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authorAndreas Müller <andreas.mueller@ost.ch>2022-04-20 10:30:56 +0200
committerAndreas Müller <andreas.mueller@ost.ch>2022-04-20 10:30:56 +0200
commit0344a846c083c11e9ed93ddc5898dd55c6dd1022 (patch)
tree284ff262d32e7e1138679b56bbc11060e4bc0db8 /buch/chapters/110-elliptisch/images
parentSingularitaeten (diff)
downloadSeminarSpezielleFunktionen-0344a846c083c11e9ed93ddc5898dd55c6dd1022.tar.gz
SeminarSpezielleFunktionen-0344a846c083c11e9ed93ddc5898dd55c6dd1022.zip
lemniscate sine stuff
Diffstat (limited to '')
-rw-r--r--buch/chapters/110-elliptisch/images/Makefile9
-rw-r--r--buch/chapters/110-elliptisch/images/jacobiplots.pdfbin56975 -> 56975 bytes
-rw-r--r--buch/chapters/110-elliptisch/images/lemniskate.pdfbin9914 -> 14339 bytes
-rw-r--r--buch/chapters/110-elliptisch/images/lemniskate.tex15
-rw-r--r--buch/chapters/110-elliptisch/images/slcl.cpp128
-rw-r--r--buch/chapters/110-elliptisch/images/slcl.pdfbin0 -> 28269 bytes
-rw-r--r--buch/chapters/110-elliptisch/images/slcl.tex88
7 files changed, 237 insertions, 3 deletions
diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile
index 68322b6..a7c9e74 100644
--- a/buch/chapters/110-elliptisch/images/Makefile
+++ b/buch/chapters/110-elliptisch/images/Makefile
@@ -5,7 +5,7 @@
#
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
lemniskate.pdf: lemniskate.tex
pdflatex lemniskate.tex
@@ -71,3 +71,10 @@ 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
diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.pdf b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
index 88cf119..f0e6e78 100644
--- a/buch/chapters/110-elliptisch/images/jacobiplots.pdf
+++ b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
Binary files differ
diff --git a/buch/chapters/110-elliptisch/images/lemniskate.pdf b/buch/chapters/110-elliptisch/images/lemniskate.pdf
index 063a3e1..9e02c3c 100644
--- a/buch/chapters/110-elliptisch/images/lemniskate.pdf
+++ b/buch/chapters/110-elliptisch/images/lemniskate.pdf
Binary files differ
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/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
new file mode 100644
index 0000000..493b5fa
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/slcl.pdf
Binary files differ
diff --git a/buch/chapters/110-elliptisch/images/slcl.tex b/buch/chapters/110-elliptisch/images/slcl.tex
new file mode 100644
index 0000000..08241ac
--- /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!20,line width=1.4pt]
+ plot[domain=0:13,samples=200] ({\x},{\dy*sin(\ts*\x)});
+\draw[color=blue!20,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) -- (13.7,0) 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] {$ \varpi\mathstrut$};
+\node at ({2*\lemniscateconstant*\dx},0) [below left] {$2\varpi\mathstrut$};
+\node at ({3*\lemniscateconstant*\dx},0) [below right] {$3\varpi\mathstrut$};
+\node at ({4*\lemniscateconstant*\dx},0) [below right] {$4\varpi\mathstrut$};
+\node at ({5*\lemniscateconstant*\dx},0) [below left] {$5\varpi\mathstrut$};
+
+\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] {$\sin \bigl(\frac{\pi}{2\varpi}s\bigr)$};
+
+\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] {$\cos\bigl(\frac{\pi}{2\varpi}s\bigr)$};
+
+\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}
+