diff options
Diffstat (limited to 'buch')
| -rw-r--r-- | buch/chapters/95-homologie/basiswahl.tex | 4 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/homologieketten.tex | 2 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/images/Makefile | 9 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/images/tetraeder.jpg | bin | 0 -> 68762 bytes | |||
| -rw-r--r-- | buch/chapters/95-homologie/images/tetraeder.pdf | bin | 0 -> 84492 bytes | |||
| -rw-r--r-- | buch/chapters/95-homologie/images/tetraeder.pov | 116 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/images/tetraeder.tex | 97 |
7 files changed, 223 insertions, 5 deletions
diff --git a/buch/chapters/95-homologie/basiswahl.tex b/buch/chapters/95-homologie/basiswahl.tex index f720c76..aacfa9f 100644 --- a/buch/chapters/95-homologie/basiswahl.tex +++ b/buch/chapters/95-homologie/basiswahl.tex @@ -100,7 +100,6 @@ in {\color{darkgreen}grün}) \end{center} Daraus kann man die Zyklen wie folgt ablesen, indem man jeweils genau eine frei wählbare Variable auf $1$ setzt: -{ \begin{align*} z_1 &= @@ -524,14 +523,13 @@ z_8 % variable 18 = 1 1 \end{pmatrix*} \end{align*} -} \begin{figure} \centering \includegraphics{chapters/95-homologie/images/homocycles.pdf} \caption{Zyklen des Randoperators $\partial_1$ im Beispiel von Seite~\pageref{buch:homologie:beispiel:gausshomo}. \label{buch:homologie:fig:homocycles}} -\end{figure} +\end{figure}% Die Zyklen sind in Abbildung~\ref{buch:homologie:fig:homocycles} {\color{red}rot} dargestellt. \end{beispiel} diff --git a/buch/chapters/95-homologie/homologieketten.tex b/buch/chapters/95-homologie/homologieketten.tex index 383c8df..1b40147 100644 --- a/buch/chapters/95-homologie/homologieketten.tex +++ b/buch/chapters/95-homologie/homologieketten.tex @@ -90,7 +90,7 @@ der entsteht, wenn man aus einem Tetraeder das innere entfernt. \begin{beispiel} \begin{figure} \centering -XXX Bild eines Tetraeders mit Bezeichnung der Ecken und Kanten +\includegraphics{chapters/95-homologie/images/tetraeder.pdf} \caption{Triangulation eines Tetraeders, die Orientierung von Kanten und Seitenflächen ist immer so gewählt, dass die Nummern der Ecken aufsteigend sind. diff --git a/buch/chapters/95-homologie/images/Makefile b/buch/chapters/95-homologie/images/Makefile index bc85c55..0a3979e 100644 --- a/buch/chapters/95-homologie/images/Makefile +++ b/buch/chapters/95-homologie/images/Makefile @@ -5,7 +5,7 @@ # all: complexbasis.pdf homocycles.pdf homoboundaries.pdf homoclasses.pdf \ gausshomoex.pdf gausshomobasis.pdf dreieck.pdf polyeder.pdf \ - approximation.pdf + approximation.pdf tetraeder.pdf dreieck.pdf: dreieck.tex pdflatex dreieck.tex @@ -37,3 +37,10 @@ approximation.pdf: approximation.tex approx.tex approx.tex: approx.m octave approx.m +tetraeder.png: tetraeder.pov + povray +A0.1 -W1920 -H1080 -O$@ $< +tetraeder.jpg: tetraeder.png Makefile + convert -extract 1080x1080+520 tetraeder.png tetraeder.jpg +tetraeder.pdf: tetraeder.tex tetraeder.jpg + pdflatex tetraeder.tex + diff --git a/buch/chapters/95-homologie/images/tetraeder.jpg b/buch/chapters/95-homologie/images/tetraeder.jpg Binary files differnew file mode 100644 index 0000000..0ec168b --- /dev/null +++ b/buch/chapters/95-homologie/images/tetraeder.jpg diff --git a/buch/chapters/95-homologie/images/tetraeder.pdf b/buch/chapters/95-homologie/images/tetraeder.pdf Binary files differnew file mode 100644 index 0000000..0a57e95 --- /dev/null +++ b/buch/chapters/95-homologie/images/tetraeder.pdf diff --git a/buch/chapters/95-homologie/images/tetraeder.pov b/buch/chapters/95-homologie/images/tetraeder.pov new file mode 100644 index 0000000..b110f96 --- /dev/null +++ b/buch/chapters/95-homologie/images/tetraeder.pov @@ -0,0 +1,116 @@ +// +// tetraeder.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.169; +#declare O = <0, 0, 0>; +#declare at = 0.02; + +camera { + location <-2, 3, -10> + look_at <0, 0.18, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +//light_source { +// <-14, 20, -50> color White +// area_light <1,0,0> <0,0,1>, 10, 10 +// adaptive 1 +// jitter +//} + +light_source { + <-41, 20, -20> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#declare v1 = <1,1,1>; +#declare v2 = <-1,1,-1>; +#declare farbe = rgbf<0.8,0.8,1.0,0.5>; + +#declare tetraederwinkel = acos(vdot(v1,v2)/(vlength(v1)*vlength(v2))); + +#declare O = < 0, 0, 0 >; +#declare A = < 0, 1, 0 >; +#declare B = < sin(tetraederwinkel), cos(tetraederwinkel), 0>; +#declare C = < sin(tetraederwinkel)*cos(2*pi/3), cos(tetraederwinkel), sin(2*pi/3)>; +#declare D = < sin(tetraederwinkel)*cos(2*pi/3), cos(tetraederwinkel), -sin(2*pi/3)>; + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.0 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 8 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 8 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to - 3 * arrowthickness * arrowdirection, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +union { + arrow(B, C, at, White) + arrow(D, C, at, White) + arrow(D, B, at, White) + arrow(B, A, at, White) + arrow(C, A, at, White) + arrow(D, A, at, White) + sphere { A, 4 * at } + sphere { B, 4 * at } + sphere { C, 4 * at } + sphere { D, 4 * at } + pigment { + color White + } + finish { + specular 0.9 + metallic + } +} + +mesh { + triangle { A, B, C } + triangle { A, C, D } + triangle { A, D, B } + triangle { B, C, D } + pigment { + color farbe + } +// finish { +// specular 0.9 +// metallic +// } +} diff --git a/buch/chapters/95-homologie/images/tetraeder.tex b/buch/chapters/95-homologie/images/tetraeder.tex new file mode 100644 index 0000000..e62770f --- /dev/null +++ b/buch/chapters/95-homologie/images/tetraeder.tex @@ -0,0 +1,97 @@ +% +% tetraeder.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,calc} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{7} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{tetraeder.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]; +}{} + +\def\knoten#1#2{ + %\fill[color=white,opacity=0.5] #1 circle[radius=0.2]; + \node at #1 {$#2$}; +} + +\knoten{(-2.2,-3.6)}{0}; +\knoten{( 3.3,-1.9)}{1}; +\knoten{(-3.4,-1.2)}{2}; +\knoten{(-0.75,3.6)}{3}; + +\def\s{0.2} + +\def\kante#1#2{ + %\fill[color=white,opacity=0.5] #1 circle[radius=0.2]; + \fill[color=white,opacity=0.5] + ($#1+(-\s,-\s)$) -- + ($#1+(+\s,-\s)$) -- + ($#1+(+\s,+\s)$) -- + ($#1+(-\s,+\s)$) -- cycle; + \node at #1 {$#2$}; +} + +\kante{(0.5,-2.8)}{k_0} +\kante{(-2.8,-2.3)}{k_1} +\kante{(-1.4,0)}{k_2} +\kante{(-0.4,-1.55)}{k_3} +\kante{(1.25,0.95)}{k_4} +\kante{(-2.08,1.1)}{k_5} + +\def\r{0.33} + +\def\flaeche#1#2{ + \fill[color=white,opacity=0.5] + ($#1+({-\r*cos(30)},{-\r*sin(30)})$) -- + ($#1+({\r*cos(30)},{-\r*sin(30)})$) -- + ($#1+(0,{\r})$) -- cycle; + \node at #1 {$#2$}; +} + +\flaeche{(-0.7,-5)}{f_0} +\draw (-0.7,-4.7) -- (-0.7,-3.25); +\draw[->,color=black!70] (-0.7,-3.06) -- (-0.7,-2.5); +\flaeche{(0.2,-0.5)}{f_1} +\flaeche{(-2.3,-0.7)}{f_2} +\coordinate (A) at (1,2.6); +\coordinate (B) at (0,1); + +\flaeche{($1.2*(A)-0.2*(B)$)}{f_3} + +\def\t{0.58} +\pgfmathparse{1-\t} +\xdef\T{\pgfmathresult} +\draw (A) -- ($\t*(A)+\T*(B)$); + +\def\t{0.48} +\pgfmathparse{1-\t} +\xdef\T{\pgfmathresult} +\draw[->,color=black!70] ($\t*(A)+\T*(B)$) -- (B); + + +\end{tikzpicture} + +\end{document} + |
