diff options
Diffstat (limited to 'buch/chapters/110-elliptisch/images')
27 files changed, 1701 insertions, 12 deletions
diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile index 68322b6..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 + sncnlimit.pdf slcl.pdf torusschnitt.pdf kegelpara.pdf lemnispara.pdf \ + ellpolnul.pdf ellall.pdf ellselection.pdf lemniskate.pdf: lemniskate.tex pdflatex lemniskate.tex @@ -71,3 +72,59 @@ jacobi12.pdf: jacobi12.tex sncnlimit.pdf: sncnlimit.tex pdflatex sncnlimit.tex +slcl: slcl.cpp + g++ -O -Wall -std=c++11 slcl.cpp -o slcl `pkg-config --cflags gsl` `pkg-config --libs gsl` + +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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differindex d11bde8..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/jacobiplots.tex b/buch/chapters/110-elliptisch/images/jacobiplots.tex index 4fc572e..fec04fc 100644 --- a/buch/chapters/110-elliptisch/images/jacobiplots.tex +++ b/buch/chapters/110-elliptisch/images/jacobiplots.tex @@ -31,7 +31,7 @@ \fill[color=gray!50] (-0.2,1.65) rectangle (7.0,2.3); \draw[line width=0.5pt] (-0.2,-6) rectangle (7.0,2.3); \begin{scope}[scale=0.5] -\node at (6.5,{\dy+2}) {$m = #1$}; +\node at (6.5,{\dy+2}) {$k^2 = #1$}; \end{scope} } \def\jacobiplot#1#2#3#4{ diff --git a/buch/chapters/110-elliptisch/images/kegelpara.pdf b/buch/chapters/110-elliptisch/images/kegelpara.pdf Binary files differnew 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/lemniskate.pdf b/buch/chapters/110-elliptisch/images/lemniskate.pdf Binary files differindex 063a3e1..9e02c3c 100644 --- a/buch/chapters/110-elliptisch/images/lemniskate.pdf +++ b/buch/chapters/110-elliptisch/images/lemniskate.pdf diff --git a/buch/chapters/110-elliptisch/images/lemniskate.tex b/buch/chapters/110-elliptisch/images/lemniskate.tex index f74a81f..fe90631 100644 --- a/buch/chapters/110-elliptisch/images/lemniskate.tex +++ b/buch/chapters/110-elliptisch/images/lemniskate.tex @@ -27,13 +27,16 @@ \draw[color=red,line width=2.0pt] plot[domain=45:\a,samples=100] ({\x}:{sqrt(2*cos(2*\x))}); -\draw[->] (-1.5,0) -- (1.5,0) coordinate[label={$x$}]; -\draw[->] (0,-0.7) -- (0,0.7) coordinate[label={right:$y$}]; +\draw[->] (-1.5,0) -- (1.7,0) coordinate[label={$X$}]; +\draw[->] (0,-0.7) -- (0,0.7) coordinate[label={right:$Y$}]; \fill[color=white] (1,0) circle[radius=0.02]; \draw (1,0) circle[radius=0.02]; +\node at ({1},0) [below] {$\displaystyle a\mathstrut$}; + \fill[color=white] (-1,0) circle[radius=0.02]; \draw (-1,0) circle[radius=0.02]; +\node at ({-1},0) [below] {$\displaystyle\llap{$-$}a\mathstrut$}; \node[color=blue] at (\a:{0.6*sqrt(2*cos(2*\a))}) [below] {$r$}; \node[color=red] at ({\b}:{sqrt(2*cos(2*\b))}) [above] {$s$}; @@ -41,6 +44,14 @@ \fill[color=white] (\a:{sqrt(2*cos(2*\a))}) circle[radius=0.02]; \draw[color=red] (\a:{sqrt(2*cos(2*\a))}) circle[radius=0.02]; +\draw ({sqrt(2)},{-0.1/\skala}) -- ({sqrt(2)},{0.1/\skala}); +\node at ({sqrt(2)},0) [below right] + {$\displaystyle a\mathstrut\sqrt{2}$}; +\draw ({-sqrt(2)},{-0.1/\skala}) -- ({-sqrt(2)},{0.1/\skala}); +\node at ({-sqrt(2)},0) [below left] + {$\displaystyle -a\mathstrut\sqrt{2}$}; + + \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 Binary files differnew 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 Binary files differindex 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.cpp b/buch/chapters/110-elliptisch/images/slcl.cpp new file mode 100644 index 0000000..8584e94 --- /dev/null +++ b/buch/chapters/110-elliptisch/images/slcl.cpp @@ -0,0 +1,128 @@ +/* + * slcl.cpp + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +#include <cstdlib> +#include <cstdio> +#include <cmath> +#include <iostream> +#include <fstream> +#include <sstream> +#include <getopt.h> +#include <vector> +#include <gsl/gsl_sf_elljac.h> + +namespace slcl { + +static struct option longopts[] { +{ "outfile", required_argument, NULL, 'o' }, +{ "a", required_argument, NULL, 'a' }, +{ "b", required_argument, NULL, 'b' }, +{ "steps", required_argument, NULL, 'n' }, +{ NULL, 0, NULL, 0 } +}; + +class plot { + typedef std::pair<double, double> point_t; + typedef std::vector<point_t> curve_t; + curve_t _sl; + curve_t _cl; + double _a; + double _b; + int _steps; +public: + double a() const { return _a; } + double b() const { return _b; } + int steps() const { return _steps; } +public: + plot(double a, double b, int steps) : _a(a), _b(b), _steps(steps) { + double l = sqrt(2); + double k = 1 / l; + double m = k * k; + double h = (b - a) / steps; + for (int i = 0; i <= steps; i++) { + double x = a + h * i; + double sn, cn, dn; + gsl_sf_elljac_e(x, m, &sn, &cn, &dn); + _sl.push_back(std::make_pair(l * x, k * sn / dn)); + _cl.push_back(std::make_pair(l * x, cn)); + } + } +private: + std::string point(const point_t p) const { + char buffer[128]; + snprintf(buffer, sizeof(buffer), "({%.4f*\\dx},{%.4f*\\dy})", + p.first, p.second); + return std::string(buffer); + } + std::string path(const curve_t& curve) const { + std::ostringstream out; + auto i = curve.begin(); + out << point(*(i++)); + do { + out << std::endl << " -- " << point(*(i++)); + } while (i != curve.end()); + out.flush(); + return out.str(); + } +public: + std::string slpath() const { + return path(_sl); + } + std::string clpath() const { + return path(_cl); + } +}; + +/** + * \brief Main function for the slcl program + */ +int main(int argc, char *argv[]) { + int longindex; + int c; + double a = 0; + double b = 10; + int steps = 100; + std::ostream *out = &std::cout; + while (EOF != (c = getopt_long(argc, argv, "a:b:o:n:", + longopts, &longindex))) + switch (c) { + case 'a': + a = std::stod(optarg); + break; + case 'b': + b = std::stod(optarg) / sqrt(2); + break; + case 'n': + steps = std::stol(optarg); + break; + case 'o': + out = new std::ofstream(optarg); + break; + } + + plot p(a, b, steps); + (*out) << "\\def\\slpath{ " << p.slpath(); + (*out) << std::endl << " }" << std::endl; + (*out) << "\\def\\clpath{ " << p.clpath(); + (*out) << std::endl << " }" << std::endl; + + out->flush(); + //out->close(); + return EXIT_SUCCESS; +} + +} // namespace slcl + +int main(int argc, char *argv[]) { + try { + return slcl::main(argc, argv); + } catch (const std::exception& e) { + std::cerr << "terminated by exception: " << e.what(); + std::cerr << std::endl; + } catch (...) { + std::cerr << "terminated by unknown exception" << std::endl; + } + return EXIT_FAILURE; +} diff --git a/buch/chapters/110-elliptisch/images/slcl.pdf b/buch/chapters/110-elliptisch/images/slcl.pdf Binary files differnew file mode 100644 index 0000000..71645e3 --- /dev/null +++ b/buch/chapters/110-elliptisch/images/slcl.pdf diff --git a/buch/chapters/110-elliptisch/images/slcl.tex b/buch/chapters/110-elliptisch/images/slcl.tex new file mode 100644 index 0000000..0af1027 --- /dev/null +++ b/buch/chapters/110-elliptisch/images/slcl.tex @@ -0,0 +1,88 @@ +% +% tikztemplate.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} +\begin{document} +\input{slcldata.tex} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +% add image content here +\def\lemniscateconstant{2.6220575542} +\pgfmathparse{(3.1415926535/2)/\lemniscateconstant} +\xdef\scalechange{\pgfmathresult} + +\pgfmathparse{\scalechange*(180/3.1415926535)} +\xdef\ts{\pgfmathresult} + +\def\dx{1} +\def\dy{3} + +\draw[line width=0.3pt] + ({\lemniscateconstant*\dx},0) + -- + ({\lemniscateconstant*\dx},{1*\dy}); +\draw[line width=0.3pt] + ({2*\lemniscateconstant*\dx},0) + -- + ({2*\lemniscateconstant*\dx},{-1*\dy}); +\draw[line width=0.3pt] + ({3*\lemniscateconstant*\dx},0) + -- + ({3*\lemniscateconstant*\dx},{-1*\dy}); +\draw[line width=0.3pt] + ({4*\lemniscateconstant*\dx},0) + -- + ({4*\lemniscateconstant*\dx},{1*\dy}); +\draw[line width=0.3pt] + ({5*\lemniscateconstant*\dx},0) + -- + ({5*\lemniscateconstant*\dx},{1*\dy}); + +\draw[color=red!40,line width=1.4pt] + plot[domain=0:13,samples=200] ({\x},{\dy*sin(\ts*\x)}); +\draw[color=blue!40,line width=1.4pt] + plot[domain=0:13,samples=200] ({\x},{\dy*cos(\ts*\x)}); + +\draw[color=red,line width=1.4pt] \slpath; +\draw[color=blue,line width=1.4pt] \clpath; + +\draw[->] (0,{-1*\dy-0.1}) -- (0,{1*\dy+0.4}) coordinate[label={right:$r$}]; +\draw[->] (-0.1,0) -- (13.6,0) coordinate[label={$s$}]; + +\foreach \i in {1,2,3,4,5}{ + \draw ({\lemniscateconstant*\i},-0.1) -- ({\lemniscateconstant*\i},0.1); +} +\node at ({\lemniscateconstant*\dx},0) [below left] {$\frac{\varpi}2\mathstrut$}; +\node at ({2*\lemniscateconstant*\dx},0) [below left] {$\varpi\mathstrut$}; +\node at ({3*\lemniscateconstant*\dx},0) [below right] {$\frac{3\varpi}2\mathstrut$}; +\node at ({4*\lemniscateconstant*\dx},0) [below right] {$2\varpi\mathstrut$}; +\node at ({5*\lemniscateconstant*\dx},0) [below left] {$\frac{5\varpi}2\mathstrut$}; + +\node[color=red] at ({1.6*\lemniscateconstant*\dx},{0.6*\dy}) + [below left] {$\operatorname{sl}(s)$}; +\node[color=red!50] at ({1.5*\lemniscateconstant*\dx},{sin(1.5*90)*\dy*0.90}) + [above right] {$\sin \bigl(\frac{\pi}{\varpi}s\bigr)$}; + +\node[color=blue] at ({1.4*\lemniscateconstant*\dx},{-0.6*\dy}) + [above right] {$\operatorname{cl}(s)$}; +\node[color=blue!50] at ({1.5*\lemniscateconstant*\dx},{cos(1.5*90)*\dy*0.90}) + [below left] {$\cos\bigl(\frac{\pi}{\varpi}s\bigr)$}; + +\draw (-0.1,{1*\dy}) -- (0.1,{1*\dy}); +\draw (-0.1,{-1*\dy}) -- (0.1,{-1*\dy}); +\node at (0,{1*\dy}) [left] {$1\mathstrut$}; +\node at (0,0) [left] {$0\mathstrut$}; +\node at (0,{-1*\dy}) [left] {$-1\mathstrut$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.pdf b/buch/chapters/110-elliptisch/images/torusschnitt.pdf Binary files differnew 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} + |