Merge branch 'AndreasFMueller:master' into master

22 files changed, 1464 insertions, 9 deletions

diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile
index a7c9e74..7636e65 100644
--- a/buch/chapters/110-elliptisch/images/Makefile
+++ b/buch/chapters/110-elliptisch/images/Makefile
@@ -5,7 +5,8 @@
 #
 all:	lemniskate.pdf ellipsenumfang.pdf unvollstaendig.pdf rechteck.pdf \
 	ellipse.pdf pendel.pdf jacobiplots.pdf jacobidef.pdf jacobi12.pdf \
-	sncnlimit.pdf slcl.pdf
+	sncnlimit.pdf slcl.pdf torusschnitt.pdf kegelpara.pdf lemnispara.pdf \
+	ellpolnul.pdf ellall.pdf ellselection.pdf
 
 lemniskate.pdf:	lemniskate.tex
 	pdflatex lemniskate.tex
@@ -78,3 +79,52 @@ slcldata.tex:	slcl
 	./slcl --outfile=slcldata.tex --a=0 --b=13.4 --steps=200
 slcl.pdf:	slcl.tex slcldata.tex
 	pdflatex slcl.tex
+
+KEGELSIZE = -W256 -H256
+KEGELSIZE = -W128 -H128
+KEGELSIZE = -W1080 -H1080
+kegelpara.png:	kegelpara.pov
+	povray +A0.1 $(KEGELSIZE) -Okegelpara.png kegelpara.pov
+
+kegelpara.jpg:	kegelpara.png Makefile
+	convert -extract 1080x1040+0+0 kegelpara.png \
+		-density 300 -units PixelsPerInch kegelpara.jpg
+
+kegelpara.pdf:	kegelpara.tex kegelpara.jpg
+	pdflatex kegelpara.tex
+
+torusschnitt.png:	torusschnitt.pov
+	povray +A0.1 -W1920 -H1080 -Otorusschnitt.png torusschnitt.pov
+
+torusschnitt.jpg:	torusschnitt.png Makefile
+	convert -extract 1640x1080+140+0 torusschnitt.png \
+		-density 300 -units PixelsPerInch torusschnitt.jpg
+
+torusschnitt.pdf:	torusschnitt.tex torusschnitt.jpg
+	pdflatex torusschnitt.tex
+
+lemnispara:	lemnispara.cpp
+	g++ -O2 -Wall -g -o lemnispara `pkg-config --cflags gsl`	\
+		lemnispara.cpp `pkg-config --libs gsl`
+
+lemnisparadata.tex:	lemnispara
+	./lemnispara
+
+lemnispara.pdf:	lemnispara.tex lemnisparadata.tex
+	pdflatex lemnispara.tex
+
+ltest:	lemnispara.pdf
+
+ellpolnul.pdf:	ellpolnul.tex ellcommon.tex
+	pdflatex ellpolnul.tex
+ellall.pdf:	ellall.tex ellcommon.tex
+	pdflatex ellall.tex
+
+rechteckpfade2.tex:	rechteck Makefile
+	./rechteck --outfile rechteckpfade2.tex --k 0.87 --vsteps=1
+ellselection.pdf:	ellselection.tex rechteckpfade2.tex
+	pdflatex ellselection.tex
+
+rechteckpfade3.tex:	rechteck
+	./rechteck --outfile rechteckpfade3.tex --k 0.70710678118654752440 \
+		--vsteps=4
diff --git a/buch/chapters/110-elliptisch/images/ellall.pdf b/buch/chapters/110-elliptisch/images/ellall.pdf
new file mode 100644
index 0000000..fd0a5dd
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellall.pdf
diff --git a/buch/chapters/110-elliptisch/images/ellall.tex b/buch/chapters/110-elliptisch/images/ellall.tex
new file mode 100644
index 0000000..b37fe12
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellall.tex
@@ -0,0 +1,215 @@
+%
+% ellpolnul.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+\input{ellcommon.tex}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+%\draw (-1,-1) rectangle (1,1);
+%\node at (-1,-1) [below left] {$0$};
+%\node at (1,-1) [below right] {$K$};
+%\node at (1,1) [above right] {$K+iK'$};
+%\node at (-1,1) [above left] {$iK'$};
+%\node at (0,0) {$u$};
+
+\fill[color=rot!10,opacity=0.5] (-5.5,-4.3) rectangle (7.3,-1.7);
+\fill[color=blau!10,opacity=0.5] (-5.5,-7.3) rectangle (7.3,-4.7);
+\fill[color=gruen!10,opacity=0.5] (-5.5,-10.3) rectangle (7.3,-7.7);
+
+\fill[color=rot!10,opacity=0.5] (-1.3,-10.5) rectangle (1.3,2.5);
+\fill[color=blau!10,opacity=0.5] (1.7,-10.5) rectangle (4.3,2.5);
+\fill[color=gruen!10,opacity=0.5] (4.7,-10.5) rectangle (7.3,2.5);
+
+\begin{scope}[xshift=1.5cm,yshift=2cm]
+\node at (0,0) {Zähler};
+\draw[<-] (-4.5,0) -- (-1,0);
+\draw[->] (1,0) -- (4.5,0);
+\node[color=black] at (-4.5,-0.4) {\Large n};
+\node[color=rot]   at (-1.5,-0.4) {\Large s};
+\node[color=blau]  at (1.5,-0.4)  {\Large c};
+\node[color=gruen] at (4.5,-0.4)  {\Large d};
+\end{scope}
+
+\begin{scope}[xshift=-5.1cm,yshift=-4.5cm]
+\node at (0,0) [rotate=90] {Nenner};
+\draw[<-] (0,-4.5) -- (0,-1);
+\draw[->] (0,1) -- (0,4.5);
+\node[color=gruen] at (0.4,-4.5) [rotate=90] {\Large d};
+\node[color=blau]  at (0.4,-1.5) [rotate=90] {\Large c};
+\node[color=rot]   at (0.4,1.5)  [rotate=90] {\Large s};
+\node[color=black] at (0.4,4.5)  [rotate=90] {\Large n};
+\end{scope}
+
+\begin{scope}[xshift=-3cm,yshift=0cm]
+\node at (0,0) {$1$};
+\draw[color=gray!20] (-1,-1) rectangle (1,1);
+\end{scope}
+
+\definecolor{sccolor}{rgb}{0.8,0.0,1.0}
+\definecolor{sdcolor}{rgb}{0.6,0.6,0.0}
+\definecolor{cdcolor}{rgb}{0.0,0.6,1.0}
+
+\begin{scope}[xshift=0cm]
+\rechteck{rot}{\operatorname{sn}(u,k)}
+\nullstelle{(-1,-1)}{rot}
+\pol{(-1,1)}{rot}
+\node at (-1,-1) {$0$};
+\node at (1,-1) {$1$};
+\node at (1,1) {$\frac1k$};
+\node at (-1,1) {$\infty$};
+\end{scope}
+
+\begin{scope}[xshift=3cm]
+\rechteck{blau}{\operatorname{cn}(u,k)}
+\nullstelle{(1,-1)}{blau}
+\pol{(-1,1)}{blau}
+\node at (-1,-1) {$1$};
+\node at (1,-1) {$0$};
+\node at (1,1) {$\frac{k'}{ik}$};
+\node at (-1,1) {$\infty$};
+\end{scope}
+
+\begin{scope}[xshift=6cm]
+\rechteck{gruen}{\operatorname{dn}(u,k)}
+\nullstelle{(1,1)}{gruen}
+\pol{(-1,1)}{gruen}
+\node at (-1,-1) {$1$};
+\node at (1,-1) {$k'$};
+\node at (1,1) {$0$};
+\node at (-1,1) {$\infty$};
+\end{scope}
+
+%
+% start row with denominator sn(u,k)
+%
+
+\begin{scope}[xshift=-3cm,yshift=-3cm]
+\rechteck{rot}{\operatorname{ns}(u,k)}
+\pol{(-1,-1)}{rot}
+\nullstelle{(-1,1)}{rot}
+\node at (-1,-1) {$\infty$};
+\node at (1,-1) {$1$};
+\node at (1,1) {$k$};
+\node at (-1,1) {$0$};
+\end{scope}
+
+\begin{scope}[xshift=0cm,yshift=-3cm]
+%\rechteck{gray}{1}
+\fill[color=white] (-1,-1) rectangle (1,1);
+\node[color=gray] at (0,0) {$1$};
+\end{scope}
+
+\begin{scope}[xshift=3cm,yshift=-3cm]
+\rechteck{sccolor}{\operatorname{cs}(u,k)}
+\pol{(-1,-1)}{sccolor}
+\nullstelle{(1,-1)}{sccolor}
+\node at (-1,-1) {$\infty$};
+\node at (1,-1) {$0$};
+\node at (1,1) {$\frac{k'}{i}$};
+\node at (-1,1) {$ $};
+\end{scope}
+
+\begin{scope}[xshift=6cm,yshift=-3cm]
+\rechteck{sdcolor}{\operatorname{ds}(u,k)}
+\pol{(-1,-1)}{sdcolor}
+\nullstelle{(1,1)}{sdcolor}
+\node at (-1,-1) {$\infty$};
+\node at (1,-1) {$k'$};
+\node at (1,1) {$0$};
+\node at (-1,1) {$ $};
+\end{scope}
+
+%
+% start row with denominator cn(u,k)
+%
+
+\begin{scope}[xshift=-3cm,yshift=-6cm]
+\rechteck{blau}{\operatorname{nc}(u,k)}
+\pol{(1,-1)}{blau}
+\nullstelle{(-1,1)}{blau}
+\node at (-1,-1) {$1$};
+\node at (-1,1) {$0$};
+\node at (1,-1) {$\infty$};
+\node at (1,1) {$\frac{ik}{k'}$};
+\end{scope}
+
+\begin{scope}[xshift=0cm,yshift=-6cm]
+\rechteck{sccolor}{\operatorname{sc}(u,k)}
+\nullstelle{(-1,-1)}{sccolor}
+\pol{(1,-1)}{sccolor}
+\node at (-1,-1) {$0$};
+\node at (1,-1) {$\infty$};
+\node at (-1,1) {$ $};
+\node at (1,1) {$\frac{i}{k'}$};
+\end{scope}
+
+\begin{scope}[xshift=3cm,yshift=-6cm]
+%\rechteck{gray}{1}
+\fill[color=white] (-1,-1) rectangle (1,1);
+\node[color=gray] at (0,0) {$1$};
+\end{scope}
+
+\begin{scope}[xshift=6cm,yshift=-6cm]
+\rechteck{cdcolor}{\operatorname{dc}(u,k)}
+\nullstelle{(1,1)}{cdcolor}
+\pol{(1,-1)}{cdcolor}
+\node at (-1,-1) {$1$};
+\node at (1,-1) {$\infty$};
+\node at (-1,1) {$k$};
+\node at (1,1) {$0$};
+\end{scope}
+
+%
+% start row with denominator dn(u,k)
+%
+
+\begin{scope}[xshift=-3cm,yshift=-9cm]
+\rechteck{gruen}{\operatorname{nd}(u,k)}
+\pol{(1,1)}{gruen}
+\nullstelle{(-1,1)}{gruen}
+\node at (-1,-1) {$1$};
+\node at (-1,1) {$0$};
+\node at (1,-1) {$\frac{1}{k'}$};
+\node at (1,1) {$\infty$};
+\end{scope}
+
+\begin{scope}[xshift=0cm,yshift=-9cm]
+\rechteck{sdcolor}{\operatorname{sd}(u,k)}
+\nullstelle{(-1,-1)}{sdcolor}
+\pol{(1,1)}{sdcolor}
+\node at (-1,-1) {$0$};
+\node at (1,-1) {$\frac{1}{k'}$};
+\node at (-1,1) {$ $};
+\node at (1,1) {$\infty$};
+\end{scope}
+
+\begin{scope}[xshift=3cm,yshift=-9cm]
+\rechteck{cdcolor}{\operatorname{cd}(u,k)}
+\pol{(1,1)}{cdcolor}
+\nullstelle{(1,-1)}{cdcolor}
+\node at (-1,-1) {$1$};
+\node at (-1,1) {$\frac1k $};
+\node at (1,-1) {$0$};
+\node at (1,1) {$\infty$};
+\end{scope}
+
+\begin{scope}[xshift=6cm,yshift=-9cm]
+%\rechteck{gray}{1}
+\fill[color=white] (-1,-1) rectangle (1,1);
+\node[color=gray] at (0,0) {$1$};
+\end{scope}
+
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/ellcommon.tex b/buch/chapters/110-elliptisch/images/ellcommon.tex
new file mode 100644
index 0000000..cd3245d
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellcommon.tex
@@ -0,0 +1,24 @@
+%
+% ellcommon.tex -- common macros/definitions for elliptic function
+%                  values display
+%
+% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\definecolor{rot}{rgb}{0.8,0,0}
+\definecolor{blau}{rgb}{0,0,1}
+\definecolor{gruen}{rgb}{0,0.6,0}
+\def\l{0.2}
+
+\def\pol#1#2{
+	\draw[color=#2!50,line width=3.0pt]
+		($#1+(-\l,-\l)$) -- ($#1+(\l,\l)$);
+	\draw[color=#2!50,line width=3.0pt]
+		($#1+(-\l,\l)$) -- ($#1+(\l,-\l)$);
+}
+\def\nullstelle#1#2{
+	\draw[color=#2!50,line width=3.0pt] #1 circle[radius=\l];
+}
+\def\rechteck#1#2{
+	\fill[color=#1!20] (-1,-1) rectangle (1,1);
+	\node[color=#1] at (0,0) {$#2\mathstrut$};
+}
diff --git a/buch/chapters/110-elliptisch/images/ellpolnul.pdf b/buch/chapters/110-elliptisch/images/ellpolnul.pdf
new file mode 100644
index 0000000..d798169
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellpolnul.pdf
diff --git a/buch/chapters/110-elliptisch/images/ellpolnul.tex b/buch/chapters/110-elliptisch/images/ellpolnul.tex
new file mode 100644
index 0000000..dfa04d3
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellpolnul.tex
@@ -0,0 +1,69 @@
+%
+% ellpolnul.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+\input{ellcommon.tex}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input rechteckpfade3.tex
+
+\pgfmathparse{2/\xmax}
+\xdef\dx{\pgfmathresult}
+\xdef\dy{\dx}
+
+\begin{scope}[xshift=-1cm,yshift=-1cm]
+\clip (0,0) rectangle (2,2);
+\netz{0.4pt}
+\draw[line width=0.4pt] (-1,0) -- (1,0);
+\end{scope}
+\fill[color=white,opacity=0.7] (-1,-1) rectangle (1,1);
+\draw (-1,-1) rectangle (1,1);
+\node at (-1,-1) [below left] {$0$};
+\node at (1,-1) [below right] {$K$};
+\node at (1,1) [above right] {$K+iK'$};
+\node at (-1,1) [above left] {$iK'$};
+\node at (0,0) {$u$};
+
+\begin{scope}[xshift=4cm]
+\rechteck{rot}{\operatorname{sn}(u,k)}
+\nullstelle{(-1,-1)}{rot}
+\pol{(-1,1)}{rot}
+\node at (-1,-1) {$0$};
+\node at (1,-1) {$1$};
+\node at (1,1) {$\frac1k$};
+\node at (-1,1) {$\infty$};
+\end{scope}
+
+\begin{scope}[xshift=7cm]
+\rechteck{blau}{\operatorname{cn}(u,k)}
+\nullstelle{(1,-1)}{blau}
+\pol{(-1,1)}{blau}
+\node at (-1,-1) {$1$};
+\node at (1,-1) {$0$};
+\node at (1,1) {$\frac{k'}{ik}$};
+\node at (-1,1) {$\infty$};
+\end{scope}
+
+\begin{scope}[xshift=10cm]
+\rechteck{gruen}{\operatorname{dn}(u,k)}
+\nullstelle{(1,1)}{gruen}
+\pol{(-1,1)}{gruen}
+\node at (-1,-1) {$1$};
+\node at (1,-1) {$k'$};
+\node at (1,1) {$0$};
+\node at (-1,1) {$\infty$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/ellselection.pdf b/buch/chapters/110-elliptisch/images/ellselection.pdf
new file mode 100644
index 0000000..7c98db1
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellselection.pdf
diff --git a/buch/chapters/110-elliptisch/images/ellselection.tex b/buch/chapters/110-elliptisch/images/ellselection.tex
new file mode 100644
index 0000000..d8afeb1
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellselection.tex
@@ -0,0 +1,141 @@
+%
+% ellselection.tex -- Wahl einer elliptischen Funktion
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\input{rechteckpfade2.tex}
+
+\def\l{0.45}
+\pgfmathparse{\l*72/2.54}
+\xdef\L{\pgfmathresult}
+
+\pgfmathparse{4.1/\xmax}
+\xdef\dx{\pgfmathresult}
+\xdef\dy{\dx}
+
+\def\sx{4.1}
+\pgfmathparse{\sx*72/2.54}
+\xdef\Sx{\pgfmathresult}
+
+\pgfmathparse{\dx*\ymax}
+\xdef\sy{\pgfmathresult}
+\pgfmathparse{\sy*72/2.54}
+\xdef\Sy{\pgfmathresult}
+
+\pgfmathparse{\sx/2-\l}
+\xdef\linksx{\pgfmathresult}
+\pgfmathparse{\sy/2-\l}
+\xdef\linksy{\pgfmathresult}
+
+\pgfmathparse{\sx/2+2*\l}
+\xdef\rechtsx{\pgfmathresult}
+\pgfmathparse{\sy/2}
+\xdef\rechtsy{\pgfmathresult}
+
+\begin{scope}
+	\clip (-\sx,-\sy) rectangle (\sx,\sy);
+	\begin{scope}[xshift={-\Sx}]
+		\hintergrund
+		\netz{0.7pt}
+	\end{scope}
+	\begin{scope}[xshift={\Sx}]
+		\hintergrund
+		\netz{0.7pt}
+	\end{scope}
+\end{scope}
+
+\fill[color=red!14,opacity=0.7] ({-\sx},0) rectangle (\sx,\sy);
+\fill[color=blue!14,opacity=0.7] ({-\sx},{-\sy}) rectangle (\sx,0);
+\fill[color=yellow!40,opacity=0.5] (0,0) rectangle (\sx,\sy);
+
+\draw (-\sx,-\sy) rectangle (\sx,\sy);
+
+\draw[->] ({-1.4*\sx},0) -- ({1.4*\sx},0) coordinate[label={$\Re u$}];
+\draw[->] (0,{-\sy-1}) -- (0,{\sy+1}) coordinate[label={right:$\Im u$}];
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\draw[->,line width=1.9pt,color=darkgreen]
+	(\sx,0) to[out=180,in=-79] (\linksx,\linksy);
+\draw[->,line width=1.9pt,color=darkgreen]
+	(\sx,{\sy-\l}) to[out=-90,in=0] (\rechtsx,\rechtsy);
+
+\def\rect#1#2{
+	\fill[color=white] (-\l,-\l) rectangle (\l,\l);
+	#2
+	\draw (-\l,-\l) rectangle (\l,\l);
+	\node at (0,0) {\Huge #1\strut};
+}
+
+\def\kreuz{
+	\begin{scope}
+	\clip ({-\l},{-\l}) rectangle ({\l},{\l});
+	\fill[color=white] ({-2*\l},{-2*\l}) rectangle ({2*\l},{2*\l});
+	\draw[color=darkgreen!30,line width=3pt] (-\l,-\l) -- (\l,\l);
+	\draw[color=darkgreen!30,line width=3pt] (-\l,\l) -- (\l,-\l);
+	\end{scope}
+}
+
+\def\kreis{
+	\begin{scope}
+	\clip ({-\l},{-\l}) rectangle ({\l},{\l});
+	\fill[color=white] ({-2*\l},{-2*\l}) rectangle ({2*\l},{2*\l});
+	\draw[color=darkgreen!30,line width=3pt]
+		(0,0) circle[radius={\l*(\L-1.5)/\L}];
+	\end{scope}
+}
+
+\begin{scope}[xshift={0},yshift={0}]
+	\rect{s}{}
+\end{scope}
+
+\begin{scope}[xshift={\Sx},yshift={0}]
+	\rect{c}{\kreis}
+\end{scope}
+
+\begin{scope}[xshift={\Sx},yshift={\Sy}]
+	\rect{d}{\kreuz}
+\end{scope}
+
+\begin{scope}[xshift={0},yshift={\Sy}]
+	\rect{n}{}
+\end{scope}
+
+\node at ({-\l+0.1},{\sy+\l-0.1})    [above left]   {$iK'\mathstrut$};
+\node at ({-\l+0.1},{-\l+0.1})       [below left]   {$0\mathstrut$};
+\node at ({\sx+\l-0.1},{-\l+0.1})    [below right]  {$K\mathstrut$};
+\node at ({\sx+\l-0.1},{\sy+\l-0.1}) [above right]  {$K+iK'\mathstrut$};
+\node at ({-\sx},0)                  [below left]   {$-K\mathstrut$};
+\node at (0,{-\sy+0.05})             [below left]   {$-iK'\mathstrut$};
+\node at ({\sx-0.1},{-\sy+0.1})      [below right]  {$K-iK'\mathstrut$};
+\node at ({-\sx+0.1},{-\sy+0.1})     [below left]   {$-K-iK'\mathstrut$};
+\node at ({-\sx+0.1},{\sy-0.1})      [above left]   {$-K+iK'\mathstrut$};
+
+\begin{scope}[xshift={-\L+0.5*\Sx},yshift={0.5*\Sy}]
+	\node at ({-\l},{\l-0.1}) [above] {Nullstelle\strut};
+	\kreis
+	\node[color=darkgreen] at (0,0) {\Huge c\strut};
+	\draw[line width=0.2pt] (-\l,-\l) rectangle (\l,\l);
+\end{scope}
+
+\begin{scope}[xshift={\L+0.5*\Sx},yshift={0.5*\Sy}]
+	\node at ({\l},{\l-0.1}) [above] {Pol\strut};
+	\kreuz
+	\node[color=darkgreen] at (0,0) {\Huge d\strut};
+	\draw[line width=0.2pt] (-\l,-\l) rectangle (\l,\l);
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.pdf b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
index f0e6e78..eb9d7f1 100644
--- a/buch/chapters/110-elliptisch/images/jacobiplots.pdf
+++ b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.pdf b/buch/chapters/110-elliptisch/images/kegelpara.pdf
new file mode 100644
index 0000000..2bbd428
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/kegelpara.pdf
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.pov b/buch/chapters/110-elliptisch/images/kegelpara.pov
new file mode 100644
index 0000000..13b66cc
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/kegelpara.pov
@@ -0,0 +1,329 @@
+//
+// kegelpara.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+#declare O = <0,0,0>;
+
+global_settings {
+        assumed_gamma 1
+}
+
+#declare imagescale = 0.08;
+
+camera {
+        location <28, 20, -40>
+        look_at <0, 0.1, 0>
+        right x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <30, 10, -40> color White
+        area_light <1,0,0> <0,0,1>, 10, 10
+        adaptive 1
+        jitter
+}
+
+sky_sphere {
+        pigment {
+                color rgb<1,1,1>
+        }
+}
+
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+arrow(<-2.6,0,0>,<2.5,0,0>,0.02,White)
+arrow(<0,-2,0>,<0,2.3,0>,0.02,White)
+arrow(<0,0,-3.2>,<0,0,3.7>,0.02,White)
+
+#declare epsilon = 0.0001;
+#declare l = 1.5;
+
+#macro Kegel(farbe)
+union {
+	difference {
+		cone { O, 0, <l, 0, 0>, l }
+		cone { O + <epsilon, 0,0>, 0, <l+epsilon, 0, 0>, l }
+	}
+	difference {
+		cone { O, 0, <-l, 0, 0>, l }
+		cone { O + <-epsilon, 0, 0>, 0, <-l-epsilon, 0, 0>, l }
+	}
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Kegelpunkt(xx, phi)
+	< xx, xx * sin(phi), xx * cos(phi) >
+#end
+
+#macro Kegelgitter(farbe, r)
+union {
+	#declare s = 0;
+	#declare smax = 2 * pi;
+	#declare sstep = pi / 6;
+	#while (s < smax - sstep/2)
+		cylinder { Kegelpunkt(l, s), Kegelpunkt(-l, s), r }
+		#declare s = s + sstep;
+	#end
+	#declare phimax = 2 * pi;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#declare xxstep = 0.5;
+	#declare xxmax = 2;
+	#declare xx = xxstep;
+	#while (xx < xxmax - xxstep/2)
+		#declare phi = 0;
+		#while (phi < phimax - phistep/2)
+			cylinder {
+				Kegelpunkt(xx, phi),
+				Kegelpunkt(xx, phi + phistep),
+				r
+			}
+			sphere { Kegelpunkt(xx, phi), r }
+			cylinder {
+				Kegelpunkt(-xx, phi),
+				Kegelpunkt(-xx, phi + phistep),
+				r
+			}
+			sphere { Kegelpunkt(-xx, phi), r }
+			#declare phi = phi + phistep;
+		#end
+		#declare xx = xx + xxstep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro F(w, r)
+	<r * cos(w), r * r/sqrt(2), r * sin(w) >
+#end
+
+#macro Paraboloid(farbe)
+mesh {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phisteps = 100;
+	#declare phistep = pi / phisteps;
+	#declare rsteps = 100;
+	#declare rmax = 1.5;
+	#declare rstep = rmax / rsteps;
+	#while (phi < phimax - phistep/2)
+		#declare r = rstep;
+		#declare h = r * r / sqrt(2);
+		triangle {
+			O, F(phi, r), F(phi + phistep, r)
+		}
+		#while (r < rmax - rstep/2)
+			// ring
+			triangle {
+				F(phi, r),
+				F(phi + phistep, r),
+				F(phi + phistep, r + rstep)
+			}
+			triangle {
+				F(phi, r),
+				F(phi + phistep, r + rstep),
+				F(phi, r + rstep)
+			}
+			#declare r = r + rstep;
+		#end
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Paraboloidgitter(farbe, gr)
+union {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phistep = pi / 6;
+
+	#declare rmax = 1.5;
+	#declare rsteps = 100;
+	#declare rstep = rmax / rsteps;
+
+	#while (phi < phimax - phistep/2)
+		#declare r = rstep;
+		#while (r < rmax - rstep/2)
+			cylinder { F(phi, r), F(phi, r + rstep), gr }
+			sphere { F(phi, r), gr }
+			#declare r = r + rstep;
+		#end
+		#declare phi = phi + phistep;
+	#end
+
+	#declare rstep = 0.2;
+	#declare r = rstep;
+
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (r < rmax)
+		#declare phi = 0;
+		#while (phi < phimax - phistep/2)
+			cylinder { F(phi, r), F(phi + phistep, r), gr }
+			sphere { F(phi, r), gr }
+			#declare phi = phi + phistep;
+		#end
+		#declare r = r + rstep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare a = sqrt(2);
+#macro G(phi,sg)
+	< a*sg*sqrt(cos(2*phi))*cos(phi), a*cos(2*phi), a*sqrt(cos(2*phi))*sin(phi)>
+#end
+
+#macro Lemniskate3D(s, farbe)
+union {
+	#declare phi = -pi / 4;
+	#declare phimax = pi / 4;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		sphere { G(phi,1), s }
+		cylinder { G(phi,1), G(phi+phistep,1), s }
+		sphere { G(phi,-1), s }
+		cylinder { G(phi,-1), G(phi+phistep,-1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare a = sqrt(2);
+#macro G2(phi,sg)
+	a * sqrt(cos(2*phi)) * < sg * cos(phi), 0, sin(phi)>
+#end
+
+#macro Lemniskate(s, farbe)
+union {
+	#declare phi = -pi / 4;
+	#declare phimax = pi / 4;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		sphere { G2(phi,1), s }
+		cylinder { G2(phi,1), G2(phi+phistep,1), s }
+		sphere { G2(phi,-1), s }
+		cylinder { G2(phi,-1), G2(phi+phistep,-1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Projektion(s, farbe)
+union {
+	#declare phistep = pi / 16;
+	#declare phi = -pi / 4 + phistep;
+	#declare phimax = pi / 4;
+	#while (phi < phimax - phistep/2)
+		cylinder { G(phi,  1), G2(phi,  1), s }
+		cylinder { G(phi, -1), G2(phi, -1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare kegelfarbe = rgbf<0.2,0.6,0.2,0.2>;
+#declare kegelgitterfarbe = rgb<0.2,0.8,0.2>;
+#declare paraboloidfarbe = rgbf<0.2,0.6,1.0,0.2>;
+#declare paraboloidgitterfarbe = rgb<0.4,1,1>;
+
+//intersection {
+//	union {
+		Paraboloid(paraboloidfarbe)
+		Paraboloidgitter(paraboloidgitterfarbe, 0.004)
+
+		Kegel(kegelfarbe)
+		Kegelgitter(kegelgitterfarbe, 0.004)
+//	}
+//	plane { <0, 0, -1>, 0.6 }
+//}
+
+
+Lemniskate3D(0.02, rgb<0.8,0.0,0.8>)
+Lemniskate(0.02, Red)
+Projektion(0.01, Yellow)
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.tex b/buch/chapters/110-elliptisch/images/kegelpara.tex
new file mode 100644
index 0000000..8fcefbf
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/kegelpara.tex
@@ -0,0 +1,41 @@
+%
+% kegelpara.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{kegelpara.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\node at (4.1,-1.4) {$X$};
+\node at (0.2,3.8) {$Z$};
+\node at (4.0,1.8) {$Y$};
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/lemnispara.cpp b/buch/chapters/110-elliptisch/images/lemnispara.cpp
new file mode 100644
index 0000000..6f4d55d
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/lemnispara.cpp
@@ -0,0 +1,126 @@
+/*
+ * lemnispara.cpp -- Display parametrisation of the lemniskate
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <cstdio>
+#include <cstdlib>
+#include <cmath>
+#include <gsl/gsl_sf_elljac.h>
+#include <iostream>
+#include <fstream>
+#include <map>
+#include <string.h>
+#include <string>
+
+const static double	s = sqrt(2);
+const static double	k = 1 / s;
+const static double	m = k * k;
+
+typedef std::pair<double, double>	point_t;
+
+point_t	operator*(const point_t& p, double s) {
+	return point_t(s * p.first, s * p.second);
+}
+
+static double	norm(const point_t& p) {
+	return hypot(p.first, p.second);
+}
+
+static point_t	normalize(const point_t& p) {
+	return p * (1/norm(p));
+}
+
+static point_t	normal(const point_t& p) {
+	return std::make_pair(p.second, -p.first);
+}
+
+class lemniscate : public point_t {
+	double	sn, cn, dn;
+public:
+	lemniscate(double t) {
+		gsl_sf_elljac_e(t, m, &sn, &cn, &dn);
+		first = s * cn * dn;
+		second = cn * sn;
+	}
+	point_t	tangent() const {
+		return std::make_pair(-s * sn * (1.5 - sn * sn),
+				dn * (1 - 2 * sn * sn));
+	}
+	point_t	unittangent() const {
+		return normalize(tangent());
+	}
+	point_t	normal() const {
+		return ::normal(tangent());
+	}
+	point_t	unitnormal() const {
+		return ::normal(unittangent());
+	}
+};
+
+std::ostream&	operator<<(std::ostream& out, const point_t& p) {
+	char	b[1024];
+	snprintf(b, sizeof(b), "({%.4f*\\dx},{%.4f*\\dy})", p.first, p.second);
+	out << b;
+	return out;
+}
+
+int	main(int argc, char *argv[]) {
+	std::ofstream	out("lemnisparadata.tex");
+
+	// the curve
+	double	tstep = 0.01;
+	double tmax = 4.05;
+	out << "\\def\\lemnispath{ ";
+	out << lemniscate(0);
+	for (double t = tstep; t < tmax; t += tstep) {
+		out << std::endl << "\t" << "-- " << lemniscate(t);
+	}
+	out << std::endl;
+	out << "}" << std::endl;
+
+	out << "\\def\\lemnispathmore{ ";
+	out << lemniscate(tmax);
+	double	tmax2 = 7.5;
+	for (double t = tmax + tstep; t < tmax2; t += tstep) {
+		out << std::endl << "\t" << "-- " << lemniscate(t);
+	}
+	out << std::endl;
+	out << "}" << std::endl;
+
+	// individual points
+	tstep = 0.2;
+	int	i = 0;
+	char	name[3];
+	strcpy(name, "L0");
+	for (double t = 0; t <= tmax; t += tstep) {
+		char	c = 'A' + i++;
+		char	buffer[128];
+		lemniscate	l(t);
+		name[0] = 'L';
+		name[1] = c;
+		out << "\\coordinate (" << name << ") at ";
+		out << l << ";" << std::endl;
+		name[0] = 'T';
+		out << "\\coordinate (" << name << ") at ";
+		out << l.unittangent() << ";" << std::endl;
+		name[0] = 'N';
+		out << "\\coordinate (" << name << ") at ";
+		out << l.unitnormal() << ";" << std::endl;
+		name[0] = 'C';
+		out << "\\def\\" << name << "{ ";
+		out << "\\node[color=red] at ($(L" << c << ")+0.06*(N" << c << ")$) ";
+		out << "[rotate={";
+		double	w = 180 * atan2(l.unitnormal().second,
+					l.unitnormal().first) / M_PI;
+		snprintf(buffer, sizeof(buffer), "%.1f", w);
+		out << buffer;
+		out << "-90}]";
+		snprintf(buffer, sizeof(buffer), "%.1f", t);
+		out << " {$\\scriptstyle " << buffer << "$};" << std::endl;
+		out << "}" << std::endl;
+	}
+
+	out.close();
+	return EXIT_SUCCESS;
+}
diff --git a/buch/chapters/110-elliptisch/images/lemnispara.pdf b/buch/chapters/110-elliptisch/images/lemnispara.pdf
new file mode 100644
index 0000000..16731d3
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/lemnispara.pdf
diff --git a/buch/chapters/110-elliptisch/images/lemnispara.tex b/buch/chapters/110-elliptisch/images/lemnispara.tex
new file mode 100644
index 0000000..c6e32d7
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/lemnispara.tex
@@ -0,0 +1,94 @@
+%
+% lemnispara.tex -- parametrization of the lemniscate
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+\def\skala{1}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+\def\dx{4}
+\def\dy{4}
+\input{lemnisparadata.tex}
+
+% add image content here
+\draw[color=red!20,line width=1.4pt] \lemnispathmore;
+\draw[color=red,line width=1.4pt] \lemnispath;
+
+\draw[->] ({-1.6*\dx},0) -- ({1.8*\dx},0) coordinate[label={$X$}];
+\draw[->] (0,{-0.7*\dy}) -- (0,{0.7*\dy}) coordinate[label={right:$Y$}];
+
+\draw ({1.5*\dx},-0.05) -- ({1.5*\dx},0.05);
+\draw ({\dx},-0.05) -- ({\dx},0.05);
+\draw ({0.5*\dx},-0.05) -- ({0.5*\dx},0.05);
+\draw ({-0.5*\dx},-0.05) -- ({-0.5*\dx},0.05);
+\draw ({-\dx},-0.05) -- ({-\dx},0.05);
+\draw ({-1.5*\dx},-0.05) -- ({-1.5*\dx},0.05);
+\draw (-0.05,{0.5*\dy}) -- (0.05,{0.5*\dy});
+\draw (-0.05,{-0.5*\dy}) -- (0.05,{-0.5*\dy});
+
+\node at ({\dx},0) [above] {$1$};
+\node at ({-\dx},0) [above] {$-1$};
+\node at ({-0.5*\dx},0) [above] {$-\frac12$};
+\node at ({0.5*\dx},0) [above] {$\frac12$};
+\node at (0,{0.5*\dy}) [left] {$\frac12$};
+\node at (0,{-0.5*\dy}) [left] {$-\frac12$};
+
+\def\s{0.02}
+
+\draw[color=red] ($(LA)-\s*(NA)$) -- ($(LA)+\s*(NA)$);
+\draw[color=red] ($(LB)-\s*(NB)$) -- ($(LB)+\s*(NB)$);
+\draw[color=red] ($(LC)-\s*(NC)$) -- ($(LC)+\s*(NC)$);
+\draw[color=red] ($(LD)-\s*(ND)$) -- ($(LD)+\s*(ND)$);
+\draw[color=red] ($(LE)-\s*(NE)$) -- ($(LE)+\s*(NE)$);
+\draw[color=red] ($(LF)-\s*(NF)$) -- ($(LF)+\s*(NF)$);
+\draw[color=red] ($(LG)-\s*(NG)$) -- ($(LG)+\s*(NG)$);
+\draw[color=red] ($(LH)-\s*(NH)$) -- ($(LH)+\s*(NH)$);
+\draw[color=red] ($(LI)-\s*(NI)$) -- ($(LI)+\s*(NI)$);
+\draw[color=red] ($(LJ)-\s*(NJ)$) -- ($(LJ)+\s*(NJ)$);
+\draw[color=red] ($(LK)-\s*(NK)$) -- ($(LK)+\s*(NK)$);
+\draw[color=red] ($(LL)-\s*(NL)$) -- ($(LL)+\s*(NL)$);
+\draw[color=red] ($(LM)-\s*(NM)$) -- ($(LM)+\s*(NM)$);
+\draw[color=red] ($(LN)-\s*(NN)$) -- ($(LN)+\s*(NN)$);
+\draw[color=red] ($(LO)-\s*(NO)$) -- ($(LO)+\s*(NO)$);
+\draw[color=red] ($(LP)-\s*(NP)$) -- ($(LP)+\s*(NP)$);
+\draw[color=red] ($(LQ)-\s*(NQ)$) -- ($(LQ)+\s*(NQ)$);
+\draw[color=red] ($(LR)-\s*(NR)$) -- ($(LR)+\s*(NR)$);
+\draw[color=red] ($(LS)-\s*(NS)$) -- ($(LS)+\s*(NS)$);
+\draw[color=red] ($(LT)-\s*(NT)$) -- ($(LT)+\s*(NT)$);
+\draw[color=red] ($(LU)-\s*(NU)$) -- ($(LU)+\s*(NU)$);
+
+\CB
+\CC
+\CD
+\CE
+\CF
+\CG
+\CH
+\CI
+\CJ
+\CK
+\CL
+\CM
+\CN
+\CO
+\CP
+\CQ
+\CR
+\CS
+\CT
+\CU
+
+\fill[color=blue] (LA) circle[radius=0.07];
+\node[color=blue] at (LA) [above right] {$S$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/rechteck.cpp b/buch/chapters/110-elliptisch/images/rechteck.cpp
index c65ae0f..b5ad0ec 100644
--- a/buch/chapters/110-elliptisch/images/rechteck.cpp
+++ b/buch/chapters/110-elliptisch/images/rechteck.cpp
@@ -163,7 +163,7 @@ curvetracer::curve_t	curvetracer::trace(const std::complex<double>& startz,
 		} catch (const toomanyiterations& x) {
 			std::cerr << "iterations exceeded after ";
 			std::cerr << result.size();
-			std::cerr << " points";
+			std::cerr << " points" << std::endl;
 			maxsteps = 0;
 		}
 	}
@@ -230,7 +230,7 @@ void	curvedrawer::operator()(const curvetracer::curve_t& curve) {
 	double	first = true;
 	for (auto z : curve) {
 		if (first) {
-			*_out << "\\draw[color=" << _color << "] ";
+			*_out << "\\draw[color=" << _color << ",line width=#1] ";
 			first = false;
 		} else {
 			*_out << std::endl << "    -- ";
@@ -244,6 +244,7 @@ static struct option	longopts[] = {
 { "outfile",	required_argument,	NULL,	'o' },
 { "k",		required_argument,	NULL,	'k' },
 { "deltax",	required_argument,	NULL,	'd' },
+{ "vsteps",	required_argument,	NULL,	'v' },
 { NULL,		0,			NULL,	 0  }
 };
 
@@ -252,7 +253,8 @@ static struct option	longopts[] = {
  */
 int	main(int argc, char *argv[]) {
 	double	k = 0.625;
-	double	deltax = 0.2;
+	double	Deltax = 0.2;
+	int	vsteps = 4;
 
 	int	c;
 	int	longindex;
@@ -261,7 +263,7 @@ int	main(int argc, char *argv[]) {
 		&longindex)))
 		switch (c) {
 		case 'd':
-			deltax = std::stod(optarg);
+			Deltax = std::stod(optarg);
 			break;
 		case 'o':
 			outfilename = std::string(optarg);
@@ -269,6 +271,9 @@ int	main(int argc, char *argv[]) {
 		case 'k':
 			k = std::stod(optarg);
 			break;
+		case 'v':
+			vsteps = std::stoi(optarg);
+			break;
 		}
 
 	double	kprime = integrand::kprime(k);
@@ -293,15 +298,21 @@ int	main(int argc, char *argv[]) {
 	curvetracer	ct(f);
 
 	// fill
+	(*cdp->out()) << "\\def\\hintergrund{" << std::endl;
 	(*cdp->out()) << "\\fill[color=red!10] ({" << (-xmax) << "*\\dx},0) "
 		<< "rectangle ({" << xmax << "*\\dx},{" << ymax << "*\\dy});"
 		<< std::endl;
 	(*cdp->out()) << "\\fill[color=blue!10] ({" << (-xmax) << "*\\dx},{"
 		<< (-ymax) << "*\\dy}) rectangle ({" << xmax << "*\\dx},0);"
 		<< std::endl;
+	(*cdp->out()) << "}" << std::endl;
+
+	// macro for grid
+	(*cdp->out()) << "\\def\\netz#1{" << std::endl;
 
 	// "circles"
 	std::complex<double>	dir(0.01, 0);
+	double	deltax = Deltax;
 	for (double im = deltax; im < 3; im += deltax) {
 		std::complex<double>	startz(0, im);
 		std::complex<double>	startw = ct.startpoint(startz);
@@ -316,9 +327,9 @@ int	main(int argc, char *argv[]) {
 	}
 
 	// imaginary axis
-	(*cdp->out()) << "\\draw[color=red] (0,0) -- (0,{" << ymax
+	(*cdp->out()) << "\\draw[color=red,line width=#1] (0,0) -- (0,{" << ymax
 		<< "*\\dy});" << std::endl;
-	(*cdp->out()) << "\\draw[color=blue] (0,0) -- (0,{" << (-ymax)
+	(*cdp->out()) << "\\draw[color=blue,line width=#1] (0,0) -- (0,{" << (-ymax)
 		<< "*\\dy});" << std::endl;
 
 	// arguments between 0 and 1
@@ -353,7 +364,8 @@ int	main(int argc, char *argv[]) {
 
 	// arguments between 1 and 1/k
 	{
-		for (double x0 = 1 + deltax; x0 < 1/k; x0 += deltax) {
+		deltax = (1/k - 1) / vsteps;
+		for (double x0 = 1 + deltax; x0 < 1/k + 0.00001; x0 += deltax) {
 			double	y0 = sqrt(1-1/(x0*x0))/kprime;
 			//std::cout << "y0 = " << y0 << std::endl;
 			double	y = gsl_sf_ellint_F(asin(y0), kprime,
@@ -389,8 +401,9 @@ int	main(int argc, char *argv[]) {
 
 	// arguments larger than 1/k
 	{
+		deltax = Deltax;
 		dir = std::complex<double>(0, 0.01);
-		double	x0 = 1;
+		double	x0 = 1/k;
 		while (x0 <= 1/k + 0.0001) { x0 += deltax; }
 		for (; x0 < 4; x0 += deltax) {
 			std::complex<double>	startz(x0);
@@ -407,6 +420,8 @@ int	main(int argc, char *argv[]) {
 		}
 	}
 
+	(*cdp->out()) << "}" << std::endl;
+
 	// border
 	(*cdp->out()) << "\\def\\xmax{" << xmax << "}" << std::endl;
 	(*cdp->out()) << "\\def\\ymax{" << ymax << "}" << std::endl;
diff --git a/buch/chapters/110-elliptisch/images/rechteck.pdf b/buch/chapters/110-elliptisch/images/rechteck.pdf
index 6209897..46f2376 100644
--- a/buch/chapters/110-elliptisch/images/rechteck.pdf
+++ b/buch/chapters/110-elliptisch/images/rechteck.pdf
diff --git a/buch/chapters/110-elliptisch/images/rechteck.tex b/buch/chapters/110-elliptisch/images/rechteck.tex
index 622a9e9..12535ba 100644
--- a/buch/chapters/110-elliptisch/images/rechteck.tex
+++ b/buch/chapters/110-elliptisch/images/rechteck.tex
@@ -18,6 +18,8 @@
 \def\dy{3}
 
 \input{rechteckpfade.tex}
+\hintergrund
+\netz{0.7pt}
 
 \begin{scope}
 	\clip ({-\xmax*\dx},{-\ymax*\dy}) rectangle ({\xmax*\dx},{\ymax*\dy});
diff --git a/buch/chapters/110-elliptisch/images/slcl.pdf b/buch/chapters/110-elliptisch/images/slcl.pdf
index c15051b..71645e3 100644
--- a/buch/chapters/110-elliptisch/images/slcl.pdf
+++ b/buch/chapters/110-elliptisch/images/slcl.pdf
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.pdf b/buch/chapters/110-elliptisch/images/torusschnitt.pdf
new file mode 100644
index 0000000..9b64ab2
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.pdf
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.pov b/buch/chapters/110-elliptisch/images/torusschnitt.pov
new file mode 100644
index 0000000..e5602df
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.pov
@@ -0,0 +1,308 @@
+//
+// kegelpara.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+#declare O = <0,0,0>;
+
+global_settings {
+        assumed_gamma 1
+}
+
+#declare imagescale = 0.060;
+
+camera {
+        location <28, 20, -40>
+        look_at <0, 0.55, 0>
+        right (16/9) * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <30, 10, -40> color White
+        area_light <1,0,0> <0,0,1>, 10, 10
+        adaptive 1
+        jitter
+}
+
+sky_sphere {
+        pigment {
+                color rgb<1,1,1>
+        }
+}
+
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+
+#macro Ticks(tl, tr) 
+union {
+	#declare s = 1;
+	#while (s <= 3.1)
+		cylinder { <-0.5*s-tl, 0, 0>, <-0.5*s+tl, 0, 0>, tr }
+		cylinder { < 0.5*s-tl, 0, 0>, < 0.5*s+tl, 0, 0>, tr }
+		#declare s = s + 1;
+	#end
+
+	#declare s = 1;
+	#while (s <= 4.1)
+		cylinder { <0, 0.5*s-tl, 0>, <0, 0.5*s+tl, 0>, tr }
+		#declare s = s + 1;
+	#end
+	#declare s = 1;
+	#while (s <= 2.1)
+		cylinder { <0,-0.5*s-tl, 0>, <0,-0.5*s+tl, 0>, tr }
+		#declare s = s + 1;
+	#end
+
+	#declare s = 1;
+	#while (s <= 4)
+		cylinder { <0, 0,  0.5*s-tl>, <0, 0,  0.5*s+tl>, tr }
+		#declare s = s + 1;
+	#end
+	#declare s = 1;
+	#while (s <= 3)
+		cylinder { <0, 0, -0.5*s-tl>, <0, 0, -0.5*s+tl>, tr }
+		#declare s = s + 1;
+	#end
+
+	pigment {
+		color White
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare epsilon = 0.001;
+#declare l = 1.5;
+
+#declare a = sqrt(2);
+#macro G2(phi,sg)
+	a * sqrt(cos(2*phi)) * < sg * cos(phi), 0, sin(phi)>
+#end
+
+#macro Lemniskate(s, farbe)
+union {
+	#declare phi = -pi / 4;
+	#declare phimax = pi / 4;
+	#declare phisteps = 100;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		sphere { G2(phi,1), s }
+		cylinder { G2(phi,1), G2(phi+phistep,1), s }
+		sphere { G2(phi,-1), s }
+		cylinder { G2(phi,-1), G2(phi+phistep,-1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Projektion(s, farbe)
+union {
+	#declare phistep = pi / 16;
+	#declare phi = -pi / 4 + phistep;
+	#declare phimax = pi / 4;
+	#while (phi < phimax - phistep/2)
+		cylinder { G(phi,  1), G2(phi,  1), s }
+		cylinder { G(phi, -1), G2(phi, -1), s }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Ebene(l, b, farbe)
+mesh {
+	triangle { <-l, 0, -b>, < l, 0, -b>, < l, 0,  b> }
+	triangle { <-l, 0, -b>, < l, 0,  b>, <-l, 0,  b> }
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Ebenengitter(l, b, s, r, farbe)
+union {
+	#declare lmax = floor(l / s);
+	#declare ll = -lmax;
+	#while (ll <= lmax)
+		cylinder { <ll * s, 0, -b>, <ll * s, 0, b>, r }
+		#declare ll = ll + 1;
+	#end
+	#declare bmax = floor(b / s);
+	#declare bb = -bmax;
+	#while (bb <= bmax)
+		cylinder { <-l, 0, bb * s>, <l, 0, bb * s>, r }
+		#declare bb = bb + 1;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare b = 0.5;
+#macro T(phi, theta)
+	b * < (2 + cos(theta)) * cos(phi), (2 + cos(theta)) * sin(phi) + 1, sin(theta) >
+#end
+
+#macro breitenkreis(theta, r)
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phisteps = 200;
+	#declare phistep = phimax / phisteps;
+	#while (phi < phimax - phistep/2)
+		cylinder { T(phi, theta), T(phi + phistep, theta), r }
+		sphere { T(phi, theta), r }
+		#declare phi = phi + phistep;
+	#end
+#end
+
+#macro laengenkreis(phi, r)
+	#declare theta = 0;
+	#declare thetamax = 2 * pi;
+	#declare thetasteps = 200;
+	#declare thetastep = thetamax / thetasteps;
+	#while (theta < thetamax - thetastep/2)
+		cylinder { T(phi, theta), T(phi, theta + thetastep), r }
+		sphere { T(phi, theta), r }
+		#declare theta = theta + thetastep;
+	#end
+#end
+
+#macro Torusgitter(farbe, r)
+union {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phistep = pi / 6;
+	#while (phi < phimax - phistep/2)
+		laengenkreis(phi, r)
+		#declare phi = phi + phistep;
+	#end
+	#declare thetamax = pi;
+	#declare thetastep = pi / 6;
+	#declare theta = thetastep;
+	#while (theta < thetamax - thetastep/2)
+		breitenkreis(theta, r)
+		breitenkreis(thetamax + theta, r)
+		#declare theta = theta + thetastep;
+	#end
+	breitenkreis(0, 1.5 * r)
+	breitenkreis(pi, 1.5 * r)
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro Torus(farbe)
+mesh {
+	#declare phi = 0;
+	#declare phimax = 2 * pi;
+	#declare phisteps = 200;
+	#declare phistep = phimax/phisteps;
+	#while (phi < phimax - phistep/2)
+		#declare theta = 0;
+		#declare thetamax = 2 * pi;
+		#declare thetasteps = 200;
+		#declare thetastep = thetamax / thetasteps;
+		#while (theta < thetamax - thetastep/2)
+			triangle {
+				T(phi,           theta),
+				T(phi + phistep, theta),
+				T(phi + phistep, theta + thetastep)
+			}
+			triangle {
+				T(phi,           theta),
+				T(phi + phistep, theta + thetastep),
+				T(phi,           theta + thetastep)
+			}
+			#declare theta = theta + thetastep;
+		#end
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare torusfarbe = rgbt<0.2,0.6,0.2,0.2>;
+#declare ebenenfarbe = rgbt<0.2,0.6,1.0,0.2>;
+
+arrow(<-2,0,0>,<2,0,0>,0.02,White)
+arrow(<0,-1.1,0>,<0,2.2,0>,0.02,White)
+arrow(<0,0,-1.7>,<0,0,2.4>,0.02,White)
+Ticks(0.007,0.036)
+
+Lemniskate(0.02, Red)
+Ebene(1.8, 1.6, ebenenfarbe)
+Ebenengitter(1.8, 1.6, 0.5, 0.005, rgb<0.4,1,1>)
+Torus(torusfarbe)
+Torusgitter(Yellow, 0.005)
+
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.tex b/buch/chapters/110-elliptisch/images/torusschnitt.tex
new file mode 100644
index 0000000..63351ad
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.tex
@@ -0,0 +1,41 @@
+%
+% torusschnitt.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{6}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=11.98cm]{torusschnitt.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\node at (4.4,-2.4) {$X$};
+\node at (3.5,0.6) {$Y$};
+\node at (0.3,3.8) {$Z$};
+
+\end{tikzpicture}
+
+\end{document}
+