
+Note that \Lcs{pstIParabolaAbsNode} will create two nodes $A$ and $B$ at most time.
+
+\begin{BDef}
+\Lcs{pstIParabolaNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstIParabolaAbsNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstIParabolaOrdNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{$y_1$}\Largb{A}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(3,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=0,PointNameSep=0.2](2,0){O}
+\pstIParabola[linecolor=blue!40](O){-\p}{-1.5}{1.5}
+\pstIParabolaNode[PosAngle=90](O){-\p}{1}{A}
+\pstIParabolaAbsNode[PosAngle=90,PointName={X_2,X_3},PosAngle={-90,90}](O){-\p}{1.5}{X2}{X3}
+\pstIParabolaOrdNode[PosAngle=-90,PointName=Y_3](O){-\p}{-1}{Y3}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstIParabolaFocusNode} is used to find the focus of the parabola,
+and the macro \Lcs{pstIParabolaDirectrixLine} is used to find the directrix line of the parabola.
+
+\begin{BDef}
+\Lcs{pstIParabolaFocusNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{F}\\
+\Lcs{pstIParabolaDirectrixLine}\OptArgs\Largr{O}\Largb{$p$}\Largb{$L_x$}\Largb{$L_y$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(3,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-30,PointNameSep=0.2](2,0){O}
+\pstIParabola[linecolor=blue!40](O){-\p}{-1.5}{1.5}
+\pstIParabolaFocusNode[linecolor=blue!40,PosAngle=120](O){-\p}{F}
+\pstIParabolaDirectrixLine[linecolor=blue!40,nodesepA=-2,nodesepB=-1,PosAngle={50,20}](O){-\p}{C}{D}
+\pstLine[linecolor=blue!40,nodesepA=-0.8,nodesepB=-2.5]{C}{F}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstIParabolaLineInter} is used to find the intersections $C$ and $D$ of the parabola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstIParabolaLineInter}\OptArgs\Largr{O}\Largb{$p$}\Largb{A}\Largb{B}\Largb{C}\Largb{D}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(3,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=0,PointNameSep=0.2](2,0){O}
+\pstIParabola[linecolor=blue!40](O){-\p}{-1.5}{1.5}
+\pstLine[linecolor=gray!40]{0,2}{1,-2}
+\pstIParabolaLineInter[linecolor=gray!40,PosAngle={70,-90}](O){-\p}{1,-2}{0,2}{P}{Q}
+\pstLine[linecolor=purple!40]{1.2,-1.5}{1.2,1.5}
+\pstIParabolaLineInter[linecolor=purple!40,PosAngle={-40,210}](O){-\p}{1.2,-1.5}{1.2,1.5}{U}{V}
+\pstLine[linecolor=green!40]{-1,0.5}{2.5,0.5}
+\pstIParabolaLineInter[linecolor=green!40,PosAngle={70,-90}](O){-\p}{-1,0.5}{2.5,0.5}{M}{N}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstIParabolaPolarNode} is used to find the polar point $T$ of chord $AB$ on Parabola $P$.
+
+\begin{BDef}
+\Lcs{pstIParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstIParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\Largr{F}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstIParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\Largr{F}\OptArg{$L_x$}\OptArg{$L_y$}\Largb{A}\Largb{B}\Largb{T}
+\end{BDef}
+
+We also use the theorem \ref{ParabolaPolarPointTheorem} to find the polar point $T$ of chord $AB$.
+If you don't know the focus $F$, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,-2)(5,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstIParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstLine[linecolor=gray!40,nodesepA=-0.5]{2,1}{4,-2}
+\pstIParabolaLineInter[PosAngle={80,-100}](O){\p}{2,1}{4,-2}{P}{Q}
+% if you don't know focus F or directrix line
+\pstIParabolaPolarNode[linecolor=purple!40,PosAngle=-90](O){\p}{P}{Q}{T}
+\end{pspicture}
+\end{LTXexample}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,-2)(5,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstIParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstIParabolaFocusNode[linecolor=red!40](O){\p}{F}
+\pstLine[linecolor=gray!40,nodesepA=-0.5]{2,1}{4,-2}
+\pstIParabolaLineInter[PosAngle={80,-100}](O){\p}{2,1}{4,-2}{P}{Q}
+% if you know focus F, but don't known directrix line
+\pstIParabolaPolarNode[linecolor=purple!40,PosAngle=-90](O){\p}(F){P}{Q}{T}
+\end{pspicture}
+\end{LTXexample}
+
+\vspace{1cm}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,-2)(5,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-130,PointNameSep=0.2](2,0){O}
+\pstIParabola[linecolor=red!40](O){\p}{-1.5}{1.5}
+\pstIParabolaFocusNode[linecolor=red!40](O){\p}{F}
+\pstIParabolaDirectrixLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PosAngle={180,180}](O){\p}{A}{B}
+\pstLine[linecolor=gray!40,nodesepA=-0.5]{2,1}{4,-2}
+\pstIParabolaLineInter[PosAngle={80,-100}](O){\p}{2,1}{4,-2}{P}{Q}
+% if you know focus F and also directrix line
+\pstIParabolaPolarNode[linecolor=purple!40,PosAngle=-90](O){\p}(F)[A][B]{P}{Q}{T}
+\end{pspicture}
+\end{LTXexample}
+
+\vspace{10pt}
+
+The macro \Lcs{pstIParabolaTangentNode} is used to find the two nodes $A$ and $B$ on the Parabola through the point $T$.
+
+\begin{BDef}
+\Lcs{pstIParabolaTangentNode}\OptArgs\Largr{O}\Largb{$p$}\Largb{T}\Largb{A}\Largb{B}
+\end{BDef}
+
+We also use the theorem \ref{ParabolaTangentPointTheorem} to find the tangent node $A$ and $B$ of outside point $T$!
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,0)(5,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-45,PointNameSep=0.2](4,2){O}
+\pstIParabola[linecolor=blue!40](O){-\p}{-1.5}{1.5}
+\pstGeonode[PosAngle=0](5,1.5){T}
+\pstIParabolaTangentNode[linecolor=red!50,PosAngle={80,-100},PointName={T_1,T_2}](O){-\p}{T}{T1}{T2}
+\pstGeonode[PosAngle=0](5,2.5){P}
+\pstIParabolaTangentNode[linecolor=red!50,PosAngle={80,90},PointName={P_1,P_2}](O){-\p}{P}{P1}{P2}
+\pstGeonode[PosAngle=0](5,2){X}
+\pstIParabolaTangentNode[linecolor=red!50,PosAngle={80,-100},PointName={X_1,X_2}](O){-\p}{X}{X1}{X2}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{General Parabola}
+The General Parabola $P$ with coordinate translation and rotation is defined by vertex $O(x_0,y_0)$,
+the half of the focus chord axis $abs(p)$, the sign of $p$ indicates the direction of the parabola,
+and the rotation angle $\theta$ of the symmetrical axis.
+
+The equation can be got from the parametric function of the parabola equation (\ref{ParametricFunctionOfStandardParabola}),
+using the rotation transform formula (\ref{RotationTransformFormula}), then we have
+\begin{equation}
+\left\{\begin{array}{l}
+x'=(t+x_0)\cos\theta-(\dfrac{t^2}{2p}+y_0)\sin\theta=x_0'+t\cos\theta-t^2\dfrac{\sin\theta}{2p}\\
+y'=(t+x_0)\sin\theta+(\dfrac{t^2}{2p}+y_0)\cos\theta=y_0'+t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+\end{array}\right.
+\end{equation}
+where the $x_0'$ and $y_0'$ are the coordinate of the given vertex O after rotation.
+So we get the parametric function of the General Parabola with coordinate translation and rotation as following:
+\begin{equation}\label{ParametricFunctionOfGeneralParabola}
+\left\{\begin{array}{l}
+x=x_0+t\cos\theta-t^2\dfrac{\sin\theta}{2p}\\
+y=y_0+t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstGeneralParabola} is used to draw a General Parabola from $x_1$ to $x_2$ with Vertex $O$,
+the half of the focus chord axis $abs(p)$.
+
+\begin{BDef}
+\Lcs{pstGeneralParabola}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$x_1$}\Largb{$x_2$}
+\end{BDef}
+
+The macro \Lcs{pstGeneralParabolaNode} is used to draw a node whose parameter is the given value $t$ on parabola,
+please refer to equation (\ref{ParametricFunctionOfGeneralParabola}).
+The macro \Lcs{pstGeneralParabolaAbsNode} is used to draw a node whose abscissa is the given value $x_1$ on parabola.
+The macro \Lcs{pstGeneralParabolaOrdNode} is used to draw a node whose ordinate is the given value $y_1$ on parabola.
+
+Note that \Lcs{pstGeneralParabolaAbsNode} and \Lcs{pstGeneralParabolaOrdNode} both create two nodes $A$ and $B$
+at most time.
+
+\begin{BDef}
+\Lcs{pstGeneralParabolaNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstGeneralParabolaAbsNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstGeneralParabolaOrdNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-1)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-40,PointNameSep=0.2](2,0){O}
+\pstGeneralParabola[linecolor=red!10](O){\p}[0]{-1.5}{1.5}
+\pstGeneralParabola[linecolor=red!15](O){\p}[10]{-1.5}{1.5}
+\pstGeneralParabola[linecolor=red!25](O){\p}[30]{-1.5}{1.5}
+\pstGeneralParabola[linecolor=red!40](O){\p}[50]{-1.5}{1.5}
+\pstGeneralParabola[linecolor=red!60](O){\p}[90]{-1.5}{1.5}
+\pstGeneralParabolaNode[PosAngle=0,linecolor=blue!60](O){\p}[30]{1.0}{A}
+\pstGeneralParabolaAbsNode[PosAngle={0,0},linecolor=blue!60](O){\p}[30]{1.0}{D}{E}
+\pstGeneralParabolaAbsNode[PosAngle={0,0},linecolor=blue!60](O){\p}[50]{1.0}{F}{G}
+\pstGeneralParabolaAbsNode[PosAngle={0,0},linecolor=blue!60](O){\p}[90]{1.0}{H}{I}
+\pstGeneralParabolaOrdNode[PosAngle={90,0},linecolor=purple!60](O){\p}[30]{0.5}{U}{V}
+\pstGeneralParabolaOrdNode[PosAngle={90,90},linecolor=purple!60](O){\p}[50]{0.5}{M}{N}
+\pstGeneralParabolaOrdNode[PosAngle={90,-90},linecolor=purple!60](O){\p}[90]{0.5}{S}{T}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralParabolaFocusNode} is used to find the focus of the parabola,
+and the macro \Lcs{pstGeneralParabolaDirectrixLine} is used to find the directrix line of the parabola.
+
+\begin{BDef}
+\Lcs{pstGeneralParabolaFocusNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{F}\\
+\Lcs{pstGeneralParabolaDirectrixLine}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$L_x$}\Largb{$L_y$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,0){O}
+\pstGeneralParabola[linecolor=red!40](O){\p}[50]{-1.5}{1.5}
+\pstGeneralParabolaFocusNode[linecolor=red!40,PosAngle=90](O){\p}[50]{F}
+\pstLineAB[linestyle=dashed,linecolor=black!25,nodesepA=-0.5,nodesepB=-2.5]{O}{F}
+\pstGeneralParabolaDirectrixLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PosAngle={-60,-60},PointName={L_1,L_2}](O){\p}[50]{L1}{L2}
+\pstGeneralParabolaNode[linecolor=red!60](O){\p}[50]{1.0}{A}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralParabolaLineInter} is used to find the intersections $C$ and $D$ of the parabola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstGeneralParabolaLineInter}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{A}\Largb{B}\Largb{C}\Largb{D}
+\end{BDef}
+
+When General Parabola becomes a Standard Parabola, the intersections with any kind of lines:
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,0){O}
+\pstGeneralParabola[linecolor=red!40](O){\p}[0]{-1.5}{1.5}
+\pstGeneralParabolaFocusNode[linecolor=red!40,PosAngle=50](O){\p}[0]{F}
+\pstLineAB[linestyle=dashed,linecolor=black!25,nodesepA=-0.2,nodesepB=-2.5]{O}{F}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.8]{1,0}{1,2}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={40,-90}](O){\p}[0]{1,0}{1,2}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=0]{0.5,0.5}{3.5,1}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={-110,-60}](O){\p}[0]{0.5,0.5}{3.5,1}{C}{D}
+\end{pspicture}
+\end{LTXexample}
+
+Here is the intersections of a real General Parabola with any kind of lines:
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-1)(3,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-60,PointNameSep=0.2](2,0){O}
+\pstGeneralParabola[linecolor=red!40](O){\p}[50]{-1.5}{1.5}
+\pstGeneralParabolaFocusNode[linecolor=red!40,PosAngle=80](O){\p}[50]{F}
+\pstLineAB[linestyle=dashed,linecolor=black!25,nodesepA=-0.2,nodesepB=-2.5]{O}{F}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.8]{1,-1}{1,3}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={-150,40}](O){\p}[50]{1,-1}{1,3}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=0.0]{-1,0}{3,2}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={90,70}](O){\p}[50]{-1,0}{3,2}{C}{D}
+% a line with gradient k=-\cos50/\sin50 parallel to OF
+\pstLineAS[linestyle=dashed,linecolor=gray!40,nodesep=-0.8,PointName=none,PointSymbol=none](0,1){50 cos 50 sin div neg}{X}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={-90,-90}](O){\p}[50]{0,1}{X}{E}{G}
+\end{pspicture}
+\end{LTXexample}
+
+When General Parabola becomes a Standard Inversion Parabola, the intersections with any kind of lines:
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-2)(3,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=0,PointNameSep=0.2](2,0){O}
+\pstGeneralParabola[linecolor=red!40](O){\p}[90]{-1.5}{1.5}
+\pstGeneralParabolaFocusNode[linecolor=red!40,PosAngle=120](O){\p}[90]{F}
+\pstLineAB[linestyle=dashed,linecolor=black!25,nodesepA=-0.2,nodesepB=-2.5]{O}{F}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.8]{1,-1}{1,2}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={-60,60}](O){\p}[90]{1,-1}{1,2}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.8]{0,-1}{2,1}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={-90,5}](O){\p}[90]{0,-1}{2,1}{C}{D}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralParabolaPolarNode} is used to find the polar point $T$ of chord $AB$ on Parabola $P$.
+
+\begin{BDef}
+\Lcs{pstGeneralParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstGeneralParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largr{F}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstGeneralParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largr{F}\OptArg{$L_x$}\OptArg{$L_y$}\Largb{A}\Largb{B}\Largb{T}
+\end{BDef}
+
+We also use the theorem \ref{ParabolaPolarPointTheorem} to find the polar point $T$ of chord $AB$.
+If you don't know the focus $F$, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-2)(3,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-60,PointNameSep=0.2](2,0){O}
+\pstGeneralParabola[linecolor=red!40](O){\p}[80]{-1.5}{1.5}
+\pstGeneralParabolaFocusNode[linecolor=red!40,PosAngle=200](O){\p}[80]{F}
+\pstGeneralParabolaDirectrixLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PosAngle={0,0},PointName={L_x,L_y}](O){\p}[80]{Lx}{Ly}
+\pstLine[linestyle=dashed,linecolor=black!25,nodesepA=-0.2,nodesepB=-2.5]{O}{F}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.4]{0.5,-1.2}{2,1}
+\pstGeneralParabolaLineInter[linecolor=red!40,PosAngle={-60,90}](O){\p}[80]{0.5,-1.2}{2,1}{A}{B}
+%\pstGeneralParabolaPolarNode[linecolor=red!40,PosAngle=-90](O){\p}[80]{A}{B}{T}
+%\pstGeneralParabolaPolarNode[linecolor=red!40,PosAngle=-90](O){\p}[80](F){A}{B}{T}
+\pstGeneralParabolaPolarNode[linecolor=red!40,PosAngle=-90](O){\p}[80](F)[Lx][Ly]{A}{B}{T}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralParabolaTangentNode} is used to find the two nodes $A$ and $B$ on the Parabola through the point $T$.
+
+\begin{BDef}
+\Lcs{pstGeneralParabolaTangentNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{T}\Largb{A}\Largb{B}
+\end{BDef}
+
+We also use the theorem \ref{ParabolaTangentPointTheorem} to find the tangent node $A$ and $B$ of outside point $T$.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-2)(3,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=0,PointNameSep=0.2](2,0){O}
+\pstGeneralParabola[linecolor=red!40](O){\p}[80]{-1.5}{1.5}
+\pstGeonode[PosAngle=0](2.5,-0.5){R}(2.5,-0.2){T}(2.5,0.6){S}
+\pstGeneralParabolaTangentNode[linecolor=red!40,PosAngle={-90,220},PointName={R_1,R_2}](O){\p}[80]{R}{R1}{R2}
+\pstGeneralParabolaTangentNode[linecolor=red!40,PosAngle={-90,170},PointName={T_1,T_2}](O){\p}[80]{T}{T1}{T2}
+\pstGeneralParabolaTangentNode[linecolor=red!40,PosAngle={-90,180},PointName={S_1,S_2}](O){\p}[80]{S}{S1}{S2}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{General Inversion Parabola}
+The General Inversion Parabola $P$ with coordinate translation and rotation is defined by vertex $O$,
+the half of the focus chord axis $abs(p)$, the sign of $p$ indicates the direction of the parabola,
+and the rotation angle $\theta$ of the symmetrical axis.
+
+The equation can be got from the parametric function of the inversion parabola (\ref{ParametricFunctionOfStandardInversionParabola}),
+using the rotation transform formula (\ref{RotationTransformFormula}), then we have
+\begin{equation}
+\left\{\begin{array}{l}
+x'=(\dfrac{t^2}{2p}+x_0)\cos\theta-(t+y_0)\sin\theta=x_0'-t\sin\theta+t^2\dfrac{\cos\theta}{2p}\\
+y'=(\dfrac{t^2}{2p}+x_0)\sin\theta+(t+y_0)\cos\theta=y_0'+t\cos\theta+t^2\dfrac{\sin\theta}{2p}
+\end{array}\right.
+\end{equation}
+where the $x_0'$ and $y_0'$ are the coordinate of the given vertex O after rotation.
+So we get the parametric function of the General Inversion Parabola with coordinate translation and rotation as following:
+\begin{equation}\label{ParametricFunctionOfGeneralInversionParabola}
+\left\{\begin{array}{l}
+x=x_0-t\sin\theta+t^2\dfrac{\cos\theta}{2p}\\
+y=y_0+t\cos\theta+t^2\dfrac{\sin\theta}{2p}
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstGeneralIParabola} is used to draw a Standard Inversion Parabola from $y_1$ to $y_2$ with Vertex $O$,
+the half of the focus chord axis $abs(p)$.
+
+\begin{BDef}
+\Lcs{pstGeneralIParabola}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$y_1$}\Largb{$y_2$}
+\end{BDef}
+
+The macro \Lcs{pstGeneralIParabolaNode} is used to draw a node whose parameter is the given value $t$ on parabola,
+please refer to equation (\ref{ParametricFunctionOfGeneralInversionParabola}).
+The macro \Lcs{pstGeneralIParabolaAbsNode} is used to draw a node whose abscissa is the given value $x_1$ on parabola.
+The macro \Lcs{pstGeneralIParabolaOrdNode} is used to draw a node whose ordinate is the given value $y_1$ on parabola.
+
+Note that \Lcs{pstGeneralIParabolaAbsNode} and \Lcs{pstGeneralIParabolaOrdNode} will create two nodes $A$ and $B$ at most time.
+
+\begin{BDef}
+\Lcs{pstGeneralIParabolaNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstGeneralIParabolaAbsNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstGeneralIParabolaOrdNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,0)(3,5)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=210,PointNameSep=0.2](0,2){O}
+\pstGeneralIParabola[linecolor=blue!10](O){\p}[0]{-1.5}{1.5}
+\pstGeneralIParabola[linecolor=blue!15](O){\p}[10]{-1.5}{1.5}
+\pstGeneralIParabola[linecolor=blue!25](O){\p}[30]{-1.5}{1.5}
+\pstGeneralIParabola[linecolor=blue!30](O){\p}[40]{-1.5}{1.5}
+\pstGeneralIParabola[linecolor=blue!40](O){\p}[50]{-1.5}{1.5}
+\pstGeneralIParabola[linecolor=blue!60](O){\p}[90]{-1.5}{1.5}
+\pstGeneralIParabolaNode[linecolor=red!60,PosAngle=90](O){\p}[30]{1.0}{A}
+\pstGeneralIParabolaNode[linecolor=red!60,PosAngle=170](O){\p}[50]{1.0}{B}
+\pstGeneralIParabolaAbsNode[linecolor=red!40,PosAngle={-45,90}](O){\p}[50]{1.0}{C}{D}
+\pstGeneralIParabolaAbsNode[linecolor=red!60,PosAngle={0,-90}](O){\p}[90]{1.0}{E}{F}
+\pstGeneralIParabolaOrdNode[linecolor=red!60,PosAngle={90,150}](O){\p}[50]{2.5}{G}{H}
+\pstGeneralIParabolaOrdNode[linecolor=blue!60,PosAngle={180,-90}](O){\p}[90]{2.5}{J}{K}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralIParabolaFocusNode} is used to find the focus of the parabola,
+and the macro \Lcs{pstGeneralIParabolaDirectrixLine} is used to find the directrix line of the parabola.
+
+\begin{BDef}
+\Lcs{pstGeneralIParabolaFocusNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{F}\\
+\Lcs{pstGeneralIParabolaDirectrixLine}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{$L_x$}\Largb{$L_y$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,0)(2,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\psset{PointName=none,nodesepA=-2,nodesepB=-1}
+\pstGeonode(0,2){O}\def\p{0.8}
+\psset{linecolor=blue!60}
+\pstGeneralIParabola(O){\p}[0]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[0]{A}
+\pstGeneralIParabolaDirectrixLine(O){\p}[0]{A1}{A2}
+\psset{linecolor=red!60}
+\pstGeneralIParabola(O){\p}[45]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[45]{B}
+\pstGeneralIParabolaDirectrixLine(O){\p}[45]{B1}{B2}
+\psset{linecolor=green!60}
+\pstGeneralIParabola(O){\p}[90]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[90]{C}
+\pstGeneralIParabolaDirectrixLine(O){\p}[90]{C1}{C2}
+\psset{linecolor=cyan!60}
+\pstGeneralIParabola(O){\p}[135]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[135]{D}
+\pstGeneralIParabolaDirectrixLine(O){\p}[135]{D1}{D2}
+\psset{linecolor=purple!60}
+\pstGeneralIParabola(O){\p}[180]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[180]{E}
+\pstGeneralIParabolaDirectrixLine(O){\p}[180]{E1}{E2}
+\psset{linecolor=yellow!60}
+\pstGeneralIParabola(O){\p}[225]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[225]{F}
+\pstGeneralIParabolaDirectrixLine(O){\p}[225]{F1}{F2}
+\psset{linecolor=black!60}
+\pstGeneralIParabola(O){\p}[270]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[270]{G}
+\pstGeneralIParabolaDirectrixLine(O){\p}[270]{G1}{G2}
+\psset{linecolor=brown!60}
+\pstGeneralIParabola(O){\p}[315]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode(O){\p}[315]{H}
+\pstGeneralIParabolaDirectrixLine(O){\p}[315]{H1}{H2}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralIParabolaLineInter} is used to find the intersections $C$ and $D$ of the parabola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstGeneralIParabolaLineInter}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{A}\Largb{B}\Largb{C}\Largb{D}
+\end{BDef}
+
+When $\theta=0$, the intersections with any kind of lines:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,-2)(5,2)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=180,PointNameSep=0.2](2,0){O}
+\pstGeneralIParabola[linecolor=red!40](O){\p}[0]{-1.5}{1.5}
+\pstLine[linestyle=dashed,linecolor=gray!40]{3,-2}{3,2}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={40,150}](O){\p}[0]{3,-2}{3,2}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40]{2,-2}{4,2}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={100,210}](O){\p}[0]{2,-2}{4,2}{C}{D}
+\pstLine[linestyle=dashed,linecolor=gray!40]{1.5,0.5}{4.5,0.5}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={120,-90}](O){\p}[0]{1.5,0.5}{4.5,0.5}{E}{F}
+\end{pspicture}
+\end{LTXexample}
+
+When $\theta=50$, the intersections with any kind of lines:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,-1)(5,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-70,PointNameSep=0.2](2,0){O}
+\pstGeneralIParabola[linecolor=red!40](O){\p}[50]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode[linecolor=red!40,PosAngle=80](O){\p}[50]{F}
+\pstLineAB[linestyle=dashed,linecolor=black!25,nodesepA=-0.2,nodesepB=-2.5]{O}{F}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.8]{3,-1}{3,3}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={-60,40}](O){\p}[50]{3,-1}{3,3}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=0.0]{2,3}{4,0}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={-10,170}](O){\p}[50]{2,3}{4,0}{C}{D}
+% a line with gradient k=\tan50 parallel to OF
+\pstLineAS[linestyle=dashed,linecolor=gray!40,nodesep=-0.8,PointName=none,PointSymbol=none](2,1){50 tan}{X}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={180,-90}](O){\p}[50]{2,1}{X}{E}{G}
+\end{pspicture}
+\end{LTXexample}
+
+When $\theta=90$, the intersections with any kind of lines:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](0,-1)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,0){O}
+\pstGeneralIParabola[linecolor=red!40](O){\p}[90]{-1.5}{1.5}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.5]{1,0}{1,2}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={180,-90}](O){\p}[90]{1,0}{1,2}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.5]{1,0}{3,1}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={-60,-90}](O){\p}[90]{1,0}{3,1}{C}{D}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-0.5]{0.8,2}{3,2}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={120,60}](O){\p}[90]{0.8,2}{3,2}{E}{G}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralIParabolaPolarNode} is used to find the polar point $T$ of chord $AB$ on Parabola $P$.
+
+\begin{BDef}
+\Lcs{pstGeneralIParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstGeneralIParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largr{F}\Largb{A}\Largb{B}\Largb{T}\\
+\Lcs{pstGeneralIParabolaPolarNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largr{F}\OptArg{$L_x$}\OptArg{$L_y$}\Largb{A}\Largb{B}\Largb{T}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,-1)(5,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=240,PointNameSep=0.4](2,0){O}
+\pstGeneralIParabola[linecolor=red!40](O){\p}[50]{-1.5}{1.5}
+\pstGeneralIParabolaFocusNode[linecolor=red!40,PosAngle=80](O){\p}[50]{F}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=0.0]{2,3}{4,0}
+\pstGeneralIParabolaLineInter[linecolor=red!40,PosAngle={-10,170}](O){\p}[50]{2,3}{4,0}{A}{B}
+\pstGeneralIParabolaPolarNode[linecolor=red!40,PosAngle=-90](O){\p}[50](F){A}{B}{T}
+\end{pspicture}
+\end{LTXexample}
+
+We also use the theorem \ref{ParabolaPolarPointTheorem} to find the polar point $T$ of chord $AB$.
+If you don't know the focus $F$, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+
+The macro \Lcs{pstGeneralIParabolaTangentNode} is used to find the two nodes $A$ and $B$ on the Parabola through the point $T$.
+
+\begin{BDef}
+\Lcs{pstGeneralIParabolaTangentNode}\OptArgs\Largr{O}\Largb{$p$}\OptArg{$\theta$}\Largb{T}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](1,-1)(5,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\p{0.4}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](2,0){O}
+\pstGeneralIParabola[linecolor=red!40](O){\p}[60]{-1.5}{1.5}
+\pstGeonode[PosAngle=-90](1,-1){R}(2,-1){T}(2.5,-1){S}
+\pstGeneralIParabolaTangentNode[linecolor=red!40,PosAngle={90,220},PointName={R_1,R_2}](O){\p}[60]{R}{R1}{R2}
+\pstGeneralIParabolaTangentNode[linecolor=red!40,PosAngle={160,60},PointName={T_1,T_2}](O){\p}[60]{T}{T1}{T2}
+\pstGeneralIParabolaTangentNode[linecolor=red!40,PosAngle={-60,40},PointName={S_1,S_2}](O){\p}[60]{S}{S1}{S2}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{Standard Hyperbola}
+The Standard Hyperbola $H$ with coordinate translation is defined by center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$.
+The equation can be written as:
+\begin{equation}\label{FunctionOfStandardHyperbola}
+\dfrac{(x-x_0)^2}{a^2}-\dfrac{(y-y_0)^2}{b^2}=1
+\end{equation}
+and the parametric function can be written as:
+\begin{equation}\label{ParametricFunctionOfStandardHyperbola}
+\left\{\begin{array}{l}
+x=a\sec\alpha+x_0\\
+y=b\tan\alpha+y_0
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstHyperbola} is used to draw a Standard Hyperbola with Center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$.
+The parameter \texttt{angleX} is used to truncate the width of the figure,
+it should be setup from 0 to 90.
+
+\begin{BDef}
+\Lcs{pstHyperbola}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{angleX}
+\end{BDef}
+
+The macro \Lcs{pstHyperbolaNode} is used to draw a node whose parameter is the given value $t$ on Hyperbola,
+please refer to equation (\ref{ParametricFunctionOfStandardHyperbola}).
+The macro \Lcs{pstHyperbolaAbsNode} is used to draw the nodes whose abscissa are the given value $x_1$ on Hyperbola.
+The macro \Lcs{pstHyperbolaOrdNode} is used to draw the nodes whose ordinate are the given value $y_1$ on Hyperbola.
+
+Note that \Lcs{pstHyperbolaAbsNode} and \Lcs{pstHyperbolaOrdNode} will create two nodes $A$ and $B$ at most time.
+
+\begin{BDef}
+\Lcs{pstHyperbolaNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstHyperbolaAbsNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstHyperbolaOrdNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](1,1){O}
+\pstHyperbola[linecolor=blue!40](O)(\a,\b)[80]
+\pstHyperbolaNode[linecolor=blue!40,PosAngle=90](O)(\a,\b){80}{A}
+\pstHyperbolaAbsNode[linecolor=blue!40,PointName={X_1,X_2},PosAngle=0](O)(\a,\b){0}{X1}{X2}
+\pstHyperbolaOrdNode[linecolor=blue!40,PointName={Y_1,Y_2},PosAngle=-90](O)(\a,\b){0}{Y1}{Y2}
+\pstHyperbola[linecolor=red!40](O)(\b,\a)[78]
+\pstHyperbolaNode[linecolor=red!40](O)(\b,\a){-75}{B}
+\pstHyperbolaAbsNode[linecolor=red!40,PointName={X_3,X_4},PosAngle=0](O)(\b,\a){0}{X3}{X4}
+\pstHyperbolaOrdNode[linecolor=red!40,PointName={Y_3,Y_4},PosAngle=-90](O)(\b,\a){0}{Y3}{Y4}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstHyperbolaFocusNode} is used to find the focus nodes of the Hyperbola,
+and the macro \Lcs{pstHyperbolaDirectrixLine} is used to find the directrix lines of the Hyperbola.
+
+\begin{BDef}
+\Lcs{pstHyperbolaFocusNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$F_1$}\Largb{$F_2$}\\
+\Lcs{pstHyperbolaDirectrixLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$L_x$}\Largb{$L_y$}\Largb{$R_x$}\Largb{$R_y$}
+\end{BDef}
+
+Note that you can use \Lcs{pstLineAS} to draw the asymptote line of the hyperbola by passing the slope gradient $k=\pm\dfrac{b}{a}$;
+or you can use the macro \Lcs{pstHyperbolaAsymptoteLine} to get them, this macro only create one node on each asymptote line,
+as the other one is the center of the hyperbola.
+
+\begin{BDef}
+\Lcs{pstHyperbolaAsymptoteLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$L_1$}\Largb{$L_2$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](1,1){O}
+\pstHyperbola[linecolor=blue!40](O)(\a,\b)[80]
+\pstHyperbolaNode[linecolor=blue!40](O)(\a,\b){80}{A}
+\pstLineAS[PointName=S_1,PosAngle=90,nodesepA=-3,nodesepB=-1.5,linecolor=blue!20]{O}{\b\space \a\space div}{S1}
+\pstLineAS[PointName=S_2,PosAngle=-90,nodesepA=-3,nodesepB=-1.5,linecolor=blue!20]{O}{\b\space \a\space div neg}{S2}
+\pstHyperbolaFocusNode[linecolor=blue!40,PointName={F_1,F_2},PosAngle={180,0}](O)(\a,\b){F1}{F2}
+\pstHyperbolaDirectrixLine[linecolor=blue!40,nodesepA=-2,nodesepB=-1,PointName={1,2,3,4},PosAngle=90,PointNameSep=0.2](O)(\a,\b){Lx}{Ly}{Rx}{Ry}
+\pstHyperbola[linecolor=red!40](O)(\b,\a)[78]
+\pstHyperbolaFocusNode[linecolor=red!40,PointName={H_1,H_2},PosAngle={180,0}](O)(\b,\a){H1}{H2}
+\pstHyperbolaDirectrixLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName={5,6,7,8},PosAngle=90,PointNameSep=0.2](O)(\b,\a){Mx}{My}{Nx}{Ny}
+\pstHyperbolaAsymptoteLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName={T_1,T_2},PosAngle=90,PointNameSep=0.2](O)(\b,\a){T1}{T2}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstHyperbolaLineInter} is used to find the intersections $C$ and $D$ of the hyperbola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstHyperbolaLineInter}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$A$}\Largb{$B$}\Largb{$C$}\Largb{$D$}
+\end{BDef}
+
+In the following example, the Line $CX$ and $CY$ are parallel to the asymptote of the hyperbola.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle={-90,90},PointNameSep=0.2](1,1){O}(1,1.5){C}
+\pstHyperbola[linecolor=blue!40](O)(\a,\b)[80]
+\pstLine[linestyle=dashed,linecolor=gray!40]{2,-1}{2,3}
+\pstHyperbolaLineInter[linecolor=blue!40,PosAngle={210,-40}](O)(\a,\b){2,-1}{2,3}{I}{J}
+\pstLineAS[linestyle=dashed,linecolor=gray!60,nodesep=-2,PosAngle=150]{1,1.5}{\b\space \a\space div}{X}
+\pstLineAS[linestyle=dashed,linecolor=gray!60,nodesep=-2,PosAngle=-10]{1,1.5}{\b\space \a\space div neg}{Y}
+\pstLineAS[linestyle=dashed,linecolor=gray!60,nodesep=-2,PosAngle=150]{1,1.5}{0.2}{Z}
+\pstHyperbolaLineInter[linecolor=blue!40,PosAngle={-10,-90}](O)(\a,\b){1,1.5}{X}{P}{Q}
+\pstHyperbolaLineInter[linecolor=blue!40,PosAngle={90,-30}](O)(\a,\b){1,1.5}{Y}{M}{N}
+\pstHyperbolaLineInter[linecolor=blue!40,PosAngle={90,90}](O)(\a,\b){1,1.5}{Z}{D}{E}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstHyperbolaPolarNode} is used to find the polar point $T$ of chord $AB$ on the hyperbola.
+
+\begin{BDef}
+\Lcs{pstHyperbolaPolarNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$A$}\Largb{$B$}\Largb{$T$}
+\end{BDef}
+
+We use the following theorem to find the polar point $T$ of chord $AB$:
+\begin{theorem}\label{HyperbolaPolarPointTheorem}
+Let $P$, $Q$ are vertex points of the hyperbola, for any chord $AB$ of hyperbola, suppose $PA$ and $BQ$ intersect at $E$,
+$PB$ and $AQ$ intersect at $F$, then the middle point $T$ of $EF$ is the polar point of chord $AB$.
+\end{theorem}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=90,PointNameSep=0.2](1,1){O}
+\pstHyperbola[linecolor=blue!40](O)(\a,\b)[80]
+\pstHyperbolaNode[linecolor=blue!40,PosAngle=80](O)(\a,\b){50}{A}
+\pstHyperbolaNode[linecolor=blue!40,PosAngle=-100](O)(\a,\b){-70}{B}
+\pstHyperbolaPolarNode[linecolor=red!40,PosAngle=-100](O)(\a,\b){A}{B}{T}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-1]{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstHyperbolaTangentNode} is used to find the tangent point $A$ and $B$ of point $T$ outside of the hyperbola.
+
+\begin{BDef}
+\Lcs{pstHyperbolaTangentNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$T$}\Largb{$A$}\Largb{$B$}
+\end{BDef}
+
+We use the following theorem to find the tangent points $A$ and $B$ of $T$:
+\begin{theorem}\label{HyperbolaTangentPointTheorem}
+Let $T$ is a point out of the hyperbola, for any two chords $TPQ$ and $TRS$ of the hyperbola, suppose $PR$ and $QS$ intersect at $X$,
+$RQ$ and $PS$ intersect at $Y$, then the intersection point $A$ and $B$ of $XY$ and the hyperbola are the tangent points from $T$.
+\end{theorem}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=90,PointNameSep=0.2](1,1){O}
+\pstHyperbola[linecolor=blue!40](O)(\a,\b)[80]
+\pstGeonode[PosAngle=-90](1.2,0.8){T}
+\pstHyperbolaTangentNode[linecolor=red!40,PosAngle={90,90},nodesep=-0.5](O)(\a,\b){T}{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{Standard Inversion Hyperbola}
+The Standard Inversion Hyperbola $H$ with coordinate translation is defined by center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$.
+The equation can be written as:
+\begin{equation}\label{FunctionOfStandardInversionHyperbola}
+\dfrac{(y-y_0)^2}{a^2}-\dfrac{(x-x_0)^2}{b^2}=1
+\end{equation}
+and the parametric function can be written as:
+\begin{equation}\label{ParametricFunctionOfStandardInversionHyperbola}
+\left\{\begin{array}{l}
+x=b\tan\alpha+x_0\\
+y=a\sec\alpha+y_0
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstIHyperbola} is used to draw a Standard Inversion Hyperbola with Center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$.
+The parameter \texttt{angleY} is used to truncate the height of the figure,
+it should be setup from 0 to 90.
+
+\begin{BDef}
+\Lcs{pstIHyperbola}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{angleY}
+\end{BDef}
+
+The macro \Lcs{pstIHyperbolaNode} is used to draw a node whose parameter is the given value $t$ on Inversion Hyperbola,
+please refer to equation (\ref{ParametricFunctionOfStandardInversionHyperbola}).
+The macro \Lcs{pstIHyperbolaAbsNode} is used to draw the nodes whose abscissa are the given value $x_1$ on Inversion Hyperbola.
+The macro \Lcs{pstIHyperbolaOrdNode} is used to draw the nodes whose ordinate are the given value $y_1$ on Inversion Hyperbola.
+
+Note that \Lcs{pstIHyperbolaAbsNode} and \Lcs{pstIHyperbolaOrdNode} will create two nodes $A$ and $B$ at most time.
+
+\begin{BDef}
+\Lcs{pstIHyperbolaNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstIHyperbolaAbsNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstIHyperbolaOrdNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}
+\pstGeonode[PosAngle=-90,PointNameSep=0.2](1,1){O}
+\pstIHyperbola[linecolor=blue!40](O)(\a,\b)[80]
+\pstIHyperbolaNode[linecolor=blue!40](O)(\a,\b){75}{A}
+\pstIHyperbolaAbsNode[linecolor=blue!40,PointName={Y_1,Y_2},PosAngle=0](O)(\a,\b){0}{Y1}{Y2}
+\pstIHyperbolaOrdNode[linecolor=red!40,PointName={X_1,X_2},PosAngle=-90](O)(\a,\b){0}{X1}{X2}
+\pstIHyperbola[linecolor=red!40](O)(\b,\a)[78]
+\pstIHyperbolaNode[linecolor=red!40](O)(\b,\a){-75}{B}
+\pstIHyperbolaAbsNode[linecolor=red!40,PointName={Y_3,Y_4},PosAngle=0](O)(\b,\a){0}{Y3}{Y4}
+\pstIHyperbolaOrdNode[linecolor=red!40,PointName={X_3,X_4},PosAngle=-90](O)(\b,\a){0}{X3}{X4}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstIHyperbolaFocusNode} is used to find the focus nodes of the Inversion Hyperbola,
+and the macro \Lcs{pstIHyperbolaDirectrixLine} is used to find the directrix lines of the Inversion Hyperbola.
+
+\begin{BDef}
+\Lcs{pstIHyperbolaFocusNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$F_1$}\Largb{$F_2$}\\
+\Lcs{pstIHyperbolaDirectrixLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$L_x$}\Largb{$L_y$}\Largb{$R_x$}\Largb{$R_y$}
+\end{BDef}
+
+Note that you can use \Lcs{pstLineAS} to draw the asymptote line of the hyperbola by passing the slope gradient $k=\pm\dfrac{a}{b}$;
+or you can use the macro \Lcs{pstIHyperbolaAsymptoteLine} to get them, this macro only create one node on each asymptote line,
+as the other one is the center of the hyperbola.
+
+\begin{BDef}
+\Lcs{pstHyperbolaAsymptoteLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$L_1$}\Largb{$L_2$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}
+\pstGeonode[PosAngle=180,PointNameSep=0.2](1,1){O}
+\pstIHyperbola[linecolor=blue!40](O)(\a,\b)[80]
+\pstIHyperbolaFocusNode[linecolor=blue!40,PointName={F_1,F_2},PosAngle={-90,90}](O)(\a,\b){F1}{F2}
+\pstIHyperbolaDirectrixLine[linecolor=blue!40,nodesepA=-2,nodesepB=-1,PointName={1,2,3,4},PosAngle=180,PointNameSep=0.2](O)(\a,\b){Lx}{Ly}{Rx}{Ry}
+\pstLineAS[PointName=S_1,PosAngle=90,nodesepA=-3,nodesepB=-1.5,linecolor=blue!20]{O}{\a\space \b\space div}{S1}
+\pstLineAS[PointName=S_2,PosAngle=-90,nodesepA=-3,nodesepB=-1.5,linecolor=blue!20]{O}{\a\space \b\space div neg}{S2}
+\pstIHyperbola[linecolor=red!40](O)(\b,\a)[78]
+\pstIHyperbolaFocusNode[linecolor=red!40,PointName={H_1,H_2},PosAngle={-90,90}](O)(\b,\a){H1}{H2}
+\pstIHyperbolaDirectrixLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName={5,6,7,8},PosAngle=0,PointNameSep=0.2](O)(\b,\a){Mx}{My}{Nx}{Ny}
+\pstIHyperbolaAsymptoteLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName={T_1,T_2},PosAngle=90,PointNameSep=0.2](O)(\b,\a){T1}{T2}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstIHyperbolaLineInter} is used to find the intersections $C$ and $D$ of the hyperbola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstIHyperbolaLineInter}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$A$}\Largb{$B$}\Largb{$C$}\Largb{$D$}
+\end{BDef}
+
+In the following example, the Line $CX$ and $CY$ are parallel to the asymptote of the hyperbola.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.35}
+\pstGeonode[PosAngle={0,180}](1,1){O}(0,1){C}
+\pstIHyperbola[linecolor=blue!40](O)(\a,\b)[82]
+\pstLine[linestyle=dashed,linecolor=gray!40]{2,-2}{2,4}
+\pstIHyperbolaLineInter[linecolor=blue!40,PosAngle={0,-30}](O)(\a,\b){2,-2}{2,4}{I}{J}
+\pstLineAS[linestyle=dashed,linecolor=gray!60,nodesep=-2,PosAngle=150]{0,1}{\a\space \b\space div}{X}
+\pstLineAS[linestyle=dashed,linecolor=gray!60,nodesep=-2,PosAngle=150]{0,1}{\a\space \b\space div neg}{Y}
+\pstLineAS[linestyle=dashed,linecolor=gray!60,nodesepA=-4,PosAngle=210]{0,1}{-3.5}{Z}
+\pstIHyperbolaLineInter[linecolor=blue!40,PosAngle={180,-100}](O)(\a,\b){0,1}{X}{P}{Q}
+\pstIHyperbolaLineInter[linecolor=blue!40,PosAngle={0,180}](O)(\a,\b){0,1}{Y}{M}{N}
+\pstIHyperbolaLineInter[linecolor=blue!40,PosAngle={190,-100}](O)(\a,\b){0,1}{Z}{D}{E}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstIHyperbolaPolarNode} is used to find the polar point $T$ of chord $AB$ on the hyperbola.
+
+\begin{BDef}
+\Lcs{pstIHyperbolaPolarNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$A$}\Largb{$B$}\Largb{$T$}
+\end{BDef}
+
+We also use the theorem \ref{HyperbolaPolarPointTheorem} to find the polar point $T$ of chord $AB$:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-1)(3,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=0,PointNameSep=0.2](1,1){O}
+\pstIHyperbola[linecolor=blue!40](O)(\a,\b)[76]
+\pstIHyperbolaNode[linecolor=blue!40,PosAngle=80](O)(\a,\b){50}{A}
+\pstIHyperbolaNode[linecolor=blue!40,PosAngle=-100](O)(\a,\b){-70}{B}
+\pstIHyperbolaPolarNode[linecolor=red!40,PosAngle=180](O)(\a,\b){A}{B}{T}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-1]{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstIHyperbolaTangentNode} is used to find the tangent point $A$ and $B$ of point $T$ outside of the hyperbola.
+
+\begin{BDef}
+\Lcs{pstIHyperbolaTangentNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\Largb{$T$}\Largb{$A$}\Largb{$B$}
+\end{BDef}
+
+We also use the following theorem \ref{HyperbolaTangentPointTheorem} to find the tangent points $A$ and $B$ of $T$.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=180](1,1){O}
+\pstIHyperbola[linecolor=blue!40](O)(\a,\b)[78]
+\pstGeonode[PosAngle=0](1.2,0.8){T}
+\pstIHyperbolaTangentNode[linecolor=red!40,PosAngle={80,-90},nodesep=-0.5](O)(\a,\b){T}{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{General Hyperbola}
+The General Hyperbola $H$ with coordinate translation and rotation is defined by center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$,
+and the rotation angle $\theta$ of the principal axis.
+The equation can be got from the parametric function of the Standard Hyperbola equation (\ref{ParametricFunctionOfStandardHyperbola}),
+using the rotation transform formula (\ref{RotationTransformFormula}), then we have
+\begin{equation}
+\left\{\begin{array}{l}
+x'=(a\sec\alpha+x_0)\cos\theta-(b\tan\alpha+y_0)\sin\theta=x_0'+a\sec\alpha\cos\theta-b\tan\alpha\sin\theta\\
+y'=(a\sec\alpha+x_0)\sin\theta+(b\tan\alpha+y_0)\cos\theta=y_0'+a\sec\alpha\sin\theta+b\tan\alpha\cos\theta
+\end{array}\right.
+\end{equation}
+where the $x_0'$ and $y_0'$ are the coordinate of the given center $O$ after rotation.
+So we get the parametric function of the General Hyperbola with coordinate translation and rotation as following:
+\begin{equation}\label{ParametricFunctionOfGeneralHyperbola}
+\left\{\begin{array}{l}
+x=x_0+a\sec\alpha\cos\theta-b\tan\alpha\sin\theta\\
+y=y_0+a\sec\alpha\sin\theta+b\tan\alpha\cos\theta
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstGeneralHyperbola} is used to draw a General Hyperbola with Center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$,
+and the rotation angle $\theta$ of the symmetrical axis.
+The parameter \texttt{angleX} is used to truncate the width of the figure,
+it should be setup from 0 to 90.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbola}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\OptArg{angleX}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=-90](1,1){O}
+\pstGeneralHyperbola[linecolor=red!20](O)(\a,\b)[0][80]
+\pstGeneralHyperbolaNode[linecolor=red!80,PosAngle=5](O)(\a,\b)[0]{0}{A}
+\pstGeneralHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeneralHyperbolaNode[linecolor=blue!40,PosAngle=10](O)(\a,\b)[40]{40}{B}
+\pstGeneralHyperbola[linecolor=green!60](O)(\a,\b)[90][80]
+\pstGeneralHyperbolaNode[linecolor=green!60,PosAngle=-90](O)(\a,\b)[90]{200}{C}
+\pstGeneralHyperbola[linecolor=purple!80](O)(\a,\b)[150][80]
+\pstGeneralHyperbolaNode[linecolor=purple!80,PosAngle=150](O)(\a,\b)[150]{50}{D}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralHyperbolaNode} is used to draw a node whose parameter is the given value $t$ on General Hyperbola,
+please refer to equation (\ref{ParametricFunctionOfGeneralHyperbola}).
+The macro \Lcs{pstGeneralHyperbolaAbsNode} is used to draw the nodes whose abscissa are the given value $x_1$ on General Hyperbola.
+The macro \Lcs{pstGeneralHyperbolaOrdNode} is used to draw the nodes whose ordinate are the given value $y_1$ on General Hyperbola.
+
+Note that \Lcs{pstGeneralHyperbolaAbsNode} and \Lcs{pstGeneralHyperbolaOrdNode} will create two nodes $A$ and $B$ at most time.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbolaNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstGeneralHyperbolaAbsNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstGeneralHyperbolaOrdNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=-90](1,1){O}
+\pstGeneralHyperbola[linecolor=purple!80](O)(\a,\b)[150][80]
+\pstGeneralHyperbolaAbsNode[linecolor=purple!80,PosAngle={200,90}](O)(\a,\b)[150]{2}{P}{Q}
+\pstGeneralHyperbolaAbsNode[linecolor=purple!80,PosAngle={-90,200}](O)(\a,\b)[150]{0}{X}{Y}
+\pstGeneralHyperbolaAbsNode[linecolor=purple!80,PosAngle={40,-40}](O)(\a,\b)[150]{0.59378}{M}{N}
+\pstLine[linestyle=dashed,linecolor=gray!40]{0.59378,-1}{0.59378,3}
+\pstGeneralHyperbolaOrdNode[linecolor=purple!80,PosAngle={200,90}](O)(\a,\b)[150]{2}{G}{H}
+\pstGeneralHyperbolaOrdNode[linecolor=purple!80,PosAngle={-90,200}](O)(\a,\b)[150]{0}{I}{J}
+\pstGeneralHyperbolaOrdNode[linecolor=purple!80,PosAngle={90,-90}](O)(\a,\b)[150]{1}{K}{L}
+\pstLine[linestyle=dashed,linecolor=gray!80,nodesep=-1.5]{K}{L}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralHyperbolaFocusNode} is used to find the focus nodes of the General Hyperbola,
+the macro \Lcs{pstGeneralHyperbolaVertexNode} is used to find the vertex nodes of the General Hyperbola,
+and the macro \Lcs{pstGeneralHyperbolaDirectrixLine} is used to find the directrix lines of the General Hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbolaFocusNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$F_1$}\Largb{$F_2$}\\
+\Lcs{pstGeneralHyperbolaVertexNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$V_1$}\Largb{$V_2$}\\
+\Lcs{pstGeneralHyperbolaDirectrixLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$L_x$}\Largb{$L_y$}\Largb{$R_x$}\Largb{$R_y$}
+\end{BDef}
+
+Note that you can use the macro \Lcs{pstGeneralHyperbolaAsymptoteLine} to get the asymptote lines, this macro only create one node on each asymptote line,
+as the other one is the center of the hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbolaAsymptoteLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$L_1$}\Largb{$L_2$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}
+\pstGeonode[PosAngle=180,PointNameSep=0.2](1,1){O}
+\pstGeneralHyperbola[linecolor=red!40](O)(\a,\b)[0][80]
+\pstGeneralHyperbolaFocusNode[linecolor=red!40,PointName={X_1,X_2},PosAngle={180,0}](O)(\a,\b)[0]{X1}{X2}
+\pstGeneralHyperbolaDirectrixLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[0]{Lx}{Ly}{Rx}{Ry}
+\pstGeneralHyperbolaAsymptoteLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[0]{L1}{L2}
+\pstGeneralHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeneralHyperbolaFocusNode[linecolor=blue!40,PointName={F_1,F_2},PosAngle={220,40}](O)(\a,\b)[40]{F1}{F2}
+\pstGeneralHyperbolaDirectrixLine[linecolor=blue!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[40]{Dx}{Dy}{Ux}{Uy}
+\pstGeneralHyperbolaAsymptoteLine[linecolor=blue!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[40]{S1}{S2}
+\pstGeneralHyperbola[linecolor=brown!40](O)(\a,\b)[90][80]
+\pstGeneralHyperbolaFocusNode[linecolor=brown!40,PointName={Y_1,Y_2},PosAngle={-90,90}](O)(\a,\b)[90]{Y1}{Y2}
+\pstGeneralHyperbolaDirectrixLine[linecolor=brown!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[90]{Tx}{Ty}{Sx}{Sy}
+\pstGeneralHyperbolaAsymptoteLine[linecolor=brown!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[90]{T1}{T2}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralHyperbolaLineInter} is used to find the intersections $C$ and $D$ of the general hyperbola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbolaLineInter}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$A$}\Largb{$B$}\Largb{$C$}\Largb{$D$}
+\end{BDef}
+
+In the following example, the lines $YY'$ and $ZZ'$ are parallel to the asymptote of the hyperbola,
+so there are only one intersection $M$ and $P$ for each line, and the second node $N$ and $Q$ are put at the origin.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=-90](1,1){O}
+\pstGeneralHyperbola[linecolor=blue!40](O)(\a,\b)[30][80]
+\pstLine[linestyle=dashed,linecolor=gray!40]{0.5,-1}{0.5,3}
+\pstGeneralHyperbolaLineInter[linecolor=blue!40,PosAngle={-30,210}](O)(\a,\b)[30]{0.5,-1}{0.5,3}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40]{-2,0}{3,3}
+\pstGeneralHyperbolaLineInter[linecolor=blue!40,PosAngle={130,-90}](O)(\a,\b)[30]{-2,0}{3,3}{C}{D}
+\pstGeonode[PosAngle={0,100}](2,0){Y}(1,1.8){Z}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=gray!40,PointName=none,PointSymbol=none]{O}{\b\space \a\space div 1 atan 30 add}{U}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=red!40,PointName=none,PointSymbol=none]{O}{\b\space \a\space div neg 1 atan 30 add}{V}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=gray!40,PosAngle=-30]{Y}{\b\space \a\space div 1 atan 30 add}{Y'}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=red!40,PosAngle=80]{Z}{\b\space \a\space div neg 1 atan 30 add}{Z'}
+\pstGeneralHyperbolaLineInter[linecolor=blue!40,PosAngle={-50,-90}](O)(\a,\b)[30]{Z}{Z'}{M}{N}
+\pstGeneralHyperbolaLineInter[linecolor=blue!40,PosAngle={30,210}](O)(\a,\b)[30]{Y}{Y'}{P}{Q}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralHyperbolaPolarNode} is used to find the polar point $T$ of chord $AB$ on the general hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbolaPolarNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$A$}\Largb{$B$}\Largb{$T$}
+\end{BDef}
+
+We also use the theorem \ref{HyperbolaPolarPointTheorem} to find the polar point $T$ of chord $AB$:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-1)(3,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=120,PointNameSep=0.2](1,1){O}
+\pstGeneralHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeneralHyperbolaNode[linecolor=blue!40,PosAngle=110](O)(\a,\b)[40]{50}{A}
+\pstGeneralHyperbolaNode[linecolor=blue!40,PosAngle=-100](O)(\a,\b)[40]{-70}{B}
+\pstGeneralHyperbolaPolarNode[linecolor=red!40,PosAngle=-90](O)(\a,\b)[40]{A}{B}{T}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-1]{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralHyperbolaTangentNode} is used to find the tangent point $A$ and $B$ of point $T$ outside of the general hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbolaTangentNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$T$}\Largb{$A$}\Largb{$B$}
+\end{BDef}
+
+We also use the following theorem \ref{HyperbolaTangentPointTheorem} to find the tangent points $A$ and $B$ of $T$.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=120](1,1){O}
+\pstGeneralHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeonode[PosAngle=-40](1.2,0.8){T}
+\pstGeneralHyperbolaTangentNode[linecolor=red!40,PosAngle={140,-90},nodesep=-0.5](O)(\a,\b)[40]{T}{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{General Inversion Hyperbola}
+The General Inversion Hyperbola $H$ with coordinate translation and rotation is defined by center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$,
+and the rotation angle $\theta$ of the principal axis.
+The equation can be got from the parametric function of the Standard Inversion Hyperbola equation (\ref{ParametricFunctionOfStandardInversionHyperbola}),
+using the rotation transform formula (\ref{RotationTransformFormula}), then we have
+\begin{equation}
+\left\{\begin{array}{l}
+x'=(b\tan\alpha+x_0)\cos\theta-(a\sec\alpha+y_0)\sin\theta=x_0'+b\tan\alpha\cos\theta-a\sec\alpha\sin\theta\\
+y'=(b\tan\alpha+x_0)\sin\theta+(a\sec\alpha+y_0)\cos\theta=y_0'+b\tan\alpha\sin\theta+a\sec\alpha\cos\theta
+\end{array}\right.
+\end{equation}
+where the $x_0'$ and $y_0'$ are the coordinate of the given center $O$ after rotation.
+So we get the parametric function of the General Inversion Hyperbola with coordinate translation and rotation as following:
+\begin{equation}\label{ParametricFunctionOfGeneralInversionHyperbola}
+\left\{\begin{array}{l}
+x=x_0+b\tan\alpha\cos\theta-a\sec\alpha\sin\theta\\
+y=y_0+b\tan\alpha\sin\theta+a\sec\alpha\cos\theta
+\end{array}\right.
+\end{equation}
+
+The macro \Lcs{pstGeneralIHyperbola} is used to draw a General Inversion Hyperbola with Center $O$,
+the half of the real axis $a$, the half of the imaginary axis $b$,
+and the rotation angle $\theta$ of the symmetrical axis.
+The parameter \texttt{angleY} is used to truncate the height of the figure,
+it should be setup from 0 to 90.
+
+\begin{BDef}
+\Lcs{pstGeneralIHyperbola}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\OptArg{angleY}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=-90](1,1){O}
+\pstGeneralIHyperbola[linecolor=red!20](O)(\a,\b)[0][80]
+\pstGeneralIHyperbolaNode[linecolor=red!80,PosAngle=-90](O)(\a,\b)[0]{0}{A}
+\pstGeneralIHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeneralIHyperbolaNode[linecolor=blue!40,PosAngle=190](O)(\a,\b)[40]{40}{B}
+\pstGeneralIHyperbola[linecolor=green!60](O)(\a,\b)[90][80]
+\pstGeneralIHyperbolaNode[linecolor=green!60,PosAngle=0](O)(\a,\b)[90]{200}{C}
+\pstGeneralIHyperbola[linecolor=purple!80](O)(\a,\b)[150][80]
+\pstGeneralIHyperbolaNode[linecolor=purple!80,PosAngle=-90](O)(\a,\b)[150]{50}{D}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralIHyperbolaNode} is used to draw a node whose parameter is the given value $t$ on General Inversion Hyperbola,
+please refer to equation (\ref{ParametricFunctionOfGeneralInversionHyperbola}).
+
+The macro \Lcs{pstGeneralIHyperbolaAbsNode} is used to draw the nodes whose abscissa are the given value $x_1$ on General Inversion Hyperbola.
+The macro \Lcs{pstGeneralIHyperbolaOrdNode} is used to draw the nodes whose ordinate are the given value $y_1$ on General Inversion Hyperbola.
+
+Note that \Lcs{pstGeneralIHyperbolaAbsNode} and \Lcs{pstGeneralIHyperbolaOrdNode} will create two nodes $A$ and $B$ at most time.
+
+\begin{BDef}
+\Lcs{pstGeneralIHyperbolaNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$t$}\Largb{A}\\
+\Lcs{pstGeneralIHyperbolaAbsNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$x_1$}\Largb{A}\Largb{B}\\
+\Lcs{pstGeneralIHyperbolaOrdNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$y_1$}\Largb{A}\Largb{B}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=180](1,1){O}
+\pstGeneralIHyperbola[linecolor=purple!80](O)(\a,\b)[150][80]
+\pstGeneralIHyperbolaAbsNode[linecolor=purple!80,PosAngle={200,90}](O)(\a,\b)[150]{2}{P}{Q}
+\pstGeneralIHyperbolaAbsNode[linecolor=purple!80,PosAngle={90,200}](O)(\a,\b)[150]{0}{X}{Y}
+\pstGeneralIHyperbolaAbsNode[linecolor=purple!80,PosAngle={40,-40}](O)(\a,\b)[150]{1}{M}{N}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-1.5]{M}{N}
+\pstGeneralIHyperbolaOrdNode[linecolor=purple!80,PosAngle={180,90}](O)(\a,\b)[150]{2}{G}{H}
+\pstGeneralIHyperbolaOrdNode[linecolor=purple!80,PosAngle={90,240}](O)(\a,\b)[150]{0}{I}{J}
+\pstGeneralIHyperbolaOrdNode[linecolor=purple!80,PosAngle={-100,-60}](O)(\a,\b)[150]{1.4063}{K}{L}
+\pstLine[linestyle=dashed,linecolor=gray!80,nodesep=-1]{I}{J}
+\pstLine[linestyle=dashed,linecolor=gray!80,nodesep=-1]{-1,1.4063}{3,1.4063}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralIHyperbolaFocusNode} is used to find the focus nodes of the General Inversion Hyperbola,
+the macro \Lcs{pstGeneralIHyperbolaVertexNode} is used to find the vertex nodes of the General Inversion Hyperbola,
+and the macro \Lcs{pstGeneralIHyperbolaDirectrixLine} is used to find the directrix lines of the General Inversion Hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralIHyperbolaFocusNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$F_1$}\Largb{$F_2$}\\
+\Lcs{pstGeneralIHyperbolaVertexNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$V_1$}\Largb{$V_2$}\\
+\Lcs{pstGeneralIHyperbolaDirectrixLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$L_x$}\Largb{$L_y$}\Largb{$R_x$}\Largb{$R_y$}
+\end{BDef}
+
+Note that you can use the macro \Lcs{pstGeneralIHyperbolaAsymptoteLine} to get the asymptote lines, this macro only create one node on each asymptote line,
+as the other one is the center of the hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralHyperbolaAsymptoteLine}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$L_1$}\Largb{$L_2$}
+\end{BDef}
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-2)(4,4)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}
+\pstGeonode[PosAngle=180,PointNameSep=0.2](1,1){O}
+\pstGeneralIHyperbola[linecolor=red!40](O)(\a,\b)[0][80]
+\pstGeneralIHyperbolaFocusNode[linecolor=red!40,PointName={X_1,X_2},PosAngle={90,-90}](O)(\a,\b)[0]{X1}{X2}
+\pstGeneralIHyperbolaDirectrixLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[0]{Lx}{Ly}{Rx}{Ry}
+\pstGeneralIHyperbolaAsymptoteLine[linecolor=red!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[0]{T1}{T2}
+\pstGeneralIHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeneralIHyperbolaFocusNode[linecolor=blue!40,PointName={F_1,F_2},PosAngle={130,-40}](O)(\a,\b)[40]{F1}{F2}
+\pstGeneralIHyperbolaDirectrixLine[linecolor=blue!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[40]{Dx}{Dy}{Ux}{Uy}
+\pstGeneralIHyperbolaAsymptoteLine[linecolor=blue!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[40]{S1}{S2}
+\pstGeneralIHyperbola[linecolor=brown!40](O)(\a,\b)[90][80]
+\pstGeneralIHyperbolaFocusNode[linecolor=brown!40,PointName={Y_1,Y_2},PosAngle={180,0}](O)(\a,\b)[90]{Y1}{Y2}
+\pstGeneralIHyperbolaDirectrixLine[linecolor=brown!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[90]{Tx}{Ty}{Sx}{Sy}
+\pstGeneralIHyperbolaAsymptoteLine[linecolor=brown!40,nodesepA=-2,nodesepB=-1,PointName=none](O)(\a,\b)[90]{R1}{R2}
+\end{pspicture}
+\end{LTXexample}
+
+
+The macro \Lcs{pstGeneralIHyperbolaLineInter} is used to find the intersections $C$ and $D$ of the general inversion hyperbola and the given line $AB$.
+
+\begin{BDef}
+\Lcs{pstGeneralIHyperbolaLineInter}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$A$}\Largb{$B$}\Largb{$C$}\Largb{$D$}
+\end{BDef}
+
+In the following example, the lines $YY'$ and $ZZ'$ are parallel to the asymptote of the hyperbola,
+so there are only one intersection $M$ and $P$ for each line, and the second node $N$ and $Q$ are put at the origin.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=225](1,1){O}
+\pstGeneralIHyperbola[linecolor=blue!40](O)(\a,\b)[30][80]
+\pstLine[linestyle=dashed,linecolor=gray!40]{-1,-1}{-1,3}
+\pstGeneralIHyperbolaLineInter[linecolor=blue!40,PosAngle={-30,210}](O)(\a,\b)[30]{-1,-1}{-1,3}{A}{B}
+\pstLine[linestyle=dashed,linecolor=gray!40]{-2,1}{3,3}
+\pstGeneralIHyperbolaLineInter[linecolor=blue!40,PosAngle={130,-90}](O)(\a,\b)[30]{-2,1}{3,3}{C}{D}
+\pstGeonode[PosAngle={-20,100}](2,0){Y}(1.8,2){Z}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=gray!40,PointName=none,PointSymbol=none]{O}{\a\space \b\space div 1 atan 30 add}{U}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=red!40,PointName=none,PointSymbol=none]{O}{\a\space \b\space div neg 1 atan 30 add}{V}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=gray!40,PosAngle=10]{Y}{\a\space \b\space div 1 atan 30 add}{Y'}
+\pstLineAA[nodesepA=-3,nodesepB=-2,linecolor=red!40,PosAngle=80]{Z}{\a\space \b\space div neg 1 atan 30 add}{Z'}
+\pstGeneralIHyperbolaLineInter[linecolor=blue!40,PosAngle={30,-90}](O)(\a,\b)[30]{Z}{Z'}{M}{N}
+\pstGeneralIHyperbolaLineInter[linecolor=blue!40,PosAngle={30,210}](O)(\a,\b)[30]{Y}{Y'}{P}{Q}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralIHyperbolaPolarNode} is used to find the polar point $T$ of chord $AB$ on the general inversion hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralIHyperbolaPolarNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$A$}\Largb{$B$}\Largb{$T$}
+\end{BDef}
+
+We also use the theorem \ref{HyperbolaPolarPointTheorem} to find the polar point $T$ of chord $AB$:
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-1,-1)(3,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=40,PointNameSep=0.2](1,1){O}
+\pstGeneralIHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeneralIHyperbolaNode[linecolor=blue!40,PosAngle=40](O)(\a,\b)[40]{50}{A}
+\pstGeneralIHyperbolaNode[linecolor=blue!40,PosAngle=-100](O)(\a,\b)[40]{-70}{B}
+\pstGeneralIHyperbolaPolarNode[linecolor=red!40,PosAngle=-90](O)(\a,\b)[40]{A}{B}{T}
+\pstLine[linestyle=dashed,linecolor=gray!40,nodesep=-1]{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
+The macro \Lcs{pstGeneralIHyperbolaTangentNode} is used to find the tangent point $A$ and $B$ of point $T$ outside of the general inversion hyperbola.
+
+\begin{BDef}
+\Lcs{pstGeneralIHyperbolaTangentNode}\OptArgs\Largr{O}\Largr{$a,\,b$}\OptArg{$\theta$}\Largb{$T$}\Largb{$A$}\Largb{$B$}
+\end{BDef}
+
+We also use the following theorem \ref{HyperbolaTangentPointTheorem} to find the tangent points $A$ and $B$ of $T$.
+
+\begin{LTXexample}[width=6cm,pos=l]
+\begin{pspicture}[showgrid=true](-2,-1)(4,3)
+\psset{dotscale=0.5}\psset{PointSymbol=*}\footnotesize
+\def\a{0.5}\def\b{0.3}\psset{PointNameSep=0.3}
+\pstGeonode[PosAngle=20](1,1){O}
+\pstGeneralIHyperbola[linecolor=blue!40](O)(\a,\b)[40][80]
+\pstGeonode[PosAngle=-40](0,1){T}
+\pstGeneralIHyperbolaTangentNode[linecolor=red!40,PosAngle={-80,120},nodesep=-0.5](O)(\a,\b)[40]{T}{A}{B}
+\end{pspicture}
+\end{LTXexample}
+
\section{Geometric Transformations}
The geometric transformations are the ideal tools to construct geometric figures. All
@@ -686,7 +2766,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PtNameMath},
\Lkeyword{CodeFig}, \Lkeyword{CodeFigColor}, and \Lkeyword{CodeFigStyle}.
@@ -699,7 +2779,7 @@
\begin{pspicture}[showgrid](-2,-2)(2,2)
\psset{CodeFig=true}
\pstGeonode[PosAngle={20,90,0}]{O}(-.6,1.5){A}(1.6,-.5){B}
-\pstSymO[CodeFigColor=blue,
+\pstSymO[CodeFigColor=blue,
PosAngle={-90,180}]{O}{A, B}[C, D]
\pstLineAB{A}{B}\pstLineAB{C}{D}
\pstLineAB{A}{D}\pstLineAB{C}{B}
@@ -713,7 +2793,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PtNameMath},
\Lkeyword{CodeFig}, \Lkeyword{CodeFigColor}, and \Lkeyword{CodeFigStyle}.
@@ -745,7 +2825,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PtNameMath}, and \Lkeyword{RotAngle}
for \Lcs{pstRotation} and \Lkeyword{AngleCoef}, \Lkeyword{RotAngle} for \Lcs{pstAngleABC}.
@@ -770,14 +2850,14 @@
\pstRotation[PosAngle=90,RotAngle=60,
CodeFig,CodeFigColor=blue,
TransformLabel=\frac{\pi}{3}]{A}{B}[C]
-\pstRotation[AngleCoef=.5,
+\pstRotation[AngleCoef=.5,
RotAngle=\pstAngleAOB{B}{A}{C},
CodeFigColor=red, CodeFig,
TransformLabel=\frac{1}{2}\widehat{BAC}]{A}{D}[E]
\end{pspicture}
\end{LTXexample}
-
+
\subsection{Translation}
\begin{BDef}
@@ -785,7 +2865,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PtNameMath}, and \Lkeyword{DistCoef}
%
@@ -822,7 +2902,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{HomCoef},
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PtNameMath}, and \Lkeyword{HomCoef}.
@@ -852,7 +2932,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PtNameMath},
\Lkeyword{CodeFig}, \Lkeyword{CodeFigColor}, and\Lkeyword{CodeFigStyle}
@@ -911,7 +2991,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PosAngle},
\Lkeyword{PointSymbol}, and \Lkeyword{PtNameMath}
%
@@ -936,12 +3016,12 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PosAngle},
\Lkeyword{PointSymbol}, \Lkeyword{PtNameMath}, \Lkeyword{DrawCirABC}, \Lkeyword{CodeFig},
\Lkeyword{CodeFigColor}, \Lkeyword{CodeFigStyle}, \Lkeyword{SegmentSymbolA},
\Lkeyword{SegmentSymbolB}, and \Lkeyword{SegmentSymbolC}.
-%
+%
Draws the circle crossing three points (the circum circle) and put its center $O$.
The effective drawing is controlled by the boolean parameter \Lkeyword{DrawCirABC}
\DefaultVal{true}. Moreover the intermediate constructs (mediator lines) can
@@ -967,7 +3047,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PosAngle},
\Lkeyword{PointSymbol}, \Lkeyword{PtNameMath}, \Lkeyword{CodeFig},
\Lkeyword{CodeFigColor}, \Lkeyword{CodeFigStyle}, and \Lkeyword{SegmentSymbol}.
@@ -1015,7 +3095,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, and \Lkeyword{PtNameMath}.
%
@@ -1067,7 +3147,7 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, and \Lkeyword{PtNameMath}.
%
@@ -1110,11 +3190,11 @@
\end{BDef}
\begin{sloppypar}
-Possible optional arguments are
+Possible optional arguments are
\Lkeyword{PointSymbol}, \Lkeyword{PosAngle},
\Lkeyword{PointName}, \Lkeyword{PointNameSep}, \Lkeyword{PtNameMath},
\Lkeyword{PointSymbolA}, \Lkeyword{PosAngleA}, \Lkeyword{PointNameA},
- \Lkeyword{PointSymbolB}, \Lkeyword{PosAngleB}, \Lkeyword{PointNameB},
+ \Lkeyword{PointSymbolB}, \Lkeyword{PosAngleB}, \Lkeyword{PointNameB},
\Lkeyword{Radius}, and \Lkeyword{Diameter}.
%
Draw the one or two intersection point(s) between the line $(AB)$ and
@@ -1177,7 +3257,7 @@
\pstCircleOA[linecolor=red]{C}{B}
\pstInterCC[PosAngleA=135, CodeFigA=true, CodeFigAarc=false,
CodeFigColor=magenta]{O}{A}{C}{B}{D}{E}
-\pstInterCC[PosAngleA=170, CodeFigA=true,
+\pstInterCC[PosAngleA=170, CodeFigA=true,
CodeFigAarc=false,
CodeFigColor=green]{B}{E}{C}{B}{F}{G}
\end{pspicture}
@@ -1304,23 +3384,23 @@
Calculates and prints the values. This is only possible on PostScript level!
-\begin{pspicture}[showgrid](-1,0)(11,8)
+\begin{pspicture}[showgrid](-1,-1)(8,7)
\def\sideC{6} \def\sideA{7} \def\sideB{8}
-\psset{PointSymbol=none,linejoin=1,linewidth=0.4pt,PtNameMath=false,labelsep=0.07,MarkAngleRadius=1.1,decimals=1,comma}
-\pstGeonode[PosAngle={90,90}](0,0){A}(\sideC;10){B}
-\psset{PointName=}
-\pstInterCC[RadiusA=\pstDistVal{\sideB},RadiusB=\pstDistVal{\sideA},PosAngle=-90,PointNameA=C]{A}{}{B}{}{C}{C-}
+\psset{unit=0.8cm,PointSymbol=none,linejoin=1,linewidth=0.4pt,PtNameMath=false,labelsep=0.07,MarkAngleRadius=1.1,decimals=1}
+\pstGeonode[PosAngle={-90,-90}](0,0){A}(\sideC;10){B}
+\pstInterCC[RadiusA=\pstDistVal{\sideB},RadiusB=\pstDistVal{\sideA},PosAngleA=90,PointNameA=C]{A}{}{B}{}{C}{C-}
\pstInterCC[RadiusA=\pstDistAB{A}{B},RadiusB=\pstDistAB{B}{C}]{C}{}{A}{}{D-}{D}
+\psset{PointName=none}
\pstInterLC[Radius=\pstDistAB{A}{C}]{C}{D}{C}{}{A'-}{A'}
\pstInterCC[RadiusA=\pstDistAB{A}{B},RadiusB=\pstDistAB{B}{C}]{A'}{}{C}{}{B'}{B'-}
-\pstInterLL[PosAngle=90,PointName=default]{B'}{C}{A}{B}{E}
+\pstInterLL[PosAngle=-90,PointName=default]{B'}{C}{A}{B}{E}
\pspolygon(A)(B)(C)
\pspolygon[fillstyle=solid,fillcolor=magenta,opacity=0.1](C)(E)(B)
%
-\psGetAngleABC[ArcColor=blue,AngleValue=true,LabelSep=0.7,arrows=->,decimals=0,PSfont=Palatino-Roman](B)(A)(C){}
-\psGetAngleABC[AngleValue=true,ArcColor=red,arrows=->,WedgeOpacity=0.6,WedgeColor=yellow!30,LabelSep=0.5](C)(B)(A){$\beta$}
-\psGetAngleABC[LabelSep=0.7,WedgeColor=green,xShift=-6,yShift=-10](A)(C)(B){$\gamma$}
-\psGetAngleABC[LabelSep=0.7,AngleArc=false,WedgeColor=green,arrows=->,xShift=-15,yShift=0](C)(E)(B){\color{blue}$\gamma$}
+\psGetAngleABC[ArcColor=blue,AngleValue=true,LabelSep=0.4,arrows=->,decimals=0,PSfont=Palatino-Roman](B)(A)(C){}
+\psGetAngleABC[AngleValue=true,ArcColor=red,arrows=->,WedgeOpacity=0.6,WedgeColor=yellow!30,LabelSep=0.4](C)(B)(A){$\beta$}
+\psGetAngleABC[LabelSep=0.4,AngleValue=true,WedgeColor=green,xShift=-6,yShift=-10](A)(C)(B){$\gamma$}
+\psGetAngleABC[LabelSep=0.4,AngleValue=true,AngleArc=false,WedgeColor=green,arrows=->,xShift=-15,yShift=0](C)(E)(B){\color{blue}$\gamma$}
\psGetAngleABC[AngleValue=true,MarkAngleRadius=1.0,LabelSep=0.5,ShowWedge=false,xShift=-5,yShift=7,arrows=->](E)(B)(C){}
%
\pcline[linestyle=none](A)(B)\nbput{\sideC}
@@ -1330,19 +3410,13 @@
\psGetDistanceAB[xShift=-17,decimals=2](E)(C){MEC}
\end{pspicture}
-
-
-
-
\begin{lstlisting}
\begin{pspicture}(-1,0)(11,8)
\psgrid[gridlabels=0pt,subgriddiv=2,gridwidth=0.4pt,subgridwidth=0.2pt,gridcolor=black!60,subgridcolor=black!40]
\def\sideC{6} \def\sideA{7} \def\sideB{8}
\psset{PointSymbol=none,linejoin=1,linewidth=0.4pt,PtNameMath=false,labelsep=0.07,MarkAngleRadius=1.1,decimals=1,comma}
-\pstGeonode[PosAngle={90,90}](0,0){A}(\sideC;10){B}
-% \pstGeonode[PosAngle={225,-75}](0,0){A}(\sideC;10){B}
-\psset{PointName=}
-\pstInterCC[RadiusA=\pstDistVal{\sideB},RadiusB=\pstDistVal{\sideA},PosAngle=-90,PointNameA=C]{A}{}{B}{}{C}{C-}
+\pstGeonode[PosAngle={-90,-90}](0,0){A}(\sideC;10){B}
+\pstInterCC[RadiusA=\pstDistVal{\sideB},RadiusB=\pstDistVal{\sideA},PosAngle=90,PointNameA=C]{A}{}{B}{}{C}{C-}
\pstInterCC[RadiusA=\pstDistAB{A}{B},RadiusB=\pstDistAB{B}{C}]{C}{}{A}{}{D-}{D}
\pstInterLC[Radius=\pstDistAB{A}{C}]{C}{D}{C}{}{A'-}{A'}
\pstInterCC[RadiusA=\pstDistAB{A}{B},RadiusB=\pstDistAB{B}{C}]{A'}{}{C}{}{B'}{B'-}
@@ -1350,10 +3424,10 @@
\pspolygon(A)(B)(C)
\pspolygon[fillstyle=solid,fillcolor=magenta,opacity=0.1](C)(E)(B)
%
-\psGetAngleABC[ArcColor=blue,AngleValue=true,LabelSep=0.7,arrows=->,decimals=0,PSfont=Palatino-Roman](B)(A)(C){}
+\psGetAngleABC[ArcColor=blue,AngleValue=true,LabelSep=0.8,arrows=->,decimals=0,PSfont=Palatino-Roman](B)(A)(C){}
\psGetAngleABC[AngleValue=true,ArcColor=red,arrows=->,WedgeOpacity=0.6,WedgeColor=yellow!30,LabelSep=0.5](C)(B)(A){$\beta$}
-\psGetAngleABC[LabelSep=0.7,WedgeColor=green,xShift=-6,yShift=-10](A)(C)(B){$\gamma$}
-\psGetAngleABC[LabelSep=0.7,AngleArc=false,WedgeColor=green,arrows=->,xShift=-15,yShift=0](C)(E)(B){\color{blue}$\gamma$}
+\psGetAngleABC[LabelSep=0.8,WedgeColor=green,xShift=-6,yShift=-10](A)(C)(B){$\gamma$}
+\psGetAngleABC[LabelSep=0.8,AngleArc=false,WedgeColor=green,arrows=->,xShift=-15,yShift=0](C)(E)(B){\color{blue}$\gamma$}
\psGetAngleABC[AngleValue=true,MarkAngleRadius=1.0,LabelSep=0.5,ShowWedge=false,xShift=-5,yShift=7,arrows=->](E)(B)(C){}
%
\pcline[linestyle=none](A)(B)\nbput{\sideC}
@@ -1697,7 +3771,7 @@
\pstBissectBAC[PointSymbol=none,PointName=none]{C}{A}{B}{AB}
\pstBissectBAC[PointSymbol=none,PointName=none]{A}{B}{C}{BB}
\pstBissectBAC[PointSymbol=none,PointName=none]{B}{C}{A}{CB}
-\pstInterLL{A}{AB}{B}{BB}{I}
+\pstInterLL{A}{AB}{B}{BB}{I}
\psset{linecolor=magenta, linestyle=dashed} \pstProjection{A}{B}{I}[I_C]
\pstLineAB{I}{I_C}\pstRightAngle[linestyle=solid]{A}{I_C}{I}
\pstProjection{A}{C}{I}[I_B]
@@ -1842,7 +3916,7 @@
\begin{LTXexample}
\begin{pspicture}[showgrid](-4,-4)(4,4)
\newcommand\Sommet{1.4142135623730951 } \newcounter{i} \setcounter{i}{1}
-\newcommand\PosFoyer{2 } \newcommand\HypAngle{0}
+\newcommand\PosFoyer{2 } \newcommand\HypAngle{0}
\newcounter{CoefDiv}\setcounter{CoefDiv}{20}
\newcounter{Inc}\setcounter{Inc}{2} \newcounter{n}\setcounter{n}{2}
\newcommand\Ri{ \PosFoyer \Sommet sub \arabic{i}\space\arabic{CoefDiv}\space div add }
@@ -1933,7 +4007,7 @@
\begin{pspicture}[showgrid](-3.5,-3.4)(3.5,4)
\HypoCyclo[3]{3}{1}{17}
\psset{linecolor=blue,linewidth=1.5\pslinewidth}
-\pstGenericCurve[GenCurvFirst=P]{N}{1}{6}
+\pstGenericCurve[GenCurvFirst=P]{N}{1}{6}
\pstGenericCurve{N}{6}{12}
\pstGenericCurve[GenCurvLast=P]{N}{12}{17}
\end{pspicture}
Modified: trunk/Master/texmf-dist/tex/generic/pst-eucl/pst-eucl.tex
===================================================================
--- trunk/Master/texmf-dist/tex/generic/pst-eucl/pst-eucl.tex 2019-10-20 17:28:18 UTC (rev 52456)
+++ trunk/Master/texmf-dist/tex/generic/pst-eucl/pst-eucl.tex 2019-10-20 20:57:52 UTC (rev 52457)
@@ -7,7 +7,7 @@
%% of the LaTeX Project Public License Distributed from CTAN
%% archives in directory macros/latex/base/lppl.txt.
%%
-%% Author : Dominique RODRIGUEZ (EN) <dominique.rodriguez at waika9.com>
+%% Author : Dominique RODRIGUEZ (EN) <dominique.rodriguez at waika9.com>
%% : hv hvoss at tug.org
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Require PSTricks and pst-node packages
@@ -14,13 +14,14 @@
\ifx\PSTnodesLoaded\endinput\else\input pst-node.tex\fi
\ifx\PSTArrowLoaded\endinput\else\input pst-arrow.tex\fi
\ifx\PSTtoolsLoaded\endinput\else\input pst-tools.tex\fi
+\ifx\PSTplotLoaded\endinput\else\input pst-plot.tex\fi
%\ifx\PSTXKeyLoaded\endinput\else \input pst-xkey\fi
\ifx\MultidoLoaded\endinput \else\input multido.tex \fi
\csname PSTEuclideLoaded\endcsname
\let\PSTEuclideLoaded\endinput
%
-\def\fileversion{1.65}
-\def\filedate{2019/08/19}
+\def\fileversion{1.66}
+\def\filedate{2019/10/20}
%%
\message{`PST-Euclide v\fileversion, \filedate\space (dr,hv)}%
%% prologue for postcript
@@ -151,12 +152,12 @@
PointSymbol=*, %PointSymbolA=*, PointSymbolB=*, PointSymbolC=*,
PointName=default, PointNameA=undef, PointNameB=undef, PointNameC=undef,
PtNameMath=true, PointNameMathSize=\textstyle, PointNameSize=\normalsize,
- PointNameSep=default, PosAngle=undef,
+ PointNameSep=default, PosAngle=undef,
PosAngleA=undef,
PosAngleB=undef, PosAngleC=undef, Mark=undef, mark=undef,SegmentSymbol=MarkHashh,
SegmentSymbolA=MarkHashh, SegmentSymbolB=MarkHash, SegmentSymbolC=MarkHashhh,
RightAngleSize=.4, RightAngleType=default, LabelAngleOffset=0, LabelSep=1,
- LabelRefPt=c, MarkAngle=undef, MarkAngleType=default, MarkAngleRadius=.4,
+ LabelRefPt=c, MarkAngle=undef, MarkAngleType=default, MarkAngleRadius=.4,
HomCoef=.5, RotAngle=60,
CurveType=none, TransformLabel=none, Central at Sym=false, DrawCirABC=true,
CodeFig=false, CodeFigColor=cyan, CodeFigStyle=dashed, CodeFigA=undef,
@@ -228,7 +229,7 @@
\edef\psk at PosAngle{\expandafter\PstParamListLasts\OldPosAngle,undef/}%
\edef\psk at PointSymbol{\expandafter\PstParamListLasts\OldPointSymbol,undef/}}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%% create a point with an associated node,
+%% Create a point with an associated node,
%% #1 -> options
%% #2 -> coordinates
%% #3 -> node name
@@ -247,7 +248,7 @@
\@ifnextchar(\Pst at Geonode@ii{\pst at MngTransformCurve\endgroup}}% DR 22032005
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%% create a point with an associated node, in a new
+%% Create a point with an associated node, in a new
%% landmark
%% #1 -> options
%% #2 -> coordinates
@@ -458,7 +459,7 @@
\ifx\psk at fillstylename\@none\else
\pscustom[linestyle=none,linewidth=0.01pt,arrows=-]{%
\psline(0,0)
- \psarc(0,0){\psk at MarkAngleRadius}{(#2)}{(#4)}
+ \psarc(0,0){\psk at MarkAngleRadius}{(#2)}{(#4)}
\psline(0,0)
}
\fi
@@ -567,49 +568,70 @@
%% #2 #3 -> 2 nodes defining the line
\def\pstLineAB{\ncline}%@ifnextchar[\Pst at LineAB{\Pst at LineAB[]}}%
%\def\Pst at LineAB[#1]#2#3{\ncline[#1]{#2}{#3}}%
-%% #2 #3 -> 2 nodes defining the center and a point on the circle
-\def\pstCircleOA{\pst at object{pstCircleOA}}%
-\def\pstCircleOA at i#1#2{%
- \bgroup\use at par%
- \rput(#1){%
- \begin at ClosedObj
+%
+%% \pstCircleOA[Options]{O}{A}[angleA][angleB]
+%% Draw a Circle with center O from angleA to angleB, going counter clockwise.
+%% The circle is defined by center $O$ and the other node $A$ on the circle,
+%% or the center $O$ and the given Radius/Diameter in options.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the node A on the circle or empty if you setup Radius/Diameter
+%% #4 -> [input] optional. start angle from angleA to angleB, going counter clockwise.
+%% #5 -> [input] optional, start angle from angleA to angleB, going counter clockwise.
+\def\pstCircleOA{\@ifnextchar[\Pst at CircleOA{\Pst at CircleOA[]}}
+\def\Pst at CircleOA[#1]#2#3{%
+ \begingroup
+ \psset{#1}%
+ \def\pst at circle@center{#2}
+ \def\pst at circle@node{#3}
+ \@ifnextchar[\pstCircleOA at i{\pstCircleOA at i[0][360]}}%
+\def\pstCircleOA at i[#1][#2]{%
+ \rput(\pst at circle@center){%
+ \begin at OpenObj
\def\pst at linetype{4}%
\addto at pscode{%
tx at EcldDict begin
- /N@#1 GetNode
+ /N@\pst at circle@center\space GetNode
\ifx\psk at Radius\@none
\ifx\psk at Diameter\@none
- 2 copy /N@#2 GetNode ABDist
- \else\psk at Diameter 2 div
+ 2 copy /N@\pst at circle@node\space GetNode ABDist
+ \else\psk at Diameter\space 2 div
\fi
\else\psk at Radius\space
\fi
end
- \psk at dimen CLW mul sub 0 360 arc closepath}%
+ %\psk at dimen CLW mul sub 0 360 arc closepath}%
+ #1 #2 arc}%
\showpointsfalse
- \end at ClosedObj
+ \end at OpenObj
}%
- \egroup%
+ \endgroup%
}%
%% #2 #3 -> 2 nodes defining a diameter of the circle
-\def\pstCircleAB{\pst at object{pstCircleAB}}%
-\def\pstCircleAB at i#1#2{%
- \bgroup\use at par%
- \Pst at MiddleAB[PointSymbol=none, PointName=none]{#2}{#1}{@MAB}{}
- \rput(#1){%
- \begin at ClosedObj
+\def\pstCircleAB{\@ifnextchar[\Pst at CircleAB{\Pst at CircleAB[]}}
+\def\Pst at CircleAB[#1]#2#3{%
+ \begingroup
+ \psset{#1}%
+ \def\pst at circle@diameter at A{#2}
+ \def\pst at circle@diameter at B{#3}
+ \@ifnextchar[\pstCircleAB at i{\pstCircleAB at i[0][360]}}%
+\def\pstCircleAB at i[#1][#2]{%
+ \Pst at MiddleAB[PointSymbol=none, PointName=none]{\pst at circle@diameter at B}{\pst at circle@diameter at A}{PST at CIRCLE@MAB}
+ \rput(\pst at circle@diameter at A){%
+ \begin at OpenObj
\def\pst at linetype{4}%
\addto at pscode{%
tx at NodeDict begin
- tx at NodeDict /N@@MAB load GetCenter
+ tx at NodeDict /N at PST@CIRCLE at MAB load GetCenter
end
2 copy
- tx at EcldDict begin /N@#2 GetNode ABDist end
- \psk at dimen CLW mul sub 0 360 arc closepath}%
+ tx at EcldDict begin /N@\pst at circle@diameter at B\space GetNode ABDist end
+ \psk at dimen\space CLW mul sub #1 #2 arc}%
\showpointsfalse
- \end at ClosedObj
+ \end at OpenObj
}%
- \egroup%
+ \endgroup%
}%
%% #2 #3 #4 -> 3 nodes defining the center and two points on the circle
\def\pstArcOAB{\pst at object{pstArcOAB}}%
@@ -1582,29 +1604,73 @@
%% Special macros for parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
-\def\pstTriangleIC{\pst at object{pstTriangleIC}}% inner circle of a triangle
-\def\pstTriangleIC at i#1#2#3{%
+%% \pstTriangleIC[Options]{A}{B}{C}[I][D]
+%% Draw the inner circle of triangle ABC
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the node A
+%% #3 -> [input] the node B
+%% #4 -> [input] the node C
+%% #5 -> [output] optional, the inner circle center I
+%% #6 -> [output] optional, the node on inner circle
+\def\pstTriangleIC{\@ifnextchar[\Pst at TriangleIC{\Pst at TriangleIC[]}}
+\def\Pst at TriangleIC[#1]#2#3#4{%
\begingroup
- \psset{PointSymbol=none,PointName=none,linestyle=none,CodeFig=false}
- \pstBissectBAC{#3}{#1}{#2}{IC_BC}
- \pstBissectBAC{#2}{#3}{#1}{IC_AB}
- \pstInterLL{#3}{IC_AB}{#1}{IC_BC}{IC_O}
- \pstProjection{#1}{#2}{IC_O}[IC_OAB]
+ \@InitListMng %
+ % we should output none point name and symbol as default
+ % to compat with the old version.
+ \psset{PointName=none,PointSymbol=none}
+ % then comes the user local options.
+ \psset{#1}%
+ \def\pst at triangle@node at a{#2}
+ \def\pst at triangle@node at b{#3}
+ \def\pst at triangle@node at c{#4}
+ \@ifnextchar[\Pst at TriangleIC@i{\Pst at TriangleIC@i[IC_O]}}
+\def\Pst at TriangleIC@i[#1]{%
+ \def\pst at triangle@inner at center{#1}
+ \@ifnextchar[\Pst at TriangleIC@j{\Pst at TriangleIC@j[IC_OAB]}}
+\def\Pst at TriangleIC@j[#1]{%
+ \begingroup
+ \psset{PointSymbol=none,PointName=none,linestyle=none,CodeFig=false}
+ \pstBissectBAC{\pst at triangle@node at c}{\pst at triangle@node at a}{\pst at triangle@node at b}{@PST at TRIANGLE@IC_BC}
+ \pstBissectBAC{\pst at triangle@node at b}{\pst at triangle@node at c}{\pst at triangle@node at a}{@PST at TRIANGLE@IC_AB}
+ \pstInterLL{\pst at triangle@node at c}{@PST at TRIANGLE@IC_AB}{\pst at triangle@node at a}{@PST at TRIANGLE@IC_BC}{\pst at triangle@inner at center}
+ \pstProjection{\pst at triangle@node at a}{\pst at triangle@node at b}{\pst at triangle@inner at center}[#1]
+ \endgroup
+ \pstCircleOA{\pst at triangle@inner at center}{#1}
+ \Pst at ManageParamList{\pst at triangle@inner at center}%
+ \Pst at ManageParamList{#1}%
\endgroup
- \begingroup
- \use at par
- \pstCircleOA{IC_O}{IC_OAB}
- \endgroup
-}
+}%
%
-\def\pstTriangleOC{\pst at object{pstTriangleOC}}% inner circle of a triangle
-\def\pstTriangleOC at i#1#2#3{%
+%% \pstTriangleOC[Options]{A}{B}{C}[O]
+%% Draw the outer circle of triangle ABC
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the node A
+%% #3 -> [input] the node B
+%% #4 -> [input] the node C
+%% #5 -> [output] optional, the outer circle center O
+\def\pstTriangleOC{\@ifnextchar[\Pst at TriangleOC{\Pst at TriangleOC[]}}
+\def\Pst at TriangleOC[#1]#2#3#4{%
\begingroup
- \addbefore at par{PointSymbol=none,PointName=none}
- \use at par
- \pstCircleABC{#1}{#2}{#3}{OC_O}
+ % we should output none point name and symbol as default
+ % to compat with the old version.
+ \psset{PointName=none,PointSymbol=none}
+ % then comes the user local options.
+ \psset{#1}%
+ \def\pst at triangle@node at a{#2}
+ \def\pst at triangle@node at b{#3}
+ \def\pst at triangle@node at c{#4}
+ \@ifnextchar[\Pst at TriangleOC@i{\Pst at TriangleOC@i[OC_O]}}
+\def\Pst at TriangleOC@i[#1]{%
+ \begingroup
+ \psset{PointSymbol=none,PointName=none,CodeFig=false}
+ \pstCircleABC{\pst at triangle@node at a}{\pst at triangle@node at b}{\pst at triangle@node at c}{#1}
+ \endgroup
+ \Pst at geonodelabel{#1}%
\endgroup
-}
+}%
%
%% Distance between two points
\def\pstDistAB#1#2{%
@@ -1704,7 +1770,7 @@
\fi
\pnode(! %
/dec \psk at decimals\space def
- \psk at PSfont findfont \psk at fontscale scalefont setfont \pst at usecolor\pslinecolor
+ \psk at PSfont findfont \psk at fontscale scalefont setfont \pst at usecolor\pslinecolor
\ifpst at psfonts
/s1 { /Symbol findfont \psk at fontscale\space scalefont setfont } bind def
\else
@@ -1730,6 +1796,6203 @@
\ignorespaces
}
%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Here are some functions to operate the Coordinate
+%% of a given node.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%% \pstAbscissa{A}
+%% return the abscissa value of node A, which can be transformed to a new abscissa by pstricks raw code.
+%% Parameters:
+%% #1 -> [input] the input node
+\def\pstAbscissa#1{%
+ tx at EcldDict begin /N@#1 GetNode pop \pst at number\psxunit div end
+}%
+%
+%% \pstOrdinate{A}
+%% return the ordinate value of node A, which can be transformed to a new ordinate by pstricks raw code.
+%% Parameters:
+%% #1 -> [input] the input node
+\def\pstOrdinate#1{%
+ tx at EcldDict begin /N@#1 GetNode exch pop \pst at number\psyunit div end
+}%
+%
+%% \pstMoveNode[Options](dx,dy){A}{B}
+%% move node A by abscissa increment dx and ordinate increment dy to the target node B.
+%% This Macro will create the new node B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> abscissa increment
+%% #3 -> ordinate increment
+%% #4 -> orignal node name
+%% #5 -> target node name
+\def\pstMoveNode{\@ifnextchar[\Pst at MoveNode{\Pst at MoveNode[]}}
+\def\Pst at MoveNode[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at MoveNode@i}
+\def\Pst at MoveNode@i(#1,#2)#3#4{%
+ \pnode(! \pstAbscissa{#3} #1 add \pstOrdinate{#3} #2 add){#4}%
+ \Pst at geonodelabel{#4}%
+ \endgroup%
+}%
+%
+%% \pstLine[Options]{node}{node}
+%% \pstLine[Options]{node}(coor)
+%% \pstLine[Options](coor){node}
+%% \pstLine[Options](coor)(coor)
+%% Create a new line with two nodes, or two coordinate or one node and one coordinate. This macro is similar with \pstLineAB, but more compatible.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the node or coordinate
+%% #3 -> [input] the node or coordinate
+\def\pstLine{\@ifnextchar[\Pst at Line{\Pst at Line[]}}
+\def\Pst at Line[#1]{%
+ \begingroup
+ \psset{#1}%
+ \@ifnextchar(\Pst at Line@i{\Pst at Line@j}}
+\def\Pst at Line@i(#1){%
+ \pnode(#1){@PSTLINE at AUXNODE@A}
+ \@ifnextchar(\Pst at Line@ii{\Pst at Line@jj}}
+\def\Pst at Line@j#1{%
+ \pnode(#1){@PSTLINE at AUXNODE@A}
+ \@ifnextchar(\Pst at Line@ii{\Pst at Line@jj}}
+\def\Pst at Line@ii(#1){%
+ \pnode(#1){@PSTLINE at AUXNODE@B}
+ \ncline{@PSTLINE at AUXNODE@A}{@PSTLINE at AUXNODE@B}
+ \endgroup}
+\def\Pst at Line@jj#1{%
+ \pnode(#1){@PSTLINE at AUXNODE@B}
+ \ncline{@PSTLINE at AUXNODE@A}{@PSTLINE at AUXNODE@B}
+ \endgroup}%
+%
+%% \pstLineAA[Options]{A}{angle}{B}
+%% \pstLineAS[Options]{A}{gradient}{B}
+%% Create a new line with Point A and the slope angle or the gradient. This macro will create the new node B on the line.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the point A on the line
+%% #3 -> [input] the slope angle or the gradient
+%% #4 -> [output] the new node B on the line
+\def\pstLineAA{\@ifnextchar[\Pst at LineAA{\Pst at LineAA[]}}
+\def\Pst at LineAA[#1]{%
+ \begingroup
+ \psset{#1}%
+ \@ifnextchar(\Pst at LineAA@i{\Pst at LineAA@j}}
+\def\Pst at LineAA@i(#1)#2#3{%
+ \pnode(#1){@PSTLINEASAUX}
+ \Pst at LineAA@k{@PSTLINEASAUX}{#2}{#3}}
+\def\Pst at LineAA@j#1#2#3{%
+ \pnode(#1){@PSTLINEASAUX}
+ \Pst at LineAA@k{@PSTLINEASAUX}{#2}{#3}}
+\def\Pst at LineAA@k#1#2#3{%
+ \pnode(!
+ \pstAbscissa{#1} #2 cos add
+ \pstOrdinate{#1} #2 sin add
+ ){#3}
+ \Pst at geonodelabel{#3}%
+ \pstLine{#1}{#3}
+ \endgroup%
+}%
+%
+\def\pstLineAS{\@ifnextchar[\Pst at LineAS{\Pst at LineAS[]}}
+\def\Pst at LineAS[#1]{%
+ \begingroup
+ \psset{#1}%
+ \@ifnextchar(\Pst at LineAS@i{\Pst at LineAS@j}}
+\def\Pst at LineAS@i(#1)#2#3{%
+ \pnode(#1){@PSTLINEASAUX}
+ \Pst at LineAS@k{@PSTLINEASAUX}{#2}{#3}}
+\def\Pst at LineAS@j#1#2#3{%
+ \pnode(#1){@PSTLINEASAUX}
+ \Pst at LineAS@k{@PSTLINEASAUX}{#2}{#3}}
+\def\Pst at LineAS@k#1#2#3{%
+ \pnode(!
+ \pstAbscissa{#1} 1 add
+ \pstOrdinate{#1} #2 add
+ ){#3}
+ \Pst at geonodelabel{#3}%
+ \pstLine{#1}{#3}
+ \endgroup%
+}%
+%
+%% \pstLineAbsNode[Options]{A}{B}{$x_0$}{C}
+%% Create a new node C on the line AB whose abscissa is the given value $x_0$.
+%% You can input $x_0$ as any number(e.g, 2.0), and use \pscalculate{} to generate the value,
+%% or use \pstAbscissa to get the abscissa of any node.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the line point A
+%% #3 -> [input] the line point B
+%% #4 -> [input] the abscissa value
+%% #5 -> [output] the target node name
+\def\pstLineAbsNode{\@ifnextchar[\Pst at LineAbsNode{\Pst at LineAbsNode[]}}
+\def\Pst at LineAbsNode[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at LineAbsNode@i}
+\def\Pst at LineAbsNode@i#1#2#3#4{%
+ \pnode(! #3 0){@LINEABSAUXA#1#2}\pnode(! #3 1){@LINEABSAUXB#1#2}%
+ \pstInterLL{#1}{#2}{@LINEABSAUXA#1#2}{@LINEABSAUXB#1#2}{#4}%
+ \Pst at geonodelabel{#4}%
+ \endgroup%
+}%
+%
+%% \pstLineOrdNode[Options]{A}{B}{$y_0$}{C}
+%% Create a new node C on the line AB whose ordinate is the given value $y_0$.
+%% You can input $y_0$ as any number(e.g, 2.0), and use \pscalculate{} to generate the value,
+%% or use \pstOrdinate to get the ordinate of any node.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the line point A
+%% #3 -> [input] the line point B
+%% #4 -> [input] the ordinate value
+%% #5 -> [output] the target node name
+\def\pstLineOrdNode{\@ifnextchar[\Pst at LineOrdNode{\Pst at LineOrdNode[]}}
+\def\Pst at LineOrdNode[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at LineOrdNode@i}
+\def\Pst at LineOrdNode@i#1#2#3#4{%
+ \pnode(! 0 #3){@LINEORDAUXA#1#2}\pnode(! 1 #3){@LINEORDAUXB#1#2}%
+ \pstInterLL{#1}{#2}{@LINEORDAUXA#1#2}{@LINEORDAUXB#1#2}{#4}%
+ \Pst at geonodelabel{#4}%
+ \endgroup%
+}%
+%
+%% \pstCircleAbsNode[Options]{O}{A}{$x_0$}{C}{D}
+%% Create the new nodes C and D on the Circle O whose abscissas are the given value $x_0$.
+%% The circle O is defined by center O and point A on the circle or Radius in parameter.
+%% You can input $x_0$ as any number(e.g, 2.0), and use \pscalculate{} to generate the value,
+%% or use \pstAbscissa to get the abscissa of any node.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the circle point A or empty with Radius parameter
+%% #4 -> [input] the abscissa value
+%% #5 -> [output] the target node name
+%% #6 -> [output] the target node name
+\def\pstCircleAbsNode{\@ifnextchar[\Pst at CircleAbsNode{\Pst at CircleAbsNode[]}}
+\def\Pst at CircleAbsNode[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at CircleAbsNode@i}
+\def\Pst at CircleAbsNode@i#1#2#3#4#5{%
+ \pnode(! #3 0){@LINEABSAUXA#1#2}\pnode(! #3 1){@LINEABSAUXB#1#2}%
+ \pstInterLC{@LINEABSAUXA#1#2}{@LINEABSAUXB#1#2}{#1}{#2}{#4}{#5}%
+ \endgroup%
+}%
+%
+%% \pstCircleOrdNode[Options]{O}{A}{$y_0$}{C}{D}
+%% Create the new nodes C and D on the Circle O whose ordinates are the given value.
+%% The circle O is defined by center O and point A on the circle or Radius in parameter.
+%% You can input $y_0$ as any number(e.g, 2.0), and use \pscalculate{} to generate the value,
+%% or use \pstOrdinate to get the ordinate of any node.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the circle point A or empty with Radius parameter
+%% #4 -> [input] the ordinate value
+%% #5 -> [output] the target node name
+%% #6 -> [output] the target node name
+\def\pstCircleOrdNode{\@ifnextchar[\Pst at CircleOrdNode{\Pst at CircleOrdNode[]}}
+\def\Pst at CircleOrdNode[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at CircleOrdNode@i}
+\def\Pst at CircleOrdNode@i#1#2#3#4#5{%
+ \pnode(! 0 #3){@LINEORDAUXA#1#2}\pnode(! 1 #3){@LINEORDAUXB#1#2}%
+ \pstInterLC{@LINEORDAUXA#1#2}{@LINEORDAUXB#1#2}{#1}{#2}{#4}{#5}%
+ \endgroup%
+}%
+%
+%% \pstCircleRotNode[Options]{O}{A}{X}
+%% Create a new node X on the Circle O whose RotAngle is the given value.
+%% The circle O is defined by center O and point A on the circle or Radius in parameter.
+%% If you not set RotAngle, the default value is $60^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the circle point A or empty with Radius parameter
+%% #4 -> [output] the target node name
+\def\pstCircleRotNode{\@ifnextchar[\Pst at CircleRotNode{\Pst at CircleRotNode[]}}
+\def\Pst at CircleRotNode[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at CircleRotNode@i}
+\def\Pst at CircleRotNode@i#1#2#3{%
+ \pnode(!
+ tx at EcldDict begin
+ /N@#1 GetNode
+ \ifx\psk at Radius\@none
+ \ifx\psk at Diameter\@none
+ 2 copy /N@#2 GetNode ABDist
+ \else\psk at Diameter 2 div
+ \fi
+ \else\psk at Radius\space
+ \fi
+ end
+ \psk at RotAngle\space sin \psk at RotAngle\space cos %
+ 2 index mul 4 index add \pst at number\psxunit\space div %
+ 5 1 roll mul add \pst at number\psyunit\space div exch pop%
+ ){#3}%
+ \Pst at geonodelabel{#3}%
+ \endgroup%
+}%
+%
+%% \pstCircleTangentLine[Options]{O}{A}{T}
+%% Draw the tangent line from A on the circle, and give the other node B on the line.
+%% The circle O is defined by center O and point A on the circle.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the circle point A.
+%% #4 -> [output] the target node name on the tangent line
+\def\pstCircleTangentLine{\@ifnextchar[\Pst at CircleTangentLine{\Pst at CircleTangentLine[]}}
+\def\Pst at CircleTangentLine[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at CircleTangentLine@i}
+\def\Pst at CircleTangentLine@i#1#2#3{%
+ \pst at getcoor{#1}\pst at tempO%
+ \pst at getcoor{#2}\pst at tempA%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ 0 index 3 index sub abs 1E-5 lt { % if y1=y0
+ 1 index 1 index 1 add
+ 6 2 roll pop pop pop pop
+ } {
+ 1 index 4 index sub abs 1E-5 lt { % if x1=x0
+ 1 index 1 add 1 index
+ 6 2 roll pop pop pop pop
+ } {
+ 1 index 4 index sub % x1-x0
+ 1 index 4 index sub % y1-y0
+ div neg 1 index add % -(x1-x0)/(y1-y0) + y1
+ 2 index 1 add % x=x1+1
+ exch 6 2 roll pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#3}%
+ \Pst at geonodelabel{#3}%
+ \pstLine{#2}{#3}
+ \endgroup%
+}%
+%
+%% \pstCircleTangentNode[Options]{O}{A}{P}{T1}{T2}
+%% Draw the tangent line from P out of the circle A(O), and give the two tangent nodes T1/T2 on the line.
+%% The circle O is defined by center O and point A on the circle or Radius in parameter.
+%% Suppose the coordinate of tangent node is (x,y), and node T is (a,b), the circle radius is r, then we have
+%% $$(y-y0)(y-b)+(x-x0)(x-a)=0$$
+%% but (x,y) is on the circle, so we have
+%% $$x=rcos\theta+x_0, y=rcos\theta+y_0$$
+%% then
+%% $$(x_0-a)\cos\theta+(b-y_0)\sin\theta=r$$
+%% if we take $e=a-x_0$, $f=b-y_0$, we have
+%% $$(e^2+f^2)\sin^\theta-2rf\sin\theta+r^2-e^2=0$$
+%% so
+%% $$\sin\theta=\dfrac{rf\pm{}e\sqrt{e^2+f^2-r^2}}{e^2+f^2}$$
+%% $$\cos\theta=\dfrac{re\mp{}f\sqrt{e^2+f^2-r^2}}{e^2+f^2}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the circle point A or empty with Radius parameter
+%% #4 -> [input] the node name T out of circle
+%% #5 -> [output] the first target name on the circle
+%% #6 -> [output] the second target name on the circle
+\def\pstCircleTangentNode{\@ifnextchar[\Pst at CircleTangentNode{\Pst at CircleTangentNode[]}}
+\def\Pst at CircleTangentNode[#1]{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \Pst at CircleTangentNode@i}
+\def\Pst at CircleTangentNode@i#1#2#3#4#5{%
+ \pst at getcoor{#1}\pst at tempO%
+ \ifx\relax#2\relax\else\pst at getcoor{#2}\pst at tempV\fi%
+ \pst at getcoor{#3}\pst at tempT%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempT \tx at UserCoor % a,b
+ \ifx\psk at Radius\@none
+ \ifx\psk at Diameter\@none
+ \pst at tempV \tx at UserCoor % nx,ny
+ 4 index sub dup mul exch 5 index sub dup mul add sqrt
+ \else\psk at Diameter\space 2 div \pst at number\psxunit div
+ \fi
+ \else\psk at Radius\space \pst at number\psxunit div
+ \fi % r
+ 2 index 5 index sub % e=a-x0
+ 2 index 5 index sub % f=b-y0
+ 1 index dup mul 1 index dup mul add % e^2+f^2
+ 0 index 4 index dup mul sub
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop 0 0
+ } {
+ sqrt % sqrt(e^2+f^2-r^2)
+ 4 index 3 index mul 4 index 2 index mul sub 2 index div % \sin\theta
+ 5 index 5 index mul 4 index 3 index mul add 3 index div % \cos\theta
+ 6 index 1 index mul 11 index add % x_1
+ 7 index 3 index mul 11 index add % y_1
+ 13 2 roll pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempT \tx at UserCoor % a,b
+ \ifx\psk at Radius\@none
+ \ifx\psk at Diameter\@none
+ \pst at tempV \tx at UserCoor % nx,ny
+ 4 index sub dup mul exch 5 index sub dup mul add sqrt
+ \else\psk at Diameter\space 2 div \pst at number\psxunit div
+ \fi
+ \else\psk at Radius\space \pst at number\psxunit div
+ \fi % r
+ 2 index 5 index sub % e=a-x0
+ 2 index 5 index sub % f=b-y0
+ 1 index dup mul 1 index dup mul add % e^2+f^2
+ 0 index 4 index dup mul sub
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop 0 0
+ } {
+ sqrt % sqrt(e^2+f^2-r^2)
+ 4 index 3 index mul 4 index 2 index mul add 2 index div % \sin\theta
+ 5 index 5 index mul 4 index 3 index mul sub 3 index div % \cos\theta
+ 6 index 1 index mul 11 index add % x_1
+ 7 index 3 index mul 11 index add % y_1
+ 13 2 roll pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLine{#3}{#4}
+ \pstLine{#3}{#5}
+ \endgroup%
+}%
+%
+%% \pstCircleExternalCommonTangent[Options]{O1}{A}{O2}{B}{T1}{T2}{T3}{T4}
+%% Find the external common tangent lines of the circle A(O1) and B(O2), mark the two tangent nodes T1/T2 on circle A(O1),
+%% and the two tangent nodes T3/T4 on circle B(O2).
+%% The circle A(O1) is defined by center O1 and point A on the circle or RadiusA/DiameterA in parameter.
+%% The circle B(O1) is defined by center O2 and point B on the circle or RadiusB/DiameterB in parameter.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the circle point A or empty with Radius parameter
+%% #4 -> [input] the node name T out of circle
+%% #5 -> [output] the first target name on the circle
+%% #6 -> [output] the second target name on the circle
+\def\pstCircleExternalCommonTangent{\@ifnextchar[\Pst at CircleExternalCommonTangent{\Pst at CircleExternalCommonTangent[]}}
+\def\Pst at CircleExternalCommonTangent[#1]{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \Pst at CircleExternCommonTangent@i}
+\def\Pst at CircleExternCommonTangent@i#1#2#3#4#5#6#7#8{%
+ % use edef to save the second Radius or Diameter.
+ \edef\pst at RadiusB@temp{\psk at RadiusB}
+ \edef\pst at DiameterB@temp{\psk at DiameterB}
+ % use RadiusA or DiameterA to find the intersection of A(O1) and O1O2.
+ \ifx\psk at RadiusA\@undef
+ \ifx\psk at DiameterA\@undef\relax\else
+ \psset{Diameter=\psk at DiameterA}
+ \fi
+ \else\psset{Radius=\psk at RadiusA}\fi
+ \pstInterLC[PointName=none,PointSymbol=none]{#1}{#3}{#1}{#2}{@PST at CIRCLE@INTER at A}{@PST at CIRCLE@INTER at C}
+ % clear the used options
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ % use RadiusB or DiameterB to find the intersection of B(O2) and O1O2.
+ \ifx\psk at RadiusB\@undef
+ \ifx\psk at DiameterB\@undef\relax\else
+ \psset{Diameter=\psk at DiameterB}
+ \fi
+ \else\psset{Radius=\psk at RadiusB}\fi
+ \pstInterLC[PointName=none,PointSymbol=none]{#1}{#3}{#3}{#4}{@PST at CIRCLE@INTER at B}{@PST at CIRCLE@INTER at D}
+ % clear the used options
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ \pstRotation[RotAngle=90,PointName=none,PointSymbol=none]{#1}{@PST at CIRCLE@INTER at A}[@PST at CIRCLE@INTER at AA]
+ \pstRotation[RotAngle=90,PointName=none,PointSymbol=none]{#3}{@PST at CIRCLE@INTER at B}[@PST at CIRCLE@INTER at BB]
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#3}{@PST at CIRCLE@INTER at AA}{@PST at CIRCLE@INTER at BB}{@PST at CIRCLE@INTER at K}
+ \pstMiddleAB[PointName=none,PointSymbol=none]{#1}{@PST at CIRCLE@INTER at K}{@PST at CIRCLE@CENTER at X}
+ \pstMiddleAB[PointName=none,PointSymbol=none]{#3}{@PST at CIRCLE@INTER at K}{@PST at CIRCLE@CENTER at Y}
+ % use RadiusA or DiameterA to find the intersection of A(O1) and K(X).
+ % Note that we should set Radius to @none and set RadiusB and DiameterB to @undef.
+ \ifx\psk at RadiusA\@undef\relax\else\psset{RadiusA=\psk at RadiusA}\fi
+ \ifx\psk at RadiusA\@undef
+ \ifx\psk at DiameterA\@undef\relax\else
+ \psset{DiameterA=\psk at DiameterA}
+ \fi
+ \else\psset{RadiusA=\psk at RadiusA}\fi
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ \let\psk at RadiusB\@undef\let\psk at DiameterB\@undef
+ \pstInterCC[PointName=none,PointSymbol=none]{#1}{#2}{@PST at CIRCLE@CENTER at X}{#1}{#5}{#6}
+ % use RadiusB or DiameterB to find the intersection of B(O2) and K(Y).
+ % Note that we should set Radius to @none and set RadiusB and DiameterB to @undef.
+ \ifx\pst at RadiusB@temp\@undef
+ \ifx\pst at DiameterB@temp\@undef\relax\else
+ \psset{DiameterA=\pst at DiameterB@temp}
+ \fi
+ \else\psset{RadiusA=\pst at RadiusB@temp}\fi
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ \let\psk at RadiusB\@undef\let\psk at DiameterB\@undef
+ \pstInterCC[PointName=none,PointSymbol=none]{#3}{#4}{@PST at CIRCLE@CENTER at Y}{#3}{#7}{#8}
+ % CodeFig:
+ %\pstLine{#1}{@PST at CIRCLE@INTER at AA}
+ %\pstLine{#3}{@PST at CIRCLE@INTER at BB}
+ %\pstLine{#1}{@PST at CIRCLE@INTER at K}
+ %\pstLine{@PST at CIRCLE@INTER at K}{@PST at CIRCLE@INTER at AA}
+ %\let\psk at Radius\@none\let\psk at Diameter\@none
+ %\pstCircleOA[linestyle=dashed,linecolor=gray!30]{@PST at CIRCLE@CENTER at X}{#1}
+ %\pstCircleOA[linestyle=dashed,linecolor=blue!30]{@PST at CIRCLE@CENTER at Y}{#3}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \Pst at ManageParamList{#8}%
+ \endgroup%
+}%
+%
+%% \pstCircleInternalCommonTangent[Options]{O1}{A}{O2}{B}{T1}{T2}{T3}{T4}
+%% Find the internal common tangent lines of the circle A(O1) and B(O2), mark the two tangent nodes T1/T2 on circle A(O1),
+%% and the two tangent nodes T3/T4 on circle B(O2).
+%% The circle A(O1) is defined by center O1 and point A on the circle or RadiusA/DiameterA in parameter.
+%% The circle B(O1) is defined by center O2 and point B on the circle or RadiusB/DiameterB in parameter.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the circle center O
+%% #3 -> [input] the circle point A or empty with Radius parameter
+%% #4 -> [input] the node name T out of circle
+%% #5 -> [output] the first target name on the circle
+%% #6 -> [output] the second target name on the circle
+\def\pstCircleInternalCommonTangent{\@ifnextchar[\Pst at CircleInternalCommonTangent{\Pst at CircleInternalCommonTangent[]}}
+\def\Pst at CircleInternalCommonTangent[#1]{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \Pst at CircleInternalCommonTangent@i}
+\def\Pst at CircleInternalCommonTangent@i#1#2#3#4#5#6#7#8{%
+ % use edef to save the second Radius or Diameter.
+ \edef\pst at RadiusB@temp{\psk at RadiusB}
+ \edef\pst at DiameterB@temp{\psk at DiameterB}
+ % use RadiusA or DiameterA to find the intersection of A(O1) and O1O2.
+ \ifx\psk at RadiusA\@undef
+ \ifx\psk at DiameterA\@undef\relax\else
+ \psset{Diameter=\psk at DiameterA}
+ \fi
+ \else\psset{Radius=\psk at RadiusA}\fi
+ \pstInterLC[PointName=none,PointSymbol=none]{#1}{#3}{#1}{#2}{@PST at CIRCLE@INTER at A}{@PST at CIRCLE@INTER at C}
+ % clear the used options
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ % use RadiusB or DiameterB to find the intersection of B(O2) and O1O2.
+ \ifx\psk at RadiusB\@undef
+ \ifx\psk at DiameterB\@undef\relax\else
+ \psset{Diameter=\psk at DiameterB}
+ \fi
+ \else\psset{Radius=\psk at RadiusB}\fi
+ \pstInterLC[PointName=none,PointSymbol=none]{#1}{#3}{#3}{#4}{@PST at CIRCLE@INTER at B}{@PST at CIRCLE@INTER at D}
+ % clear the used options
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ \pstRotation[RotAngle=90,PointName=none,PointSymbol=none]{#1}{@PST at CIRCLE@INTER at A}[@PST at CIRCLE@INTER at AA]
+ \pstRotation[RotAngle=-90,PointName=none,PointSymbol=none]{#3}{@PST at CIRCLE@INTER at B}[@PST at CIRCLE@INTER at BB]
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#3}{@PST at CIRCLE@INTER at AA}{@PST at CIRCLE@INTER at BB}{@PST at CIRCLE@INTER at K}
+ \pstMiddleAB[PointName=none,PointSymbol=none]{#1}{@PST at CIRCLE@INTER at K}{@PST at CIRCLE@CENTER at X}
+ \pstMiddleAB[PointName=none,PointSymbol=none]{#3}{@PST at CIRCLE@INTER at K}{@PST at CIRCLE@CENTER at Y}
+ % use RadiusA or DiameterA to find the intersection of A(O1) and K(X).
+ % Note that we should set Radius to @none and set RadiusB and DiameterB to @undef.
+ \ifx\psk at RadiusA\@undef\relax\else\psset{RadiusA=\psk at RadiusA}\fi
+ \ifx\psk at RadiusA\@undef
+ \ifx\psk at DiameterA\@undef\relax\else
+ \psset{DiameterA=\psk at DiameterA}
+ \fi
+ \else\psset{RadiusA=\psk at RadiusA}\fi
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ \let\psk at RadiusB\@undef\let\psk at DiameterB\@undef
+ \pstInterCC[PointName=none,PointSymbol=none]{#1}{#2}{@PST at CIRCLE@CENTER at X}{#1}{#5}{#6}
+ % use RadiusB or DiameterB to find the intersection of B(O2) and K(Y).
+ % Note that we should set Radius to @none and set RadiusB and DiameterB to @undef.
+ \ifx\pst at RadiusB@temp\@undef
+ \ifx\pst at DiameterB@temp\@undef\relax\else
+ \psset{DiameterA=\pst at DiameterB@temp}
+ \fi
+ \else\psset{RadiusA=\pst at RadiusB@temp}\fi
+ \let\psk at Radius\@none\let\psk at Diameter\@none
+ \let\psk at RadiusB\@undef\let\psk at DiameterB\@undef
+ \pstInterCC[PointName=none,PointSymbol=none]{#3}{#4}{@PST at CIRCLE@CENTER at Y}{#3}{#7}{#8}
+ % CodeFig:
+ %\pstLine{#1}{@PST at CIRCLE@INTER at AA}
+ %\pstLine{#3}{@PST at CIRCLE@INTER at BB}
+ %\pstLine{#1}{@PST at CIRCLE@INTER at K}
+ %\pstLine{@PST at CIRCLE@INTER at K}{@PST at CIRCLE@INTER at AA}
+ %\let\psk at Radius\@none\let\psk at Diameter\@none
+ %\pstCircleOA[linestyle=dashed,linecolor=gray!30]{@PST at CIRCLE@CENTER at X}{#1}
+ %\pstCircleOA[linestyle=dashed,linecolor=blue!30]{@PST at CIRCLE@CENTER at Y}{#3}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \Pst at ManageParamList{#8}%
+ \endgroup%
+}%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Here are some functions to operate the conic curves.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% 1. Standard Ellipse with coordinate translation
+%% ----------------------------------------------------------
+%% The Ellipse E is defined by center O, the half of the major axis $max(abs(a),abs(b))$,
+%% the half of the minor axis $min(abs(a),abs(b))$, the equation as following:
+%% \begin{equation}\label{FunctionOfStandardEllipse}
+%% \dfrac{(x-x_o)^2}{a^2}+\dfrac{(y-y_o)^2}{b^2}=1
+%% \end{equation}
+%% Sometimes we use the parametric function of the ellipse:
+%% \begin{equation}\label{ParametricFunctionOfEllipse}
+%% \left\{\begin{array}{l}
+%% x=a\cos\alpha+x_o\\
+%% y=b\sin\alpha+y_0
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstEllipse[Options](O)(a,b)[angleA][angleB]
+%% Draw a Ellipse with center O from angleA to angleB, going counter clockwise,
+%% the half of the major axis $max(abs(a),abs(b))$, and the half of the minor axis $min(abs(a),abs(b))$.
+%% This macro is like \psellipse and \psellipticarc in \PST.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] start angle from angleA to angleB, going counter clockwise.
+%% #5 -> [input] start angle from angleA to angleB, going counter clockwise.
+\def\pstEllipse{\@ifnextchar[\Pst at Ellipse{\Pst at Ellipse[]}}
+\def\Pst at Ellipse[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at Ellipse@i{\Pst at Ellipse@i[0][360]}}%
+\def\Pst at Ellipse@i[#1][#2]{%
+ \parametricplot{#1}{#2}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 3 index 2 index t cos mul add % x0+a\cos{t}
+ 3 index 2 index t sin mul add % y0+b\sin{t}
+ 6 2 roll pop pop pop pop
+ }%
+ \endgroup%
+}%
+%
+%% \pstEllipseNode[Options](O)(a,b){t}{P}
+%% Create a new node P on the Ellipse E whose parameter is the given value $t$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the parametric argument t.
+%% #5 -> [output] the target node name.
+\def\pstEllipseNode{\@ifnextchar[\Pst at pstEllipseNode{\Pst at pstEllipseNode[]}}
+\def\Pst at pstEllipseNode[#1](#2)(#3)#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 3 index 2 index #4 cos mul add % x0+a\cos{t}
+ 3 index 2 index #4 sin mul add % y0+b\sin{t}
+ 6 2 roll pop pop pop pop
+ ){#5}%
+ \Pst at geonodelabel{#5}%
+ \endgroup%
+}%
+%
+%% \pstEllipseRotNode[Options](O)(a,b){P}
+%% Create a new node P on the Ellipse E whose RotAngle is the given value.
+%% If you not set RotAngle, the default value is $60^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [output] the target node name
+\def\pstEllipseRotNode{\@ifnextchar[\Pst at EllipseRotNode{\Pst at EllipseRotNode[]}}
+\def\Pst at EllipseRotNode[#1]{%
+ \begingroup
+ \psset{#1}%
+ \Pst at EllipseRotNode@i}
+\def\Pst at EllipseRotNode@i(#1)(#2)#3{%
+ \pst at getcoor{#1}\pst at tempO%
+ \pst at getcoor{#2}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 3 index 2 index \psk at RotAngle\space cos mul add % x0+a\cos{t}
+ 3 index 2 index \psk at RotAngle\space sin mul add % y0+b\sin{t}
+ 6 2 roll pop pop pop pop
+ ){#3}%
+ \Pst at geonodelabel{#3}%
+ \endgroup%
+}%
+%
+%% \pstEllipseAbsNode[Options](O)(a,b){x_1}{A}{B}
+%% Create the two nodes A and B whose abscissas are the given value $x_1$ on the Ellipse E.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the given abscissa value $x_1$
+%% #5 -> [output] the first target node name $A$
+%% #6 -> [output] the second target node name $B$
+\def\pstEllipseAbsNode{\@ifnextchar[\Pst at EllipseAbsNode{\Pst at EllipseAbsNode[]}}
+\def\Pst at EllipseAbsNode[#1]{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \Pst at EllipseAbsNode@i}
+\def\Pst at EllipseAbsNode@i(#1)(#2)#3#4#5{%
+ \pst at getcoor{#1}\pst at tempO%
+ \pst at getcoor{#2}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #3 % x_1
+ 2 index dup mul 1 index 6 index sub dup mul sub % a^2-(x_1-x_o)^2
+ dup 0 lt {
+ pop pop pop pop pop pop 0 0
+ }{
+ sqrt 2 index mul 3 index div % b/a sqrt(a^2-(x_1-x_o)^2)
+ 4 index exch sub % y_o-ditto
+ 6 2 roll pop pop pop pop
+ } ifelse
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #3 % x_1
+ 2 index dup mul 1 index 6 index sub dup mul sub % a^2-(x_1-x_o)^2
+ dup 0 lt {
+ pop pop pop pop pop pop 0 0
+ }{
+ sqrt 2 index mul 3 index div % b/a sqrt(a^2-(x_1-x_o)^2)
+ 4 index add % y_o+ditto
+ 6 2 roll pop pop pop pop
+ } ifelse
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstEllipseOrdNode[Options](O)(a,b){y_1}{A}{B}
+%% Create the two nodes A and B whose ordinates are the given value $y_1$ on the Ellipse E.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the given ordinate value $y_1$
+%% #5 -> [output] the first target node name $A$
+%% #6 -> [output] the second target node name $B$
+\def\pstEllipseOrdNode{\@ifnextchar[\Pst at EllipseOrdNode{\Pst at EllipseOrdNode[]}}
+\def\Pst at EllipseOrdNode[#1]{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \Pst at EllipseOrdNode@i}
+\def\Pst at EllipseOrdNode@i(#1)(#2)#3#4#5{%
+ \pst at getcoor{#1}\pst at tempO%
+ \pst at getcoor{#2}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #3 % y_1
+ 1 index dup mul 1 index 5 index sub dup mul sub % b^2-(y_1-y_o)^2
+ dup 0 lt {
+ pop pop pop pop pop pop 0 0
+ }{
+ sqrt 3 index mul 2 index div % a/b sqrt(b^2-(y_1-y_o)^2)
+ 5 index exch sub % x_o-ditto
+ exch 6 2 roll pop pop pop pop
+ } ifelse
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ #3 % y_1
+ 1 index dup mul 1 index 5 index sub dup mul sub % b^2-(y_1-y_o)^2
+ dup 0 lt {
+ pop pop pop pop pop pop 0 0
+ }{
+ sqrt 3 index mul 2 index div % a/b sqrt(b^2-(y_1-y_o)^2)
+ 5 index add % x_o+ditto
+ exch 6 2 roll pop pop pop pop
+ } ifelse
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstEllipseFocusNode[Options](O)(a,b){$F_1$}{$F_2$}
+%% Get the two focus $F_1$ and $F_2$ point of the Ellipse E and create two new nodes.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [output] the left/down node name
+%% #5 -> [output] the right/up node name
+\def\pstEllipseFocusNode{\@ifnextchar[\Pst at EllipseFocusNode{\Pst at EllipseFocusNode[]}}
+\def\Pst at EllipseFocusNode[#1]{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \Pst at EllipseFocusNode@i}
+\def\Pst at EllipseFocusNode@i(#1)(#2)#3#4{%
+ \pst at getcoor{#1}\pst at tempO%
+ \pst at getcoor{#2}\pst at tempR%
+ \pnode(!
+ \pst at tempO % x0,y0
+ \pst at tempR % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index exch sub % x0-c
+ 3 index 6 2 roll pop pop pop pop % Left(x0-c,y0)
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 3 index exch sub % y0-c
+ 4 index exch 6 2 roll pop pop pop pop % Down(x0,y0-c)
+ } ifelse
+ \tx at UserCoor %
+ ){#3}%
+ \pnode(!
+ \pst at tempO % x0,y0
+ \pst at tempR % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index add % x0+c
+ 3 index 6 2 roll pop pop pop pop % Right(x0+c,y0)
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 3 index add % y0+c
+ 4 index exch 6 2 roll pop pop pop pop % Up(x0,y0+c)
+ } ifelse
+ \tx at UserCoor %
+ ){#4}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstEllipseDirectrixLine[Options](O)(a,b){Lx}{Ly}{Rx}{Ry}
+%% Get the two directrix line of the Ellipse E and create two new nodes on each of them.
+%% the nodes Lx Ly Rx Ry lie on the tangent line of the vertex on the other axis.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [output] the A node name on the left/down directrix line
+%% #5 -> [output] the B node name on the left/down directrix line
+%% #6 -> [output] the A node name on the right/up directrix line
+%% #7 -> [output] the B node name on the right/up directrix line
+\def\pstEllipseDirectrixLine{\@ifnextchar[\Pst at EllipseDirectrixLine{\Pst at EllipseDirectrixLine[]}}
+\def\Pst at EllipseDirectrixLine[#1]{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1} %
+ \Pst at EllipseDirectrixLine@i}
+\def\Pst at EllipseDirectrixLine@i(#1)(#2)#3#4#5#6{%
+ \pst at getcoor{#1}\pst at tempO%
+ \pst at getcoor{#2}\pst at tempR%
+ \pnode(!
+ \pst at tempO % x0,y0
+ \pst at tempR % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% left directrix line perpendicular to the haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 2 index dup mul exch div % a^2/c
+ 4 index exch sub % x0-a^2/c
+ 3 index 2 index sub % y0-b
+ 6 2 roll pop pop pop pop % First A(x0-a^2/c,y0-b)
+ }{% down directrix line perpendicular to the vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 1 index dup mul exch div % b^2/c
+ 3 index exch sub % y0-b^2/c
+ 4 index 3 index sub % x0-a
+ exch 6 2 roll pop pop pop pop % Fisrt A(x0-a,y0-b^2/c)
+ } ifelse
+ \tx at UserCoor %
+ ){#3}%
+ \pnode(!
+ \pst at tempO % x0,y0
+ \pst at tempR % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% left directrix line perpendicular to the haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 2 index dup mul exch div % a^2/c
+ 4 index exch sub % x0-a^2/c
+ 3 index 2 index add % y0+b
+ 6 2 roll pop pop pop pop % First B(x0-a^2/c,y0+b)
+ }{% down directrix line perpendicular to the vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 1 index dup mul exch div % b^2/c
+ 3 index exch sub % y0-b^2/c
+ 4 index 3 index add % x0+a
+ exch 6 2 roll pop pop pop pop % Fisrt B(x0+a,y0-b^2/c)
+ } ifelse
+ \tx at UserCoor %
+ ){#4}%
+ \pnode(!
+ \pst at tempO % x0,y0
+ \pst at tempR % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% right directrix line perpendicular to the haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 2 index dup mul exch div % a^2/c
+ 4 index add % x0+a^2/c
+ 3 index 2 index sub % y0-b
+ 6 2 roll pop pop pop pop % Second A(x0+a^2/c,y0-b)
+ }{% up directrix line perpendicular to the vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 1 index dup mul exch div % b^2/c
+ 3 index add % y0+b^2/c
+ 4 index 3 index sub % x0-a
+ exch 6 2 roll pop pop pop pop % Second A(x0-a,y0+b^2/c)
+ } ifelse
+ \tx at UserCoor %
+ ){#5}%
+ \pnode(!
+ \pst at tempO % x0,y0
+ \pst at tempR % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% right directrix line perpendicular to the haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 2 index dup mul exch div % a^2/c
+ 4 index add % x0+a^2/c
+ 3 index 2 index add % y0+b
+ 6 2 roll pop pop pop pop % Second B(x0+a^2/c,y0+b)
+ }{% up directrix line perpendicular to the vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 1 index dup mul exch div % b^2/c
+ 3 index add % y0+b^2/c
+ 4 index 3 index add % x0+a
+ exch 6 2 roll pop pop pop pop % Second B(x0+a,y0+b^2/c)
+ } ifelse
+ \tx at UserCoor %
+ ){#6}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \pstLineAB{#3}{#4}
+ \pstLineAB{#5}{#6}
+ \endgroup%
+}%
+%
+%% \pstEllipseLineInter[Options](O)(a,b){A}{B}{C}{D}
+%% Get the two intersection $C$ and $D$ of the Ellipse E and the given line AB.
+%% We can represent the line AB as the following function when line AB is not vertical:
+%% $$y=kx+t$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, t=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% the intersection points are:
+%% $$x1=x_o-\dfrac{a^2km+ab\sqrt{w}}{u}, x2=x_o-\dfrac{a^2km-ab\sqrt{w}}{u}$$
+%% $$y_{1,2}=kx_{1,2}+t$$
+%% where,
+%% $$u=a^2k^2+b^2, m=kx_o-y_o+t, w=u-m^2$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the node name A on the given line
+%% #5 -> [input] the node name B on the given line
+%% #6 -> [output] the first intersection node name
+%% #7 -> [output] the second intersection node name
+\def\pstEllipseLineInter{\@ifnextchar[\Pst at EllipseLineInter{\Pst at EllipseLineInter[]}}
+\def\Pst at EllipseLineInter[#1](#2)(#3)#4#5#6#7{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pst at getcoor{#4}\pst at tempA%
+ \pst at getcoor{#5}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ pop pop pop
+ 2 index dup mul 1 index 6 index sub dup mul sub % a^2-(x_1-x_o)^2
+ dup 0 lt {
+ pop pop pop pop pop pop 0 0
+ }{
+ sqrt 2 index mul 3 index div % b/a sqrt(a^2-(x_1-x_o)^2)
+ 4 index exch sub % y_o-ditto
+ 6 2 roll pop pop pop pop
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ 7 index dup mul 2 index dup mul mul 7 index dup mul add % u
+ 2 index 11 index mul 10 index sub 2 index add % m
+ 1 index 1 index dup mul sub dup 0 lt { % w
+ pop pop pop pop pop pop pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 10 index mul 9 index mul 10 index dup mul 2 index mul 5 index mul add 2 index div 12 index exch sub % x1
+ dup 5 index mul 4 index add % y1
+ 14 2 roll pop pop pop pop pop pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#6}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ pop pop pop
+ 2 index dup mul 1 index 6 index sub dup mul sub % a^2-(x_1-x_o)^2
+ dup 0 lt {
+ pop pop pop pop pop pop 0 0
+ }{
+ sqrt 2 index mul 3 index div % b/a sqrt(a^2-(x_1-x_o)^2)
+ 4 index add % y_o+ditto
+ 6 2 roll pop pop pop pop
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ 7 index dup mul 2 index dup mul mul 7 index dup mul add % u
+ 2 index 11 index mul 10 index sub 2 index add % m
+ 1 index 1 index dup mul sub dup 0 lt { % w
+ pop pop pop pop pop pop pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 10 index mul 9 index mul 10 index dup mul 2 index mul 5 index mul sub 2 index div 12 index add % x2
+ dup 5 index mul 4 index add % y2
+ 14 2 roll pop pop pop pop pop pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#7}%
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \endgroup%
+}%
+%
+%% \pstEllipsePolarNode[Options](O)(a,b){A}{B}{T}
+%% Draw the every tangent line through the point $A$ and $B$ on the Ellipse E and get the insection node T of the two tangent lines. We call T as the polar point of chord AB.
+%% We use the intersection of two polar lines to get the tangent line. The proposition can be represented as:
+%% Give chord AB on the ellipse, we draw any other two chords PQ and RS, AB and PQ intersect at I, AQ and BP intersect at X, AP and BQ intersect at Y, we call XY is the polar line of point I. Also AB and RS intersect at J, AR and BS intersect at M, AS and BR intersect at N, we call MN is the polar line of point J. Then the intersection T of XY and MN is the polar point of chord AB, i.e. TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the node name A on the ellipse
+%% #5 -> [input] the node name B on the ellipse
+%% #6 -> [output] the polar point of chord AB
+\def\pstEllipsePolarNode{\@ifnextchar[\Pst at EllipsePolarNode{\Pst at EllipsePolarNode[]}}
+\def\Pst at EllipsePolarNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \psset{#1}%
+ \pstEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=71](#2)(#3){@PSTELLIPSE at POLARAUX@P}
+ \pstEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=-31](#2)(#3){@PSTELLIPSE at POLARAUX@Q}
+ \pstEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=-122](#2)(#3){@PSTELLIPSE at POLARAUX@R}
+ \pstEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=13](#2)(#3){@PSTELLIPSE at POLARAUX@S}
+ \pstInterLL[PointName=none,PointSymbol=none]{#4}{@PSTELLIPSE at POLARAUX@Q}{#5}{@PSTELLIPSE at POLARAUX@P}{@PSTELLIPSE at POLARAUX@X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#4}{@PSTELLIPSE at POLARAUX@P}{#5}{@PSTELLIPSE at POLARAUX@Q}{@PSTELLIPSE at POLARAUX@Y}
+ \pstInterLL[PointName=none,PointSymbol=none]{#4}{@PSTELLIPSE at POLARAUX@R}{#5}{@PSTELLIPSE at POLARAUX@S}{@PSTELLIPSE at POLARAUX@M}
+ \pstInterLL[PointName=none,PointSymbol=none]{#4}{@PSTELLIPSE at POLARAUX@S}{#5}{@PSTELLIPSE at POLARAUX@R}{@PSTELLIPSE at POLARAUX@N}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PSTELLIPSE at POLARAUX@X}{@PSTELLIPSE at POLARAUX@Y}{@PSTELLIPSE at POLARAUX@M}{@PSTELLIPSE at POLARAUX@N}{#6}
+ \Pst at geonodelabel{#6}%
+ \pstLineAB{#4}{#6}
+ \pstLineAB{#5}{#6}
+ \endgroup%
+}%
+%
+%% \pstEllipseTangentNode[Options](O)(a,b){T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the Ellipse E and get the node A and B on the Ellipse.
+%% We use the following proposition to find the tangent node of T:
+%% Give point T outside of the ellipse, we draw any other two chords TPQ and TRS, PS and QR intersect at I, PR and QS intersect at X, XI and Ellipse intersect at A and B, then TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the node name T outside the ellipse
+%% #5 -> [output] the node name A on the ellipse
+%% #6 -> [output] the node name B on the ellipse
+\def\pstEllipseTangentNode{\@ifnextchar[\Pst at EllipseTangentNode{\Pst at EllipseTangentNode[]}}
+\def\Pst at EllipseTangentNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pstEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=71](#2)(#3){@PSTELLIPSE at TANGENTAUX@P0}
+ \pstEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=31](#2)(#3){@PSTELLIPSE at TANGENTAUX@R0}
+ \pstEllipseLineInter[PointName=none,PointSymbol=none](#2)(#3){#4}{@PSTELLIPSE at TANGENTAUX@P0}{@PSTELLIPSE at TANGENTAUX@P}{@PSTELLIPSE at TANGENTAUX@Q}
+ \pstEllipseLineInter[PointName=none,PointSymbol=none](#2)(#3){#4}{@PSTELLIPSE at TANGENTAUX@R0}{@PSTELLIPSE at TANGENTAUX@R}{@PSTELLIPSE at TANGENTAUX@S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PSTELLIPSE at TANGENTAUX@P}{@PSTELLIPSE at TANGENTAUX@S}{@PSTELLIPSE at TANGENTAUX@Q}{@PSTELLIPSE at TANGENTAUX@R}{@PSTELLIPSE at TANGENTAUX@I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PSTELLIPSE at TANGENTAUX@P}{@PSTELLIPSE at TANGENTAUX@R}{@PSTELLIPSE at TANGENTAUX@Q}{@PSTELLIPSE at TANGENTAUX@S}{@PSTELLIPSE at TANGENTAUX@X}
+ \pstEllipseLineInter[PointName=none,PointSymbol=none](#2)(#3){@PSTELLIPSE at TANGENTAUX@X}{@PSTELLIPSE at TANGENTAUX@I}{#5}{#6}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \pstLineAB{#4}{#5}
+ \pstLineAB{#4}{#6}
+ \endgroup%
+}%
+%
+% 2. General Ellipse with coordinate translation and rotation
+%% ----------------------------------------------------------
+%% The General Ellipse E is defined by center O, the half of the major axis $max(abs(a),abs(b))$,
+%% the half of the minor axis $min(abs(a),abs(b))$, and the rotation angle $\theta$ of the major axis.
+%%
+%% The equation can be got from the parametric function of the ellipse \ref{ParametricFunctionOfEllipse},
+%% using the rotation transform formula:
+%% \begin{equation}\label{RotationTransformFormula}
+%% \left\{\begin{array}{l}
+%% x'=x\cos\theta-y\sin\theta\\
+%% y'=x\sin\theta+y\cos\theta
+%% \end{array}\right.
+%% \end{equation}
+%% then we have
+%% \begin{equation}
+%% \left\{\begin{array}{l}
+%% x'=(a\cos\alpha+x_o)\cos\theta-(b\sin\alpha+y_o)\sin\theta=a\cos\alpha\cos\theta-b\sin\alpha\sin\theta+x_o'\\
+%% y'=(a\cos\alpha+x_o)\sin\theta+(b\sin\alpha+y_o)\cos\theta=a\cos\alpha\sin\theta+b\sin\alpha\cos\theta+y_o'
+%% \end{array}\right.
+%% \end{equation}
+%% where the $x_o'$ and $y_o'$ are the coordinate of the given center O after rotation.
+%% So we get the parametric function of the general Ellipse with coordinate translation and rotation as following:
+%% \begin{equation}\label{ParametricFunctionOfGeneralEllipse}
+%% \left\{\begin{array}{l}
+%% x=a\cos\alpha\cos\theta-b\sin\alpha\sin\theta+x_o\\
+%% y=a\cos\alpha\sin\theta+b\sin\alpha\cos\theta+y_o
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstGeneralEllipse[Options](O)(a,b)[rotation][angleA][angleB]
+%% Draw a General Ellipse with center O from angleA to angleB, going counter clockwise,
+%% the half of the major axis $max(abs(a),abs(b))$, the half of the minor axis $min(abs(a),abs(b))$,
+%% and the rotation angle $\theta$ of the major axis.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [input] start angle from angleA to angleB, going counter clockwise.
+%% #6 -> [input] start angle from angleA to angleB, going counter clockwise.
+\def\pstGeneralEllipse{\@ifnextchar[\Pst at GeneralEllipse{\Pst at GeneralEllipse[]}}
+\def\Pst at GeneralEllipse[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipse@i{\Pst at GeneralEllipse[0]}}%
+\def\Pst at GeneralEllipse@i[#1]{%
+ \def\PST at GENERALELLIPSE@ROTATION{#1}%
+ \@ifnextchar[\Pst at GeneralEllipse@ii{\Pst at GeneralEllipse@ii[0][360]}}%
+\def\Pst at GeneralEllipse@ii[#1][#2]{%
+ \parametricplot{#1}{#2}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ \PST at GENERALELLIPSE@ROTATION\space dup cos exch sin % \cos\theta \sin\theta
+ 3 index 2 index mul % a\cos\theta
+ 3 index 2 index mul % b\sin\theta
+ 5 index 3 index mul % a\sin\theta
+ 5 index 5 index mul % b\cos\theta
+ 10 4 roll pop pop pop pop % a\cos\theta b\sin\theta a\sin\theta b\cos\theta x0 y0
+ 5 index t cos mul % a\cos\theta\cos{t}
+ 5 index t sin mul sub 2 index add % a\cos\theta\cos{t}-b\sin\theta\sin{t}+x0
+ 4 index t cos mul % a\sin\theta\cos{t}
+ 4 index t sin mul add 2 index add % a\sin\theta\cos{t}+b\cos\theta\sin{t}+y0
+ 8 2 roll pop pop pop pop pop pop
+ }
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseNode[Options](O)(a,b)[rotation]{t}{A}
+%% Get the new node A whose parameter is the given value $t$ on the General Ellipse E.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [input] the parameter value $t$.
+%% #6 -> [output] the new node name on the ellipse
+\def\pstGeneralEllipseNode{\@ifnextchar[\Pst at GeneralEllipseNode{\Pst at GeneralEllipseNode[]}}
+\def\Pst at GeneralEllipseNode[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipseNode@i{\Pst at GeneralEllipseNode@i[0]}}
+\def\Pst at GeneralEllipseNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ #2 cos #2 sin %
+ 5 index 4 index mul 2 index mul % a\cos\theta\cos\alpha
+ 5 index 4 index mul 2 index mul % b\sin\theta\sin\alpha
+ sub 8 index add % a\cos\theta\cos\alpha - b\sin\theta\sin\alpha + x_o
+ 6 index 4 index mul 3 index mul % a\sin\theta\cos\alpha
+ 6 index 6 index mul 3 index mul % b\cos\theta\sin\alpha
+ add 8 index add % a\sin\theta\cos\alpha + b\cos\theta\sin\alpha + y_o
+ 10 2 roll pop pop pop pop pop pop pop pop
+ ){#3}%
+ \Pst at geonodelabel{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseRotNode[Options,RotAngle=<degree>](O)(a,b)[rotation]{A}
+%% Get the new node A whose RotAngle is the given value on the General Ellipse E.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% If you not set RotAngle, the default value is $60^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [output] the new node name on the ellipse
+\def\pstGeneralEllipseRotNode{\@ifnextchar[\Pst at GeneralEllipseRotNode{\Pst at GeneralEllipseRotNode[]}}
+\def\Pst at GeneralEllipseRotNode[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipseRotNode@i{\Pst at GeneralEllipseRotNode@i[0]}}
+\def\Pst at GeneralEllipseRotNode@i[#1]#2{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ \psk at RotAngle\space cos \psk at RotAngle\space sin %
+ 5 index 4 index mul 2 index mul % a\cos\theta\cos\alpha
+ 5 index 4 index mul 2 index mul % b\sin\theta\sin\alpha
+ sub 8 index add % a\cos\theta\cos\alpha - b\sin\theta\sin\alpha + x_o
+ 6 index 4 index mul 3 index mul % a\sin\theta\cos\alpha
+ 6 index 6 index mul 3 index mul % b\cos\theta\sin\alpha
+ add 8 index add % a\sin\theta\cos\alpha + b\cos\theta\sin\alpha + y_o
+ 10 2 roll pop pop pop pop pop pop pop pop
+ ){#2}%
+ \Pst at geonodelabel{#2}%
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseAbsNode[Options](O)(a,b)[rotation]{x_1}{A}{B}
+%% Create the node A and B whose abbscissa are the given value $x_1$ on the General Ellipse E.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% When x equals $x_1$, we have
+%% $$y_{1,2}=gc_{1,2}+hs_{1,2}+y_o$$
+%% where,
+%% $$c_1=\dfrac{de+f\sqrt{w}}{u}, s_1=\dfrac{-df+e\sqrt{w}}{u}$$
+%% $$c_2=\dfrac{de-f\sqrt{w}}{u}, s_2=\dfrac{-df-e\sqrt{w}}{u}$$
+%% $$e=a\cos\theta,f=b\sin\theta,g=a\sin\theta,h=b\cos\theta$$
+%% $$d=x_1-x_o,u=e^2+f^2,w=u-d^2$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [input] the given abscissa value $x_1$.
+%% #6 -> [output] the first intersection node $A$
+%% #7 -> [output] the second intersection node $B$
+\def\pstGeneralEllipseAbsNode{\@ifnextchar[\Pst at GeneralEllipseAbsNode{\Pst at GeneralEllipseAbsNode[]}}
+\def\Pst at GeneralEllipseAbsNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipseAbsNode@i{\Pst at GeneralEllipseAbsNode@i[0]}}
+\def\Pst at GeneralEllipseAbsNode@i[#1]#2#3#4{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % d=x_1-x_o
+ 4 index dup mul 4 index dup mul add % u=e^2+f^2
+ 0 index 2 index dup mul sub dup 0 lt { % w=u-d^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ %% c_1=(de+f sqrt(w))/u, s_1=(-df+e sqrt(w))/u
+ 2 index 7 index mul 6 index 2 index mul add 2 index div % c1
+ 3 index 7 index mul 8 index 3 index mul exch sub 3 index div % s1
+ %% y_{1,2}=gc_{1,2}+hs_{1,2}+y_o
+ 6 index 2 index mul 6 index 2 index mul add 14 index add % y1
+ #2 exch 17 2 roll pop pop pop pop pop
+ pop pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ ){#3}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % d=x_1-x_o
+ 4 index dup mul 4 index dup mul add % u=e^2+f^2
+ 0 index 2 index dup mul sub dup 0 lt { % w=u-d^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ %% c_2=(de-f sqrt(w))/u, s_2=(-df-e sqrt(w))/u
+ 2 index 7 index mul 6 index 2 index mul sub 2 index div % c2
+ 3 index 7 index mul 8 index 3 index mul add neg 3 index div % s2
+ %% y_{1,2}=gc_{1,2}+hs_{1,2}+y_o
+ 6 index 2 index mul 6 index 2 index mul add 14 index add % y2
+ #2 exch 17 2 roll pop pop pop pop pop
+ pop pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ ){#4}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseOrdNode[Options](O)(a,b)[rotation]{y_1}{A}{B}
+%% Create the node A and B whose ordinates are the given value $y_1$ on the General Ellipse E.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% when y equals $y_1$, we have
+%% x_{1,2}=ec_{1,2}-fs_{1,2}+x_o
+%% where,
+%% $$c_1=\dfrac{dg+h\sqrt{w}}{u}, s_1=\dfrac{dh-g\sqrt{w}}{u}$$
+%% $$c_2=\dfrac{dg-h\sqrt(w}}{u}, s_2=\dfrac{dh+g\sqrt{w}}{u}$$
+%% $$e=a\cos\theta,f=b\sin\theta,g=a\sin\theta,h=b\cos\theta$$
+%% $$d=y_1-y_o,u=h^2+g^2,w=u-d^2$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [input] the given ordinate value $y_1$.
+%% #6 -> [output] the first intersection node $A$
+%% #7 -> [output] the second intersection node $B$
+\def\pstGeneralEllipseOrdNode{\@ifnextchar[\Pst at GeneralEllipseOrdNode{\Pst at GeneralEllipseOrdNode[]}}
+\def\Pst at GeneralEllipseOrdNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipseOrdNode@i{\Pst at GeneralEllipseOrdNode@i[0]}}
+\def\Pst at GeneralEllipseOrdNode@i[#1]#2#3#4{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % d=y_1-y_o
+ 2 index dup mul 2 index dup mul add % u=h^2+g^2
+ 0 index 2 index dup mul sub dup 0 lt { % w=u-d^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ %% c_1=(dg+h sqrt(w))/u, s_1=(dh-g sqrt(w))/u
+ 2 index 5 index mul 4 index 2 index mul add 2 index div % c1
+ 3 index 5 index mul 6 index 3 index mul sub 3 index div % s1
+ %% x_{1,2}=ec_{1,2}-fs_{1,2}+x_o
+ 8 index 2 index mul 8 index 2 index mul sub 15 index add % x1
+ #2 17 2 roll pop pop pop pop pop
+ pop pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ ){#3}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % d=y_1-y_o
+ 2 index dup mul 2 index dup mul add % u=h^2+g^2
+ 0 index 2 index dup mul sub dup 0 lt { % w=u-d^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ %% c_2=(dg-h sqrt(w))/u, s_2=(dh+g sqrt(w))/u
+ 2 index 5 index mul 4 index 2 index mul sub 2 index div % c2
+ 3 index 5 index mul 6 index 3 index mul add 3 index div % s2
+ %% x_{1,2}=ec_{1,2}-fs_{1,2}+x_o
+ 8 index 2 index mul 8 index 2 index mul sub 15 index add % x2
+ #2 17 2 roll pop pop pop pop pop
+ pop pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ ){#4}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseFocusNode[Options](O)(a,b)[rotation]{$F_1$}{$F_2$}
+%% Get the two focus $F_1$ and $F_2$ point of the General Ellipse E and create two new nodes.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [output] the left/down node name
+%% #6 -> [output] the right/up node name
+\def\pstGeneralEllipseFocusNode{\@ifnextchar[\Pst at GeneralEllipseFocusNode{\Pst at GeneralEllipseFocusNode[]}}
+\def\Pst at GeneralEllipseFocusNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipseFocusNode@i{\Pst at GeneralEllipseFocusNode@i[0]}}
+\def\Pst at GeneralEllipseFocusNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index 1 index #1 cos mul sub % x_o-c\cos\alpha
+ 4 index 2 index #1 sin mul sub % y_o-c\sin\alpha
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 4 index 1 index #1 sin mul add % x_o+c\sin\alpha
+ 4 index 2 index #1 cos mul sub % y_o-c\cos\alpha
+ } ifelse
+ 7 2 roll pop pop pop pop pop
+ ){#2}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index 1 index #1 cos mul add % x_o+c\cos\alpha
+ 4 index 2 index #1 sin mul add % y_o+c\sin\alpha
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 4 index 1 index #1 sin mul sub % x_o-c\sin\alpha
+ 4 index 2 index #1 cos mul add % y_o+c\cos\alpha
+ } ifelse
+ 7 2 roll pop pop pop pop pop
+ ){#3}%
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseDirectrixLine[Options](O)(a,b)[rotation]{Lx}{Ly}{Rx}{Ry}
+%% Get the two directrix line of the General Ellipse E and create two new nodes for each one.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% the nodes Lx Ly Rx Ry lie on the tangent line of the vertex on the other axis,
+%% they can be got after the orig nodes rotation $\theta$ about the center $O$, we have
+%% \begin{equation}\label{ExtensionRotationTransformFormula}
+%% \left\{\begin{array}{l}
+%% x=x_o+(x-x_o)\cos\theta-(y-y_o)\sin\theta\\
+%% y=y_o+(x-x_o)\sin\theta+(y-y_o)\cos\theta
+%% \end{array}\right.
+%% \end{equation}
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [output] the A node name on the left/down directrix line
+%% #6 -> [output] the B node name on the left/down directrix line
+%% #7 -> [output] the A node name on the right/up directrix line
+%% #8 -> [output] the B node name on the right/up directrix line
+\def\pstGeneralEllipseDirectrixLine{\@ifnextchar[\Pst at GeneralEllipseDirectrixLine{\Pst at GeneralEllipseDirectrixLine[]}}
+\def\Pst at GeneralEllipseDirectrixLine[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipseDirectrixLine@i{\Pst at GeneralEllipseDirectrixLine@i[0]}}
+\def\Pst at GeneralEllipseDirectrixLine@i[#1]#2#3#4#5{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index 3 index dup mul 2 index div sub % x0-a^2/c
+ 4 index 3 index sub
+ 7 2 roll pop pop pop % (x0-a^2/c,y0-b)
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 4 index 3 index sub % x0-a
+ 4 index 3 index dup mul 3 index div sub
+ 7 2 roll pop pop pop % (x0-a,y0-b^2/c)
+ } ifelse
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index 3 index dup mul 2 index div sub % x0-a^2/c
+ 4 index 3 index add
+ 7 2 roll pop pop pop % (x0-a^2/c,y0+b)
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 4 index 3 index add % x0+a
+ 4 index 3 index dup mul 3 index div sub
+ 7 2 roll pop pop pop % (x0+a,y0-b^2/c)
+ } ifelse
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index 3 index dup mul 2 index div add % x0+a^2/c
+ 4 index 3 index sub
+ 7 2 roll pop pop pop % (x0+a^2/c,y0-b)
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 4 index 3 index sub % x0-a
+ 4 index 3 index dup mul 3 index div add
+ 7 2 roll pop pop pop % (x0-a,y0+b^2/c)
+ } ifelse
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor % a,b
+ abs exch abs exch % |a|,|b|
+ 2 copy gt {% Focus on haxis
+ 1 index dup mul 1 index dup mul sub sqrt % c=sqrt(a^2-b^2)
+ 4 index 3 index dup mul 2 index div add % x0+a^2/c
+ 4 index 3 index add
+ 7 2 roll pop pop pop % (x0+a^2/c,y0+b)
+ }{% Focus on vaxis
+ 0 index dup mul 2 index dup mul sub sqrt % c=sqrt(b^2-a^2)
+ 4 index 3 index add % x0+a
+ 4 index 3 index dup mul 3 index div add
+ 7 2 roll pop pop pop % (x0+a,y0+b^2/c)
+ } ifelse
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#5}%
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLineAB{#2}{#3}
+ \pstLineAB{#4}{#5}
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseLineInter[Options](O)(a,b)[rotation]{A}{B}{C}{D}
+%% Get the two intersection $C$ and $D$ of the General Ellipse E and the given line AB.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% We can represent the line AB as the following function when line AB is not vertical:
+%% $$y=kx+t$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1},t=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% The intersections are:
+%% $$x_{1,2}=ec_{1,2}-fs_{1,2}+x_o,y_{1,2}=kx_{1,2}+t$$
+%% where,
+%% $$e=a\cos\theta,f=b\sin\theta,g=a\sin\theta,h=b\cos\theta$$
+%% $$c_1=\dfrac{mA+B\sqrt{w}}{A^2+B^2}$$
+%% $$s_1=\dfrac{mB-A\sqrt{w}}{A^2+B^2}$$
+%% $$c_2=\dfrac{mA-B\sqrt{w}}{A^2+B^2}$$
+%% $$s_2=\dfrac{mB+A\sqrt{w}}{A^2+B^2}$$
+%% $$w=A^2+B^2-m^2, m=kx_o-y_o+t, A=g-ke, B=h+kf$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [input] the node name A on the given line
+%% #6 -> [input] the node name B on the given line
+%% #7 -> [output] the first intersection node name
+%% #8 -> [output] the second intersection node name
+\def\pstGeneralEllipseLineInter{\@ifnextchar[\Pst at GeneralEllipseLineInter{\Pst at GeneralEllipseLineInter[]}}
+\def\Pst at GeneralEllipseLineInter[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralEllipseLineInter@i{\Pst at GeneralEllipseLineInter@i[0]}}
+\def\Pst at GeneralEllipseLineInter@i[#1]#2#3#4#5{%
+ \pst at getcoor{#2}\pst at tempA%
+ \pst at getcoor{#3}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ pop pop pop pop
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ \pst at tempA \tx at UserCoor pop % x1
+ 9 index sub % d=x_1-x_o
+ 4 index dup mul 4 index dup mul add % u=e^2+f^2
+ 0 index 2 index dup mul sub dup 0 lt { % w=u-d^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ %% c_1=(de+f sqrt(w))/u, s_1=(-df+e sqrt(w))/u
+ 2 index 7 index mul 6 index 2 index mul add 2 index div % c1
+ 3 index 7 index mul 8 index 3 index mul exch sub 3 index div % s1
+ %% y_{1,2}=gc_{1,2}+hs_{1,2}+y_o
+ 6 index 2 index mul 6 index 2 index mul add 14 index add % y1
+ \pst at tempA \tx at UserCoor pop % x1
+ exch 17 2 roll pop pop pop pop pop
+ pop pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ }{
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 9 index 2 index mul % e=a\cos\theta
+ 9 index 2 index mul % f=b\sin\theta
+ 11 index 3 index mul % g=a\sin\theta
+ 11 index 5 index mul % h=b\cos\theta
+ 1 index 8 index 5 index mul sub % A=g-ke
+ 1 index 9 index 5 index mul add % B=h+kf
+ 1 index dup mul 1 index dup mul add % C=A^2+B^2
+ 10 index 19 index mul 18 index sub 10 index add % m=kx_o-y_o+t
+ 1 index 1 index dup mul sub dup 0 lt { % w=C-m^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ 1 index 5 index mul 4 index 2 index mul add 3 index div % c1
+ 2 index 5 index mul 6 index 3 index mul sub 4 index div % s1
+ %% x_{1,2}=ec_{1,2}-fs_{1,2}+x_o,y_{1,2}=kx_{1,2}+t
+ 10 index 2 index mul 10 index 2 index mul sub 23 index add % x1
+ dup 16 index mul 15 index add % y1
+ 25 2 roll pop pop
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ pop pop pop pop
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ \pst at tempA \tx at UserCoor pop % x1
+ 9 index sub % d=x_1-x_o
+ 4 index dup mul 4 index dup mul add % u=e^2+f^2
+ 0 index 2 index dup mul sub dup 0 lt { % w=u-d^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ %% c_2=(de-f sqrt(w))/u, s_2=(-df-e sqrt(w))/u
+ 2 index 7 index mul 6 index 2 index mul sub 2 index div % c2
+ 3 index 7 index mul 8 index 3 index mul add neg 3 index div % s2
+ %% y_{1,2}=gc_{1,2}+hs_{1,2}+y_o
+ 6 index 2 index mul 6 index 2 index mul add 14 index add % y2
+ \pst at tempA \tx at UserCoor pop % x1
+ exch 17 2 roll pop pop pop pop pop
+ pop pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ }{
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 9 index 2 index mul % e=a\cos\theta
+ 9 index 2 index mul % f=b\sin\theta
+ 11 index 3 index mul % g=a\sin\theta
+ 11 index 5 index mul % h=b\cos\theta
+ 1 index 8 index 5 index mul sub % A=g-ke
+ 1 index 9 index 5 index mul add % B=h+kf
+ 1 index dup mul 1 index dup mul add % C=A^2+B^2
+ 10 index 19 index mul 18 index sub 10 index add % m=kx_o-y_o+t
+ 1 index 1 index dup mul sub dup 0 lt { % w=C-m^2
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt % sqrt(w)
+ 1 index 5 index mul 4 index 2 index mul sub 3 index div % c2
+ 2 index 5 index mul 6 index 3 index mul add 4 index div % s2
+ %% x_{1,2}=ec_{1,2}-fs_{1,2}+x_o,y_{1,2}=kx_{1,2}+t
+ 10 index 2 index mul 10 index 2 index mul sub 23 index add % x2
+ dup 16 index mul 15 index add % y2
+ 25 2 roll pop pop
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipsePolarNode[Options](O)(a,b)[rotation]{A}{B}{T}
+%% Draw the every tangent line through the point $A$ and $B$ on the General Ellipse E and get the insection node T of the two tangent lines.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% We call T as the polar point of chord AB, which can be got by the following proposition:
+%% Give chord AB on the ellipse, we draw any other two chords PQ and RS, AB and PQ intersect at I, AQ and BP intersect at X, AP and BQ intersect at Y, we call XY is the polar line of point I. Also AB and RS intersect at J, AR and BS intersect at M, AS and BR intersect at N, we call MN is the polar line of point J. Then the intersection T of XY and MN is the polar point of chord AB, i.e. TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [input] the node name A on the ellipse
+%% #6 -> [input] the node name B on the ellipse
+%% #7 -> [output] the polar point of chord AB
+\def\pstGeneralEllipsePolarNode{\@ifnextchar[\Pst at GeneralEllipsePolarNode{\Pst at GeneralEllipsePolarNode[]}}
+\def\Pst at GeneralEllipsePolarNode[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \def\PST at GENERALELLIPSE@CENTER{#2}%
+ \def\PST at GENERALELLIPSE@RADIUS{#3}%
+ \@ifnextchar[\Pst at GeneralEllipsePolarNode@i{\Pst at GeneralEllipsePolarNode@i[0]}}
+\def\Pst at GeneralEllipsePolarNode@i[#1]#2#3#4{%
+ \pstGeneralEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=71](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{@PST at GENERALELLIPSE@POLARAUX at P}
+ \pstGeneralEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=-31](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{@PST at GENERALELLIPSE@POLARAUX at Q}
+ \pstGeneralEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=-122](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{@PST at GENERALELLIPSE@POLARAUX at R}
+ \pstGeneralEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=13](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{@PST at GENERALELLIPSE@POLARAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{#2}{@PST at GENERALELLIPSE@POLARAUX at Q}{#3}{@PST at GENERALELLIPSE@POLARAUX at P}{@PST at GENERALELLIPSE@POLARAUX at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#2}{@PST at GENERALELLIPSE@POLARAUX at P}{#3}{@PST at GENERALELLIPSE@POLARAUX at Q}{@PST at GENERALELLIPSE@POLARAUX at Y}
+ \pstInterLL[PointName=none,PointSymbol=none]{#2}{@PST at GENERALELLIPSE@POLARAUX at R}{#3}{@PST at GENERALELLIPSE@POLARAUX at S}{@PST at GENERALELLIPSE@POLARAUX at M}
+ \pstInterLL[PointName=none,PointSymbol=none]{#2}{@PST at GENERALELLIPSE@POLARAUX at S}{#3}{@PST at GENERALELLIPSE@POLARAUX at R}{@PST at GENERALELLIPSE@POLARAUX at N}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALELLIPSE@POLARAUX at X}{@PST at GENERALELLIPSE@POLARAUX at Y}{@PST at GENERALELLIPSE@POLARAUX at M}{@PST at GENERALELLIPSE@POLARAUX at N}{#4}
+ \Pst at geonodelabel{#4}%
+ \pstLineAB{#2}{#4}
+ \pstLineAB{#3}{#4}
+ \endgroup%
+}%
+%
+%% \pstGeneralEllipseTangentNode[Options](O)(a,b)[rotation]{T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the General Ellipse E and get the node A and B on the Ellipse.
+%% If you not input rotation angle, the default value is $0^\circ$, which is same as \pstEllipse.
+%% We use the following proposition to find the tangent node of T:
+%% Give point T outside of the ellipse, we draw any other two chords TPQ and TRS, PS and QR intersect at I, PR and QS intersect at X, XI and Ellipse intersect at A and B, then TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the ellipse center O
+%% #3 -> [input] the horizontal and vertical radii
+%% #4 -> [input] the rotation angle $\theta$ of the major axis.
+%% #5 -> [input] the node name T outside the ellipse
+%% #6 -> [output] the node name A on the ellipse
+%% #7 -> [output] the node name B on the ellipse
+\def\pstGeneralEllipseTangentNode{\@ifnextchar[\Pst at GeneralEllipseTangentNode{\Pst at GeneralEllipseTangentNode[]}}
+\def\Pst at GeneralEllipseTangentNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \def\PST at GENERALELLIPSE@CENTER{#2}%
+ \def\PST at GENERALELLIPSE@RADIUS{#3}%
+ \@ifnextchar[\Pst at GeneralEllipseTangentNode@i{\Pst at GeneralEllipseTangentNode@i[0]}}
+\def\Pst at GeneralEllipseTangentNode@i[#1]#2#3#4{%
+ \pstGeneralEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=71](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{@PST at GENERALELLIPSE@TANGENTAUX at P0}
+ \pstGeneralEllipseRotNode[PointName=none,PointSymbol=none,RotAngle=31](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{@PST at GENERALELLIPSE@TANGENTAUX at R0}
+ \pstGeneralEllipseLineInter[PointName=none,PointSymbol=none](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{#2}{@PST at GENERALELLIPSE@TANGENTAUX at P0}{@PST at GENERALELLIPSE@TANGENTAUX at P}{@PST at GENERALELLIPSE@TANGENTAUX at Q}
+ \pstGeneralEllipseLineInter[PointName=none,PointSymbol=none](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{#2}{@PST at GENERALELLIPSE@TANGENTAUX at R0}{@PST at GENERALELLIPSE@TANGENTAUX at R}{@PST at GENERALELLIPSE@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALELLIPSE@TANGENTAUX at P}{@PST at GENERALELLIPSE@TANGENTAUX at S}{@PST at GENERALELLIPSE@TANGENTAUX at Q}{@PST at GENERALELLIPSE@TANGENTAUX at R}{@PST at GENERALELLIPSE@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALELLIPSE@TANGENTAUX at P}{@PST at GENERALELLIPSE@TANGENTAUX at R}{@PST at GENERALELLIPSE@TANGENTAUX at Q}{@PST at GENERALELLIPSE@TANGENTAUX at S}{@PST at GENERALELLIPSE@TANGENTAUX at X}
+ \pstGeneralEllipseLineInter[PointName=none,PointSymbol=none](\PST at GENERALELLIPSE@CENTER)(\PST at GENERALELLIPSE@RADIUS)[#1]{@PST at GENERALELLIPSE@TANGENTAUX at X}{@PST at GENERALELLIPSE@TANGENTAUX at I}{#3}{#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \pstLineAB{#2}{#3}
+ \pstLineAB{#2}{#4}
+ \endgroup%
+}%
+%
+% 3. Standard Parabola with coordinate translation
+%% ----------------------------------------------------------
+%% The Parabola P is defined by vertex O, the half of the focus chord axis $abs(p)$, the sign of $p$ indicates the direction of the parabola.
+%% The equation can be written as:
+%% \begin{equation}\label{FunctionOfStandardParabola}
+%% (x-x0)^2=2p(y-y0)
+%% \end{equation}
+%% and the parametric function can be written as:
+%% \begin{equation}\label{ParametricFunctionOfStandardParabola}
+%% \left\{\begin{array}{l}
+%% x=t+x_o\\
+%% y=\dfrac{t^2}{2p}+y_o
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstParabola[Options](O){p}{x0}{x1}
+%% Draw a Parabola from x0 to x1 with Vertex O, the half of the focus chord axis $abs(p)$, the sign of $p$ indicates the direction of the parabola.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the start of independent variable $x$
+%% #5 -> [input] the end of independent variable $x$
+\def\pstParabola{\@ifnextchar[\Pst at Parabola{\Pst at Parabola[]}}
+\def\Pst at Parabola[#1](#2)#3#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \parametricplot{#4}{#5}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 % p
+ 2 index t add % t+x0
+ 2 index t dup mul 3 index 2 mul div add % t^2/(2p)+y0
+ 5 2 roll pop pop pop % t+x0,t^2/(2p)+y0
+ }
+ \endgroup%
+}%
+%
+%% \pstParabolaNode[Options](O){p}{$t$}{A}
+%% Draw the node whose parameter is the given value $t$ on the Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the given parameter value $t$
+%% #5 -> [output] the node $A$ on the parabola
+\def\pstParabolaNode{\@ifnextchar[\Pst at ParabolaNode{\Pst at ParabolaNode[]}}
+\def\Pst at ParabolaNode[#1](#2)#3#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 % p
+ 2 index #4 add % t+x0
+ 2 index #4 dup mul 3 index 2 mul div add % t^2/(2p)+y0
+ 5 2 roll pop pop pop % t+x0,t^2/(2p)+y0
+ ){#5}
+ \Pst at geonodelabel{#5}%
+ \endgroup%
+}%
+%
+%% \pstParabolaAbsNode[Options](O){p}{$x_1$}{A}
+%% Draw the node whose abscissa is the given value $x_1$ on the Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the given abscissa value $x_1$
+%% #5 -> [output] the node $A$ on the parabola
+\def\pstParabolaAbsNode{\@ifnextchar[\Pst at ParabolaAbsNode{\Pst at ParabolaAbsNode[]}}
+\def\Pst at ParabolaAbsNode[#1](#2)#3#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 #4 % p x_1
+ 2 index 1 index 5 index sub dup mul 3 index 2 mul div add % y0+(x_1-x_0)^2/(2p)
+ 5 2 roll pop pop pop
+ ){#5}
+ \Pst at geonodelabel{#5}%
+ \endgroup%
+}%
+%
+%% \pstParabolaOrdNode[Options](O){p}{$y_1$}{A}{B}
+%% Draw the nodes whose ordinate is the given value $y_1$ on the Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the given abscissa value $x_1$
+%% #5 -> [output] the first node $A$ on the parabola
+%% #6 -> [output] the second node $B$ on the parabola
+\def\pstParabolaOrdNode{\@ifnextchar[\Pst at ParabolaOrdNode{\Pst at ParabolaOrdNode[]}}
+\def\Pst at ParabolaOrdNode[#1](#2)#3#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 #4 % p y_1
+ 2 index sub mul 2 mul dup 0 lt {
+ pop pop pop 0 0
+ }{
+ sqrt 2 index exch sub
+ #4 4 2 roll pop pop
+ } ifelse
+ ){#5}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 #4 % p y_1
+ 2 index sub mul 2 mul dup 0 lt {
+ pop pop pop 0 0
+ }{
+ sqrt 2 index add
+ #4 4 2 roll pop pop
+ } ifelse
+ ){#6}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \endgroup%
+}%
+%
+%% \pstParabolaFocusNode[Options](O){p}{F}
+%% Draw the focus node of a Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [output] the focus node $F$
+\def\pstParabolaFocusNode{\@ifnextchar[\Pst at ParabolaFocusNode{\Pst at ParabolaFocusNode[]}}
+\def\Pst at ParabolaFocusNode[#1](#2)#3#4{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 %p
+ 2 index % x0
+ 2 index 2 index 2 div add % y0+p/2
+ 5 2 roll pop pop pop
+ ){#4}
+ \Pst at geonodelabel{#4}%
+ \endgroup%
+}%
+%
+%% \pstParabolaDirectrixLine[Options](O){p}{LA}{LB}
+%% Draw the directrix line of a Parabola P.
+%% The node LA also lies on the symmetrical axis line, and node LB is another one on the directrix line.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [output] the first node $A$ on the directrix line
+%% #5 -> [output] the second node $B$ on the directrix line
+\def\pstParabolaDirectrixLine{\@ifnextchar[\Pst at ParabolaDirectrixLine{\Pst at ParabolaDirectrixLine[]}}
+\def\Pst at ParabolaDirectrixLine[#1](#2)#3#4#5{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 %p
+ 2 index % x0
+ 2 index 2 index 2 div sub % y0-p/2
+ 5 2 roll pop pop pop
+ ){#4}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 %p
+ 2 index 1 add % x0+1
+ 2 index 2 index 2 div sub % y0-p/2
+ 5 2 roll pop pop pop
+ ){#5}
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLineAB{#4}{#5}
+ \endgroup%
+}%
+%
+%% \pstParabolaLineInter[Options](O){p}{A}{B}{C}{D}
+%% Find the intersections C and D of line AB and the Parabola P.
+%% We can represent the line AB as the following function when line AB is not vertical:
+%% $$y=kx+t$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1},t=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% The intersections are:
+%% $$x_{1,2}=e\pm\sqrt{w}+x_o, y_{1,2}=kx_{1,2}+t$$
+%% where,
+%% $$e=pk, w=e^2+2f, f=pm, m=kx_o-y_o+t$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the node $A$ on the given line
+%% #5 -> [input] the node $B$ on the given line
+%% #6 -> [output] the first intersection node $C$
+%% #7 -> [output] the second intersection node $D$
+\def\pstParabolaLineInter{\@ifnextchar[\Pst at ParabolaLineInter{\Pst at ParabolaLineInter[]}}
+\def\Pst at ParabolaLineInter[#1](#2)#3#4#5#6#7{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#4}\pst at tempA%
+ \pst at getcoor{#5}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ 3 index 6 index sub dup mul #3 2 mul div 5 index add % y=(x1-x0)^2/(2p)+y0
+ 2 index exch 8 2 roll pop pop pop pop pop pop
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ #3 %p
+ 2 index 1 index mul % e=pk
+ 3 index 10 index mul 9 index sub 3 index add % m=kx_o-y_o+t
+ 2 index 1 index mul % f=pm
+ 2 index dup mul 1 index 2 mul add % w=e^2+2f
+ dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(w)
+ %% x_{1,2}=e \pm sqrt(w)+x_o,y_{1,2}=kx_{1,2}+t
+ 3 index 1 index sub 13 index add % x1
+ 7 index 1 index mul 7 index add % y1
+ 15 2 roll
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#6}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ #3 %p
+ 2 index 1 index mul % e=pk
+ 3 index 10 index mul 9 index sub 3 index add % m=kx_o-y_o+t
+ 2 index 1 index mul % f=pm
+ 2 index dup mul 1 index 2 mul add % w=e^2+2f
+ dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(w)
+ %% x_{1,2}=e \pm sqrt(w)+x_o,y_{1,2}=kx_{1,2}+t
+ 3 index 1 index add 13 index add % x2
+ 7 index 1 index mul 7 index add % y2
+ 15 2 roll
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#7}
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \endgroup%
+}%
+%
+%% \pstParabolaPolarNode[Options](O){p}(F)[L1][L2]{A}{B}{T}
+%% Find the polar point of chord AB on Parabola P.
+%% We use the following proposition to find the polar point of chord AB:
+%% Give any chord AB, drawing two focal chord AFC and BFD, where F is the focus, then drawing FX which is perpendicular to AFC at point F, and intersect with the directrix line at X; also drawing FY which is perpendicular to BFD at point F, and intersect with the directrix line at Y. Then the intersection T of AX and BY is the polar point of chord AB.
+%% If you don't know the focus F, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] optional, the focus node/coordinate F of the parabola.
+%% #5 -> [input] optional, the node/coordinate L1 on the directrix line of the parabola.
+%% #6 -> [input] optional, the node/coordinate L2 on the directrix line of the parabola.
+%% #7 -> [input] the node A on the parabola.
+%% #8 -> [input] the node B on the parabola.
+%% #9 -> [output] the polar node T of chord AB.
+\def\pstParabolaPolarNode{\@ifnextchar[\Pst at ParabolaPolarNode{\Pst at ParabolaPolarNode[]}}
+\def\Pst at ParabolaPolarNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \def\pst at parabola@vertex{#2}%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar(\Pst at ParabolaPolarNode@i{\Pst at ParabolaPolarNode@j}}%
+\def\Pst at ParabolaPolarNode@i(#1){%
+ \def\pst at parabola@focus{#1}%
+ \@ifnextchar[\Pst at ParabolaPolarNode@ii{\Pst at ParabolaPolarNode@jj}}%
+\def\Pst at ParabolaPolarNode@j{%
+ \pstParabolaFocusNode[PointName=none,PointSymbol=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{@PST at PARABOLA@FOCUS}
+ \Pst at ParabolaPolarNode@i(@PST at PARABOLA@FOCUS)}%
+\def\Pst at ParabolaPolarNode@jj{%
+ \pstParabolaDirectrixLine[PointName=none,PointSymbol=none,linestyle=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{@PST at PARABOLA@LA}{@PST at PARABOLA@LB}
+ \Pst at ParabolaPolarNode@ii[@PST at PARABOLA@LA][@PST at PARABOLA@LB]}%
+\def\Pst at ParabolaPolarNode@ii[#1][#2]#3#4#5{%
+ \pstParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{#3}{\pst at parabola@focus}{@PST at PARABOLA@FOCALCHORDAFC at A}{@PST at PARABOLA@FOCALCHORDAFC at C}
+ \pstParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{#4}{\pst at parabola@focus}{@PST at PARABOLA@FOCALCHORDBFD at B}{@PST at PARABOLA@FOCALCHORDBFD at D}
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at parabola@focus}{@PST at PARABOLA@FOCALCHORDAFC at A}[@PST at PARABOLA@FOCALCHORDAFC at AA]
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at parabola@focus}{@PST at PARABOLA@FOCALCHORDBFD at B}[@PST at PARABOLA@FOCALCHORDBFD at BB]
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at parabola@focus}{@PST at PARABOLA@FOCALCHORDAFC at AA}{@PST at PARABOLA@FOCALCHORD at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at parabola@focus}{@PST at PARABOLA@FOCALCHORDBFD at BB}{@PST at PARABOLA@FOCALCHORD at Y}
+ \pstInterLL{#3}{@PST at PARABOLA@FOCALCHORD at X}{#4}{@PST at PARABOLA@FOCALCHORD at Y}{#5}
+ \Pst at geonodelabel{#5}%
+ \pstLineAB{#3}{#5}
+ \pstLineAB{#4}{#5}
+ \endgroup
+}%
+%
+%% \pstParabolaTangentNode[Options](O){p}{T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the Parabola P and get the node A and B on the Parabola.
+%% We use the following proposition to find the tangent node of T:
+%% Give point T outside of the parabola, we draw any other two chords TPQ and TRS, PS and QR intersect at I, PR and QS intersect at X, XI and Parabola intersect at A and B, then TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the node name T outside the parabola
+%% #5 -> [output] the tangent node name A on the parabola
+%% #6 -> [output] the tangent node name B on the parabola
+\def\pstParabolaTangentNode{\@ifnextchar[\Pst at ParabolaTangentNode{\Pst at ParabolaTangentNode[]}}
+\def\Pst at ParabolaTangentNode[#1](#2)#3#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pstParabolaAbsNode[PointName=none,PointSymbol=none](#2){#3}{\pstAbscissa{#4} \pstAbscissa{#2} le {\pstAbscissa{#2} #3 abs 2 div add}{\pstAbscissa{#2} #3 abs 2 div sub} ifelse}{@PST at PARABOLA@TANGENTAUX at P0}
+ \pstParabolaAbsNode[PointName=none,PointSymbol=none](#2){#3}{\pstAbscissa{#4} \pstAbscissa{#2} le {\pstAbscissa{#2} #3 abs 4 div add}{\pstAbscissa{#2} #3 abs 4 div sub} ifelse}{@PST at PARABOLA@TANGENTAUX at R0}
+ \pstParabolaLineInter[PointName=none,PointSymbol=none](#2){#3}{#4}{@PST at PARABOLA@TANGENTAUX at P0}{@PST at PARABOLA@TANGENTAUX at P}{@PST at PARABOLA@TANGENTAUX at Q}
+ \pstParabolaLineInter[PointName=none,PointSymbol=none](#2){#3}{#4}{@PST at PARABOLA@TANGENTAUX at R0}{@PST at PARABOLA@TANGENTAUX at R}{@PST at PARABOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at PARABOLA@TANGENTAUX at P}{@PST at PARABOLA@TANGENTAUX at S}{@PST at PARABOLA@TANGENTAUX at Q}{@PST at PARABOLA@TANGENTAUX at R}{@PST at PARABOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at PARABOLA@TANGENTAUX at P}{@PST at PARABOLA@TANGENTAUX at R}{@PST at PARABOLA@TANGENTAUX at Q}{@PST at PARABOLA@TANGENTAUX at S}{@PST at PARABOLA@TANGENTAUX at X}
+ \pstParabolaLineInter(#2){#3}{@PST at PARABOLA@TANGENTAUX at X}{@PST at PARABOLA@TANGENTAUX at I}{#5}{#6}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \pstLineAB{#4}{#5}
+ \pstLineAB{#4}{#6}
+ \endgroup%
+}%
+%
+% 4. Standard Inversion Parabola with coordinate translation
+%% ----------------------------------------------------------
+%% The Inversion Parabola P is defined by vertex O, the half of the focus chord axis $abs(p)$, the sign of $p$ indicates the direction of the parabola.
+%% The equation can be written as:
+%% \begin{equation}\label{StandardInversionParabola}
+%% (y-y0)^2=2p(x-x0)
+%% \end{equation}
+%% and the parametric function can be written as:
+%% \begin{equation}\label{ParametricFunctionOfStandardInversionParabola}
+%% \left\{\begin{array}{l}
+%% x=\dfrac{t^2}{2p}+x_o\\
+%% y=t+y_o
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstIParabola[Options](O){p}{y0}{y1}
+%% Draw a Inversion Parabola from y0 to y1 with Vertex O, the half of the focus chord axis $abs{p}$, the sign of $p$ indicates the direction of the parabola.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the start of independent variable $y$
+%% #5 -> [input] the end of independent variable $y$
+\def\pstIParabola{\@ifnextchar[\Pst at IParabola{\Pst at IParabola[]}}
+\def\Pst at IParabola[#1](#2)#3#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \parametricplot{#4}{#5}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 % p
+ 1 index t add % t+y0
+ 3 index t dup mul 3 index 2 mul div add % t^2/(2p)+x0
+ 5 2 roll pop pop pop % t+y0,t^2/(2p)+x0
+ exch % 2pt^2+x0,t+y0
+ }
+ \endgroup%
+}%
+%
+%% \pstIParabolaNode[Options](O){p}{t}{A}
+%% Draw the node whose paramater is the given value $t$ on the Inversion Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the given parameter value $t$
+%% #5 -> [output] the node $A$ on the parabola
+\def\pstIParabolaNode{\@ifnextchar[\Pst at IParabolaNode{\Pst at IParabolaNode[]}}
+\def\Pst at IParabolaNode[#1](#2)#3#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 % p
+ 1 index #4 add % t+y0
+ 3 index #4 dup mul 3 index 2 mul div add % t^2/(2p)+x0
+ 5 2 roll pop pop pop % t+y0,t^2/(2p)+x0
+ exch % 2pt^2+x0,t+y0
+ ){#5}
+ \Pst at geonodelabel{#5}%
+ \endgroup%
+}%
+%
+%% \pstIParabolaOrdNode[Options](O){p}{$y_1$}{A}
+%% Draw the node whose ordinate is the given value $y_1$ on the Inversion Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the given ordinate value $y_1$
+%% #5 -> [output] the node $A$ on the parabola
+\def\pstIParabolaOrdNode{\@ifnextchar[\Pst at IParabolaOrdNode{\Pst at IParabolaOrdNode[]}}
+\def\Pst at IParabolaOrdNode[#1](#2)#3#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 #4 % p y_1
+ 3 index 1 index 4 index sub dup mul 3 index 2 mul div add % x0+(y_1-y_0)^2/(2p)
+ exch 5 2 roll pop pop pop
+ ){#5}
+ \Pst at geonodelabel{#5}%
+ \endgroup%
+}%
+%
+%% \pstIParabolaAbsNode[Options](O){p}{$x_1$}{A}{B}
+%% Draw the nodes whose abscissa is the given value $x_1$ on the Inversion Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the given abscissa value $x_1$
+%% #5 -> [output] the first node $A$ on the parabola
+%% #6 -> [output] the second node $B$ on the parabola
+\def\pstIParabolaAbsNode{\@ifnextchar[\Pst at IParabolaAbsNode{\Pst at IParabolaAbsNode[]}}
+\def\Pst at IParabolaAbsNode[#1](#2)#3#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 #4 % p x_1
+ 3 index sub mul 2 mul dup 0 lt {
+ pop pop pop 0 0
+ }{
+ sqrt 1 index exch sub
+ #4 exch 4 2 roll pop pop
+ } ifelse
+ ){#5}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 #4 % p x_1
+ 3 index sub mul 2 mul dup 0 lt {
+ pop pop pop 0 0
+ }{
+ sqrt 1 index add
+ #4 exch 4 2 roll pop pop
+ } ifelse
+ ){#6}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \endgroup%
+}%
+%
+%% \pstIParabolaFocusNode[Options](O){p}{F}
+%% Draw the focus node of a Inversion Parabola P.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [output] the focus node $F$
+\def\pstIParabolaFocusNode{\@ifnextchar[\Pst at IParabolaFocusNode{\Pst at IParabolaFocusNode[]}}
+\def\Pst at IParabolaFocusNode[#1](#2)#3#4{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 %p
+ 2 index 1 index 2 div add % x0+p/2
+ 2 index % y0
+ 5 2 roll pop pop pop
+ ){#4}
+ \Pst at geonodelabel{#4}%
+ \endgroup%
+}%
+%
+%% \pstIParabolaDirectrixLine[Options](O){p}{LA}{LB}
+%% Draw the directrix line of a Inversion Parabola P.
+%% The node LA also lies on the symmetrical axis line, and node LB is another one on the directrix line.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [output] the first node $A$ on the directrix line
+%% #5 -> [output] the second node $B$ on the directrix line
+\def\pstIParabolaDirectrixLine{\@ifnextchar[\Pst at IParabolaDirectrixLine{\Pst at IParabolaDirectrixLine[]}}
+\def\Pst at IParabolaDirectrixLine[#1](#2)#3#4#5{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 %p
+ 2 index 1 index 2 div sub % x0-p/2
+ 2 index % y0
+ 5 2 roll pop pop pop
+ ){#4}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ #3 %p
+ 2 index 1 index 2 div sub % x0-p/2
+ 2 index 1 add % y0+1
+ 5 2 roll pop pop pop
+ ){#5}
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLineAB{#4}{#5}
+ \endgroup%
+}%
+%
+%% \pstIParabolaLineInter[Options](O){p}{A}{B}{C}{D}
+%% Find the intersections C and D of line AB and the Inversion Parabola P.
+%% We can represent the line AB as the following function when line AB is not vertical:
+%% $$y=kx+t$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, t=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% The intersections are:
+%% $$x_{1,2}=\dfrac{e\pm\sqrt{w}{k^2+x_o}, y_{1,2}=kx_{1,2}+t$$
+%% where,
+%% $$e=p-km, w=p^2-2pkm, m=kx_o-y_o+t$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the node $A$ on the given line
+%% #5 -> [input] the node $B$ on the given line
+%% #6 -> [output] the first intersection node $C$
+%% #7 -> [output] the second intersection node $D$
+\def\pstIParabolaLineInter{\@ifnextchar[\Pst at IParabolaLineInter{\Pst at IParabolaLineInter[]}}
+\def\Pst at IParabolaLineInter[#1](#2)#3#4#5#6#7{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#4}\pst at tempA%
+ \pst at getcoor{#5}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 2 index 1 index sub abs 1E-5 lt { % if the line AB is horizontal
+ 2 index 5 index sub dup mul #3 2 mul div 6 index add % x=(y1-y0)^2/(2p)+x0
+ 3 index 8 2 roll pop pop pop pop pop pop
+ } {
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ % y_{1,2}=y0 \pm sqrt(2p(x1-x0))
+ 4 index #3 2 mul 5 index 8 index sub mul dup 0 lt {
+ pop pop pop pop pop pop pop pop 0 0
+ }{
+ sqrt sub % y1
+ 2 index exch 8 2 roll pop pop pop pop pop pop
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ #3 %p
+ 2 index 9 index mul 8 index sub 2 index add % m=kx_o-y_o+t
+ 1 index 4 index 2 index mul sub % e=p-km
+ 2 index dup mul 3 index 2 mul 3 index mul 6 index mul sub % w=p^2-2pkm
+ dup 0 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(w)
+ %% x_{1,2}=(e \pm sqrt(w))/k^2+x_o,y_{1,2}=kx_{1,2}+t
+ 1 index 1 index sub 6 index dup mul div 12 index add % x1
+ 6 index 1 index mul 6 index add % y1
+ 14 2 roll
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#6}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 2 index 1 index sub abs 1E-5 lt { % if the line AB is horizontal
+ pop pop pop pop pop pop 0 0
+ } {
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ % y_{1,2}=y0 \pm sqrt(2p(x1-x0))
+ 4 index #3 2 mul 5 index 8 index sub mul dup 0 lt {
+ pop pop pop pop pop pop pop pop 0 0
+ }{
+ sqrt add % y2
+ 2 index exch 8 2 roll pop pop pop pop pop pop
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % t
+ #3 %p
+ 2 index 9 index mul 8 index sub 2 index add % m=kx_o-y_o+t
+ 1 index 4 index 2 index mul sub % e=p-km
+ 2 index dup mul 3 index 2 mul 3 index mul 6 index mul sub % w=p^2-2pkm
+ dup 0 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(w)
+ %% x_{1,2}=(e \pm sqrt(w))/k^2+x_o,y_{1,2}=kx_{1,2}+t
+ 1 index 1 index add 6 index dup mul div 12 index add % x1
+ 6 index 1 index mul 6 index add % y1
+ 14 2 roll
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#7}
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \endgroup%
+}%
+%
+%% \pstIParabolaPolarNode[Options](O){p}(F)[L1][L2]{A}{B}{T}
+%% Find the polar point of chord AB on Inversion Parabola P.
+%% We use the following proposition to find the polar point of chord AB:
+%% Give any chord AB, drawing two focal chord AFC and BFD, where F is the focus, then drawing FX which is perpendicular to AFC at point F, and intersect with the directrix line at X; also drawing FY which is perpendicular to BFD at point F, and intersect with the directrix line at Y. Then the intersection T of AX and BY is the polar point of chord AB.
+%% If you don't know the focus F, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] optional, the focus node/coordinate F of the parabola.
+%% #5 -> [input] optional, the node/coordinate L1 on the directrix line of the parabola.
+%% #6 -> [input] optional, the node/coordinate L2 on the directrix line of the parabola.
+%% #7 -> [input] the node A on the parabola.
+%% #8 -> [input] the node B on the parabola.
+%% #9 -> [output] the polar node T of chord AB.
+\def\pstIParabolaPolarNode{\@ifnextchar[\Pst at IParabolaPolarNode{\Pst at IParabolaPolarNode[]}}
+\def\Pst at IParabolaPolarNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \def\pst at parabola@vertex{#2}%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar(\Pst at IParabolaPolarNode@i{\Pst at IParabolaPolarNode@j}}%
+\def\Pst at IParabolaPolarNode@i(#1){%
+ \def\pst at parabola@focus{#1}%
+ \@ifnextchar[\Pst at IParabolaPolarNode@ii{\Pst at IParabolaPolarNode@jj}}%
+\def\Pst at IParabolaPolarNode@j{%
+ \pstIParabolaFocusNode[PointName=none,PointSymbol=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{@PST at IPARABOLA@FOCUS}
+ \Pst at IParabolaPolarNode@i(@PST at IPARABOLA@FOCUS)}%
+\def\Pst at IParabolaPolarNode@jj{%
+ \pstIParabolaDirectrixLine[PointName=none,PointSymbol=none,linestyle=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{@PST at IPARABOLA@LA}{@PST at IPARABOLA@LB}
+ \Pst at IParabolaPolarNode@ii[@PST at IPARABOLA@LA][@PST at IPARABOLA@LB]}%
+\def\Pst at IParabolaPolarNode@ii[#1][#2]#3#4#5{%
+ \pstIParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{#3}{\pst at parabola@focus}{@PST at IPARABOLA@FOCALCHORDAFC at A}{@PST at IPARABOLA@FOCALCHORDAFC at C}
+ \pstIParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at parabola@vertex){\pst at parabola@semifocalchord}{#4}{\pst at parabola@focus}{@PST at IPARABOLA@FOCALCHORDBFD at B}{@PST at IPARABOLA@FOCALCHORDBFD at D}
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at parabola@focus}{@PST at IPARABOLA@FOCALCHORDAFC at A}[@PST at IPARABOLA@FOCALCHORDAFC at AA]
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at parabola@focus}{@PST at IPARABOLA@FOCALCHORDBFD at B}[@PST at IPARABOLA@FOCALCHORDBFD at BB]
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at parabola@focus}{@PST at IPARABOLA@FOCALCHORDAFC at AA}{@PST at IPARABOLA@FOCALCHORD at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at parabola@focus}{@PST at IPARABOLA@FOCALCHORDBFD at BB}{@PST at IPARABOLA@FOCALCHORD at Y}
+ \pstInterLL{#3}{@PST at IPARABOLA@FOCALCHORD at X}{#4}{@PST at IPARABOLA@FOCALCHORD at Y}{#5}
+ \Pst at geonodelabel{#5}%
+ \pstLineAB{#3}{#5}
+ \pstLineAB{#4}{#5}
+ \endgroup
+}%
+%
+%% \pstIParabolaTangentNode[Options](O){p}{T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the Inversion Parabola P and get the node A and B on the Inversion Parabola.
+%% We use the following proposition to find the tangent node of T:
+%% Give point T outside of the parabola, we draw any other two chords TPQ and TRS, PS and QR intersect at I, PR and QS intersect at X, XI and Parabola intersect at A and B, then TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the node name T outside the parabola
+%% #5 -> [output] the tangent node name A on the parabola
+%% #6 -> [output] the tangent node name B on the parabola
+\def\pstIParabolaTangentNode{\@ifnextchar[\Pst at IParabolaTangentNode{\Pst at IParabolaTangentNode[]}}
+\def\Pst at IParabolaTangentNode[#1](#2)#3#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pstIParabolaOrdNode[PointName=none,PointSymbol=none](#2){#3}{\pstOrdinate{#4} \pstOrdinate{#2} le {\pstOrdinate{#2} #3 abs 2 div add}{\pstOrdinate{#2} #3 abs 2 div sub} ifelse}{@PST at IPARABOLA@TANGENTAUX at P0}
+ \pstIParabolaOrdNode[PointName=none,PointSymbol=none](#2){#3}{\pstOrdinate{#4} \pstOrdinate{#2} le {\pstOrdinate{#2} #3 abs 4 div add}{\pstOrdinate{#2} #3 abs 4 div sub} ifelse}{@PST at IPARABOLA@TANGENTAUX at R0}
+ \pstIParabolaLineInter[PointName=none,PointSymbol=none](#2){#3}{#4}{@PST at IPARABOLA@TANGENTAUX at P0}{@PST at IPARABOLA@TANGENTAUX at P}{@PST at IPARABOLA@TANGENTAUX at Q}
+ \pstIParabolaLineInter[PointName=none,PointSymbol=none](#2){#3}{#4}{@PST at IPARABOLA@TANGENTAUX at R0}{@PST at IPARABOLA@TANGENTAUX at R}{@PST at IPARABOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at IPARABOLA@TANGENTAUX at P}{@PST at IPARABOLA@TANGENTAUX at S}{@PST at IPARABOLA@TANGENTAUX at Q}{@PST at IPARABOLA@TANGENTAUX at R}{@PST at IPARABOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at IPARABOLA@TANGENTAUX at P}{@PST at IPARABOLA@TANGENTAUX at R}{@PST at IPARABOLA@TANGENTAUX at Q}{@PST at IPARABOLA@TANGENTAUX at S}{@PST at IPARABOLA@TANGENTAUX at X}
+ \pstIParabolaLineInter(#2){#3}{@PST at IPARABOLA@TANGENTAUX at X}{@PST at IPARABOLA@TANGENTAUX at I}{#5}{#6}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \pstLineAB{#4}{#5}
+ \pstLineAB{#4}{#6}
+ \endgroup%
+}%
+%
+% 5. General Parabola with coordinate translation and rotation
+%% ----------------------------------------------------------
+%% The General Parabola P is defined by vertex O, the half of the focus chord axis $abs(p)$, the sign of $p$ indicates the direction of the parabola,
+%% and the rotation angle $\theta$ of the symmetrical axis.
+%%
+%% The equation can be got from the parametric function of the parabola \ref{ParametricFunctionOfStandardParabola},
+%% using the rotation transform formula \ref{RotationTransformFormula}, then we have
+%% \begin{equation}
+%% \left\{\begin{array}{l}
+%% x'=(t+x_o)\cos\theta-(\dfrac{t^2}{2p}+y_o)\sin\theta=x_o'+t\cos\theta-t^2\dfrac{\sin\theta}{2p}\\
+%% y'=(t+x_o)\sin\theta+(\dfrac{t^2}{2p}+y_o)\cos\theta=y_o'+t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+%% \end{array}\right.
+%% \end{equation}
+%% where the $x_o'$ and $y_o'$ are the coordinate of the given vertex O after rotation.
+%% So we get the parametric function of the General Parabola with coordinate translation and rotation as following:
+%% \begin{equation}\label{ParametricFunctionOfGeneralParabola}
+%% \left\{\begin{array}{l}
+%% x=x_o+t\cos\theta-t^2\dfrac{\sin\theta}{2p}\\
+%% y=y_o+t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstGeneralParabola[Options](O){p}[rotation]{x0}{x1}
+%% Draw a General Parabola from x0 to x1 with Vertex O, the half of the focus chord axis $p$, the sign of $p$ indicates the direction of the parabola,
+%% and the rotation angle $\theta$ of the symmetrical axis.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the start of independent variable $x$
+%% #6 -> [input] the end of independent variable $x$
+\def\pstGeneralParabola{\@ifnextchar[\Pst at GeneralParabola{\Pst at GeneralParabola[]}}
+\def\Pst at GeneralParabola[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO %
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabola@i{\Pst at GeneralParabola@i[0]}}%
+\def\Pst at GeneralParabola@i[#1]#2#3{%
+ \parametricplot{#2}{#3}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 4 index 2 index t mul add % x_o+t\cos\theta
+ 3 index 2 mul t dup mul exch div % t^2/(2p)
+ 2 index mul sub % x_o+t\cos\theta-t^2\dfrac{\sin\theta}{2p}
+ 4 index 2 index t mul add % y_o+t\sin\theta
+ 4 index 2 mul t dup mul exch div % t^2/(2p)
+ 4 index mul add % y_o+t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+ 7 2 roll pop pop pop pop pop
+ }
+ \endgroup%
+}%
+%
+%% \pstGeneralParabolaNode[Options](O){p}[rotation]{t}{A}
+%% Create a new node on the given General Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the parameter t to get the node on the parabola.
+%% #6 -> [output] the target node A on the parabola.
+\def\pstGeneralParabolaNode{\@ifnextchar[\Pst at GeneralParabolaNode{\Pst at GeneralParabolaNode[]}}
+\def\Pst at GeneralParabolaNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO %
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaNode@i{\Pst at GeneralParabolaNode@i[0]}}%
+\def\Pst at GeneralParabolaNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 4 index 2 index #2 mul add % x_o+t\cos\theta
+ 3 index 2 mul #2 dup mul exch div % t^2/(2p)
+ 2 index mul sub % x_o+t\cos\theta-t^2\dfrac{\sin\theta}{2p}
+ 4 index 2 index #2 mul add % y_o+t\sin\theta
+ 4 index 2 mul #2 dup mul exch div % t^2/(2p)
+ 4 index mul add % y_o+t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+ 7 2 roll pop pop pop pop pop
+ ){#3}
+ \Pst at geonodelabel{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralParabolaAbsNode[Options](O){p}[rotation]{$x_1$}{A}{B}
+%% Draw the nodes whose abscissas are the given value $x_1$ on the General Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% We use the parametric function \ref{ParametricFunctionOfGeneralParabola}, when $x$ equals $x_1$, we have
+%% $$t=\dfrac{p\cos\theta \pm \sqrt{p^2\cos^2\theta-2p(x_1-x_o)\sin\theta}}{\sin\theta} or x_1-x_o when \sin\theta=0$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the given abscissa value $x_1$
+%% #6 -> [output] the target node A on the parabola.
+%% #7 -> [output] the target node B on the parabola.
+\def\pstGeneralParabolaAbsNode{\@ifnextchar[\Pst at GeneralParabolaAbsNode{\Pst at GeneralParabolaAbsNode[]}}
+\def\Pst at GeneralParabolaAbsNode[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaAbsNode@i{\Pst at GeneralParabolaAbsNode@i[0]}}%
+\def\Pst at GeneralParabolaAbsNode@i[#1]#2#3#4{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space #2 % p x_1
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ dup abs 1E-5 lt { % like ZeroEq
+ 2 index 6 index sub % t
+ 5 index 1 index dup mul 6 index 2 mul div add % y=y_o+t^2/(2p)
+ 4 index exch 9 2 roll pop pop pop pop pop pop pop
+ } {
+ 3 index dup mul 2 index dup mul mul % p^2\cos^2\theta
+ 4 index 2 mul 4 index 8 index sub mul 2 index mul % 2p(x_1-x_o)\sin\theta
+ sub dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 4 index 3 index mul exch sub 1 index div % t_1
+ 5 index 1 index 3 index mul add 1 index dup mul 6 index 2 mul div 4 index mul add % y_1
+ 4 index exch 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#3}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space #2 % p x_1
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ dup abs 1E-5 lt { % like ZeroEq
+ pop pop pop pop pop pop 0 0
+ } {
+ 3 index dup mul 2 index dup mul mul % p^2\cos^2\theta
+ 4 index 2 mul 4 index 8 index sub mul 2 index mul % 2p(x_1-x_o)\sin\theta
+ sub dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 4 index 3 index mul add 1 index div % t_2
+ 5 index 1 index 3 index mul add 1 index dup mul 6 index 2 mul div 4 index mul add % y_2
+ 4 index exch 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralParabolaOrdNode[Options](O){p}[rotation]{$y_1$}{A}{B}
+%% Draw the nodes whose ordinates are the given value $y_1$ on the General Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% We use the parametric function \ref{ParametricFunctionOfGeneralParabola}, when y equals $y_1$, we have
+%% $$t=\dfrac{-p\sin\theta \pm \sqrt{p^2\sin^2\theta+2p(y_1-y_0)\cos\theta}{\cos\theta} or y_1-y_o when \cos\theta=0$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the given ordinate value $y_1$
+%% #6 -> [output] the target node A on the parabola.
+%% #7 -> [output] the target node B on the parabola.
+\def\pstGeneralParabolaOrdNode{\@ifnextchar[\Pst at GeneralParabolaOrdNode{\Pst at GeneralParabolaOrdNode[]}}
+\def\Pst at GeneralParabolaOrdNode[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaOrdNode@i{\Pst at GeneralParabolaOrdNode@i[0]}}%
+\def\Pst at GeneralParabolaOrdNode@i[#1]#2#3#4{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space #2 % p y_1
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 1 index abs 1E-5 lt { % if \cos\theta=0
+ 2 index 5 index sub % t
+ 6 index 1 index dup mul 6 index 2 mul div sub % x=x_o-t^2/(2p)
+ 4 index 9 2 roll pop pop pop pop pop pop pop
+ } {
+ 3 index dup mul 1 index dup mul mul % p^2\sin^2\theta
+ 4 index 2 mul 4 index 7 index sub mul 3 index mul % 2p(y_1-y_o)\cos\theta
+ add dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 4 index 2 index mul add neg 2 index div % t_1
+ 6 index 1 index 4 index mul add 1 index dup mul 6 index 2 mul div 3 index mul sub % x_1
+ 4 index 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#3}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space #2 % p y_1
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 1 index abs 1E-5 lt { % if \cos\theta=0
+ pop pop pop pop pop pop 0 0
+ } {
+ 3 index dup mul 1 index dup mul mul % p^2\sin^2\theta
+ 4 index 2 mul 4 index 7 index sub mul 3 index mul % 2p(y_1-y_o)\cos\theta
+ add dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 4 index 2 index mul sub 2 index div % t_2
+ 6 index 1 index 4 index mul add 1 index dup mul 6 index 2 mul div 3 index mul sub % x_2
+ 4 index 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralParabolaFocusNode[Options](O){p}[rotation]{F}
+%% Draw the focus nodes of the General Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the focus node A of the parabola.
+\def\pstGeneralParabolaFocusNode{\@ifnextchar[\Pst at GeneralParabolaFocusNode{\Pst at GeneralParabolaFocusNode[]}}
+\def\Pst at GeneralParabolaFocusNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaFocusNode@i{\Pst at GeneralParabolaFocusNode@i[0]}}%
+\def\Pst at GeneralParabolaFocusNode@i[#1]#2{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space 2 div % p/2
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 4 index 3 index 2 index mul sub % x=x0-p/2\sin\theta
+ 4 index 4 index 4 index mul add % y=y0+p/2\cos\theta
+ 7 2 roll pop pop pop pop pop
+ ){#2}
+ \Pst at geonodelabel{#2}%
+ \endgroup%
+}%
+%
+%% \pstGeneralParabolaDirectrixLine[Options](O){p}[rotation]{LA}{LB}
+%% Draw the directrix line of the General Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% The node LA also lies on the symmetrical axis line, and node LB is another one on the directrix line.
+%% they can be got after the orig nodes rotation $\theta$ about the vertex $O$,
+%% please refer to the extension rotation formula \ref{ExtensionRotationTransformFormula}.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the first node LA on the directrix line.
+%% #6 -> [output] the second node LB on the directrix line.
+\def\pstGeneralParabolaDirectrixLine{\@ifnextchar[\Pst at GeneralParabolaDirectrixLine{\Pst at GeneralParabolaDirectrixLine[]}}
+\def\Pst at GeneralParabolaDirectrixLine[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaDirectrixLine@i{\Pst at GeneralParabolaDirectrixLine@i[0]}}%
+\def\Pst at GeneralParabolaDirectrixLine@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space 2 div % p/2
+ 2 index 2 index 2 index sub 5 2 roll pop
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space 2 div % p/2
+ 2 index 1 add 2 index 2 index sub 5 2 roll pop
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \pstLineAB{#2}{#3}
+ \endgroup%
+}%
+%
+%% \pstGeneralParabolaLineInter[Options](O){p}[rotation]{A}{B}{C}{D}
+%% Find the intersections C and D of line AB and the General Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%%
+%% Case 1. When line AB is vertical, i.e, $x_1=x_2$, we have
+%% Case 1.1 When $\sin\theta=0$, we have
+%% $$t=x_1-x_o$$
+%% and $x_{C}=x_1,y_{C}=y_o+\dfrac{t^2}{2p}$, but D is not defined.
+%% Case 1.2 When $\sin\theta\neq0$, we have
+%% $$t_{1,2}=\dfrac{p\cos\theta \pm \sqrt{p^2\cos^2\theta-2p(x_1-x_o)\sin\theta}}{\sin\theta}$$
+%% and $x_{C,D}=x_1,y_{C,D}=y_o+t_{1,2}\sin\theta+\dfrac{t_{1,2}^2}{2p}\cos\theta$.
+%% Case 2. When line AB is not vertical, we can represent the line AB as the following function:
+%% $$y=kx+d$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, d=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}
+%% refer to equation (\ref{ParametricFunctionOfGeneralParabola}), we have
+%% $$ft^2-2pet-2pm=0$$
+%% where
+%% $$m=kx_o-y_o+d,e=k\cos\theta-\sin\theta,f=k\sin\theta+\cos\theta$$
+%% Case 2.1 When $f=k\sin\theta+\cos\theta=0$, which gives $\sin\theta\neq0$ and $k=-\dfrac{\cos\theta}{\sin\theta}$,
+%% at this time, we have $t=m\sin\theta$ and then
+%% $$x_{C}=x_o+t\cos\theta-\dfrac{t^2}{2p}\sin\theta, y_{C}=y_o+t\sin\theta+\dfrac{t^2}{2p}\cos\theta$$
+%% but D is not defined.
+%% Case 2.2 When $f=k\sin\theta+\cos\theta\neq0$, we have $t_{1,2}=\dfrac{pe\pm\sqrt{(pe)^2+2fmp}}{f}$, and then
+%% $$x_{C,D}=x_o+t_{1,2}\cos\theta-\dfrac{t_{1,2}^2}{2p}\sin\theta, y_{C,D}=y_o+t_{1,2}\sin\theta+\dfrac{t_{1,2}^2}{2p}\cos\theta$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the node $A$ on the given line
+%% #6 -> [input] the node $B$ on the given line
+%% #7 -> [output] the first intersection node $C$
+%% #8 -> [output] the second intersection node $D$
+\def\pstGeneralParabolaLineInter{\@ifnextchar[\Pst at GeneralParabolaLineInter{\Pst at GeneralParabolaLineInter[]}}
+\def\Pst at GeneralParabolaLineInter[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaLineInter@i{\Pst at GeneralParabolaLineInter@i[0]}}%
+\def\Pst at GeneralParabolaLineInter@i[#1]#2#3#4#5{%
+ \pst at getcoor{#2}\pst at tempA%
+ \pst at getcoor{#3}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ dup abs 1E-5 lt { % like ZeroEq % if \sin\theta=0
+ 5 index 8 index sub % t=x1-x0
+ 7 index 1 index dup mul \pst at parabola@semifocalchord\space 2 mul div add % y=y_o+t^2/(2p)
+ 7 index exch 9 2 roll pop pop pop pop pop pop pop
+ } {
+ \pst at parabola@semifocalchord\space %p
+ 2 index 1 index mul % p\cos\theta
+ dup dup mul 2 index 2 mul 9 index 12 index sub mul 4 index mul sub % p^2\cos^2\theta-2p(x_1-x_o)\sin\theta
+ dup 0 lt {
+ pop pop pop pop pop
+ pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt 1 index exch sub 3 index div % t_1
+ 9 index 1 index 5 index mul add 1 index dup mul 4 index 2 mul div 6 index mul add % y_1
+ 9 index exch 11 2 roll pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ \pst at parabola@semifocalchord\space %p
+ 2 index 9 index mul 8 index sub 2 index add % m=kx_o-y_o+t
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 5 index 2 index mul 1 index sub % e=k\cos\theta-\sin\theta
+ 6 index 2 index mul 3 index add % f=k\sin\theta+\cos\theta
+ % (aaaaaa----) print pstack (aaaaaa===) print
+ dup abs 1E-5 lt { % like ZeroEq
+ 4 index 3 index mul % t=m\sin\theta
+ 14 index 1 index 6 index mul add 1 index dup mul 8 index 2 mul div 5 index mul sub % x_1
+ 14 index 2 index 6 index mul add 2 index dup mul 9 index 2 mul div 7 index mul add % y_1
+ % (xxxxxx----) print pstack (xxxxxx===) print
+ 17 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ pop pop pop
+ } {
+ 5 index 2 index mul % pe
+ dup dup mul 2 index 7 index mul 8 index 2 mul mul add % (pe)^2+2fmp
+ dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index exch sub 2 index div % t_1
+ 15 index 1 index 7 index mul add 1 index dup mul 9 index 2 mul div 6 index mul sub % x_1
+ 15 index 2 index 7 index mul add 2 index dup mul 10 index 2 mul div 8 index mul add % y_1
+ % (bbbbbb----) print pstack (bbbbbb===) print
+ 18 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ dup abs 1E-5 lt { % like ZeroEq % if \sin\theta=0
+ pop pop pop pop pop pop pop pop 0 0
+ } {
+ \pst at parabola@semifocalchord\space %p
+ 2 index 1 index mul % p\cos\theta
+ dup dup mul 2 index 2 mul 9 index 12 index sub mul 4 index mul sub % p^2\cos^2\theta-2p(x_1-x_o)\sin\theta
+ dup 0 lt {
+ pop pop pop pop pop
+ pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt 1 index add 3 index div % t_2
+ 9 index 1 index 5 index mul add 1 index dup mul 4 index 2 mul div 6 index mul add % y_2
+ 9 index exch 11 2 roll pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ \pst at parabola@semifocalchord\space %p
+ 2 index 9 index mul 8 index sub 2 index add % m=kx_o-y_o+d
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 5 index 2 index mul 1 index sub % e=k\cos\theta-\sin\theta
+ 6 index 2 index mul 3 index add % f=k\sin\theta+\cos\theta
+ dup abs 1E-5 lt { % like ZeroEq
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 5 index 2 index mul % pe
+ dup dup mul 2 index 7 index mul 8 index 2 mul mul add % (pe)^2+2fmp
+ dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index add 2 index div % t_2
+ 15 index 1 index 7 index mul add 1 index dup mul 9 index 2 mul div 6 index mul sub % x_2
+ 15 index 2 index 7 index mul add 2 index dup mul 10 index 2 mul div 8 index mul add % y_2
+ % (ccccccc----) print pstack (ccccccc===) print
+ 18 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#5}
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstGeneralParabolaPolarNode[Options](O){p}[rotation](F)[L1][L2]{A}{B}{T}
+%% Find the polar point of chord AB on General Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% We use the following proposition to find the polar point of chord AB:
+%% Give any chord AB, drawing two focal chord AFC and BFD, where F is the focus, then drawing FX which is perpendicular to AFC at point F, and intersect with the directrix line at X; also drawing FY which is perpendicular to BFD at point F, and intersect with the directrix line at Y. Then the intersection T of AX and BY is the polar point of chord AB.
+%% If you don't know the focus F, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] optional, the focus node/coordinate F of the parabola.
+%% #6 -> [input] optional, the node/coordinate L1 on the directrix line of the parabola.
+%% #7 -> [input] optional, the node/coordinate L2 on the directrix line of the parabola.
+%% #8 -> [input] the node A on the parabola.
+%% #9 -> [input] the node B on the parabola.
+%% #10 -> [output] the polar node T of chord AB.
+\def\pstGeneralParabolaPolarNode{\@ifnextchar[\Pst at GeneralParabolaPolarNode{\Pst at GeneralParabolaPolarNode[]}}
+\def\Pst at GeneralParabolaPolarNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \def\pst at generalparabola@vertex{#2}%
+ \def\pst at generalparabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaPolarNode@i{\Pst at GeneralParabolaPolarNode@i[0]}}%
+\def\Pst at GeneralParabolaPolarNode@i[#1]{%
+ \def\pst at generalparabola@gradientangle{#1}%
+ \@ifnextchar(\Pst at GeneralParabolaPolarNode@j{\Pst at GeneralParabolaPolarNode@k}}%
+\def\Pst at GeneralParabolaPolarNode@j(#1){%
+ \def\pst at generalparabola@focus{#1}%
+ \@ifnextchar[\Pst at GeneralParabolaPolarNode@jj{\Pst at GeneralParabolaPolarNode@kk}}%
+\def\Pst at GeneralParabolaPolarNode@k{%
+ \pstGeneralParabolaFocusNode[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{@PST at GENERALPARABOLA@FOCUS}
+ \Pst at GeneralParabolaPolarNode@j(@PST at GENERALPARABOLA@FOCUS)}%
+\def\Pst at GeneralParabolaPolarNode@kk{%
+ \pstGeneralParabolaDirectrixLine[linestyle=none,PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{@PST at GENERALPARABOLA@DIRECTRIXLINEA}{@PST at GENERALPARABOLA@DIRECTRIXLINEB}
+ \Pst at GeneralParabolaPolarNode@jj[@PST at GENERALPARABOLA@DIRECTRIXLINEA][@PST at GENERALPARABOLA@DIRECTRIXLINEB]}%
+\def\Pst at GeneralParabolaPolarNode@jj[#1][#2]#3#4#5{%
+ \pstGeneralParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{#3}{\pst at generalparabola@focus}{@PST at GENERALPARABOLA@FOCALCHORDAFC at A}{@PST at GENERALPARABOLA@FOCALCHORDAFC at C}
+ \pstGeneralParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{#4}{\pst at generalparabola@focus}{@PST at GENERALPARABOLA@FOCALCHORDBFD at B}{@PST at GENERALPARABOLA@FOCALCHORDBFD at D}
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at generalparabola@focus}{@PST at GENERALPARABOLA@FOCALCHORDAFC at A}[@PST at GENERALPARABOLA@FOCALCHORDAFC at AA]
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at generalparabola@focus}{@PST at GENERALPARABOLA@FOCALCHORDBFD at B}[@PST at GENERALPARABOLA@FOCALCHORDBFD at BB]
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at generalparabola@focus}{@PST at GENERALPARABOLA@FOCALCHORDAFC at AA}{@PST at GENERALPARABOLA@FOCALCHORD at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at generalparabola@focus}{@PST at GENERALPARABOLA@FOCALCHORDBFD at BB}{@PST at GENERALPARABOLA@FOCALCHORD at Y}
+ \pstInterLL{#3}{@PST at GENERALPARABOLA@FOCALCHORD at X}{#4}{@PST at GENERALPARABOLA@FOCALCHORD at Y}{#5}
+ \Pst at geonodelabel{#5}%
+ \pstLineAB{#3}{#5}
+ \pstLineAB{#4}{#5}
+ \endgroup
+}%
+%
+%% \pstGeneralParabolaTangentNode[Options](O){p}[rotation]{T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the General Parabola P and get the node A and B on the General Parabola.
+%% We use the following proposition to find the tangent node of T:
+%% Give point T outside of the parabola, we draw any other two chords TPQ and TRS, PS and QR intersect at I, PR and QS intersect at X, XI and GeneralParabola intersect at A and B, then TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the given node T outside the parabola
+%% #6 -> [output] the tangent node name A on the parabola
+%% #7 -> [output] the tangent node name B on the parabola
+\def\pstGeneralParabolaTangentNode{\@ifnextchar[\Pst at GeneralParabolaTangentNode{\Pst at GeneralParabolaTangentNode[]}}
+\def\Pst at GeneralParabolaTangentNode[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \def\pst at generalparabola@vertex{#2}%
+ \def\pst at generalparabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralParabolaTangentNode@i{\Pst at GeneralParabolaTangentNode@i[0]}}%
+\def\Pst at GeneralParabolaTangentNode@i[#1]#2#3#4{%
+ \pstGeneralParabolaAbsNode[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{\pstAbscissa{#2} \pstAbscissa{\pst at generalparabola@vertex} le {\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 2 div add}{\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 2 div sub} ifelse}{@PST at GENERALPARABOLA@TANGENTAUX at P0}{@PST at GENERALPARABOLA@TANGENTAUX at P1}
+ \pstGeneralParabolaAbsNode[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{\pstAbscissa{#2} \pstAbscissa{\pst at generalparabola@vertex} le {\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 4 div add}{\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 4 div sub} ifelse}{@PST at GENERALPARABOLA@TANGENTAUX at R0}{@PST at GENERALPARABOLA@TANGENTAUX at R1}
+ \pstGeneralParabolaLineInter[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{#2}{@PST at GENERALPARABOLA@TANGENTAUX at P0}{@PST at GENERALPARABOLA@TANGENTAUX at P}{@PST at GENERALPARABOLA@TANGENTAUX at Q}
+ \pstGeneralParabolaLineInter[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{#2}{@PST at GENERALPARABOLA@TANGENTAUX at R0}{@PST at GENERALPARABOLA@TANGENTAUX at R}{@PST at GENERALPARABOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALPARABOLA@TANGENTAUX at P}{@PST at GENERALPARABOLA@TANGENTAUX at S}{@PST at GENERALPARABOLA@TANGENTAUX at Q}{@PST at GENERALPARABOLA@TANGENTAUX at R}{@PST at GENERALPARABOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALPARABOLA@TANGENTAUX at P}{@PST at GENERALPARABOLA@TANGENTAUX at R}{@PST at GENERALPARABOLA@TANGENTAUX at Q}{@PST at GENERALPARABOLA@TANGENTAUX at S}{@PST at GENERALPARABOLA@TANGENTAUX at X}
+ \pstGeneralParabolaLineInter(\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{@PST at GENERALPARABOLA@TANGENTAUX at X}{@PST at GENERALPARABOLA@TANGENTAUX at I}{#3}{#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \pstLineAB{#2}{#3}
+ \pstLineAB{#2}{#4}
+ \endgroup%
+}%
+%
+% 6. General Inversion Parabola with coordinate translation and rotation
+%% ----------------------------------------------------------
+%% The General Inversion Parabola P is defined by vertex O, the half of the focus chord axis $abs(p)$, the sign of $p$ indicates the direction of the parabola,
+%% and the rotation angle $\theta$ of the symmetrical axis.
+%%
+%% The equation can be got from the parametric function of the inversion parabola \ref{ParametricFunctionOfStandardInversionParabola},
+%% using the rotation transform formula \ref{RotationTransformFormula}, then we have
+%% \begin{equation}
+%% \left\{\begin{array}{l}
+%% x'=(\dfrac{t^2}{2p}+x_o)\cos\theta-(t+y_o)\sin\theta=x_o'-t\sin\theta+t^2\dfrac{\cos\theta}{2p}\\
+%% y'=(\dfrac{t^2}{2p}+x_o)\sin\theta+(t+y_o)\cos\theta=y_o'+t\cos\theta+t^2\dfrac{\sin\theta}{2p}
+%% \end{array}\right.
+%% \end{equation}
+%% where the $x_o'$ and $y_o'$ are the coordinate of the given vertex O after rotation.
+%% So we get the parametric function of the General Inversion Parabola with coordinate translation and rotation as following:
+%% \begin{equation}\label{ParametricFunctionOfGeneralInversionParabola}
+%% \left\{\begin{array}{l}
+%% x=x_o-t\sin\theta+t^2\dfrac{\cos\theta}{2p}\\
+%% y=y_o+t\cos\theta+t^2\dfrac{\sin\theta}{2p}
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstGeneralIParabola[Options](O){p}[rotation]{y0}{y1}
+%% Draw a General Inversion Parabola from y0 to y1 with Vertex O, the half of the focus chord axis $p$,
+%% and the rotation angle $\theta$ of the symmetrical axis.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the start of independent variable $y$
+%% #6 -> [input] the end of independent variable $y$
+\def\pstGeneralIParabola{\@ifnextchar[\Pst at GeneralIParabola{\Pst at GeneralIParabola[]}}
+\def\Pst at GeneralIParabola[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO %
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabola@i{\Pst at GeneralIParabola@i[0]}}%
+\def\Pst at GeneralIParabola@i[#1]#2#3{%
+ \parametricplot{#2}{#3}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 4 index 1 index t mul sub % x_o-t\sin\theta
+ 3 index 2 mul t dup mul exch div % t^2/(2p)
+ 3 index mul add % x_o-t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+ 4 index 3 index t mul add % y_o+t\cos\theta
+ 4 index 2 mul t dup mul exch div % t^2/(2p)
+ 3 index mul add % y_o+t\cos\theta+t^2\dfrac{\sin\theta}{2p}
+ 7 2 roll pop pop pop pop pop
+ }
+ \endgroup%
+}%
+%
+%% \pstGeneralIParabolaNode[Options](O){p}[rotation]{t}{A}
+%% Create a new node $A$ whose parameter is the the given value $t$ on the given General Inversion Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the parameter t to get the node on the parabola.
+%% #6 -> [output] the target node A on the parabola.
+\def\pstGeneralIParabolaNode{\@ifnextchar[\Pst at GeneralIParabolaNode{\Pst at GeneralIParabolaNode[]}}
+\def\Pst at GeneralIParabolaNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO %
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaNode@i{\Pst at GeneralIParabolaNode@i[0]}}%
+\def\Pst at GeneralIParabolaNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 4 index 1 index #2 mul sub % x_o-t\sin\theta
+ 3 index 2 mul #2 dup mul exch div % t^2/(2p)
+ 3 index mul add % x_o-t\sin\theta+t^2\dfrac{\cos\theta}{2p}
+ 4 index 3 index #2 mul add % y_o+t\cos\theta
+ 4 index 2 mul #2 dup mul exch div % t^2/(2p)
+ 3 index mul add % y_o+t\cos\theta+t^2\dfrac{\sin\theta}{2p}
+ 7 2 roll pop pop pop pop pop
+ ){#3}
+ \Pst at geonodelabel{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIParabolaAbsNode[Options](O){p}[rotation]{x_1}{A}{B}
+%% Find the nodes $A$ and $B$ whose abscissa are the the given value $x_1$ on the given General Inversion Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% when $x=x_1$, we have
+%% $$t^2\dfrac{\cos\theta}{2p}-t\sin\theta-(x_1-x_o)=0$$
+%% when $\cos\theta=0$, we have $t=-(x_1-x_o)$, so
+%% $$x=x_1, y=y_o+\dfrac{t^2}{2p}$$
+%% when $\cos\theta\neq0$, we have
+%% $$t_{1,2}=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{\sin\theta\pm\sqrt{\sin^2\theta+4\cos\theta/(2p)(x_1-x_o)}}{2\cos\theta/(2p)}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the abscissa $x_1$ to get the node on the parabola.
+%% #6 -> [output] the first node A on the parabola.
+%% #7 -> [output] the second node B on the parabola.
+\def\pstGeneralIParabolaAbsNode{\@ifnextchar[\Pst at GeneralIParabolaAbsNode{\Pst at GeneralIParabolaAbsNode[]}}
+\def\Pst at GeneralIParabolaAbsNode[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO %
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaAbsNode@i{\Pst at GeneralIParabolaAbsNode@i[0]}}%
+\def\Pst at GeneralIParabolaAbsNode@i[#1]#2#3#4{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 1 index abs 1E-5 lt { % if \cos\theta=0
+ 4 index #2 sub % t=x0-x1
+ 4 index 1 index dup mul 5 index 2 mul div add % y=y0+t^2/(2p)
+ #2 exch 8 2 roll pop pop pop pop pop pop
+ } {
+ 1 index 3 index div % \cos\theta/p
+ #2 6 index sub 1 index mul 2 mul 2 index dup mul add % 2(x1-x0)\cos\theta/p+\sin^2\theta
+ dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 2 index exch sub 1 index div % t1
+ 5 index 1 index 5 index mul add 1 index dup mul 4 index mul 6 index 2 mul div add % y1
+ #2 exch 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#3}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 1 index abs 1E-5 lt { % if \cos\theta=0
+ pop pop pop pop pop 0 0
+ } {
+ 1 index 3 index div % \cos\theta/p
+ #2 6 index sub 1 index mul 2 mul 2 index dup mul add % 2(x1-x0)\cos\theta/p+\sin^2\theta
+ dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 2 index add 1 index div % t2
+ 5 index 1 index 5 index mul add 1 index dup mul 4 index mul 6 index 2 mul div add % y2
+ #2 exch 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIParabolaOrdNode[Options](O){p}[rotation]{y_1}{A}{B}
+%% Find the nodes $A$ and $B$ whose ordinate are the the given value $y_1$ on the given General Inversion Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% when $y=y_1$, we have
+%% $$t^2\dfrac{\sin\theta}{2p}+t\cos\theta-(y_1-y_o)=0$$
+%% when $\sin\theta=0$, we have $t=(y_1-y_o)$, so
+%% $$y=y_1, x=x_o+\dfrac{t^2}{2p}$$
+%% when $\sin\theta\neq0$, we have
+%% $$t_{1,2}=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-\cos\theta\pm\sqrt{\cos^2\theta+4\sin\theta/(2p)(y_1-y_o)}}{2\sin\theta/(2p)}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the ordinate $y_1$ to get the node on the parabola.
+%% #6 -> [output] the first node A on the parabola.
+%% #7 -> [output] the second node B on the parabola.
+\def\pstGeneralIParabolaOrdNode{\@ifnextchar[\Pst at GeneralIParabolaOrdNode{\Pst at GeneralIParabolaOrdNode[]}}
+\def\Pst at GeneralIParabolaOrdNode[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO %
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaOrdNode@i{\Pst at GeneralIParabolaOrdNode@i[0]}}%
+\def\Pst at GeneralIParabolaOrdNode@i[#1]#2#3#4{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 0 index abs 1E-5 lt { % if \sin\theta=0
+ #2 4 index sub % t=y1-y0
+ 5 index 1 index dup mul 5 index 2 mul div add % x=x0+t^2/(2p)
+ #2 exch 8 2 roll pop pop pop pop pop pop
+ } {
+ 0 index 3 index div % \sin\theta/p
+ #2 5 index sub 1 index mul 2 mul 3 index dup mul add % 2(y1-y0)\sin\theta/p+\cos^2\theta
+ dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 3 index add neg 1 index div % t1
+ 6 index 1 index 4 index mul sub 1 index dup mul 5 index mul 6 index 2 mul div add % x1
+ #2 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#3}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space % p
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 0 index abs 1E-5 lt { % if \sin\theta=0
+ #2 4 index sub % t=y1-y0
+ 5 index 1 index dup mul 5 index 2 mul div add % x=x0+t^2/(2p)
+ #2 exch 8 2 roll pop pop pop pop pop pop
+ } {
+ 0 index 3 index div % \sin\theta/p
+ #2 5 index sub 1 index mul 2 mul 3 index dup mul add % 2(y1-y0)\sin\theta/p+\cos^2\theta
+ dup 0 lt {
+ pop pop pop pop pop pop pop 0 0
+ } {
+ sqrt 3 index sub 1 index div % t2
+ 6 index 1 index 4 index mul sub 1 index dup mul 5 index mul 6 index 2 mul div add % x2
+ #2 9 2 roll pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ ){#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIParabolaFocusNode[Options](O){p}[rotation]{F}
+%% Find the focus node $F$ of the given General Inversion Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the focus node F of the parabola.
+\def\pstGeneralIParabolaFocusNode{\@ifnextchar[\Pst at GeneralIParabolaFocusNode{\Pst at GeneralIParabolaFocusNode[]}}
+\def\Pst at GeneralIParabolaFocusNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO %
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaFocusNode@i{\Pst at GeneralIParabolaFocusNode@i[0]}}%
+\def\Pst at GeneralIParabolaFocusNode@i[#1]#2{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space 2 div % p/2
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 4 index 3 index 3 index mul add % x=x0+p/2\cos\theta
+ 4 index 4 index 3 index mul add % y=y0+p/2\sin\theta
+ 7 2 roll pop pop pop pop pop
+ ){#2}
+ \Pst at geonodelabel{#2}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIParabolaDirectrixLine[Options](O){p}[rotation]{LA}{LB}
+%% Draw the directrix line of the General Inversion Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% The node LA also lies on the symmetrical axis line, and node LB is another one on the directrix line.
+%% they can be got after the orig nodes rotation $\theta$ about the vertex $O$,
+%% please refer to the extension rotation formula \ref{ExtensionRotationTransformFormula}.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the first node LA on the directrix line.
+%% #6 -> [output] the second node LB on the directrix line.
+\def\pstGeneralIParabolaDirectrixLine{\@ifnextchar[\Pst at GeneralIParabolaDirectrixLine{\Pst at GeneralIParabolaDirectrixLine[]}}
+\def\Pst at GeneralIParabolaDirectrixLine[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaDirectrixLine@i{\Pst at GeneralIParabolaDirectrixLine@i[0]}}%
+\def\Pst at GeneralIParabolaDirectrixLine@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space 2 div % p/2
+ 2 index 1 index sub 2 index 5 2 roll pop
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at parabola@semifocalchord\space 2 div % p/2
+ 2 index 1 index sub 2 index 1 add 5 2 roll pop
+ #1 cos #1 sin % x y x0 y0 \cos\theta \sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \pstLineAB{#2}{#3}
+ \endgroup%
+}%
+%
+%% \pstGeneralIParabolaLineInter[Options](O){p}[rotation]{A}{B}{C}{D}
+%% Find the intersections C and D of line AB and the General Imversion Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%%
+%% Case 1. When line AB is vertical, i.e, $x_1=x_2$, we have
+%% $$t^2\dfrac{\cos\theta}{2p}-t\sin\theta-(x_1-x_o)=0$$
+%% Case 1.1 When $\cos\theta=0$, we have $t=x_o-x_1$,and then
+%% $$x_{C}=x_1,y_{C}=y_o+\dfrac{t^2}{2p}$$
+%% but D is not defined.
+%%
+%% Case 1.2 When $\cos\theta\neq0$, we have
+%% $$t_{1,2}=\dfrac{p\sin\theta \pm \sqrt{p^2\sin^2\theta+2p(x_1-x_o)\cos\theta}}{\cos\theta}$$
+%% and then
+%% $$x_{C,D}=x_1,y_{C,D}=y_o+t_{1,2}\sin\theta+\dfrac{t_{1,2}^2}{2p}\cos\theta$$
+%%
+%% Case 2. When line AB is not vertical, we can represent the line AB as the following function:
+%% $$y=kx+d$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, d=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% refer to equation (\ref{ParametricFunctionOfGeneralInversionParabola}), we have
+%% $$ft^2+2pet-2pm=0$$
+%% where
+%% $$m=kx_o-y_o+d,e=\cos\theta+k\sin\theta,f=\sin\theta-k\cos\theta$$
+%%
+%% Case 2.1 When $f=0$, which gives $\sin\theta\neq0$ and $k=\dfrac{\sin\theta}{\cos\theta}$,
+%% at this time, we have $t=m\cos\theta$, and then
+%% $$x_{C}=x_o-t\sin\theta+t^2\dfrac{\cos\theta}{2p},y_{C}=y_o+t\cos\theta+t^2\dfrac{\sin\theta}{2p}$$
+%% but D is not defined.
+%% Case 2.2 When $f\neq0$, we have
+%% $$t_{1,2}=\dfrac{-pe\pm\sqrt{(pe)^2+2fmp}}{f}$$
+%% and then
+%% $$x_{C,D}=x_o-t_{1,2}\sin\theta+t_{1,2}^2\dfrac{\cos\theta}{2p},y_{C,D}=y_o+t_{1,2}\cos\theta+t_{1,2}^2\dfrac{\sin\theta}{2p}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the node $A$ on the given line
+%% #6 -> [input] the node $B$ on the given line
+%% #7 -> [output] the first intersection node $C$
+%% #8 -> [output] the second intersection node $D$
+\def\pstGeneralIParabolaLineInter{\@ifnextchar[\Pst at GeneralIParabolaLineInter{\Pst at GeneralIParabolaLineInter[]}}
+\def\Pst at GeneralIParabolaLineInter[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \def\pst at parabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaLineInter@i{\Pst at GeneralIParabolaLineInter@i[0]}}%
+\def\Pst at GeneralIParabolaLineInter@i[#1]#2#3#4#5{%
+ \pst at getcoor{#2}\pst at tempA%
+ \pst at getcoor{#3}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 1 index abs 1E-5 lt { % like ZeroEq % if \cos\theta=0
+ 7 index 6 index sub % t=x0-x1
+ 7 index 1 index dup mul \pst at parabola@semifocalchord\space 2 mul div add % y=y_o+t^2/(2p)
+ 7 index exch 9 2 roll pop pop pop pop pop pop pop
+ } {
+ \pst at parabola@semifocalchord\space %p
+ 1 index 1 index mul % p\sin\theta
+ dup dup mul 2 index 2 mul 9 index 12 index sub mul 5 index mul add % p^2\sin^2\theta+2p(x_1-x_o)\cos\theta
+ dup 0 lt {
+ pop pop pop pop pop
+ pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt 1 index exch sub 4 index div % t_1
+ 9 index 1 index 6 index mul add 1 index dup mul 4 index 2 mul div 5 index mul add % y_1
+ 9 index exch 11 2 roll pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ \pst at parabola@semifocalchord\space %p
+ 2 index 9 index mul 8 index sub 2 index add % m=kx_o-y_o+d
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 5 index 1 index mul 2 index add % e=k\sin\theta+\cos\theta
+ 6 index 3 index mul 2 index exch sub % f=\sin\theta-k\cos\theta
+ dup abs 1E-5 lt { % like ZeroEq
+ 4 index 4 index mul % t=m\cos\theta
+ 14 index 1 index 5 index mul sub 1 index dup mul 8 index 2 mul div 6 index mul add % x_1
+ 14 index 2 index 7 index mul add 2 index dup mul 9 index 2 mul div 6 index mul add % y_1
+ 17 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ pop pop pop
+ } {
+ 5 index 2 index mul % pe
+ dup dup mul 2 index 7 index mul 8 index 2 mul mul add % (pe)^2+2fmp
+ dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index add neg 2 index div % t_1
+ 15 index 1 index 6 index mul sub 1 index dup mul 9 index 2 mul div 7 index mul add % x_1
+ 15 index 2 index 8 index mul add 2 index dup mul 10 index 2 mul div 7 index mul add % y_1
+ 18 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 1 index abs 1E-5 lt { % like ZeroEq % if \cos\theta=0
+ pop pop pop pop pop pop pop pop 0 0
+ } {
+ \pst at parabola@semifocalchord\space %p
+ 1 index 1 index mul % p\sin\theta
+ dup dup mul 2 index 2 mul 9 index 12 index sub mul 5 index mul add % p^2\sin^2\theta+2p(x_1-x_o)\cos\theta
+ dup 0 lt {
+ pop pop pop pop pop
+ pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt 1 index add 4 index div % t_2
+ 9 index 1 index 6 index mul add 1 index dup mul 4 index 2 mul div 5 index mul add % y_2
+ 9 index exch 11 2 roll pop pop pop pop
+ pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ \pst at parabola@semifocalchord\space %p
+ 2 index 9 index mul 8 index sub 2 index add % m=kx_o-y_o+t
+ #1 dup cos exch sin % \cos\theta \sin\theta
+ 5 index 1 index mul 2 index add % e=k\sin\theta+\cos\theta
+ 6 index 3 index mul 2 index exch sub % f=\sin\theta-k\cos\theta
+ dup abs 1E-5 lt { % like ZeroEq
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ 0 0
+ } {
+ 5 index 2 index mul % pe
+ dup dup mul 2 index 7 index mul 8 index 2 mul mul add % (pe)^2+2fmp
+ dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index sub 2 index div % t_2
+ 15 index 1 index 6 index mul sub 1 index dup mul 9 index 2 mul div 7 index mul add % x_2
+ 15 index 2 index 8 index mul add 2 index dup mul 10 index 2 mul div 7 index mul add % y_2
+ 18 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#5}
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIParabolaPolarNode[Options](O){p}[rotation](F)[L1][L2]{A}{B}{T}
+%% Find the polar point of chord AB on General Inversion Parabola P.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% We use the following proposition to find the polar point of chord AB:
+%% Give any chord AB, drawing two focal chord AFC and BFD, where F is the focus, then drawing FX which is perpendicular to AFC at point F, and intersect with the directrix line at X; also drawing FY which is perpendicular to BFD at point F, and intersect with the directrix line at Y. Then the intersection T of AX and BY is the polar point of chord AB.
+%% If you don't know the focus F, or the directrix line, we will find them automated, otherwise you can pass them to this macro.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] optional, the focus node/coordinate F of the parabola.
+%% #6 -> [input] optional, the node/coordinate L1 on the directrix line of the parabola.
+%% #7 -> [input] optional, the node/coordinate L2 on the directrix line of the parabola.
+%% #8 -> [input] the node A on the parabola.
+%% #9 -> [input] the node B on the parabola.
+%% #10 -> [output] the polar node T of chord AB.
+\def\pstGeneralIParabolaPolarNode{\@ifnextchar[\Pst at GeneralIParabolaPolarNode{\Pst at GeneralIParabolaPolarNode[]}}
+\def\Pst at GeneralIParabolaPolarNode[#1](#2)#3{%
+ \begingroup
+ \psset{#1}%
+ \def\pst at generalparabola@vertex{#2}%
+ \def\pst at generalparabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaPolarNode@i{\Pst at GeneralIParabolaPolarNode@i[0]}}%
+\def\Pst at GeneralIParabolaPolarNode@i[#1]{%
+ \def\pst at generalparabola@gradientangle{#1}%
+ \@ifnextchar(\Pst at GeneralIParabolaPolarNode@j{\Pst at GeneralIParabolaPolarNode@k}}%
+\def\Pst at GeneralIParabolaPolarNode@j(#1){%
+ \def\pst at generalparabola@focus{#1}%
+ \@ifnextchar[\Pst at GeneralIParabolaPolarNode@jj{\Pst at GeneralIParabolaPolarNode@kk}}%
+\def\Pst at GeneralIParabolaPolarNode@k{%
+ \pstGeneralIParabolaFocusNode[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{@PST at GENERALIPARABOLA@FOCUS}
+ \Pst at GeneralIParabolaPolarNode@j(@PST at GENERALIPARABOLA@FOCUS)}%
+\def\Pst at GeneralIParabolaPolarNode@kk{%
+ \pstGeneralIParabolaDirectrixLine[linestyle=none,PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{@PST at GENERALIPARABOLA@DIRECTRIXLINEA}{@PST at GENERALIPARABOLA@DIRECTRIXLINEB}
+ \Pst at GeneralIParabolaPolarNode@jj[@PST at GENERALIPARABOLA@DIRECTRIXLINEA][@PST at GENERALIPARABOLA@DIRECTRIXLINEB]}%
+\def\Pst at GeneralIParabolaPolarNode@jj[#1][#2]#3#4#5{%
+ \pstGeneralIParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{#3}{\pst at generalparabola@focus}{@PST at GENERALIPARABOLA@FOCALCHORDAFC at A}{@PST at GENERALIPARABOLA@FOCALCHORDAFC at C}
+ \pstGeneralIParabolaLineInter[PointName=none,PointSymbol=none,linestyle=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[\pst at generalparabola@gradientangle]{#4}{\pst at generalparabola@focus}{@PST at GENERALIPARABOLA@FOCALCHORDBFD at B}{@PST at GENERALIPARABOLA@FOCALCHORDBFD at D}
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at generalparabola@focus}{@PST at GENERALIPARABOLA@FOCALCHORDAFC at A}[@PST at GENERALIPARABOLA@FOCALCHORDAFC at AA]
+ \pstRotation[PointName=none,PointSymbol=none,RotAngle=90]{\pst at generalparabola@focus}{@PST at GENERALIPARABOLA@FOCALCHORDBFD at B}[@PST at GENERALIPARABOLA@FOCALCHORDBFD at BB]
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at generalparabola@focus}{@PST at GENERALIPARABOLA@FOCALCHORDAFC at AA}{@PST at GENERALIPARABOLA@FOCALCHORD at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#1}{#2}{\pst at generalparabola@focus}{@PST at GENERALIPARABOLA@FOCALCHORDBFD at BB}{@PST at GENERALIPARABOLA@FOCALCHORD at Y}
+ \pstInterLL{#3}{@PST at GENERALIPARABOLA@FOCALCHORD at X}{#4}{@PST at GENERALIPARABOLA@FOCALCHORD at Y}{#5}
+ \Pst at geonodelabel{#5}%
+ \pstLineAB{#3}{#5}
+ \pstLineAB{#4}{#5}
+ \endgroup
+}%
+%
+%% \pstGeneralIParabolaTangentNode[Options](O){p}[rotation]{T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the General Inversion Parabola P and get the node A and B on the General Inversion Parabola.
+%% We use the following proposition to find the tangent node of T:
+%% Give point T outside of the parabola, we draw any other two chords TPQ and TRS, PS and QR intersect at I, PR and QS intersect at X, XI and GeneralIParabola intersect at A and B, then TA is the tangent line through A and TB is the tangent line through B.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the parabola vertex O
+%% #3 -> [input] the half of focal chord $p$
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the given node T outside the parabola
+%% #6 -> [output] the tangent node name A on the parabola
+%% #7 -> [output] the tangent node name B on the parabola
+\def\pstGeneralIParabolaTangentNode{\@ifnextchar[\Pst at GeneralIParabolaTangentNode{\Pst at GeneralIParabolaTangentNode[]}}
+\def\Pst at GeneralIParabolaTangentNode[#1](#2)#3{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \def\pst at generalparabola@vertex{#2}%
+ \def\pst at generalparabola@semifocalchord{#3}%
+ \@ifnextchar[\Pst at GeneralIParabolaTangentNode@i{\Pst at GeneralIParabolaTangentNode@i[0]}}%
+\def\Pst at GeneralIParabolaTangentNode@i[#1]#2#3#4{%
+ \pstGeneralIParabolaAbsNode[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{\pstAbscissa{#2} \pstAbscissa{\pst at generalparabola@vertex} le {\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 2 div add}{\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 2 div sub} ifelse}{@PST at GENERALIPARABOLA@TANGENTAUX at P0}{@PST at GENERALIPARABOLA@TANGENTAUX at P1}
+ \pstGeneralIParabolaAbsNode[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{\pstAbscissa{#2} \pstAbscissa{\pst at generalparabola@vertex} le {\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 4 div add}{\pstAbscissa{\pst at generalparabola@vertex} \pst at generalparabola@semifocalchord\space abs 4 div sub} ifelse}{@PST at GENERALIPARABOLA@TANGENTAUX at R0}{@PST at GENERALIPARABOLA@TANGENTAUX at R1}
+ \pstGeneralIParabolaLineInter[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{#2}{@PST at GENERALIPARABOLA@TANGENTAUX at P0}{@PST at GENERALIPARABOLA@TANGENTAUX at P}{@PST at GENERALIPARABOLA@TANGENTAUX at Q}
+ \pstGeneralIParabolaLineInter[PointName=none,PointSymbol=none](\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{#2}{@PST at GENERALIPARABOLA@TANGENTAUX at R0}{@PST at GENERALIPARABOLA@TANGENTAUX at R}{@PST at GENERALIPARABOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALIPARABOLA@TANGENTAUX at P}{@PST at GENERALIPARABOLA@TANGENTAUX at S}{@PST at GENERALIPARABOLA@TANGENTAUX at Q}{@PST at GENERALIPARABOLA@TANGENTAUX at R}{@PST at GENERALIPARABOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALIPARABOLA@TANGENTAUX at P}{@PST at GENERALIPARABOLA@TANGENTAUX at R}{@PST at GENERALIPARABOLA@TANGENTAUX at Q}{@PST at GENERALIPARABOLA@TANGENTAUX at S}{@PST at GENERALIPARABOLA@TANGENTAUX at X}
+ \pstGeneralIParabolaLineInter(\pst at generalparabola@vertex){\pst at generalparabola@semifocalchord}[#1]{@PST at GENERALIPARABOLA@TANGENTAUX at X}{@PST at GENERALIPARABOLA@TANGENTAUX at I}{#3}{#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \pstLineAB{#2}{#3}
+ \pstLineAB{#2}{#4}
+ \endgroup%
+}%
+%
+%
+% 7. Standard Hyperbola with coordinate translation
+%% ----------------------------------------------------------
+%% The Standard Hyperbola H is defined by center O, the half of the real axis $a$, the half of the imaginary axis $b$.
+%% The equation can be written as:
+%% \begin{equation}\label{FunctionOfStandardHyperbola}
+%% \dfrac{(x-x0)^2}{a^2}-\dfrac{(y-y0)^2}{b^2}=1
+%% \end{equation}
+%% and the parametric function can be written as:
+%% \begin{equation}\label{ParametricFunctionOfStandardHyperbola}
+%% \left\{\begin{array}{l}
+%% x=a\sec\alpha+x_o\\
+%% y=b\tan\alpha+y_o
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstHyperbola[Options](O)(a,b)[maxAngleX]
+%% Draw a Hyperbola with center O, the half of the real axis $abs(a)$, and the half of the imaginary axis $abs(b)$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the maximal angle to draw the branch.
+\def\pstHyperbola{\@ifnextchar[\Pst at Hyperbola{\Pst at Hyperbola[]}}
+\def\Pst at Hyperbola[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at Hyperbola@i{\Pst at Hyperbola@i[85]}}%
+\def\Pst at Hyperbola@i[#1]{%
+ \pst at cnth=#1\pst at cntg=180\pst at cntm=180\pst at cntn=360
+ \ifnum\pst at cnth<0
+ \loop\advance\pst at cnth by 90
+ \ifnum\pst at cnth<0
+ \repeat
+ \fi
+ \ifnum\pst at cnth>90
+ \loop\advance\pst at cnth by -90
+ \ifnum\pst at cnth>90
+ \repeat
+ \fi
+ \ifnum\pst at cnth>85
+ \advance\pst at cnth by -5
+ \fi
+ \advance\pst at cntg by -\pst at cnth
+ \advance\pst at cntm by \pst at cnth
+ \advance\pst at cntn by -\pst at cnth
+ \Pst at Hyperbola@ii[0][\number\pst at cnth]%
+ \Pst at Hyperbola@ii[\number\pst at cntg][180]%
+ \Pst at Hyperbola@ii[180][\number\pst at cntm]%
+ \Pst at Hyperbola@ii[\number\pst at cntn][360]%
+ \endgroup%
+}%
+\def\Pst at Hyperbola@ii[#1][#2]{%
+ \parametricplot{#1}{#2}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ t dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ } {
+ 5 index 4 index 3 index div add % x0+a\sec{t}
+ 5 index 4 index 3 index mul 4 index div add % y0+b\tan{t}
+ 8 2 roll pop pop pop pop pop pop
+ } ifelse
+ }%
+}%
+%
+%% \pstHyperbolaNode[Options](O)(a,b){t}{P}
+%% Create a new node P on the Hyperbola E whose parameter is the given value $t$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the parametric argument t.
+%% #5 -> [output] the target node name.
+\def\pstHyperbolaNode{\@ifnextchar[\Pst at HyperbolaNode{\Pst at HyperbolaNode[]}}
+\def\Pst at HyperbolaNode[#1](#2)(#3)#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ }{
+ 5 index 4 index 3 index div add % x0+a\sec{t}
+ 5 index 4 index 3 index 5 index div mul add % y0+b\tan{t}
+ } ifelse
+ 8 2 roll pop pop pop pop pop pop
+ ){#5}%
+ \Pst at geonodelabel{#5}%
+ \endgroup%
+}%
+%
+%% \pstHyperbolaAbsNode[Options](O)(a,b){x_1}{A}{B}
+%% Create a new node P on the Hyperbola E whose abscissa is the given value $x_1$.
+%% when $x=x_1$, we have
+%% $$\dfrac{(x_1-x_0)^2}{a^2}-\dfrac{(y-y_0)^2}{b^2}=1$$
+%% $$(y-y_0)^2=\dfrac{b^2(x_1-x_0)^2-a^2b^2}{a^2}$$
+%% $$y=y_0\pm\dfrac{b}{a}\sqrt{(x_1-x_0)^2-a^2}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the abbscissa value $x_1$.
+%% #5 -> [output] the first target node name.
+%% #6 -> [output] the first second node name.
+\def\pstHyperbolaAbsNode{\@ifnextchar[\Pst at HyperbolaAbsNode{\Pst at HyperbolaAbsNode[]}}
+\def\Pst at HyperbolaAbsNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 4 index sub dup mul 2 index dup mul sub % (x_1-x_0)^2-a^2
+ dup 0 lt {
+ pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index mul 2 index div 3 index exch sub % y1
+ #4 exch 6 2 roll pop pop pop pop
+ } ifelse
+ ){#5}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 4 index sub dup mul 2 index dup mul sub % (x_1-x_0)^2-a^2
+ dup 0 lt {
+ pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index mul 2 index div 3 index add % y2
+ #4 exch 6 2 roll pop pop pop pop
+ } ifelse
+ ){#6}%
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \endgroup%
+}%
+%
+%% \pstHyperbolaOrdNode[Options](O)(a,b){y_1}{A}{B}
+%% Create a new node P on the Hyperbola E whose ordinate is the given value $y_1$.
+%% when $y=y_1$, we have
+%% $$\dfrac{(x-x_0)^2}{a^2}-\dfrac{(y_1-y_0)^2}{b^2}=1$$
+%% $$(x-x_0)^2=\dfrac{a^2(y_1-y_0)^2+a^2b^2}{b^2}$$
+%% $$x=x_0\pm\dfrac{a}{b}\sqrt{(y_1-y_0)^2+b^2}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the ordinate value $y_1$.
+%% #5 -> [output] the first target node name.
+%% #6 -> [output] the first second node name.
+\def\pstHyperbolaOrdNode{\@ifnextchar[\Pst at HyperbolaOrdNode{\Pst at HyperbolaOrdNode[]}}
+\def\Pst at HyperbolaOrdNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 3 index sub dup mul 1 index dup mul add % (y_1-y_0)^2+b^2
+ sqrt 2 index mul 1 index div 4 index exch sub % x1
+ #4 6 2 roll pop pop pop pop
+ ){#5}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 3 index sub dup mul 1 index dup mul add % (y_1-y_0)^2+b^2
+ sqrt 2 index mul 1 index div 4 index add % x2
+ #4 6 2 roll pop pop pop pop
+ ){#6}%
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \endgroup%
+}%
+%
+%% \pstHyperbolaFocusNode[Options](O)(a,b){F1}{F2}
+%% Create the two focus node F1 and F2 of the Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [output] the first focus node name.
+%% #5 -> [output] the first focus node name.
+\def\pstHyperbolaFocusNode{\@ifnextchar[\Pst at HyperbolaFocusNode{\Pst at HyperbolaFocusNode[]}}
+\def\Pst at HyperbolaFocusNode[#1](#2)(#3)#4#5{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ dup mul exch dup mul add sqrt % c
+ 2 index exch sub 1 index % x0-c,y0
+ 4 2 roll pop pop
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ dup mul exch dup mul add sqrt % c
+ 2 index add 1 index % x0+c,y0
+ 6 2 roll pop pop pop pop
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstHyperbolaDirectrixLine[Options](O)(a,b){Lx}{Ly}{Rx}{Ry}
+%% Draw the two directrix lines L1 and L2 of the Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [output] the first node name on the first directrix line.
+%% #5 -> [output] the second node name on the first directrix line.
+%% #6 -> [output] the first node name on the second directrix line.
+%% #7 -> [output] the second node name on the second directrix line.
+\def\pstHyperbolaDirectrixLine{\@ifnextchar[\Pst at HyperbolaDirectrixLine{\Pst at HyperbolaDirectrixLine[]}}
+\def\Pst at HyperbolaDirectrixLine[#1](#2)(#3)#4#5#6#7{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 4 index exch sub 3 index % x0-a^2/c,y0
+ 6 2 roll pop pop pop pop
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 4 index exch sub 3 index 1 add % x0-a^2/c,y0+1
+ 6 2 roll pop pop pop pop
+ ){#5}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 4 index add 3 index % x0+a^2/c,y0
+ 6 2 roll pop pop pop pop
+ ){#6}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 4 index add 3 index 1 add % x0+a^2/c,y0+1
+ 6 2 roll pop pop pop pop
+ ){#7}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \pstLineAB{#4}{#5}%
+ \pstLineAB{#6}{#7}%
+ \endgroup%
+}%
+%
+%% \pstHyperbolaAsymptoteLine[Options](O)(a,b){L1}{L2}
+%% Draw the two asymptote lines L1 and L2 of the Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [output] the second node name on the first asymptote line, the first node is the center O.
+%% #5 -> [output] the second node name on the second asymptote line, the first node is the center O.
+\def\pstHyperbolaAsymptoteLine{\@ifnextchar[\Pst at HyperbolaAsymptoteLine{\Pst at HyperbolaAsymptoteLine[]}}
+\def\Pst at HyperbolaAsymptoteLine[#1](#2)(#3)#4#5{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 add % x=x0+1
+ 4 index 2 index add % y=y0+b/a
+ 7 2 roll pop pop pop pop pop
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 add % x=x0+1
+ 4 index 2 index sub % y=y0-b/a
+ 7 2 roll pop pop pop pop pop
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLineAB{#2}{#4}%
+ \pstLineAB{#2}{#5}%
+ \endgroup%
+}%
+%
+%% \pstHyperbolaLineInter[Options](O)(a,b){A}{B}{C}{D}
+%% Find the two intersection nodes C and D of the Hyperbola H and line AB.
+%%
+%% Case 1. When line AB is vertical, i.e, $x_1=x_2$, we have
+%% $$y=y_0\pm\dfrac{b}{a}\sqrt{(x_1-x_0)^2-a^2}$$
+%%
+%% Case 2. When line AB is not vertical, we can represent the line AB as the following function:
+%% $$y=kx+d$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, d=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% refer to equation (\ref{FunctionOfStandardHyperbola}), we have
+%% $$(b^2-k^2a^2)X^2-2a^2kmX-a^2(m^2+b^2)=0$$
+%% $$Y=kX+m$$
+%% where
+%% $$m=kx_o-y_o+d,X=x-x_0,Y=y-y_0$$
+%%
+%% Case 2.1 When $b^2-k^2a^2=0$, which gives $k=\pm\dfrac{b}{a}\neq0$,
+%% at this time, when $m=0$, the line AB becomes the asymptote of the hyperbola,
+%% so there is none intersection any more; else we have
+%% $$x_{C}=x_o-\dfrac{m^2+b^2}{2km},y_{C}=kx_{C}+d$$
+%% but D is not defined.
+%%
+%% Case 2.2 When $b^2-k^2a^2\neq0$, we have
+%% $$x_{C,D}=x_0+\dfrac{a^2km\pm{}ab\sqrt{m^2+b^2-k^2a^2}}{b^2-k^2a^2},y_{C,D}=kx_{C,D}+d$$
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the first node name on the given line AB.
+%% #5 -> [input] the second node name on the given line AB.
+%% #6 -> [output] the first intersection node.
+%% #7 -> [output] the second intersection node.
+\def\pstHyperbolaLineInter{\@ifnextchar[\Pst at HyperbolaLineInter{\Pst at HyperbolaLineInter[]}}
+\def\Pst at HyperbolaLineInter[#1](#2)(#3)#4#5#6#7{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pst at getcoor{#4}\pst at tempA%
+ \pst at getcoor{#5}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ \pst at tempA \tx at UserCoor % x_1,y_1
+ \pst at tempB \tx at UserCoor % x_2,y_2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ %% $$y=y_0\pm\dfrac{b}{a}\sqrt{(x_1-x_0)^2-a^2}$$
+ 3 index 8 index sub dup mul 6 index dup mul sub
+ dup 0 lt {
+ pop pop pop pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt 5 index mul 6 index div 7 index exch sub % y1
+ 4 index exch % x1
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ 7 index dup mul 9 index 4 index mul dup mul sub % b^2-k^2a^2
+ 0 index abs 1E-5 lt {
+ 1 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ %% $$x_{C}=x_o-\dfrac{m^2+b^2}{2km},y_{C}=kx_{C}+d$$
+ 1 index dup mul 9 index dup mul add 2 index 5 index mul 2 mul div % (m^2+b^2)/2km
+ 12 index exch sub % x1
+ 4 index 1 index mul 4 index add % y1
+ 14 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } {
+ 1 index dup mul 1 index add % m^2+b^2-k^2a^2
+ dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 10 index mul 9 index mul 10 index dup mul 5 index mul 3 index mul exch sub 1 index div 12 index add % x1
+ 4 index 1 index mul 4 index add % y1
+ 14 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#6}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ \pst at tempA \tx at UserCoor % x_1,y_1
+ \pst at tempB \tx at UserCoor % x_2,y_2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ %% $$y=y_0\pm\dfrac{b}{a}\sqrt{(x_1-x_0)^2-a^2}$$
+ 3 index 8 index sub dup mul 6 index dup mul sub
+ dup 0 lt {
+ pop pop pop pop pop pop pop pop
+ 0 0
+ } {
+ sqrt 5 index mul 6 index div 7 index add % y2
+ 4 index exch % x2
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ 7 index dup mul 9 index dup mul 4 index dup mul mul sub % b^2-k^2a^2
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ 1 index dup mul 1 index add % m^2+b^2-k^2a^2
+ dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 10 index mul 9 index mul 10 index dup mul 5 index mul 3 index mul add 1 index div 12 index add % x2
+ 4 index 1 index mul 4 index add % y2
+ 14 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#7}%
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \endgroup%
+}%
+%
+%% \pstHyperbolaPolarNode[Options](O)(a,b){A}{B}{T}
+%% Find the polar point of chord AB on hyperbola H.
+%% We use the following proposition to find the polar point of chord AB:
+%% Let $P$, $Q$ are vertex points of the hyperbola, for any chord $AB$ of hyperbola, $PA$ and $BQ$ intersect at $E$, $PB$ and $AQ$ intersect at $F$, then the middle point $M$ of $EF$ is the polar point of chord $AB$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the node A on the hyperbola.
+%% #5 -> [input] the node B on the hyperbola.
+%% #6 -> [output] the polar node T of chord AB.
+\def\pstHyperbolaPolarNode{\@ifnextchar[\Pst at HyperbolaPolarNode{\Pst at HyperbolaPolarNode[]}}
+\def\Pst at HyperbolaPolarNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \psset{#1}%
+ \pstHyperbolaOrdNode[PointName=none,PointSymbol=none](#2)(#3){\pstOrdinate{#2}}{@PST at HYPERBOLA@VERTEXA}{@PST at HYPERBOLA@VERTEXB}
+ \pstInterLL[PointName=none,PointSymbol=none]{#4}{@PST at HYPERBOLA@VERTEXA}{#5}{@PST at HYPERBOLA@VERTEXB}{@PST at HYPERBOLA@INTER at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#5}{@PST at HYPERBOLA@VERTEXA}{#4}{@PST at HYPERBOLA@VERTEXB}{@PST at HYPERBOLA@INTER at Y}
+ \pstMiddleAB{@PST at HYPERBOLA@INTER at X}{@PST at HYPERBOLA@INTER at Y}{#6}
+ \Pst at geonodelabel{#6}%
+ \pstLineAB{#4}{#6}
+ \pstLineAB{#5}{#6}
+ \endgroup
+}%
+%
+%% \pstHyperbolaTangentNode[Options](O)(a,b){T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the Hyperbola H and get the node A and B on the Hyperbola.
+%% We use the following proposition to find the tangent points $A$ and $B$ of $T$:
+%% Let $T$ is a point out of the hyperbola, we give any two chords $TPQ$ and $TRS$ of the hyperbola, $PR$ and $QS$ intersect at $X$, $RQ$ and $PS$ intersect at $Y$,
+%% then the intersection point $A$ and $B$ of $XY$ and the hyperbola are the tangent points from $T$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the given node T outside the hyperbola
+%% #5 -> [output] the tangent node name A on the hyperbola
+%% #6 -> [output] the tangent node name B on the hyperbola
+\def\pstHyperbolaTangentNode{\@ifnextchar[\Pst at HyperbolaTangentNode{\Pst at HyperbolaTangentNode[]}}
+\def\Pst at HyperbolaTangentNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pstHyperbolaOrdNode[PointName=none,PointSymbol=none](#2)(#3){\pstOrdinate{#4} 0.3 add}{@PST at HYPERBOLA@TANGENTAUX at P0}{@PST at HYPERBOLA@TANGENTAUX at P1}
+ \pstHyperbolaOrdNode[PointName=none,PointSymbol=none](#2)(#3){\pstOrdinate{#4} 0.3 sub}{@PST at HYPERBOLA@TANGENTAUX at R0}{@PST at HYPERBOLA@TANGENTAUX at R1}
+ \pstHyperbolaLineInter[PointName=none,PointSymbol=none](#2)(#3){#4}{@PST at HYPERBOLA@TANGENTAUX at P0}{@PST at HYPERBOLA@TANGENTAUX at P}{@PST at HYPERBOLA@TANGENTAUX at Q}
+ \pstHyperbolaLineInter[PointName=none,PointSymbol=none](#2)(#3){#4}{@PST at HYPERBOLA@TANGENTAUX at R0}{@PST at HYPERBOLA@TANGENTAUX at R}{@PST at HYPERBOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at HYPERBOLA@TANGENTAUX at P}{@PST at HYPERBOLA@TANGENTAUX at S}{@PST at HYPERBOLA@TANGENTAUX at Q}{@PST at HYPERBOLA@TANGENTAUX at R}{@PST at HYPERBOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at HYPERBOLA@TANGENTAUX at P}{@PST at HYPERBOLA@TANGENTAUX at R}{@PST at HYPERBOLA@TANGENTAUX at Q}{@PST at HYPERBOLA@TANGENTAUX at S}{@PST at HYPERBOLA@TANGENTAUX at X}
+ \pstHyperbolaLineInter(#2)(#3){@PST at HYPERBOLA@TANGENTAUX at X}{@PST at HYPERBOLA@TANGENTAUX at I}{#5}{#6}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \pstLineAB{#4}{#5}
+ \pstLineAB{#4}{#6}
+ \endgroup%
+}%
+%
+% 8. Standard Inversion Hyperbola with coordinate translation
+%% ----------------------------------------------------------
+%% The Standard Inversion Hyperbola H is defined by center O, the half of the real axis $a$, the half of the imaginary axis $b$.
+%% The equation can be written as:
+%% \begin{equation}\label{FunctionOfStandardInversionHyperbola}
+%% \dfrac{(y-y0)^2}{a^2}-\dfrac{(x-x0)^2}{b^2}=1
+%% \end{equation}
+%% and the parametric function can be written as:
+%% \begin{equation}\label{ParametricFunctionOfStandardInversionHyperbola}
+%% \left\{\begin{array}{l}
+%% x=b\tan\alpha+x_o\\
+%% y=a\sec\alpha+y_o
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstIHyperbola[Options](O)(a,b)[maxAngleY]
+%% Draw a Inversion Hyperbola with center O, the half of the real axis $abs(a)$, and the half of the imaginary axis $abs(b)$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the maximal angle to draw the branch.
+\def\pstIHyperbola{\@ifnextchar[\Pst at IHyperbola{\Pst at IHyperbola[]}}
+\def\Pst at IHyperbola[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at IHyperbola@i{\Pst at IHyperbola@i[85]}}%
+\def\Pst at IHyperbola@i[#1]{%
+ \pst at cnth=#1\pst at cntg=180\pst at cntm=180\pst at cntn=360
+ \ifnum\pst at cnth<0
+ \loop\advance\pst at cnth by 90
+ \ifnum\pst at cnth<0
+ \repeat
+ \fi
+ \ifnum\pst at cnth>90
+ \loop\advance\pst at cnth by -90
+ \ifnum\pst at cnth>90
+ \repeat
+ \fi
+ \ifnum\pst at cnth>85
+ \advance\pst at cnth by -5
+ \fi
+ \advance\pst at cntg by -\pst at cnth
+ \advance\pst at cntm by \pst at cnth
+ \advance\pst at cntn by -\pst at cnth
+ \Pst at IHyperbola@ii[0][\number\pst at cnth]%
+ \Pst at IHyperbola@ii[\number\pst at cntg][180]%
+ \Pst at IHyperbola@ii[180][\number\pst at cntm]%
+ \Pst at IHyperbola@ii[\number\pst at cntn][360]%
+ \endgroup%
+}%
+\def\Pst at IHyperbola@ii[#1][#2]{%
+ \parametricplot{#1}{#2}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ t dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ } {
+ 5 index 3 index 2 index mul 3 index div add % x0+b\tan{t}
+ 5 index 5 index 4 index div add % y0+a\sec{t}
+ 8 2 roll pop pop pop pop pop pop
+ } ifelse
+ }%
+}%
+%
+%% \pstIHyperbolaNode[Options](O)(a,b){t}{P}
+%% Create a new node P on the Inversion Hyperbola E whose parameter is the given value $t$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the parametric argument t.
+%% #5 -> [output] the target node name.
+\def\pstIHyperbolaNode{\@ifnextchar[\Pst at IHyperbolaNode{\Pst at IHyperbolaNode[]}}
+\def\Pst at IHyperbolaNode[#1](#2)(#3)#4#5{%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ }{
+ 5 index 3 index 2 index mul 3 index div add % x0+b\tan{t}
+ 5 index 5 index 4 index div add % y0+a\sec{t}
+ 8 2 roll pop pop pop pop pop pop
+ } ifelse
+ ){#5}%
+ \Pst at geonodelabel{#5}%
+ \endgroup%
+}%
+%
+%% \pstIHyperbolaAbsNode[Options](O)(a,b){x_1}{A}{B}
+%% Create a new node P on the Inversion Hyperbola E whose abscissa is the given value $x_1$.
+%% when $x=x_1$, we have
+%% $$\dfrac{(y-y_0)^2}{a^2}-\dfrac{(x_1-x_0)^2}{b^2}=1$$
+%% $$(y-y_0)^2=\dfrac{a^2(x_1-x_0)^2+a^2b^2}{b^2}$$
+%% $$y=y_0\pm\dfrac{a}{b}\sqrt{(x_1-x_0)^2+b^2}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the abbscissa value $x_1$.
+%% #5 -> [output] the first target node name.
+%% #6 -> [output] the first second node name.
+\def\pstIHyperbolaAbsNode{\@ifnextchar[\Pst at IHyperbolaAbsNode{\Pst at IHyperbolaAbsNode[]}}
+\def\Pst at IHyperbolaAbsNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 4 index sub dup mul 1 index dup mul add % (x_1-x_0)^2+b^2
+ sqrt 2 index mul 1 index div 3 index exch sub % y1
+ #4 exch 6 2 roll pop pop pop pop
+ ){#5}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 4 index sub dup mul 1 index dup mul add % (x_1-x_0)^2+b^2
+ sqrt 2 index mul 1 index div 3 index add % y1
+ #4 exch 6 2 roll pop pop pop pop
+ ){#6}%
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \endgroup%
+}%
+%
+%% \pstIHyperbolaOrdNode[Options](O)(a,b){y_1}{A}{B}
+%% Create a new node P on the Inversion Hyperbola E whose ordinate is the given value $y_1$.
+%% when $y=y_1$, we have
+%% $$\dfrac{(y_1-y_0)^2}{a^2}-\dfrac{(x-x_0)^2}{b^2}=1$$
+%% $$(x-x_0)^2=\dfrac{b^2(y_1-y_0)^2-a^2b^2}{a^2}$$
+%% $$x=x_0\pm\dfrac{b}{a}\sqrt{(y_1-y_0)^2-a^2}$$
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the abbscissa value $x_1$.
+%% #5 -> [output] the first target node name.
+%% #6 -> [output] the first second node name.
+\def\pstIHyperbolaOrdNode{\@ifnextchar[\Pst at IHyperbolaOrdNode{\Pst at IHyperbolaOrdNode[]}}
+\def\Pst at IHyperbolaOrdNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 3 index sub dup mul 2 index dup mul sub % (y_1-y_0)^2-a^2
+ dup 0 lt {
+ pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index mul 2 index div 4 index exch sub % x1
+ #4 6 2 roll pop pop pop pop
+ } ifelse
+ ){#5}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #4 3 index sub dup mul 2 index dup mul sub % (y_1-y_0)^2-a^2
+ dup 0 lt {
+ pop pop pop pop pop 0 0
+ } {
+ sqrt 1 index mul 2 index div 4 index add % x2
+ #4 6 2 roll pop pop pop pop
+ } ifelse
+ ){#6}%
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \endgroup%
+}%
+%
+%% \pstIHyperbolaFocusNode[Options](O)(a,b){F1}{F2}
+%% Create the two focus node F1 and F2 of the Inversion Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [output] the first focus node name.
+%% #5 -> [output] the first focus node name.
+\def\pstIHyperbolaFocusNode{\@ifnextchar[\Pst at IHyperbolaFocusNode{\Pst at IHyperbolaFocusNode[]}}
+\def\Pst at IHyperbolaFocusNode[#1](#2)(#3)#4#5{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ dup mul exch dup mul add sqrt % c
+ 1 index exch sub 2 index exch % x0,y0-c
+ 4 2 roll pop pop
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ dup mul exch dup mul add sqrt % c
+ 1 index add 2 index exch % x0,y0+c
+ 4 2 roll pop pop
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstIHyperbolaDirectrixLine[Options](O)(a,b){Lx}{Ly}{Rx}{Ry}
+%% Draw the two directrix lines L1 and L2 of the Inversion Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [output] the first node name on the first directrix line.
+%% #5 -> [output] the second node name on the first directrix line.
+%% #6 -> [output] the first node name on the second directrix line.
+%% #7 -> [output] the second node name on the second directrix line.
+\def\pstIHyperbolaDirectrixLine{\@ifnextchar[\Pst at IHyperbolaDirectrixLine{\Pst at IHyperbolaDirectrixLine[]}}
+\def\Pst at IHyperbolaDirectrixLine[#1](#2)(#3)#4#5#6#7{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 3 index exch sub 4 index exch % x0,y0-a^2/c
+ 6 2 roll pop pop pop pop
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 3 index exch sub 4 index 1 add exch % x0+1,y0-a^2/c
+ 6 2 roll pop pop pop pop
+ ){#5}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 3 index add 4 index exch % x0,y0+a^2/c
+ 6 2 roll pop pop pop pop
+ ){#6}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 1 index dup mul dup 2 index dup mul add sqrt div % a^2/c
+ 3 index add 4 index 1 add exch % x0+1,y0+a^2/c
+ 6 2 roll pop pop pop pop
+ ){#7}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \pstLineAB{#4}{#5}%
+ \pstLineAB{#6}{#7}%
+ \endgroup%
+}%
+%
+%% \pstIHyperbolaAsymptoteLine[Options](O)(a,b){L1}{L2}
+%% Draw the two asymptote lines L1 and L2 of the Inversion Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [output] the second node name on the first asymptote line, the first node is the center O.
+%% #5 -> [output] the second node name on the second asymptote line, the first node is the center O.
+\def\pstIHyperbolaAsymptoteLine{\@ifnextchar[\Pst at IHyperbolaAsymptoteLine{\Pst at IHyperbolaAsymptoteLine[]}}
+\def\Pst at IHyperbolaAsymptoteLine[#1](#2)(#3)#4#5{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 index sub % x=x0-b/a
+ 4 index 1 add % y=y0+1
+ 7 2 roll pop pop pop pop pop
+ ){#4}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 index add % x=x0+b/a
+ 4 index 1 add % y=y0+1
+ 7 2 roll pop pop pop pop pop
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLineAB{#2}{#4}%
+ \pstLineAB{#2}{#5}%
+ \endgroup%
+}%
+%
+%% \pstIHyperbolaLineInter[Options](O)(a,b){A}{B}{C}{D}
+%% Find the two intersection nodes C and D of the Inversion Hyperbola H and line AB.
+%%
+%% Case 1. When line AB is vertical, i.e, $x_1=x_2$, we have
+%% $$y=y_0\pm\dfrac{a}{b}\sqrt{(x_1-x_0)^2+b^2}$$
+%%
+%% Case 2. When line AB is not vertical, we can represent the line AB as the following function:
+%% $$y=kx+d$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, d=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% refer to equation (\ref{FunctionOfStandardInversionHyperbola}), we have
+%% $$(b^2k^2-a^2)X^2+2b^2kmX+b^2(m^2-a^2)=0$$
+%% $$Y=kX+m$$
+%% where
+%% $$m=kx_o-y_o+d,X=x-x_0,Y=y-y_0$$
+%%
+%% Case 2.1 When $b^2k^2-a^2=0$, which gives $k=\pm\dfrac{a}{b}\neq0$,
+%% at this time, when $m=0$, the line AB becomes the asymptote of the hyperbola,
+%% so there is none intersection any more; else we have
+%% $$x_{C}=x_o-\dfrac{m^2-a^2}{2km},y_{C}=kx_{C}+d$$
+%% but D is not defined.
+%%
+%% Case 2.2 When $b^2k^2-a^2\neq0$, we have
+%% $$x_{C,D}=x_0+\dfrac{-b^2km\pm{}ab\sqrt{m^2+b^2k^2-a^2}}{b^2k^2-a^2},y_{C,D}=kx_{C,D}+d$$
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the first node name on the given line AB.
+%% #5 -> [input] the second node name on the given line AB.
+%% #6 -> [output] the first intersection node.
+%% #7 -> [output] the second intersection node.
+\def\pstIHyperbolaLineInter{\@ifnextchar[\Pst at IHyperbolaLineInter{\Pst at IHyperbolaLineInter[]}}
+\def\Pst at IHyperbolaLineInter[#1](#2)(#3)#4#5#6#7{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \pst at getcoor{#4}\pst at tempA%
+ \pst at getcoor{#5}\pst at tempB%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ \pst at tempA \tx at UserCoor % x_1,y_1
+ \pst at tempB \tx at UserCoor % x_2,y_2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ %% $$y=y_0\pm\dfrac{a}{b}\sqrt{(x_1-x_0)^2+b^2}$$
+ 3 index 8 index sub dup mul 5 index dup mul add
+ sqrt 6 index mul 5 index div 7 index exch sub % y1
+ 4 index exch % x1
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ 7 index dup mul 3 index dup mul mul 9 index dup mul sub % b^2k^2-a^2
+ 0 index abs 1E-5 lt {
+ 1 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ %% $$x_{C}=x_o-\dfrac{m^2-a^2}{2km},y_{C}=kx_{C}+d$$
+ 1 index dup mul 10 index dup mul sub 2 index 5 index mul 2 mul div % (m^2-a^2)/2km
+ 12 index exch sub % x1
+ 4 index 1 index mul 4 index add % y1
+ 14 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } {
+ 1 index dup mul 1 index add % m^2+b^2k^2-a^2
+ dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 10 index mul 9 index mul 9 index dup mul 5 index mul 3 index mul add neg 1 index div 12 index add % x1
+ 4 index 1 index mul 4 index add % y1
+ 14 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#6}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ \pst at tempA \tx at UserCoor % x_1,y_1
+ \pst at tempB \tx at UserCoor % x_2,y_2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ %% $$y=y_0\pm\dfrac{a}{b}\sqrt{(x_1-x_0)^2+b^2}$$
+ 3 index 8 index sub dup mul 5 index dup mul add
+ sqrt 6 index mul 5 index div 7 index add % y2
+ 4 index exch % x2
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ 7 index dup mul 3 index dup mul mul 9 index dup mul sub % b^2k^2-a^2
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ 0 0
+ } {
+ 1 index dup mul 1 index add % m^2+b^2k^2-a^2
+ dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt 10 index mul 9 index mul 9 index dup mul 5 index mul 3 index mul sub 1 index div 12 index add % x2
+ 4 index 1 index mul 4 index add % y2
+ 14 2 roll pop pop pop pop pop pop
+ pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#7}%
+ \Pst at ManageParamList{#6}%
+ \Pst at ManageParamList{#7}%
+ \endgroup%
+}%
+%
+%% \pstIHyperbolaPolarNode[Options](O)(a,b){A}{B}{T}
+%% Find the polar point of chord AB on Inversion Hyperbola H.
+%% We use the following proposition to find the polar point of chord AB:
+%% Let $P$, $Q$ are vertex points of the hyperbola, for any chord $AB$ of hyperbola, $PA$ and $BQ$ intersect at $E$, $PB$ and $AQ$ intersect at $F$, then the middle point $M$ of $EF$ is the polar point of chord $AB$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the node A on the hyperbola.
+%% #5 -> [input] the node B on the hyperbola.
+%% #6 -> [output] the polar node T of chord AB.
+\def\pstIHyperbolaPolarNode{\@ifnextchar[\Pst at IHyperbolaPolarNode{\Pst at IHyperbolaPolarNode[]}}
+\def\Pst at IHyperbolaPolarNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \psset{#1}%
+ \pstIHyperbolaAbsNode[PointName=none,PointSymbol=none](#2)(#3){\pstAbscissa{#2}}{@PST at IHYPERBOLA@VERTEXA}{@PST at IHYPERBOLA@VERTEXB}
+ \pstInterLL[PointName=none,PointSymbol=none]{#4}{@PST at IHYPERBOLA@VERTEXA}{#5}{@PST at IHYPERBOLA@VERTEXB}{@PST at IHYPERBOLA@INTER at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#5}{@PST at IHYPERBOLA@VERTEXA}{#4}{@PST at IHYPERBOLA@VERTEXB}{@PST at IHYPERBOLA@INTER at Y}
+ \pstMiddleAB{@PST at IHYPERBOLA@INTER at X}{@PST at IHYPERBOLA@INTER at Y}{#6}
+ \Pst at geonodelabel{#6}%
+ \pstLineAB{#4}{#6}
+ \pstLineAB{#5}{#6}
+ \endgroup
+}%
+%
+%% \pstIHyperbolaTangentNode[Options](O)(a,b){T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the Inversion Hyperbola H and get the node A and B on the Inversion Hyperbola.
+%% We use the following proposition to find the tangent points $A$ and $B$ of $T$:
+%% Let $T$ is a point out of the hyperbola, we give any two chords $TPQ$ and $TRS$ of the hyperbola, $PR$ and $QS$ intersect at $X$, $RQ$ and $PS$ intersect at $Y$,
+%% then the intersection point $A$ and $B$ of $XY$ and the hyperbola are the tangent points from $T$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the given node T outside the hyperbola
+%% #5 -> [output] the tangent node name A on the hyperbola
+%% #6 -> [output] the tangent node name B on the hyperbola
+\def\pstIHyperbolaTangentNode{\@ifnextchar[\Pst at IHyperbolaTangentNode{\Pst at IHyperbolaTangentNode[]}}
+\def\Pst at IHyperbolaTangentNode[#1](#2)(#3)#4#5#6{%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pstIHyperbolaAbsNode[PointName=none,PointSymbol=none](#2)(#3){\pstAbscissa{#4} 0.3 add}{@PST at IHYPERBOLA@TANGENTAUX at P0}{@PST at IHYPERBOLA@TANGENTAUX at P1}
+ \pstIHyperbolaAbsNode[PointName=none,PointSymbol=none](#2)(#3){\pstAbscissa{#4} 0.3 sub}{@PST at IHYPERBOLA@TANGENTAUX at R0}{@PST at IHYPERBOLA@TANGENTAUX at R1}
+ \pstIHyperbolaLineInter[PointName=none,PointSymbol=none](#2)(#3){#4}{@PST at IHYPERBOLA@TANGENTAUX at P0}{@PST at IHYPERBOLA@TANGENTAUX at P}{@PST at IHYPERBOLA@TANGENTAUX at Q}
+ \pstIHyperbolaLineInter[PointName=none,PointSymbol=none](#2)(#3){#4}{@PST at IHYPERBOLA@TANGENTAUX at R0}{@PST at IHYPERBOLA@TANGENTAUX at R}{@PST at IHYPERBOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at IHYPERBOLA@TANGENTAUX at P}{@PST at IHYPERBOLA@TANGENTAUX at S}{@PST at IHYPERBOLA@TANGENTAUX at Q}{@PST at IHYPERBOLA@TANGENTAUX at R}{@PST at IHYPERBOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at IHYPERBOLA@TANGENTAUX at P}{@PST at IHYPERBOLA@TANGENTAUX at R}{@PST at IHYPERBOLA@TANGENTAUX at Q}{@PST at IHYPERBOLA@TANGENTAUX at S}{@PST at IHYPERBOLA@TANGENTAUX at X}
+ \pstIHyperbolaLineInter(#2)(#3){@PST at IHYPERBOLA@TANGENTAUX at X}{@PST at IHYPERBOLA@TANGENTAUX at I}{#5}{#6}
+ \Pst at ManageParamList{#5}%
+ \Pst at ManageParamList{#6}%
+ \pstLineAB{#4}{#5}
+ \pstLineAB{#4}{#6}
+ \endgroup%
+}%
+%
+% 9. General Hyperbola with coordinate translation and rotation
+%% ----------------------------------------------------------
+%% The General Hyperbola H is defined by center O, the half of the real axis $a$, the half of the imaginary axis $b$,
+%% and the rotation angle $\theta$ of the principal axis.
+%% The equation can be got from the parametric function of the Standard Hyperbola \ref{ParametricFunctionOfStandardHyperbola},
+%% using the rotation transform formula \ref{RotationTransformFormula}, then we have
+%% \begin{equation}
+%% \left\{\begin{array}{l}
+%% x'=(a\sec\alpha+x_o)\cos\theta-(b\tan\alpha+y_o)\sin\theta=x_o'+a\sec\alpha\cos\theta-b\tan\alpha\sin\theta\\
+%% y'=(a\sec\alpha+x_o)\sin\theta+(b\tan\alpha+y_o)\cos\theta=y_o'+a\sec\alpha\sin\theta+b\tan\alpha\cos\theta
+%% \end{array}\right.
+%% \end{equation}
+%% where the $x_o'$ and $y_o'$ are the coordinate of the given center O after rotation.
+%% So we get the parametric function of the General Hyperbola with coordinate translation and rotation as following:
+%% \begin{equation}\label{ParametricFunctionOfGeneralHyperbola}
+%% \left\{\begin{array}{l}
+%% x=x_o+a\sec\alpha\cos\theta-b\tan\alpha\sin\theta\\
+%% y=y_o+a\sec\alpha\sin\theta+b\tan\alpha\cos\theta
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstGeneralHyperbola[Options](O)(a,b)[rotation][maxAngleX]
+%% Draw a General Hyperbola with center O, the half of the real axis $abs(a)$, the half of the imaginary axis $abs(b)$,
+%% and the rotation angle $\theta$ of the symmetrical axis.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the maximal angle to draw the branch.
+\def\pstGeneralHyperbola{\@ifnextchar[\Pst at GeneralHyperbola{\Pst at GeneralHyperbola[]}}
+\def\Pst at GeneralHyperbola[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbola@i{\Pst at GeneralHyperbola@i[0]}}%
+\def\Pst at GeneralHyperbola@i[#1]{%
+ \def\pst at hyperbola@rotation{#1}%
+ \@ifnextchar[\Pst at GeneralHyperbola@j{\Pst at GeneralHyperbola@j[85]}}%
+\def\Pst at GeneralHyperbola@j[#1]{%
+ \pst at cnth=#1\pst at cntg=180\pst at cntm=180\pst at cntn=360
+ \ifnum\pst at cnth<0
+ \loop\advance\pst at cnth by 90
+ \ifnum\pst at cnth<0
+ \repeat
+ \fi
+ \ifnum\pst at cnth>90
+ \loop\advance\pst at cnth by -90
+ \ifnum\pst at cnth>90
+ \repeat
+ \fi
+ \ifnum\pst at cnth>85
+ \advance\pst at cnth by -5
+ \fi
+ \advance\pst at cntg by -\pst at cnth
+ \advance\pst at cntm by \pst at cnth
+ \advance\pst at cntn by -\pst at cnth
+ \Pst at GeneralHyperbola@k[0][\number\pst at cnth]%
+ \Pst at GeneralHyperbola@k[\number\pst at cntg][180]%
+ \Pst at GeneralHyperbola@k[180][\number\pst at cntm]%
+ \Pst at GeneralHyperbola@k[\number\pst at cntn][360]%
+ \endgroup%
+}%
+\def\Pst at GeneralHyperbola@k[#1][#2]{%
+ \parametricplot{#1}{#2}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ t dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ } {
+ \pst at hyperbola@rotation\space dup cos exch sin % cos\theta sin\theta
+ %% x=x_o+a\sec\alpha\cos\theta-b\tan\alpha\sin\theta
+ 7 index 6 index 5 index div 3 index mul add
+ 5 index 4 index mul 5 index div 2 index mul sub
+ %% y=y_o+a\sec\alpha\sin\theta+b\tan\alpha\cos\theta
+ 7 index 7 index 6 index div 3 index mul add
+ 6 index 5 index mul 6 index div 4 index mul add
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } ifelse
+ }%
+}%
+%
+%% \pstGeneralHyperbolaNode[Options](O)(a,b)[rotation]{t}{A}
+%% Draw a node whose parameter value is the given value t on the General Hyperbola.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the parameter value t.
+%% #6 -> [output] the target node name.
+\def\pstGeneralHyperbolaNode{\@ifnextchar[\Pst at GeneralHyperbolaNode{\Pst at GeneralHyperbolaNode[]}}
+\def\Pst at GeneralHyperbolaNode[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaNode@i{\Pst at GeneralHyperbolaNode@i[0]}}%
+\def\Pst at GeneralHyperbolaNode@i[#1]#2#3{%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #2 dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ } {
+ #1 dup cos exch sin % cos\theta sin\theta
+ %% x=x_o+a\sec\alpha\cos\theta-b\tan\alpha\sin\theta
+ 7 index 6 index 5 index div 3 index mul add
+ 5 index 4 index mul 5 index div 2 index mul sub
+ %% y=y_o+a\sec\alpha\sin\theta+b\tan\alpha\cos\theta
+ 7 index 7 index 6 index div 3 index mul add
+ 6 index 5 index mul 6 index div 4 index mul add
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } ifelse
+ ){#3}%
+ \Pst at geonodelabel{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaAbsNode[Options](O)(a,b)[rotation]{x_1}{A}{B}
+%% Draw the nodes whose abscissa value are the given value x_1 on the General Hyperbola.
+%%
+%% set $e=a\cos\theta$, $f=b\sin\theta$, $g=a\sin\theta$, $h=b\cos\theta$, then we have
+%% $$x=x_0+e\sec\alpha-f\tan\alpha, y=y_0+g\sec\alpha+h\tan\alpha$$
+%% when $x=x_1$, we get
+%% $$e\sec\alpha-f\tan\alpha=x_1-x_0$$
+%% set $n=x_1-x_0$, we have
+%% $$n\cos\alpha+f\sin\alpha=e$$
+%% then
+%% $$(n^2+f^2)\sin^2\alpha-2ef\sin\alpha+e^2-n^2=0$$
+%% if $n^2+f^2=0$, we have $n=f=0$, i.e, $x_1=x_0$ and $\sin\theta=0$, but the last equation gives $e=0$, which is not possible.
+%% so $n^2+f^2\neq0$, we get
+%% $$\sin_{1,2}\alpha=\dfrac{ef\pm{}n\sqrt{n^2+f^2-e^2}}{n^2+f^2}$$
+%% and
+%% $$\cos_{1,2}\alpha=\dfrac{en\mp{}f\sqrt{n^2+f^2-e^2}}{n^2+f^2}$$
+%% where $\cos\alpha$ can not be zero, but when $f=\pm{}e$, we have $\cos\alpha=0$, we should skip it.
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the abscissa value x_1.
+%% #6 -> [output] the first target node name.
+%% #7 -> [output] the second target node name.
+\def\pstGeneralHyperbolaAbsNode{\@ifnextchar[\Pst at GeneralHyperbolaAbsNode{\Pst at GeneralHyperbolaAbsNode[]}}
+\def\Pst at GeneralHyperbolaAbsNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaAbsNode@i{\Pst at GeneralHyperbolaAbsNode@i[0]}}%
+\def\Pst at GeneralHyperbolaAbsNode@i[#1]#2#3#4{%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 10 index sub % n=x1-x0
+ 0 index dup mul 4 index dup mul add % n^2+f^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 6 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+f^2-e^2)
+ %% \sin_{1,2}\alpha=\dfrac{ef\pm{}n\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 2 index 1 index mul 7 index 7 index mul exch sub 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{en\mp{}f\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 6 index 2 index mul 8 index 5 index mul add 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 9 index 2 index div add 8 index 3 index mul 2 index div sub % x_1=x_0+e\sec\alpha-f\tan\alpha
+ 14 index 8 index 3 index div add 7 index 4 index mul 3 index div add % y_1=y_0+g\sec\alpha+h\tan\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#3}%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 10 index sub % n=x1-x0
+ 0 index dup mul 4 index dup mul add % n^2+f^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 6 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+f^2-e^2)
+ %% \sin_{1,2}\alpha=\dfrac{ef\pm{}n\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 2 index 1 index mul 7 index 7 index mul add 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{en\mp{}f\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 6 index 2 index mul 8 index 5 index mul exch sub 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 9 index 2 index div add 8 index 3 index mul 2 index div sub % x_2=x_0+e\sec\alpha-f\tan\alpha
+ 14 index 8 index 3 index div add 7 index 4 index mul 3 index div add % y_2=y_0+g\sec\alpha+h\tan\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaOrdNode[Options](O)(a,b)[rotation]{y_1}{A}{B}
+%% Draw the nodes whose ordinate value are the given value y_1 on the General Hyperbola.
+%%
+%% set $e=a\cos\theta$, $f=b\sin\theta$, $g=a\sin\theta$, $h=b\cos\theta$, then we have
+%% $$x=x_0+e\sec\alpha-f\tan\alpha, y=y_0+g\sec\alpha+h\tan\alpha$$
+%% when $y=y_1$, we get
+%% $$g\sec\alpha+h\tan\alpha=y_1-y_0$$
+%% set $m=y_1-y_0$, we have
+%% $$m\cos\alpha-h\sin\alpha=g$$
+%% then
+%% $$(m^2+h^2)\sin^2\alpha+2gh\sin\alpha+g^2-m^2=0$$
+%% if $m^2+h^2=0$, we have $m=h=0$, i.e, $y_1=y_0$ and $\cos\theta=0$, but the last equation gives $e=0$, which is not possible.
+%% so $m^2+h^2\neq0$, we get
+%% $$\sin_{1,2}\alpha=\dfrac{-gh\pm{}m\sqrt{m^2+h^2-g^2}}{m^2+h^2}$$
+%% and
+%% $$\cos_{1,2}\alpha=\dfrac{gm\pm{}h\sqrt{m^2+h^2-g^2}}{m^2+h^2}$$
+%% where $\cos\alpha$ can not be zero, but when $h=\pm{}g$, we have $\cos\alpha=0$, we should skip it.
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the ordinate value y_1.
+%% #6 -> [output] the first target node name.
+%% #7 -> [output] the second target node name.
+\def\pstGeneralHyperbolaOrdNode{\@ifnextchar[\Pst at GeneralHyperbolaOrdNode{\Pst at GeneralHyperbolaOrdNode[]}}
+\def\Pst at GeneralHyperbolaOrdNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaOrdNode@i{\Pst at GeneralHyperbolaOrdNode@i[0]}}%
+\def\Pst at GeneralHyperbolaOrdNode@i[#1]#2#3#4{%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % m=y1-y0
+ 0 index dup mul 2 index dup mul add % m^2+h^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 4 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(m^2+h^2-g^2)
+ %% \sin_{1,2}\alpha=\dfrac{-gh\pm{}m\sqrt{m^2+h^2-g^2}}{m^2+h^2}
+ 2 index 1 index mul 5 index 5 index mul add neg 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{gm\pm{}h\sqrt{m^2+h^2-g^2}}{m^2+h^2}
+ 4 index 2 index mul 6 index 5 index mul exch sub 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 9 index 2 index div add 8 index 3 index mul 2 index div sub % x_1=x_0+e\sec\alpha-f\tan\alpha
+ 14 index 8 index 3 index div add 7 index 4 index mul 3 index div add % y_1=y_0+g\sec\alpha+h\tan\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#3}%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % m=y1-y0
+ 0 index dup mul 2 index dup mul add % m^2+h^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 4 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(m^2+h^2-g^2)
+ %% \sin_{1,2}\alpha=\dfrac{-gh\pm{}m\sqrt{m^2+h^2-g^2}}{m^2+h^2}
+ 2 index 1 index mul 5 index 5 index mul sub 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{gm\pm{}h\sqrt{m^2+h^2-g^2}}{m^2+h^2}
+ 4 index 2 index mul 6 index 5 index mul add 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 9 index 2 index div add 8 index 3 index mul 2 index div sub % x_1=x_0+e\sec\alpha-f\tan\alpha
+ 14 index 8 index 3 index div add 7 index 4 index mul 3 index div add % y_1=y_0+g\sec\alpha+h\tan\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaFocusNode[Options](O)(a,b)[rotation]{F1}{F2}
+%% Draw the focus nodes of the General Hyperbola H.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the focus node F1 of the Hyperbola.
+%% #6 -> [output] the focus node F2 of the Hyperbola.
+\def\pstGeneralHyperbolaFocusNode{\@ifnextchar[\Pst at GeneralHyperbolaFocusNode{\Pst at GeneralHyperbolaFocusNode[]}}
+\def\Pst at GeneralHyperbolaFocusNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaFocusNode@i{\Pst at GeneralHyperbolaFocusNode@i[0]}}%
+\def\Pst at GeneralHyperbolaFocusNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index dup mul 3 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 6 index 1 index 4 index mul sub % x=x0-c\cos\theta
+ 6 index 2 index 4 index mul sub % y=y0-c\sin\theta
+ 9 2 roll pop pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index dup mul 3 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 6 index 1 index 4 index mul add % x=x0+c\cos\theta
+ 6 index 2 index 4 index mul add % y=y0+c\sin\theta
+ 9 2 roll pop pop pop pop pop pop pop
+ ){#3}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaVertexNode[Options](O)(a,b)[rotation]{V1}{V2}
+%% Draw the vertex nodes of the General Hyperbola H.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the vertex node V1 of the Hyperbola.
+%% #6 -> [output] the vertex node V2 of the Hyperbola.
+\def\pstGeneralHyperbolaVertexNode{\@ifnextchar[\Pst at GeneralHyperbolaVertexNode{\Pst at GeneralHyperbolaVertexNode[]}}
+\def\Pst at GeneralHyperbolaVertexNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaVertexNode@i{\Pst at GeneralHyperbolaVertexNode@i[0]}}%
+\def\Pst at GeneralHyperbolaVertexNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 5 index 4 index 3 index mul sub % x=x0-a\cos\theta
+ 5 index 5 index 3 index mul sub % y=y0-a\sin\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 5 index 4 index 3 index mul add % x=x0+a\cos\theta
+ 5 index 5 index 3 index mul add % y=y0+a\sin\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaDirectrixLine[Options](O)(a,b)[rotation]{Lx}{Ly}{Rx}{Ry}
+%% Draw the two directrix lines of the General Hyperbola H.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the first node Lx on the first directrix line.
+%% #6 -> [output] the second node Lx on the first directrix line.
+%% #7 -> [output] the first node Lx on the second directrix line.
+%% #8 -> [output] the second node Lx on the second directrix line.
+\def\pstGeneralHyperbolaDirectrixLine{\@ifnextchar[\Pst at GeneralHyperbolaDirectrixLine{\Pst at GeneralHyperbolaDirectrixLine[]}}
+\def\Pst at GeneralHyperbolaDirectrixLine[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaDirectrixLine@i{\Pst at GeneralHyperbolaDirectrixLine@i[0]}}%
+\def\Pst at GeneralHyperbolaDirectrixLine@i[#1]#2#3#4#5{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 5 index exch sub 4 index % x0-a^2/c,y0
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 5 index exch sub 4 index 1 add % x0-a^2/c,y0+1
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 5 index add 4 index % x0+a^2/c,y0
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#4}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 5 index add 4 index 1 add % x0+a^2/c,y0+1
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#5}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLineAB{#2}{#3}%
+ \pstLineAB{#4}{#5}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaAsymptoteLine[Options](O)(a,b)[rotation]{L1}{L2}
+%% Draw the two asymptote lines L1 and L2 of the General Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the second node name on the first asymptote line, the first node is the center O.
+%% #6 -> [output] the second node name on the second asymptote line, the first node is the center O.
+\def\pstGeneralHyperbolaAsymptoteLine{\@ifnextchar[\Pst at GeneralHyperbolaAsymptoteLine{\Pst at GeneralHyperbolaAsymptoteLine[]}}
+\def\Pst at GeneralHyperbolaAsymptoteLine[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \def\pst at hyperbola@center{#2}
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaAsymptoteLine@i{\Pst at GeneralHyperbolaAsymptoteLine@i[0]}}%
+\def\Pst at GeneralHyperbolaAsymptoteLine@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 add % x=x0+1
+ 4 index 2 index add % y=y0+b/a
+ 7 2 roll pop pop pop % x y x_0 y_0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 add % x=x0+1
+ 4 index 2 index sub % y=y0-b/a
+ 7 2 roll pop pop pop % x y x_0 y_0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}%
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \pstLineAB{\pst at hyperbola@center}{#2}%
+ \pstLineAB{\pst at hyperbola@center}{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaLineInter[Options](O)(a,b)[rotation]{A}{B}{C}{D}
+%% Find the intersection nodes $C$ and $D$ of the given line AB with the General Hyperbola H.
+%%
+%% set $e=a\cos\theta$, $f=b\sin\theta$, $g=a\sin\theta$, $h=b\cos\theta$, then we have
+%% $$x=x_0+e\sec\alpha-f\tan\alpha, y=y_0+g\sec\alpha+h\tan\alpha$$
+%% when line AB is vertical, the solve is like as pstGeneralHyperbolaLineInter,
+%% else we can represent the line AB as the following function:
+%% $$y=kx+d$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, d=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% Let $X=x-x_0$, $Y=y-y_0$, then we have $Y=kX+m$, where $m=kx_0-y_0+d$, then
+%% $$g\sec\alpha+h\tan\alpha=k(e\sec\alpha-f\tan\alpha)+m$$
+%% $$(kf+h)\sin\alpha=m\cos\alpha+ke-g$$
+%% $$(kf+h)^2\sin^2\alpha=m^2\cos^2\alpha+2m(ke-g)\cos\alpha+(ke-g)^2$$
+%% $$(m^2+(kf+h)^2)\cos^2\alpha-2m(g-ke)\cos\alpha+(g-ke)^2-(kf+h)^2=0$$
+%% when $m^2+(kf+h)^2=0$, we have $m=kf+h=0$, at this time $ke-g=0$, we get
+%% $$k=-\dfrac{\cos\theta}{\sin\theta}=\dfrac{\sin\theta}{\cos\theta}$$
+%% which is not possible.
+%% let $p=kf+h$, $q=g-ke$, so we have
+%% $$(m^2+p^2)\cos^2\alpha-2mq\cos\alpha+q^2-p^2=0$$
+%% then
+%% $$\cos_{1,2}\alpha=\dfrac{mq\pm{}p\sqrt{m^2+p^2-q^2}}{m^2+p^2},\sin_{1,2}\alpha=\dfrac{-qp\pm{}m\sqrt{m^2+p^2-q^2}}{m^2+p^2}$$
+%% where $\cos\alpha$ can not be zero, we should skip it.
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the first node on the given line.
+%% #6 -> [input] the second node on the given line.
+%% #7 -> [output] the first intersection node name.
+%% #8 -> [output] the second intersection node name.
+\def\pstGeneralHyperbolaLineInter{\@ifnextchar[\Pst at GeneralHyperbolaLineInter{\Pst at GeneralHyperbolaLineInter[]}}
+\def\Pst at GeneralHyperbolaLineInter[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralHyperbolaLineInter@i{\Pst at GeneralHyperbolaLineInter@i[0]}}%
+\def\Pst at GeneralHyperbolaLineInter@i[#1]#2#3#4#5{%
+ \pst at getcoor{#2}\pst at tempA%
+ \pst at getcoor{#3}\pst at tempB%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % cos\theta sin\theta
+ 7 index 2 index mul % e=a\cos\theta
+ 7 index 2 index mul % f=b\sin\theta
+ 9 index 3 index mul % g=a\sin\theta
+ 9 index 5 index mul % h=b\cos\theta
+ 7 index 14 index sub % n=x1-x0
+ 0 index dup mul 4 index dup mul add % n^2+f^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 6 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+f^2-e^2)
+ %% \sin_{1,2}\alpha=\dfrac{ef\pm{}n\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 2 index 1 index mul 7 index 7 index mul exch sub 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{en\mp{}f\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 6 index 2 index mul 8 index 5 index mul add 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 18 index 9 index 2 index div add 8 index 3 index mul 2 index div sub % x_1=x_0+e\sec\alpha-f\tan\alpha
+ 18 index 8 index 3 index div add 7 index 4 index mul 3 index div add % y_1=y_0+g\sec\alpha+h\tan\alpha
+ 21 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ #1 dup cos exch sin % cos\theta sin\theta
+ 10 index 2 index mul % e=a\cos\theta
+ 10 index 2 index mul % f=b\sin\theta
+ 12 index 3 index mul % g=a\sin\theta
+ 12 index 5 index mul % h=b\cos\theta
+ 0 index 9 index 4 index mul add % p=h+kf
+ 2 index 10 index 6 index mul sub % q=g-ke
+ 8 index dup mul 2 index dup mul add % m^2+p^2
+ dup 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ 0 0
+ } {
+ 0 index 2 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt % sqrt(m^2+p^2-q^2)
+ % \sin_{1,2}\alpha=\dfrac{-qp\pm{}m\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 10 index 1 index mul 4 index 4 index mul sub 2 index div % \sin\alpha
+ % \cos_{1,2}\alpha=\dfrac{mq\pm{}p\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 4 index 2 index mul 12 index 5 index mul add 3 index div % \cos\alpha
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop 0 0
+ } {
+ % x=x_0+e\sec\alpha-f\tan\alpha, y=y_0+g\sec\alpha+h\tan\alpha
+ 22 index 10 index 2 index div add 9 index 3 index mul 2 index div sub % x_1
+ 22 index 9 index 3 index div add 8 index 4 index mul 3 index div add % y_1
+ 25 2 roll
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % cos\theta sin\theta
+ 7 index 2 index mul % e=a\cos\theta
+ 7 index 2 index mul % f=b\sin\theta
+ 9 index 3 index mul % g=a\sin\theta
+ 9 index 5 index mul % h=b\cos\theta
+ 7 index 14 index sub % n=x1-x0
+ 0 index dup mul 4 index dup mul add % n^2+f^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 6 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+f^2-e^2)
+ %% \sin_{1,2}\alpha=\dfrac{ef\pm{}n\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 2 index 1 index mul 7 index 7 index mul add 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{en\mp{}f\sqrt{n^2+f^2-e^2}}{n^2+f^2}
+ 6 index 2 index mul 8 index 5 index mul exch sub 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 18 index 9 index 2 index div add 8 index 3 index mul 2 index div sub % x_2=x_0+e\sec\alpha-f\tan\alpha
+ 18 index 8 index 3 index div add 7 index 4 index mul 3 index div add % y_2=y_0+g\sec\alpha+h\tan\alpha
+ 21 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ #1 dup cos exch sin % cos\theta sin\theta
+ 10 index 2 index mul % e=a\cos\theta
+ 10 index 2 index mul % f=b\sin\theta
+ 12 index 3 index mul % g=a\sin\theta
+ 12 index 5 index mul % h=b\cos\theta
+ 0 index 9 index 4 index mul add % p=h+kf
+ 2 index 10 index 6 index mul sub % q=g-ke
+ 8 index dup mul 2 index dup mul add % m^2+p^2
+ dup 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ 0 0
+ } {
+ 0 index 2 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt % sqrt(m^2+p^2-q^2)
+ % \sin_{1,2}\alpha=\dfrac{-qp\pm{}m\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 10 index 1 index mul 4 index 4 index mul add neg 2 index div % \sin\alpha
+ % \cos_{1,2}\alpha=\dfrac{mq\pm{}p\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 4 index 2 index mul 12 index 5 index mul exch sub 3 index div % \cos\alpha
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop 0 0
+ } {
+ % x=x_0+e\sec\alpha-f\tan\alpha, y=y_0+g\sec\alpha+h\tan\alpha
+ 22 index 10 index 2 index div add 9 index 3 index mul 2 index div sub % x_1
+ 22 index 9 index 3 index div add 8 index 4 index mul 3 index div add % y_1
+ 25 2 roll
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstGeneralHyperbolaPolarNode[Options](O)(a,b)[rotation]{A}{B}{T}
+%% Find the polar point of chord AB on General Hyperbola H.
+%% We use the following proposition to find the polar point of chord AB:
+%% Let $P$, $Q$ are vertex points of the hyperbola, for any chord $AB$ of hyperbola, $PA$ and $BQ$ intersect at $E$, $PB$ and $AQ$ intersect at $F$, then the middle point $M$ of $EF$ is the polar point of chord $AB$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the node A on the hyperbola.
+%% #6 -> [input] the node B on the hyperbola.
+%% #7 -> [output] the polar node T of chord AB.
+\def\pstGeneralHyperbolaPolarNode{\@ifnextchar[\Pst at GeneralHyperbolaPolarNode{\Pst at GeneralHyperbolaPolarNode[]}}
+\def\Pst at GeneralHyperbolaPolarNode[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \def\pst at hyperbola@center{#2}
+ \def\pst at hyperbola@radii{#3}
+ \@ifnextchar[\Pst at GeneralHyperbolaPolarNode@i{\Pst at GeneralHyperbolaPolarNode@i[0]}}%
+\def\Pst at GeneralHyperbolaPolarNode@i[#1]#2#3#4{%
+ \pstGeneralHyperbolaVertexNode[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{@PST at GENERALHYPERBOLA@VERTEXA}{@PST at GENERALHYPERBOLA@VERTEXB}
+ \pstInterLL[PointName=none,PointSymbol=none]{#2}{@PST at GENERALHYPERBOLA@VERTEXA}{#3}{@PST at GENERALHYPERBOLA@VERTEXB}{@PST at GENERALHYPERBOLA@INTER at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#3}{@PST at GENERALHYPERBOLA@VERTEXA}{#2}{@PST at GENERALHYPERBOLA@VERTEXB}{@PST at GENERALHYPERBOLA@INTER at Y}
+ \pstMiddleAB{@PST at GENERALHYPERBOLA@INTER at X}{@PST at GENERALHYPERBOLA@INTER at Y}{#4}
+ \Pst at geonodelabel{#4}%
+ \pstLineAB{#2}{#4}
+ \pstLineAB{#3}{#4}
+ \endgroup
+}%
+%
+%% \pstGeneralHyperbolaTangentNode[Options](O)(a,b)[rotation]{T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the General Hyperbola H and get the node A and B on the General Hyperbola.
+%% We use the following proposition to find the tangent points $A$ and $B$ of $T$:
+%% Let $T$ is a point out of the hyperbola, we give any two chords $TPQ$ and $TRS$ of the hyperbola, $PR$ and $QS$ intersect at $X$, $RQ$ and $PS$ intersect at $Y$,
+%% then the intersection point $A$ and $B$ of $XY$ and the hyperbola are the tangent points from $T$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the given node T outside the hyperbola
+%% #6 -> [output] the tangent node name A on the hyperbola
+%% #7 -> [output] the tangent node name B on the hyperbola
+\def\pstGeneralHyperbolaTangentNode{\@ifnextchar[\Pst at GeneralHyperbolaTangentNode{\Pst at GeneralHyperbolaTangentNode[]}}
+\def\Pst at GeneralHyperbolaTangentNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \def\pst at hyperbola@center{#2}
+ \def\pst at hyperbola@radii{#3}
+ \@ifnextchar[\Pst at GeneralHyperbolaTangentNode@i{\Pst at GeneralHyperbolaTangentNode@i[0]}}%
+\def\Pst at GeneralHyperbolaTangentNode@i[#1]#2#3#4{%
+ \pstGeneralHyperbolaNode[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{10}{@PST at GENERALHYPERBOLA@TANGENTAUX at P0}
+ \pstGeneralHyperbolaNode[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{-10}{@PST at GENERALHYPERBOLA@TANGENTAUX at R0}
+ \pstGeneralHyperbolaLineInter[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{#2}{@PST at GENERALHYPERBOLA@TANGENTAUX at P0}{@PST at GENERALHYPERBOLA@TANGENTAUX at P}{@PST at GENERALHYPERBOLA@TANGENTAUX at Q}
+ \pstGeneralHyperbolaLineInter[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{#2}{@PST at GENERALHYPERBOLA@TANGENTAUX at R0}{@PST at GENERALHYPERBOLA@TANGENTAUX at R}{@PST at GENERALHYPERBOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALHYPERBOLA@TANGENTAUX at P}{@PST at GENERALHYPERBOLA@TANGENTAUX at S}{@PST at GENERALHYPERBOLA@TANGENTAUX at Q}{@PST at GENERALHYPERBOLA@TANGENTAUX at R}{@PST at GENERALHYPERBOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALHYPERBOLA@TANGENTAUX at P}{@PST at GENERALHYPERBOLA@TANGENTAUX at R}{@PST at GENERALHYPERBOLA@TANGENTAUX at Q}{@PST at GENERALHYPERBOLA@TANGENTAUX at S}{@PST at GENERALHYPERBOLA@TANGENTAUX at X}
+ \pstGeneralHyperbolaLineInter(\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{@PST at GENERALHYPERBOLA@TANGENTAUX at X}{@PST at GENERALHYPERBOLA@TANGENTAUX at I}{#3}{#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \pstLineAB{#2}{#3}
+ \pstLineAB{#2}{#4}
+ \endgroup%
+}%
+%
+% 10. General Inversion Hyperbola with coordinate translation and rotation
+%% ----------------------------------------------------------
+%% The General Inversion Hyperbola H is defined by center O, the half of the real axis $a$, the half of the imaginary axis $b$,
+%% and the rotation angle $\theta$ of the principal axis.
+%% The equation can be got from the parametric function of the Standard Inversion Hyperbola \ref{ParametricFunctionOfStandardInversionHyperbola},
+%% using the rotation transform formula \ref{RotationTransformFormula}, then we have
+%% \begin{equation}
+%% \left\{\begin{array}{l}
+%% x'=(b\tan\alpha+x_o)\cos\theta-(a\sec\alpha+y_o)\sin\theta=x_o'+b\tan\alpha\cos\theta-a\sec\alpha\sin\theta\\
+%% y'=(b\tan\alpha+x_o)\sin\theta+(a\sec\alpha+y_o)\cos\theta=y_o'+b\tan\alpha\sin\theta+a\sec\alpha\cos\theta
+%% \end{array}\right.
+%% \end{equation}
+%% where the $x_o'$ and $y_o'$ are the coordinate of the given center O after rotation.
+%% So we get the parametric function of the General Inversion Hyperbola with coordinate translation and rotation as following:
+%% \begin{equation}\label{ParametricFunctionOfGeneralInversionHyperbola}
+%% \left\{\begin{array}{l}
+%% x=x_o+b\tan\alpha\cos\theta-a\sec\alpha\sin\theta\\
+%% y=y_o+b\tan\alpha\sin\theta+a\sec\alpha\cos\theta
+%% \end{array}\right.
+%% \end{equation}
+%
+%% \pstGeneralIHyperbola[Options](O)(a,b)[rotation][maxAngleX]
+%% Draw a General Inversion Hyperbola with center O, the half of the real axis $abs(a)$, the half of the imaginary axis $abs(b)$,
+%% and the rotation angle $\theta$ of the symmetrical axis.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the maximal angle to draw the branch.
+\def\pstGeneralIHyperbola{\@ifnextchar[\Pst at GeneralIHyperbola{\Pst at GeneralIHyperbola[]}}
+\def\Pst at GeneralIHyperbola[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbola@i{\Pst at GeneralIHyperbola@i[0]}}%
+\def\Pst at GeneralIHyperbola@i[#1]{%
+ \def\pst at hyperbola@rotation{#1}%
+ \@ifnextchar[\Pst at GeneralIHyperbola@j{\Pst at GeneralIHyperbola@j[85]}}%
+\def\Pst at GeneralIHyperbola@j[#1]{%
+ \pst at cnth=#1\pst at cntg=180\pst at cntm=180\pst at cntn=360
+ \ifnum\pst at cnth<0
+ \loop\advance\pst at cnth by 90
+ \ifnum\pst at cnth<0
+ \repeat
+ \fi
+ \ifnum\pst at cnth>90
+ \loop\advance\pst at cnth by -90
+ \ifnum\pst at cnth>90
+ \repeat
+ \fi
+ \ifnum\pst at cnth>85
+ \advance\pst at cnth by -5
+ \fi
+ \advance\pst at cntg by -\pst at cnth
+ \advance\pst at cntm by \pst at cnth
+ \advance\pst at cntn by -\pst at cnth
+ \Pst at GeneralIHyperbola@k[0][\number\pst at cnth]%
+ \Pst at GeneralIHyperbola@k[\number\pst at cntg][180]%
+ \Pst at GeneralIHyperbola@k[180][\number\pst at cntm]%
+ \Pst at GeneralIHyperbola@k[\number\pst at cntn][360]%
+ \endgroup%
+}%
+\def\Pst at GeneralIHyperbola@k[#1][#2]{%
+ \parametricplot{#1}{#2}{%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ t dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ } {
+ \pst at hyperbola@rotation\space dup cos exch sin % cos\theta sin\theta
+ %% x=x_o+b\tan\alpha\cos\theta-a\sec\alpha\sin\theta
+ 7 index 5 index 5 index div 4 index mul 3 index mul add
+ 6 index 5 index div 2 index mul sub
+ %% y=y_o+b\tan\alpha\sin\theta+a\sec\alpha\cos\theta
+ 7 index 6 index 5 index mul 6 index div 3 index mul add
+ 7 index 6 index div 4 index mul add
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } ifelse
+ }%
+}%
+%
+%% \pstGeneralIHyperbolaNode[Options](O)(a,b)[rotation]{t}{A}
+%% Draw a node whose parameter value is the given value t on the General Inversion Hyperbola.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the parameter value t.
+%% #6 -> [output] the target node name.
+\def\pstGeneralIHyperbolaNode{\@ifnextchar[\Pst at GeneralIHyperbolaNode{\Pst at GeneralIHyperbolaNode[]}}
+\def\Pst at GeneralIHyperbolaNode[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaNode@i{\Pst at GeneralIHyperbolaNode@i[0]}}%
+\def\Pst at GeneralIHyperbolaNode@i[#1]#2#3{%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #2 dup cos exch sin % cos{t} sin{t}
+ 1 index abs 1E-5 lt {
+ pop pop pop pop
+ } {
+ #1 dup cos exch sin % cos\theta sin\theta
+ %% x=x_o+b\tan\alpha\cos\theta-a\sec\alpha\sin\theta
+ 7 index 5 index 5 index div 4 index mul 3 index mul add
+ 6 index 5 index div 2 index mul sub
+ %% y=y_o+b\tan\alpha\sin\theta+a\sec\alpha\cos\theta
+ 7 index 6 index 5 index mul 6 index div 3 index mul add
+ 7 index 6 index div 4 index mul add
+ 10 2 roll pop pop pop pop pop pop pop pop
+ } ifelse
+ ){#3}%
+ \Pst at geonodelabel{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIHyperbolaAbsNode[Options](O)(a,b)[rotation]{x_1}{A}{B}
+%% Draw the nodes whose abscissa value are the given value x_1 on the General Inversion Hyperbola.
+%%
+%% set $e=a\cos\theta$, $f=b\sin\theta$, $g=a\sin\theta$, $h=b\cos\theta$, then we have
+%% $$x=x_0+h\tan\alpha-g\sec\alpha, y=y_0+f\tan\alpha+e\sec\alpha$$
+%% when $x=x_1$, we get
+%% $$h\tan\alpha-g\sec\alpha=x_1-x_0$$
+%% set $n=x_1-x_0$, we have
+%% $$h\sin\alpha-n\cos\alpha=g$$
+%% then
+%% $$(n^2+h^2)\cos^2\alpha+2ng\cos\alpha+g^2-h^2=0$$
+%% if $n^2+h^2=0$, we have $n=h=0$, i.e, $x_1=x_0$ and $\cos\theta=0$, but the last equation gives $g=0$, which is not possible.
+%% so $n^2+h^2\neq0$, we get
+%% $$\cos_{1,2}\alpha=\dfrac{-ng\pm{}h\sqrt{n^2+h^2-g^2}}{n^2+h^2}$$
+%% and
+%% $$\sin_{1,2}\alpha=\dfrac{gh\pm{}n\sqrt{n^2+h^2-g^2}}{n^2+h^2}$$
+%% where $\cos\alpha$ can not be zero, but when $h=\pm{}g$, we have $\cos\alpha=0$, we should skip it.
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the abscissa value x_1.
+%% #6 -> [output] the first target node name.
+%% #7 -> [output] the second target node name.
+\def\pstGeneralIHyperbolaAbsNode{\@ifnextchar[\Pst at GeneralIHyperbolaAbsNode{\Pst at GeneralIHyperbolaAbsNode[]}}
+\def\Pst at GeneralIHyperbolaAbsNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaAbsNode@i{\Pst at GeneralIHyperbolaAbsNode@i[0]}}%
+\def\Pst at GeneralIHyperbolaAbsNode@i[#1]#2#3#4{%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 10 index sub % n=x1-x0
+ 0 index dup mul 2 index dup mul add % n^2+h^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 4 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+h^2-g^2)
+ %% \sin_{1,2}\alpha=\dfrac{gh\pm{}n\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 2 index 1 index mul 5 index 5 index mul exch sub 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{-ng\pm{}h\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 4 index 2 index mul 4 index 7 index mul add neg 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 6 index 3 index mul 2 index div add 7 index 2 index div sub % x_1=x_0+h\tan\alpha-g\sec\alpha
+ 14 index 9 index 4 index mul 3 index div add 10 index 3 index div add % y_1=y_0+f\tan\alpha+e\sec\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#3}%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 10 index sub % n=x1-x0
+ 0 index dup mul 2 index dup mul add % n^2+h^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 4 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+h^2-g^2)
+ %% \sin_{1,2}\alpha=\dfrac{gh\pm{}n\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 2 index 1 index mul 5 index 5 index mul add 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{-ng\pm{}h\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 4 index 2 index mul 4 index 7 index mul sub 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 6 index 3 index mul 2 index div add 7 index 2 index div sub % x_2=x_0+h\tan\alpha-g\sec\alpha
+ 14 index 9 index 4 index mul 3 index div add 10 index 3 index div add % y_2=y_0+f\tan\alpha+e\sec\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIHyperbolaOrdNode[Options](O)(a,b)[rotation]{y_1}{A}{B}
+%% Draw the nodes whose ordinate value are the given value y_1 on the General Inversion Hyperbola.
+%%
+%% set $e=a\cos\theta$, $f=b\sin\theta$, $g=a\sin\theta$, $h=b\cos\theta$, then we have
+%% $$x=x_0+h\tan\alpha-g\sec\alpha, y=y_0+f\tan\alpha+e\sec\alpha$$
+%% when $y=y_1$, we get
+%% $$f\tan\alpha+e\sec\alpha=y_1-y_0$$
+%% set $m=y_1-y_0$, we have
+%% $$m\cos\alpha-f\sin\alpha=e$$
+%% then
+%% $$(m^2+f^2)\sin^2\alpha+2fe\sin\alpha+e^2-m^2=0$$
+%% if $m^2+f^2=0$, we have $m=f=0$, i.e, $y_1=y_0$ and $\sin\theta=0$, but the last equation gives $e=0$, which is not possible.
+%% so $m^2+f^2\neq0$, we get
+%% $$\sin_{1,2}\alpha=\dfrac{-fe\pm{}m\sqrt{m^2+f^2-e^2}}{m^2+f^2}$$
+%% and
+%% $$\cos_{1,2}\alpha=\dfrac{em\pm{}f\sqrt{m^2+f^2-e^2}}{m^2+f^2}$$
+%% where $\cos\alpha$ can not be zero, but when $f=\pm{}e$, we have $\cos\alpha=0$, we should skip it.
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the ordinate value y_1.
+%% #6 -> [output] the first target node name.
+%% #7 -> [output] the second target node name.
+\def\pstGeneralIHyperbolaOrdNode{\@ifnextchar[\Pst at GeneralIHyperbolaOrdNode{\Pst at GeneralIHyperbolaOrdNode[]}}
+\def\Pst at GeneralIHyperbolaOrdNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaOrdNode@i{\Pst at GeneralIHyperbolaOrdNode@i[0]}}%
+\def\Pst at GeneralIHyperbolaOrdNode@i[#1]#2#3#4{%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % m=y1-y0
+ 0 index dup mul 4 index dup mul add % m^2+f^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 6 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(m^2+f^2-e^2)
+ %% \sin_{1,2}\alpha=\dfrac{-fe\pm{}m\sqrt{m^2+f^2-e^2}}{m^2+f^2}
+ 2 index 1 index mul 7 index 7 index mul add neg 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{em\pm{}f\sqrt{m^2+f^2-e^2}}{m^2+f^2}
+ 6 index 2 index mul 8 index 5 index mul exch sub 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 6 index 3 index mul 2 index div add 7 index 2 index div sub % x_1=x_0+h\tan\alpha-g\sec\alpha
+ 14 index 9 index 4 index mul 3 index div add 10 index 3 index div add % y_1=y_0+f\tan\alpha+e\sec\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#3}%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 2 index mul % e=a\cos\theta
+ 3 index 2 index mul % f=b\sin\theta
+ 5 index 3 index mul % g=a\sin\theta
+ 5 index 5 index mul % h=b\cos\theta
+ #2 9 index sub % m=y1-y0
+ 0 index dup mul 4 index dup mul add % m^2+f^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 6 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(m^2+f^2-e^2)
+ %% \sin_{1,2}\alpha=\dfrac{-fe\pm{}m\sqrt{m^2+f^2-e^2}}{m^2+f^2}
+ 2 index 1 index mul 7 index 7 index mul sub 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{em\pm{}f\sqrt{m^2+f^2-e^2}}{m^2+f^2}
+ 6 index 2 index mul 8 index 5 index mul add 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 14 index 6 index 3 index mul 2 index div add 7 index 2 index div sub % x_2=x_0+h\tan\alpha-g\sec\alpha
+ 14 index 9 index 4 index mul 3 index div add 10 index 3 index div add % y_2=y_0+f\tan\alpha+e\sec\alpha
+ 17 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIHyperbolaFocusNode[Options](O)(a,b)[rotation]{F1}{F2}
+%% Draw the focus nodes of the General Inversion Hyperbola H.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the focus node F1 of the Hyperbola.
+%% #6 -> [output] the focus node F2 of the Hyperbola.
+\def\pstGeneralIHyperbolaFocusNode{\@ifnextchar[\Pst at GeneralIHyperbolaFocusNode{\Pst at GeneralIHyperbolaFocusNode[]}}
+\def\Pst at GeneralIHyperbolaFocusNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaFocusNode@i{\Pst at GeneralIHyperbolaFocusNode@i[0]}}%
+\def\Pst at GeneralIHyperbolaFocusNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index dup mul 3 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 6 index 1 index 3 index mul sub % x=x0-c\sin\theta
+ 6 index 2 index 5 index mul add % y=y0+c\cos\theta
+ 9 2 roll pop pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index dup mul 3 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 6 index 1 index 3 index mul add % x=x0+c\sin\theta
+ 6 index 2 index 5 index mul sub % y=y0-c\cos\theta
+ 9 2 roll pop pop pop pop pop pop pop
+ ){#3}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIHyperbolaVertexNode[Options](O)(a,b)[rotation]{V1}{V2}
+%% Draw the vertex nodes of the General Inversion Hyperbola H.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the vertex node V1 of the Hyperbola.
+%% #6 -> [output] the vertex node V2 of the Hyperbola.
+\def\pstGeneralIHyperbolaVertexNode{\@ifnextchar[\Pst at GeneralIHyperbolaVertexNode{\Pst at GeneralIHyperbolaVertexNode[]}}
+\def\Pst at GeneralIHyperbolaVertexNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaVertexNode@i{\Pst at GeneralIHyperbolaVertexNode@i[0]}}%
+\def\Pst at GeneralIHyperbolaVertexNode@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 5 index 4 index 2 index mul sub % x=x0-a\sin\theta
+ 5 index 5 index 4 index mul add % y=y0+a\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ #1 dup cos exch sin % cos\theta sin\theta
+ 5 index 4 index 2 index mul add % x=x0+a\sin\theta
+ 5 index 5 index 4 index mul sub % y=y0-a\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIHyperbolaDirectrixLine[Options](O)(a,b)[rotation]{Lx}{Ly}{Rx}{Ry}
+%% Draw the two directrix lines of the General Inversion Hyperbola H.
+%% If you not input the rotation angle, the default value is $0^\circ$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the first node Lx on the first directrix line.
+%% #6 -> [output] the second node Lx on the first directrix line.
+%% #7 -> [output] the first node Lx on the second directrix line.
+%% #8 -> [output] the second node Lx on the second directrix line.
+\def\pstGeneralIHyperbolaDirectrixLine{\@ifnextchar[\Pst at GeneralIHyperbolaDirectrixLine{\Pst at GeneralIHyperbolaDirectrixLine[]}}
+\def\Pst at GeneralIHyperbolaDirectrixLine[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaDirectrixLine@i{\Pst at GeneralIHyperbolaDirectrixLine@i[0]}}%
+\def\Pst at GeneralIHyperbolaDirectrixLine@i[#1]#2#3#4#5{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 4 index exch sub 5 index exch % x0,y0-a^2/c
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 4 index exch sub 5 index 1 add exch % x0+1,y0-a^2/c
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 4 index add 5 index exch % x0,y0+a^2/c
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#4}
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ 1 index dup mul 1 index dup mul add sqrt % c=sqrt(a^2+b^2)
+ 2 index dup mul 1 index div 4 index add 5 index 1 add exch % x0+1,y0+a^2/c
+ 7 2 roll pop pop pop % x,y,x0,y0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#5}
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \pstLineAB{#2}{#3}%
+ \pstLineAB{#4}{#5}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIHyperbolaAsymptoteLine[Options](O)(a,b)[rotation]{L1}{L2}
+%% Draw the two asymptote lines L1 and L2 of the General Inversion Hyperbola H.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [output] the second node name on the first asymptote line, the first node is the center O.
+%% #6 -> [output] the second node name on the second asymptote line, the first node is the center O.
+\def\pstGeneralIHyperbolaAsymptoteLine{\@ifnextchar[\Pst at GeneralIHyperbolaAsymptoteLine{\Pst at GeneralIHyperbolaAsymptoteLine[]}}
+\def\Pst at GeneralIHyperbolaAsymptoteLine[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \def\pst at hyperbola@center{#2}
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaAsymptoteLine@i{\Pst at GeneralIHyperbolaAsymptoteLine@i[0]}}%
+\def\Pst at GeneralIHyperbolaAsymptoteLine@i[#1]#2#3{%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 index sub % x=x0-b/a
+ 4 index 1 add % y=y0+1
+ 7 2 roll pop pop pop % x y x_0 y_0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#2}%
+ \pnode(!
+ \pst at tempO \tx at UserCoor % x_o,y_o
+ \pst at tempR \tx at UserCoor % a,b
+ 0 index 2 index div % b/a
+ 4 index 1 index add % x=x0+b/a
+ 4 index 1 add % y=y0+1
+ 7 2 roll pop pop pop % x y x_0 y_0
+ #1 dup cos exch sin % cos\theta sin\theta
+ 3 index 6 index 5 index sub 3 index mul add 5 index 4 index sub 2 index mul sub % x0+(x-x0)\cos\theta-(y-y0)\sin\theta
+ 3 index 7 index 6 index sub 3 index mul add 6 index 5 index sub 4 index mul add % y0+(x-x0)\sin\theta+(y-y0)\cos\theta
+ 8 2 roll pop pop pop pop pop pop
+ ){#3}%
+ \Pst at ManageParamList{#2}%
+ \Pst at ManageParamList{#3}%
+ \pstLineAB{\pst at hyperbola@center}{#2}%
+ \pstLineAB{\pst at hyperbola@center}{#3}%
+ \endgroup%
+}%
+%
+%
+%% \pstGeneralIHyperbolaLineInter[Options](O)(a,b)[rotation]{A}{B}{C}{D}
+%% Find the intersection nodes $C$ and $D$ of the given line AB with the General Inversion Hyperbola H.
+%%
+%% set $e=a\cos\theta$, $f=b\sin\theta$, $g=a\sin\theta$, $h=b\cos\theta$, then we have
+%% $$x=x_0+h\tan\alpha-g\sec\alpha, y=y_0+f\tan\alpha+e\sec\alpha$$
+%% when line AB is vertical, the solve is like as pstGeneralIHyperbolaAbsNode,
+%% else we can represent the line AB as the following function:
+%% $$y=kx+d$$ where $$k=\dfrac{y_2-y_1}{x_2-x_1}, d=\dfrac{x_2y_1-x_1y_2}{x_2-x_1}$$
+%% Let $X=x-x_0$, $Y=y-y_0$, then we have $Y=kX+m$, where $m=kx_0-y_0+d$, then
+%% $$f\tan\alpha+e\sec\alpha=k(h\tan\alpha-g\sec\alpha)+m$$
+%% $$(kh-f)\sin\alpha+m\cos\alpha=kg+e$$
+%% $$(kh-f)^2\sin^2\alpha=m^2\cos^2\alpha-2m(kg+e)\cos\alpha+(kg+e)^2$$
+%% $$(m^2+(kh-f)^2)\cos^2\alpha-2m(kg+e)\cos\alpha+(kg+e)^2-(kh-f)^2=0$$
+%% when $m^2+(kh-f)^2=0$, we have $m=kh-f=0$, at this time $kg+e=0$, we get
+%% $$k=\dfrac{\sin\theta}{\cos\theta}=-\dfrac{\cos\theta}{\sin\theta}$$
+%% which is not possible.
+%%
+%% let $p=kh-f$, $q=kg+e$, so we have
+%% $$(m^2+p^2)\cos^2\alpha-2mq\cos\alpha+q^2-p^2=0$$
+%% then
+%% $$\cos_{1,2}\alpha=\dfrac{mq\pm{}p\sqrt{m^2+p^2-q^2}}{m^2+p^2},\sin_{1,2}\alpha=\dfrac{qp\mp{}m\sqrt{m^2+p^2-q^2}}{m^2+p^2}$$
+%% where $\cos\alpha$ can not be zero, we should skip it.
+%%
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the first node on the given line.
+%% #6 -> [input] the second node on the given line.
+%% #7 -> [output] the first intersection node name.
+%% #8 -> [output] the second intersection node name.
+\def\pstGeneralIHyperbolaLineInter{\@ifnextchar[\Pst at GeneralIHyperbolaLineInter{\Pst at GeneralIHyperbolaLineInter[]}}
+\def\Pst at GeneralIHyperbolaLineInter[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \pst at getcoor{#2}\pst at tempO%
+ \pst at getcoor{#3}\pst at tempR%
+ \@ifnextchar[\Pst at GeneralIHyperbolaLineInter@i{\Pst at GeneralIHyperbolaLineInter@i[0]}}%
+\def\Pst at GeneralIHyperbolaLineInter@i[#1]#2#3#4#5{%
+ \pst at getcoor{#2}\pst at tempA%
+ \pst at getcoor{#3}\pst at tempB%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % cos\theta sin\theta
+ 7 index 2 index mul % e=a\cos\theta
+ 7 index 2 index mul % f=b\sin\theta
+ 9 index 3 index mul % g=a\sin\theta
+ 9 index 5 index mul % h=b\cos\theta
+ 7 index 14 index sub % n=x1-x0
+ 0 index dup mul 2 index dup mul add % n^2+h^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 4 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+h^2-g^2)
+ %% \sin_{1,2}\alpha=\dfrac{gh\pm{}n\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 2 index 1 index mul 5 index 5 index mul exch sub 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{-ng\pm{}h\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 4 index 2 index mul 4 index 7 index mul add neg 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 18 index 6 index 3 index mul 2 index div add 7 index 2 index div sub % x_1=x_0+h\tan\alpha-g\sec\alpha
+ 18 index 9 index 4 index mul 3 index div add 10 index 3 index div add % y_1=y_0+f\tan\alpha+e\sec\alpha
+ 21 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ #1 dup cos exch sin % cos\theta sin\theta
+ 10 index 2 index mul % e=a\cos\theta
+ 10 index 2 index mul % f=b\sin\theta
+ 12 index 3 index mul % g=a\sin\theta
+ 12 index 5 index mul % h=b\cos\theta
+ 8 index 1 index mul 3 index sub % p=kh-f
+ 9 index 3 index mul 5 index add % q=kg+e
+ 8 index dup mul 2 index dup mul add % m^2+p^2
+ dup 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ 0 0
+ } {
+ 0 index 2 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt % sqrt(m^2+p^2-q^2)
+ % \sin_{1,2}\alpha=\dfrac{qp\mp{}m\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 10 index 1 index mul 4 index 4 index mul add 2 index div % \sin\alpha
+ % \cos_{1,2}\alpha=\dfrac{mq\pm{}p\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 4 index 2 index mul 12 index 5 index mul exch sub 3 index div % \cos\alpha
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop 0 0
+ } {
+ % x=x_0+h\tan\alpha-g\sec\alpha, y=y_0+f\tan\alpha+e\sec\alpha
+ 22 index 8 index 2 index div sub 7 index 3 index mul 2 index div add % x_1
+ 22 index 11 index 3 index div add 10 index 4 index mul 3 index div add % y_1
+ 25 2 roll
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#4}%
+ \pnode(!%
+ \pst at tempO \tx at UserCoor % x0,y0
+ \pst at tempR \tx at UserCoor abs exch abs exch % |a|,|b|
+ \pst at tempA \tx at UserCoor % x1,y1
+ \pst at tempB \tx at UserCoor % x2,y2
+ 3 index 2 index sub abs 1E-5 lt { % if the line AB is vertical
+ #1 dup cos exch sin % cos\theta sin\theta
+ 7 index 2 index mul % e=a\cos\theta
+ 7 index 2 index mul % f=b\sin\theta
+ 9 index 3 index mul % g=a\sin\theta
+ 9 index 5 index mul % h=b\cos\theta
+ 7 index 14 index sub % n=x1-x0
+ 0 index dup mul 2 index dup mul add % n^2+h^2
+ 0 index abs 1E-5 lt {
+ pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ 0 index 4 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop
+ pop pop pop pop pop pop 0 0
+ } {
+ sqrt % sqrt(n^2+h^2-g^2)
+ %% \sin_{1,2}\alpha=\dfrac{gh\pm{}n\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 2 index 1 index mul 5 index 5 index mul add 2 index div % sin
+ %% \cos_{1,2}\alpha=\dfrac{-ng\pm{}h\sqrt{n^2+h^2-g^2}}{n^2+h^2}
+ 4 index 2 index mul 4 index 7 index mul sub 3 index div % cos
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop 0 0
+ } {
+ 18 index 6 index 3 index mul 2 index div add 7 index 2 index div sub % x_2=x_0+h\tan\alpha-g\sec\alpha
+ 18 index 9 index 4 index mul 3 index div add 10 index 3 index div add % y_2=y_0+f\tan\alpha+e\sec\alpha
+ 21 2 roll pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } {
+ 0 index 3 index sub 2 index 5 index sub div % k
+ 2 index 4 index mul 2 index 6 index mul sub 3 index 6 index sub div % d
+ 1 index 10 index mul 9 index sub 1 index add % m=kx_o-y_o+d
+ #1 dup cos exch sin % cos\theta sin\theta
+ 10 index 2 index mul % e=a\cos\theta
+ 10 index 2 index mul % f=b\sin\theta
+ 12 index 3 index mul % g=a\sin\theta
+ 12 index 5 index mul % h=b\cos\theta
+ 8 index 1 index mul 3 index sub % p=kh-f
+ 9 index 3 index mul 5 index add % q=kg+e
+ 8 index dup mul 2 index dup mul add % m^2+p^2
+ dup 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ 0 0
+ } {
+ 0 index 2 index dup mul sub dup 0 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop 0 0
+ } {
+ sqrt % sqrt(m^2+p^2-q^2)
+ % \sin_{1,2}\alpha=\dfrac{qp\mp{}m\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 10 index 1 index mul 4 index 4 index mul exch sub 2 index div % \sin\alpha
+ % \cos_{1,2}\alpha=\dfrac{mq\pm{}p\sqrt{m^2+p^2-q^2}}{m^2+p^2}
+ 4 index 2 index mul 12 index 5 index mul add 3 index div % \cos\alpha
+ dup abs 1E-5 lt {
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop 0 0
+ } {
+ % x=x_0+h\tan\alpha-g\sec\alpha, y=y_0+f\tan\alpha+e\sec\alpha
+ 22 index 8 index 2 index div sub 7 index 3 index mul 2 index div add % x_2
+ 22 index 11 index 3 index div add 10 index 4 index mul 3 index div add % y_2
+ 25 2 roll
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop pop pop pop pop pop pop pop
+ pop pop pop
+ } ifelse
+ } ifelse
+ } ifelse
+ } ifelse
+ ){#5}%
+ \Pst at ManageParamList{#4}%
+ \Pst at ManageParamList{#5}%
+ \endgroup%
+}%
+%
+%% \pstGeneralIHyperbolaPolarNode[Options](O)(a,b)[rotation]{A}{B}{T}
+%% Find the polar point of chord AB on General Inversion Hyperbola H.
+%% We use the following proposition to find the polar point of chord AB:
+%% Let $P$, $Q$ are vertex points of the hyperbola, for any chord $AB$ of hyperbola, $PA$ and $BQ$ intersect at $E$, $PB$ and $AQ$ intersect at $F$, then the middle point $M$ of $EF$ is the polar point of chord $AB$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the node A on the hyperbola.
+%% #6 -> [input] the node B on the hyperbola.
+%% #7 -> [output] the polar node T of chord AB.
+\def\pstGeneralIHyperbolaPolarNode{\@ifnextchar[\Pst at GeneralIHyperbolaPolarNode{\Pst at GeneralIHyperbolaPolarNode[]}}
+\def\Pst at GeneralIHyperbolaPolarNode[#1](#2)(#3){%
+ \begingroup
+ \psset{#1}%
+ \def\pst at hyperbola@center{#2}
+ \def\pst at hyperbola@radii{#3}
+ \@ifnextchar[\Pst at GeneralIHyperbolaPolarNode@i{\Pst at GeneralIHyperbolaPolarNode@i[0]}}%
+\def\Pst at GeneralIHyperbolaPolarNode@i[#1]#2#3#4{%
+ \pstGeneralIHyperbolaVertexNode[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{@PST at GENERALIHYPERBOLA@VERTEXA}{@PST at GENERALIHYPERBOLA@VERTEXB}
+ \pstInterLL[PointName=none,PointSymbol=none]{#2}{@PST at GENERALIHYPERBOLA@VERTEXA}{#3}{@PST at GENERALIHYPERBOLA@VERTEXB}{@PST at GENERALIHYPERBOLA@INTER at X}
+ \pstInterLL[PointName=none,PointSymbol=none]{#3}{@PST at GENERALIHYPERBOLA@VERTEXA}{#2}{@PST at GENERALIHYPERBOLA@VERTEXB}{@PST at GENERALIHYPERBOLA@INTER at Y}
+ \pstMiddleAB{@PST at GENERALIHYPERBOLA@INTER at X}{@PST at GENERALIHYPERBOLA@INTER at Y}{#4}
+ \Pst at geonodelabel{#4}%
+ \pstLineAB{#2}{#4}
+ \pstLineAB{#3}{#4}
+ \endgroup
+}%
+%
+%% \pstGeneralIHyperbolaTangentNode[Options](O)(a,b)[rotation]{T}{A}{B}
+%% Draw the two tangent lines through the point $T$ to the General Inversion Hyperbola H and get the node A and B on the General Inversion Hyperbola.
+%% We use the following proposition to find the tangent points $A$ and $B$ of $T$:
+%% Let $T$ is a point out of the hyperbola, we give any two chords $TPQ$ and $TRS$ of the hyperbola, $PR$ and $QS$ intersect at $X$, $RQ$ and $PS$ intersect at $Y$,
+%% then the intersection point $A$ and $B$ of $XY$ and the hyperbola are the tangent points from $T$.
+%% Parameters:
+%% #1 -> options
+%% #2 -> [input] the hyperbola center O
+%% #3 -> [input] the radii of real and imaginary axis
+%% #4 -> [input] the rotation angle $\theta$ of the symmetrical axis.
+%% #5 -> [input] the given node T outside the hyperbola
+%% #6 -> [output] the tangent node name A on the hyperbola
+%% #7 -> [output] the tangent node name B on the hyperbola
+\def\pstGeneralIHyperbolaTangentNode{\@ifnextchar[\Pst at GeneralIHyperbolaTangentNode{\Pst at GeneralIHyperbolaTangentNode[]}}
+\def\Pst at GeneralIHyperbolaTangentNode[#1](#2)(#3){%
+ \begingroup
+ \@InitListMng %
+ \psset{#1}%
+ \def\pst at hyperbola@center{#2}
+ \def\pst at hyperbola@radii{#3}
+ \@ifnextchar[\Pst at GeneralIHyperbolaTangentNode@i{\Pst at GeneralIHyperbolaTangentNode@i[0]}}%
+\def\Pst at GeneralIHyperbolaTangentNode@i[#1]#2#3#4{%
+ \pstGeneralIHyperbolaNode[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{10}{@PST at GENERALIHYPERBOLA@TANGENTAUX at P0}
+ \pstGeneralIHyperbolaNode[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{-10}{@PST at GENERALIHYPERBOLA@TANGENTAUX at R0}
+ \pstGeneralIHyperbolaLineInter[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{#2}{@PST at GENERALIHYPERBOLA@TANGENTAUX at P0}{@PST at GENERALIHYPERBOLA@TANGENTAUX at P}{@PST at GENERALIHYPERBOLA@TANGENTAUX at Q}
+ \pstGeneralIHyperbolaLineInter[PointName=none,PointSymbol=none](\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{#2}{@PST at GENERALIHYPERBOLA@TANGENTAUX at R0}{@PST at GENERALIHYPERBOLA@TANGENTAUX at R}{@PST at GENERALIHYPERBOLA@TANGENTAUX at S}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALIHYPERBOLA@TANGENTAUX at P}{@PST at GENERALIHYPERBOLA@TANGENTAUX at S}{@PST at GENERALIHYPERBOLA@TANGENTAUX at Q}{@PST at GENERALIHYPERBOLA@TANGENTAUX at R}{@PST at GENERALIHYPERBOLA@TANGENTAUX at I}
+ \pstInterLL[PointName=none,PointSymbol=none]{@PST at GENERALIHYPERBOLA@TANGENTAUX at P}{@PST at GENERALIHYPERBOLA@TANGENTAUX at R}{@PST at GENERALIHYPERBOLA@TANGENTAUX at Q}{@PST at GENERALIHYPERBOLA@TANGENTAUX at S}{@PST at GENERALIHYPERBOLA@TANGENTAUX at X}
+ \pstGeneralIHyperbolaLineInter(\pst at hyperbola@center)(\pst at hyperbola@radii)[#1]{@PST at GENERALIHYPERBOLA@TANGENTAUX at X}{@PST at GENERALIHYPERBOLA@TANGENTAUX at I}{#3}{#4}
+ \Pst at ManageParamList{#3}%
+ \Pst at ManageParamList{#4}%
+ \pstLineAB{#2}{#3}
+ \pstLineAB{#2}{#4}
+ \endgroup%
+}%
+%
\catcode`\@=\PstAtCode\relax
%
\endinput
Modified: trunk/Master/tlpkg/libexec/ctan2tds
===================================================================
--- trunk/Master/tlpkg/libexec/ctan2tds 2019-10-20 17:28:18 UTC (rev 52456)
+++ trunk/Master/tlpkg/libexec/ctan2tds 2019-10-20 20:57:52 UTC (rev 52457)
@@ -1962,6 +1962,7 @@
'pst-poker', '\.eps|' . $standardtex,
'pst-text', 'pst-(char|text)\.(tex|sty)',
'ptex-base', '\.(ini|lib|src)$|(ascii-jplain|kinsoku|ptex).tex$',
+ 'ptex-manual', 'NULL', # keep ptex-manual.sty in doc
'punk', 'NULL', # have separate punk-latex
'qobitree', 'qobitree.tex',
'r_und_s', '\.tex|\.sty',
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