added images

documents/GeoTopo/Kapitel2.tex

@@ -432,7 +432,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Mitschrieb vom 21.11.2013                                         %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{korollar}
+\begin{korollar}\label{kor:regular-surface-mannigfaltigkeit}
     Jede reguläre Fläche $S \subseteq \mdr^3$ ist eine 2-dimensionale,
     differenzierbare Mannigfaltigkeit.
 \end{korollar}
@@ -441,7 +441,13 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
     \todo{Hier muss ich nochmals drüberlesen.}
     \underline{z.Z.:} $F_j^{-1} \circ F_i$ ist Diffeomorphismus
 
-    \todo[inline]{Bild $F_j^{-1} \circ F_i$}
+    \begin{figure}[htp]
+        \centering
+        \input{figures/topology-parametric-surface-mapping.tex}
+        \caption{Reguläre Fläche $S$ zum Beweis von Korollar~\ref{kor:regular-surface-mannigfaltigkeit}}
+        \label{fig:parametric-surface-mapping}
+    \end{figure}
+    
 
     \underline{Idee:} Finde differenzierbare Funktion $\tilde{F_j^{-1}}$
     in Umgebung $W$ von $s$, sodass $\tilde{F_j^{-1}}|_{S \cap W} = F_j^{-1}$.
@@ -819,8 +825,6 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
         \item \Obda{} sei $0 \in P$ und $P \subseteq \fB_1(0)$. Projeziere
               $0P$ von $0$ aus auf $\partial \fB_1(0) = S^2$.
               Erhalte Triangulierung von $S^2$.
-
-              \todo[inline]{Bild von rundem Wuerfel}
         \item Sind $P_1$ und $P_2$ konvexe Polygone und $T_1, T_2$
               die zugehörigen Triangulierungen von $S^2$, so gibt es 
               eine eine Triangulierungen $T$, die sowohl um $T_1$ als
@@ -855,7 +859,9 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
 
     Dann gilt: $d_{n-1} \circ d_n = 0$
 
-    \todo[inline]{Skizze von Dreieck}
+    \input{figures/topology-oriented-triangle.tex}
+
+    $a < b < c$
 
     $d_2 \sigma = e_1 - e_2 + e_3 = c - b - (c-a) + b - a = 0$
 \end{korollar}

documents/GeoTopo/figures/topology-oriented-triangle.tex

@@ -0,0 +1,12 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (a)[point,label={[label distance=0cm]210:$a$}] at (210:1cm) {};
+    \node (b)[point,label={[label distance=0cm]-45:$b$}] at (330:1cm) {};
+    \node (c)[point,label={[label distance=0cm]90:$c$}] at (90:1cm) {};
+
+    \node (sigma) at (0,0) {$\sigma$};
+
+    \draw[->, very thick] (a) edge node[label=below:$e_3$]  {} (b);
+    \draw[->, very thick] (b) edge node[label=right:$e_1$]  {} (c);
+    \draw[->, very thick] (c) edge node[label=left:$e_2$]  {} (a);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-parametric-surface-mapping.tex

@@ -0,0 +1,25 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
+    \draw (0,0) ellipse (2cm and 1cm);
+    \def\ringa{(-0.3,0) circle (0.5cm)}
+    \def\ringb{(+0.3,0) circle (0.5cm)}
+
+    \draw \ringa;
+    \draw[red] \ringb;
+
+    %\node at (-1,0.3) {$U_i$};
+    %\node at (+1,0.3) {$U_j$};
+    \node at (-1.9,-2) {$U_i$};
+    \node[red] at (+1.9,-2) {$U_j$};
+    \node at (+2.0,0.7) {$S$};
+    \node[point,label={[label distance=-0.1cm]90:$s$}] at (0,0) {};
+
+
+    \path[<-] (-0.35,0)  edge [bend angle=10,bend right] node[label={[label distance=0.1cm]210:$F_i$}] {} (-1,-1.5);
+    \path[<-,red] (+0.35,0)  edge [bend angle=10,bend left]  node[label={[label distance=0.1cm]-30:$F_j$}] {} (+1,-1.5);
+
+    \draw (-1,-2) circle (0.5cm);
+    \draw[red] (+1,-2) circle (0.5cm);
+
+    \path[->, green, thick] (-0.3,-2) edge node[label=below:$\scriptstyle F_j^{-1} \circ F_i$] {} (0.3,-2);
+\end{tikzpicture}

tikz/topology-oriented-triangle/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-oriented-triangle
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-oriented-triangle/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-oriented-triangle.png)

tikz/topology-oriented-triangle/topology-oriented-triangle.tex

@@ -0,0 +1,19 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+\usetikzlibrary{calc,shadings}
+\usepackage{pgfplots}
+
+\begin{document}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (a)[point,label={[label distance=0cm]210:$a$}] at (210:1cm) {};
+    \node (b)[point,label={[label distance=0cm]-45:$b$}] at (330:1cm) {};
+    \node (c)[point,label={[label distance=0cm]90:$c$}] at (90:1cm) {};
+
+    \node (sigma) at (0,0) {$\sigma$};
+
+    \draw[->, very thick] (a) edge node[label=below:$e_3$]  {} (b);
+    \draw[->, very thick] (b) edge node[label=right:$e_1$]  {} (c);
+    \draw[->, very thick] (c) edge node[label=left:$e_2$]  {} (a);
+\end{tikzpicture}
+\end{document}

tikz/topology-parametric-surface-mapping/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-parametric-surface-mapping
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-parametric-surface-mapping/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-parametric-surface-mapping.png)

tikz/topology-parametric-surface-mapping/topology-parametric-surface-mapping.tex

@@ -0,0 +1,33 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{amsmath,amssymb}% math symbols / fonts
+\usepackage{tikz}
+\usepackage{tqft}
+\usetikzlibrary{patterns}
+
+\begin{document}
+    \begin{tikzpicture}
+        \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
+        \draw (0,0) ellipse (2cm and 1cm);
+        \def\ringa{(-0.3,0) circle (0.5cm)}
+        \def\ringb{(+0.3,0) circle (0.5cm)}
+
+        \draw \ringa;
+        \draw[red] \ringb;
+
+        %\node at (-1,0.3) {$U_i$};
+        %\node at (+1,0.3) {$U_j$};
+        \node at (-1.9,-2) {$U_i$};
+        \node[red] at (+1.9,-2) {$U_j$};
+        \node at (+2.0,0.7) {$S$};
+        \node[point,label={[label distance=-0.1cm]90:$s$}] at (0,0) {};
+
+
+        \path[<-] (-0.35,0)  edge [bend angle=10,bend right] node[label={[label distance=0.1cm]210:$F_i$}] {} (-1,-1.5);
+        \path[<-,red] (+0.35,0)  edge [bend angle=10,bend left]  node[label={[label distance=0.1cm]-30:$F_j$}] {} (+1,-1.5);
+
+        \draw (-1,-2) circle (0.5cm);
+        \draw[red] (+1,-2) circle (0.5cm);
+
+        \path[->, green, thick] (-0.3,-2) edge node[label=below:$\scriptstyle F_j^{-1} \circ F_i$] {} (0.3,-2);
+    \end{tikzpicture}
+\end{document}