From ca7aa88b2a8edd58d8d8e7dedb85d9f441b701e4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 18 Feb 2022 17:28:23 +0100 Subject: new stuff about elliptic functions --- buch/chapters/110-elliptisch/images/Makefile | 8 +- buch/chapters/110-elliptisch/images/jacobi12.pdf | Bin 0 -> 22051 bytes buch/chapters/110-elliptisch/images/jacobi12.tex | 210 +++++++++++++++++++++ buch/chapters/110-elliptisch/images/jacobidef.pdf | Bin 0 -> 23654 bytes buch/chapters/110-elliptisch/images/jacobidef.tex | 100 ++++++++++ .../chapters/110-elliptisch/images/jacobiplots.pdf | Bin 57192 -> 57192 bytes 6 files changed, 317 insertions(+), 1 deletion(-) create mode 100644 buch/chapters/110-elliptisch/images/jacobi12.pdf create mode 100644 buch/chapters/110-elliptisch/images/jacobi12.tex create mode 100644 buch/chapters/110-elliptisch/images/jacobidef.pdf create mode 100644 buch/chapters/110-elliptisch/images/jacobidef.tex (limited to 'buch/chapters/110-elliptisch/images') diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile index f3f783f..d1e0afe 100644 --- a/buch/chapters/110-elliptisch/images/Makefile +++ b/buch/chapters/110-elliptisch/images/Makefile @@ -4,7 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # all: lemniskate.pdf ellipsenumfang.pdf unvollstaendig.pdf rechteck.pdf \ - ellipse.pdf pendel.pdf jacobiplots.pdf + ellipse.pdf pendel.pdf jacobiplots.pdf jacobidef.pdf jacobi12.pdf lemniskate.pdf: lemniskate.tex pdflatex lemniskate.tex @@ -60,3 +60,9 @@ jacobipaths.tex: jacobi Makefile jacobiplots.pdf: jacobiplots.tex jacobipaths.tex pdflatex jacobiplots.tex + +jacobidef.pdf: jacobidef.tex + pdflatex jacobidef.tex + +jacobi12.pdf: jacobi12.tex + pdflatex jacobi12.tex diff --git a/buch/chapters/110-elliptisch/images/jacobi12.pdf b/buch/chapters/110-elliptisch/images/jacobi12.pdf new file mode 100644 index 0000000..46f9883 Binary files /dev/null and b/buch/chapters/110-elliptisch/images/jacobi12.pdf differ diff --git a/buch/chapters/110-elliptisch/images/jacobi12.tex b/buch/chapters/110-elliptisch/images/jacobi12.tex new file mode 100644 index 0000000..aec9f04 --- /dev/null +++ b/buch/chapters/110-elliptisch/images/jacobi12.tex @@ -0,0 +1,210 @@ +% +% 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} +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\definecolor{zaehlerblau}{rgb}{0,0.8,0.8} +\definecolor{zaehlergruen}{rgb}{0.0,0.4,0.2} +\definecolor{zaehlerrot}{rgb}{1.0,0.0,0.0} +\definecolor{zaehlerschwarz}{rgb}{1.0,0.6,0.2} + +\definecolor{nennerblau}{rgb}{0.2,0.4,1} +\definecolor{nennergruen}{rgb}{0,0.6,0} +\definecolor{nennerrot}{rgb}{0.8,0,0.4} +\definecolor{nennerschwarz}{rgb}{1,0.8,0} + + +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} +\def\skala{10} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\a{1.6} +\def\w{41} +\pgfmathparse{atan(sin(\w)/(\a*cos(\w)))} +\xdef\v{\pgfmathresult} +\pgfmathparse{(\w+\v)/2+2} +\xdef\u{\pgfmathresult} + +\pgfmathparse{cos(\w)} +\xdef\cn{\pgfmathresult} +\pgfmathparse{sin(\w)} +\xdef\sn{\pgfmathresult} +\pgfmathparse{sqrt(\a*\a*cos(\w)*cos(\w)+sin(\w)*sin(\w))} +\xdef\r{\pgfmathresult} +\pgfmathparse{\r/\a} +\xdef\dn{\pgfmathresult} + +\pgfmathparse{1/\sn} +\xdef\ns{\pgfmathresult} +\pgfmathparse{1/\cn} +\xdef\nc{\pgfmathresult} +\pgfmathparse{1/\dn} +\xdef\nd{\pgfmathresult} + +\pgfmathparse{\sn/\cn} +\xdef\sc{\pgfmathresult} +\pgfmathparse{\cn/\sn} +\xdef\cs{\pgfmathresult} + +\pgfmathparse{\sn/\dn} +\xdef\sd{\pgfmathresult} +\pgfmathparse{\dn/\sn} +\xdef\ds{\pgfmathresult} + +\pgfmathparse{\cn/\dn} +\xdef\cd{\pgfmathresult} +\pgfmathparse{\dn/\cn} +\xdef\dc{\pgfmathresult} + + +\def\zaehler#1#2#3#4#5#6{ +\begin{scope}[rotate=#3] + \fill[color=#4!50,opacity=0.5] + ({#1-2/\skala},{#6-0.2/\skala}) + rectangle + ({#2+5.5/\skala},{#6+0.6/\skala}); + \node[color=#4] at ({#2+1/\skala},{#6+0.2/\skala}) + [right,rotate=#3] {#5}; +\end{scope} +} + +\begin{scope} +\clip ({\cn-1.5/\skala},{-0.7/\skala}) + rectangle + ({\cs+3.4/\skala},{(\cs+3.7/\skala)*tan(\w)}); + +\zaehler{1}{\ns}{\w}{zaehlerschwarz}{Zähler $1$}{0.2/\skala} +\zaehler{\dn}{\ds}{\v}{zaehlergruen}{Zähler $\operatorname{dn}(u,k)$}{0} +\zaehler{\cn}{\cs}{0}{zaehlerblau}{Zähler $\operatorname{cn}(u,k)$}{(-0.3/\skala)} + +\fill[color=zaehlerrot!50,opacity=0.5] + ({\cn-0.2},{(\cn-0.2)*tan(\v)+1.0/\skala}) + -- + ({\cs+1.2},{(\cs+1.2)*tan(\v)+1.0/\skala}) + -- + ({\cs+1.2},{(\cs+1.2)*tan(\w)-0.15/\skala}) + -- + ({\cn-0.2},{(\cn-0.2)*tan(\w)-0.15/\skala}) + -- + cycle; +\end{scope} + +\begin{scope}[rotate=\u] +\node[color=zaehlerrot] at (0:{(\ds+\ns)/2+1/\skala}) + [right,rotate={\u-3}] + {Zähler $\operatorname{sn}(u,k)$}; +\end{scope} + +\def\rechteck#1#2#3{ + \fill[color=#2!50,opacity=0.5] ({#1-0.55/\skala},{-0.6/\skala}) + rectangle ({#1+0.55/\skala},{1+3.2/\skala}); + \node[color=#2] at ({#1-0/\skala},{1+0.8/\skala}) [right,rotate=90] {#3}; +} + +\rechteck{\cn}{nennerschwarz}{Nenner $1$} +\rechteck{\cd}{nennergruen}{Nenner $\operatorname{dn}(u,k)$} +\rechteck{1}{nennerblau}{Nenner $\operatorname{cn}(u,k)$} +\rechteck{\cs}{nennerrot}{Nenner $\operatorname{sn}(u,k)$} + +\draw[color=gray!50] (\cn,0) -- (\cn,1); +\draw[color=gray!50] (\cd,0) -- (\cd,1); +\draw[color=gray!50] (1,0) -- (1,1); +\draw[color=gray!50] (\cs,0) -- (\cs,1); + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius={0.10/\skala}]; + \fill[color=white] #1 circle[radius={0.06/\skala}]; +} + +\draw[->] ({-0.1/\skala},0) -- ({\cs+0.8/\skala},0) + coordinate[label={$x$}]; +\draw[->] (0,{-0.1/\skala}) -- (0,{1+3.0/\skala}) + coordinate[label={right:$y$}]; + +\draw[color=gray] (0,1) -- (\cs,1); +\draw[color=gray!50,line width=1.4pt] + plot[domain=0:90,samples=100] ({cos(\x)},{sin(\x)/\a}); + +\draw[line width=1.4pt] (1,0) arc (0:{\w-4.5}:1); +\draw[line width=1.4pt,color=gray!40] ({\w-4.5}:1) arc ({\w-4.5}:\w:1); +\draw[line width=1.4pt] ({\w}:1) arc (\w:90:1); + +\draw (1,0) -- (1,\sc); +\node at (1,\sc) [above right,rotate=-90] {$\operatorname{sc}(u,k)\mathstrut$}; + +\draw (0,0) -- (\w:\ns); +\draw (0,0) -- (\v:\ds); + +\node at ($0.95*(\w:\ns)$) + [above,rotate=\w] {$\operatorname{ns}(u,k)\mathstrut$}; +\node at ($0.95*(\w:\nc)$) + [above,rotate=\w] {$\operatorname{nc}(u,k)\mathstrut$}; + +\node at (\w:1) [above left,rotate=\w] {$1\mathstrut$}; +\node at (\v:1) [above left,rotate=\v] {$1\mathstrut$}; + + +\draw[color=red,line width=1.4pt] (\cn,0) -- (\cn,\sn); +\node[color=red] at (\cn,\sn) + [above right,rotate=-90] {$\operatorname{sn}(u,k)\mathstrut$}; + +\fill[color=darkgreen!50] (0,0) -- (0:0.35) arc (0:\v:0.35) -- cycle; +\node[color=darkgreen] at ({\v/2}:0.28) {$\vartheta$}; +\draw[color=darkgreen,line width=1.4pt] (0,0) -- (\v:\dn); +\node[color=darkgreen] at ($1.0*(\v:\dn)$) + [above left,rotate=\v] {$\operatorname{dn}(u,k)\mathstrut$}; + +\fill[color=gray!50] (0,0) -- ({0.2},0) arc (0:\w:0.2) -- cycle; +\node[color=black] at ({\w/2}:0.13) {$\varphi$}; +\draw[color=black] (0,0) -- (\cs,1); + + +\draw[color=blue,line width=1.4pt] (0,0) -- (\cn,0); +\node[color=blue] at (\cn,0) + [below] {$\operatorname{cn}(u,k)\mathstrut$}; + +\draw[color=gray!50,line width=1pt] (\cd,0) -- (\cd,\sd); +\node at (\cd,0) [below] {$\operatorname{cd}(u,k)\mathstrut$}; +\node at (\cd,\sd) [above right,rotate=-90] {$\operatorname{sd}(u,k)\mathstrut$}; + +\draw[color=gray!50,line width=1pt] (\cs,0) -- (\cs,1); + +\punkt{(\cn,\sn)}{red} +\punkt{(\cn,0)}{blue} +\punkt{(\v:\dn)}{darkgreen} + +\punkt{(1,\sc)}{black} +\punkt{(\cs,1)}{black} + +\punkt{(\cd,0)}{black} +\punkt{(\cd,\sd)}{black} + +\punkt{(\w:\nd)}{black} +\punkt{(\cs,0)}{black} +\punkt{(\v:\ds)}{black} +\punkt{(\v:\dc)}{black} + +\node at (\w:\nd) [above left,rotate=\w] {$\operatorname{nd}(u,k)\mathstrut$}; +\node at (\cs,0) [below] {$\operatorname{cs}(u,k)\mathstrut$}; +\node at (\v:\ds) [above left,rotate=\v] {$\operatorname{ds}(u,k)\mathstrut$}; +\node at (\v:\dc) [above left,rotate=\v] {$\operatorname{dc}(u,k)\mathstrut$}; +\node at (1,0) [below] {$1\mathstrut$}; +\node at (\cs,1) [above right,rotate=-90] {$1\mathstrut$}; + +\draw ({-0.1/\skala},1) -- ({0.1/\skala},1); +\node at ({-0.1/\skala},1) [left] {$1$}; +\draw ({-0.1/\skala},{1/\a}) -- ({0.1/\skala},{1/\a}); +\node at ({-0.1/\skala},{1/\a}) [left] {$\displaystyle\frac1a$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/110-elliptisch/images/jacobidef.pdf b/buch/chapters/110-elliptisch/images/jacobidef.pdf new file mode 100644 index 0000000..9b4ab67 Binary files /dev/null and b/buch/chapters/110-elliptisch/images/jacobidef.pdf differ diff --git a/buch/chapters/110-elliptisch/images/jacobidef.tex b/buch/chapters/110-elliptisch/images/jacobidef.tex new file mode 100644 index 0000000..855b330 --- /dev/null +++ b/buch/chapters/110-elliptisch/images/jacobidef.tex @@ -0,0 +1,100 @@ +% +% 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,calc} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{document} +\def\skala{8} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\a{1.6} +\def\w{40} +\pgfmathparse{atan(sin(\w)/(\a*cos(\w)))} +\xdef\v{\pgfmathresult} + +\pgfmathparse{cos(\w)} +\xdef\cn{\pgfmathresult} +\pgfmathparse{sin(\w)} +\xdef\sn{\pgfmathresult} +\pgfmathparse{sqrt(\a*\a*cos(\w)*cos(\w)+sin(\w)*sin(\w))} +\xdef\r{\pgfmathresult} +\pgfmathparse{\r/\a} +\xdef\dn{\pgfmathresult} + +\draw[color=red,line width=1.4pt] ({\cn},0) -- ({\cn},{\sn}); +\node[color=red] at ({\cn},{0.3*\sn}) + [above,rotate=90] {$\operatorname{sn}(u,k)$}; +\node[color=red] at ({\a*\cn},{0.3*\sn}) + [above,rotate=90] {$\operatorname{sn}(u,k)$}; + +\draw[->,line width=1pt] ({-0.1/\skala},0) -- ({\a+0.4/\skala},0) + coordinate[label={$x$}]; +\draw[->,line width=1pt] (0,{-0.1/\skala}) -- (0,{1+0.3/\skala}) + coordinate[label={right:$y$}]; + +\draw[color=gray,line width=1.4pt] + plot[domain=-0.5:90.5,samples=100] ({cos(\x)},{sin(\x)}); +\draw[color=gray!50,line width=1.4pt] + plot[domain=-0.5:90.5,samples=100] ({cos(\x)},{sin(\x)/\a}); +\draw[color=black,line width=1.4pt] + plot[domain=-0.5:90.5,samples=100] ({\a*cos(\x)},{sin(\x)}); + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius={0.10/\skala}]; + \fill[color=white] #1 circle[radius={0.06/\skala}]; +} + +\fill[color=darkgreen!50] (0,0) -- (0:0.35) arc (0:\v:0.35) -- cycle; +%\draw[color=darkgreen] (0:0.35) arc (0:\v:0.35); +\node[color=darkgreen] at ({\v/2}:0.28) {$\vartheta$}; + +\draw[color=red!40,line width=1.4pt] ({\a*\cn},0) -- ({\a*\cn},{\sn}); + +\draw[color=darkgreen!40,line width=1.4pt] (0,0) -- ({\a*\cn},{\sn}); +\draw[color=darkgreen,line width=1.4pt] (0,0) -- ({\cn},{\sn/\a}); +\node[color=darkgreen] at ({0.6*\cn},{0.6*\sn/\a}) + [below,rotate=\v] {$\operatorname{dn}(u,k)$}; +\punkt{({\cn},{\sn/\a})}{darkgreen} +\node[color=darkgreen] at ({0.87*\a*\cn},{0.87*\sn}) + [above,rotate=\v] {$r=a\operatorname{dn}(u,k)$}; + +\fill[color=gray!50] (0,0) -- (0:0.2) arc (0:\w:0.2) -- cycle; +\node at ({\w/2}:0.15) {$\varphi$}; + + +\draw (0,0) -- (\w:1); + +\punkt{({\a*\cn},{\sn})}{black} +\punkt{({\cn},{\sn})}{red!50} + +\node at ({\a*\cn},{\sn}) + [above right] {$P=(a\cos\varphi,\sin\varphi)$}; +\node at ($(\w:1)+({-0.1/\skala},0)$) + [left] {$Q=(\cos\varphi,\sin\varphi)$}; + +\node at (\a,0) [below] {$a\mathstrut$}; +\node at ({-0.1/\skala},1) [left] {$1$}; +\node at ({-0.1/\skala},{1/\a}) [left] {$\displaystyle\frac{1}{a}$}; +\draw ({-0.1/\skala},{1/\a}) -- ({0.1/\skala},{1/\a}); + +\draw[color=blue!40,line width=1.4pt] (0,0) -- ({\a*\cn},0); +\draw ({\a*\cn},{-0.1/\skala}) -- ({\a*\cn},{0.1/\skala}); +\node[color=blue] at ({\a*\cn},{-0.1/\skala}) + [below] {$a\operatorname{cn}(u,k)$}; +\draw[color=blue,line width=1.4pt] (0,0) -- ({\cn},0); +\node[color=blue] at ({0.5*\cn},0) [below] {$\operatorname{cn}(u,k)$}; +\punkt{({cos(\w)},0)}{blue} + +\node at (0,0) [below left] {$O$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.pdf b/buch/chapters/110-elliptisch/images/jacobiplots.pdf index 47870ef..8ebd501 100644 Binary files a/buch/chapters/110-elliptisch/images/jacobiplots.pdf and b/buch/chapters/110-elliptisch/images/jacobiplots.pdf differ -- cgit v1.2.1