new image: tetraeder

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
new 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
new 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}
+