added many images

documents/GeoTopo/Arbeitszeit.txt

@@ -3,4 +3,4 @@ in dem erstellen dieses Skripts steckt
 
 Datum      | Uhrzeit
 ---------------------------------------------------------------------
-03.12.2013 | 11:00 - 12:00
+03.12.2013 | 11:00 - 12:00, 13:10 - 15:00

documents/GeoTopo/Kapitel2.tex

@@ -867,7 +867,9 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
 \end{korollar}
 
 \begin{beispiel}
-    \input{figures/topology-oriented-triangle.tex}
+    \begin{figure}[h!]
+        \input{figures/topology-oriented-triangle.tex}
+    \end{figure}
 
     $a < b < c$
 
@@ -907,17 +909,19 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
     \end{enumerate}
 \end{definition}
 
+\begin{minipage}{\textwidth}%don't break this theorem!
 \begin{satz}
     Für jeden endlichen Simplizialkomplex $K$ der Dimension $d$ gilt:
 
     \[\sum_{k=0}^d (-1)^k b_k (K) = \sum_{k=0}^d (-1)^k a_k(K) = \chi(K) \]
 \end{satz}
+\end{minipage}
 
 \begin{bemerkung}
     Es gilt \underline{nicht} $a_k = b_k\;\forall k \in \mdn_0$.
 \end{bemerkung}
 
-\begin{beweis}
+\begin{beweis}\leavevmode
     \begin{itemize}
         \item Dimensionsformel für $d_n$: $a_n = \dim Z_n + \dim B_{n-1}$ für $n \geq 1$
         \item Dimensionsformel für $Z_n \rightarrow H_n = Z_n / B_n: \dim Z_n = b_n + \dim B_n$

documents/GeoTopo/Kapitel3.tex

@@ -3,6 +3,21 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \chapter{Fundamentalgruppe und Überlagerungen}
 \section{Homotopie von Wegen}
+\begin{figure}[ht]
+    \centering
+    \subfloat[$\gamma_1$ und $\gamma_2$ sind homotop, da man sie 
+             \enquote{zueinander verschieben} kann.]{
+        \input{figures/topology-homotop-paths.tex}
+        \label{fig:homotope-wege-anschaulich}
+    }\hspace{1em}%
+    \subfloat[$\gamma_1$ und $\gamma_2$ sind wegen dem Hindernis nicht homotop.]{
+        \input{figures/topology-non-homotop-paths.tex}
+        \label{fig:nicht-homotope-wege-anschaulich}
+    }
+    \label{Formen}
+    \caption{Beispiele für Wege $\gamma_1$ und $\gamma_2$}
+\end{figure}
+
 \begin{definition}
     Sei $X$ ein topologischer Raum, $a, b \in X$, 
     $\gamma_1, \gamma_2: [0,1] \rightarrow X$ Wege von $a$ nach $b$,
@@ -26,7 +41,7 @@
     Wege in $X$ von $a$ nach $b$.
 \end{korollar}
 
-\begin{beweis}
+\begin{beweis}\leavevmode
     \begin{itemize}
         \item reflexiv: $H(t,s) = \gamma(t)$ für alle $t,s \in I \times I$
         \item symmetrisch: $H'(t,s) = H(t,1-s)$ für alle $t,s \in I \times I$
@@ -34,8 +49,8 @@
               nach $\gamma_2$ bzw. von $\gamma_2$ nach $\gamma_3$.
 
               Dann sei $H(t,s) := \begin{cases}
-              H'(t, 2s)    &\text{, falls } 0 \leq s \leq \frac{1}{2}\\
-              H''(t, 2s-1) &\text{, falls } \frac{1}{2} \leq s \leq 1\end{cases}$
+              H'(t, 2s)    &\text{falls } 0 \leq s \leq \frac{1}{2}\\
+              H''(t, 2s-1) &\text{falls } \frac{1}{2} \leq s \leq 1\end{cases}$
 
               $\Rightarrow$ $H$ ist stetig und Homotopie von $\gamma_1$ nach 
               $\gamma_2$
@@ -43,7 +58,46 @@
     $\qed$
 \end{beweis}
 
-\todo[inline]{Noch ca. eine halbe seite mit 3 Beispielen}
+\begin{beispiel}
+    \begin{enumerate}[label=\arabic*)]
+        \item Sei $X = S^1$. $\gamma_1$ und $\gamma_2$ aus 
+              Abb.~\ref{fig:circle-two-paths} nicht homöotop.
+              \begin{figure}
+                \centering
+                \input{figures/topology-circle-two-paths.tex}
+                \caption{Kreis mit zwei Wegen}
+                \label{fig:circle-two-paths}
+              \end{figure}
+        \item Sei $X = T^2$. $\gamma_1, \gamma_2$ und $\gamma_3$
+              aus Abb.~\ref{fig:torus-three-paths} sind paarweise
+              nicht homöotop.
+              \begin{figure}
+                \centering
+                \input{figures/todo.tex}
+                \caption{Torus mit drei Wegen}
+                \label{fig:torus-three-paths}
+              \end{figure}
+        \item Sei $X = \mdr^2$ und $a=b=(0,0)$. 
+
+              Je zwei Wege im $\mdr^2$ mit Anfangs- und Enpunkt $(0,0)$
+              sind homöotop.
+
+              \begin{figure}
+                \centering
+                \input{figures/topology-paths-in-r2.tex}
+                \caption{Zwei Wege im $\mdr^2$ mit Anfangs- und Enpunkt $(0,0)$}
+                \label{fig:torus-three-paths}
+              \end{figure}
+
+              Sei $\gamma_0: I \rightarrow \mdr^2$ der konstante Weg
+              $\gamma_0(t) = 0 \; \forall t \in I$. Sei
+              $\gamma(0) = \gamma(1) = 0$.
+
+              $H(t,s) := (1-s) \gamma(t)$ ist stetig, 
+              $H(t,0) = \gamma(t)\; \forall t \in I$ und
+              $H(t,1) = 0 \; \forall t \in I$
+    \end{enumerate}
+\end{beispiel}
 
 % Die Übungsaufgaben sollen ganz am Ende des Kapitels sein.
 \input{Kapitel3-UB}

documents/GeoTopo/figures/topology-circle-two-paths.tex

@@ -0,0 +1,10 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \draw [red,  thick,domain=90:-90,  samples=100] plot ({cos(\x)}, {sin(\x)});
+    \draw [green,thick,domain=-90:-270,samples=100] plot ({cos(\x)}, {sin(\x)});
+    \node (a)[point,label=90:$a$] at (0,-1cm) {};
+    \node (b)[point,label=-90:$b$] at (0, 1cm) {};
+
+    \node at (1,0) [anchor=180, red] {$\gamma_1$};
+    \node at (-1,0) [anchor=0, green] {$\gamma_2$};
+\end{tikzpicture}

documents/GeoTopo/figures/topology-homotop-paths.tex

@@ -0,0 +1,16 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (a)[point,label=180:$a$] at (0,0) {};
+    \node (b)[point,label=0:$b$]   at (3, 0) {};
+    \draw [rounded corners] (a) .. controls (0.8,0.8) .. (1.5,0.8) .. controls (2.2,1) .. (b);
+    \draw [rounded corners] (a) .. controls (0.6,0.6) .. (1.3,0.6) .. controls (2.0,0.8) .. (b);
+    \draw [rounded corners] (a) .. controls (0.3,0.3) .. (1.0,0.3) .. controls (2.0,0.4) .. (b);
+    \draw [rounded corners] (a) -- (b);
+    \draw [rounded corners] (a) .. controls (1,-0.8) .. (2,-0.4) .. controls (2.2,-0.9) .. (b);
+    \draw [rounded corners] (a) .. controls (1,-0.6) .. (2,-0.2) .. controls (2.2,-0.5) .. (b);
+    \draw [rounded corners] (a) .. controls (1,-0.3) .. (2,-0.05) .. controls (2.2,-0.2) .. (b);
+    \draw [rounded corners,->, thick, red] (a) .. controls (1,1) .. (2,1) .. controls (2.5,1) .. (b);
+    \draw [rounded corners,->, thick, blue] (a) .. controls (1,-1) .. (2,-0.5) .. controls (2.2,-1) .. (b);
+    \node at (1,1.2) [red] {$\gamma_1$};
+    \node at (0.5,-0.8) [blue] {$\gamma_2$};
+\end{tikzpicture}

documents/GeoTopo/figures/topology-non-homotop-paths.tex

@@ -0,0 +1,10 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (a)[point,label=180:$a$] at (0,0) {};
+    \node (b)[point,label=0:$b$]   at (3, 0) {};
+    \draw[orange,pattern color=orange,pattern=north east lines] (1.5,0) circle (0.3cm);
+    \draw [rounded corners,->, thick, red] (a) .. controls (1,1) .. (2,1) .. controls (2.5,1) .. (b);
+    \draw [rounded corners,->, thick, blue] (a) .. controls (1,-1) .. (2,-0.5) .. controls (2.2,-1) .. (b);
+    \node at (1,1.2) [red] {$\gamma_1$};
+    \node at (0.5,-0.8) [blue] {$\gamma_2$};
+\end{tikzpicture}

documents/GeoTopo/figures/topology-paths-in-r2.tex

@@ -0,0 +1,27 @@
+\begin{tikzpicture}
+    \begin{axis}[
+        legend pos=south west,
+        axis x line=middle,
+        axis y line=middle,
+        %grid = major,
+        width=9cm,
+        height=9cm,
+        grid style={dashed, gray!30},
+        xmin=-2,     % start the diagram at this x-coordinate
+        xmax= 1,    % end   the diagram at this x-coordinate
+        ymin=-1,     % start the diagram at this y-coordinate
+        ymax= 5,   % end   the diagram at this y-coordinate
+        %axis background/.style={fill=white},
+        %xlabel=$x$,
+        %ylabel=$y$,
+        ticks=none,
+        %tick align=outside,
+        %minor tick num=-3,
+        enlargelimits=true,
+        tension=0.08]
+        \addplot[mark=none, orange, smooth cycle, thick, tension=1, dashed] coordinates {%
+   (0,0) (-1,1) (-2,2) (-1,3) (0, 3) (1, 4)};
+        \addplot[mark=none, blue, smooth cycle, thick, tension=3] coordinates {%
+   (0,0) (-1,1) (-2,2) (-1,3) (0, 3) (1, 4)};
+    \end{axis} 
+\end{tikzpicture}

tikz/topology-circle-two-paths/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-circle-two-paths
+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-circle-two-paths/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-circle-two-paths.png)

tikz/topology-circle-two-paths/topology-circle-two-paths.tex

@@ -0,0 +1,18 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+
+\usepackage{pgfplots}
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \draw [red,  thick,domain=90:-90,  samples=100] plot ({cos(\x)}, {sin(\x)});
+    \draw [green,thick,domain=-90:-270,samples=100] plot ({cos(\x)}, {sin(\x)});
+    \node (a)[point,label=90:$a$] at (0,-1cm) {};
+    \node (b)[point,label=-90:$b$] at (0, 1cm) {};
+
+    \node at (1,0) [anchor=180, red] {$\gamma_1$};
+    \node at (-1,0) [anchor=0, green] {$\gamma_2$};
+
+\end{tikzpicture}
+\end{document}

tikz/topology-homotop-paths/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-homotop-paths
+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-homotop-paths/Readme.md

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

tikz/topology-homotop-paths/topology-homotop-paths.tex

@@ -0,0 +1,23 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+
+\usepackage{pgfplots}
+\usepackage{tikz}
+
+\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=180:$a$] at (0,0) {};
+    \node (b)[point,label=0:$b$]   at (3, 0) {};
+    \draw [rounded corners] (a) .. controls (0.8,0.8) .. (1.5,0.8) .. controls (2.2,1) .. (b);
+    \draw [rounded corners] (a) .. controls (0.6,0.6) .. (1.3,0.6) .. controls (2.0,0.8) .. (b);
+    \draw [rounded corners] (a) .. controls (0.3,0.3) .. (1.0,0.3) .. controls (2.0,0.4) .. (b);
+    \draw [rounded corners] (a) -- (b);
+    \draw [rounded corners] (a) .. controls (1,-0.8) .. (2,-0.4) .. controls (2.2,-0.9) .. (b);
+    \draw [rounded corners] (a) .. controls (1,-0.6) .. (2,-0.2) .. controls (2.2,-0.5) .. (b);
+    \draw [rounded corners] (a) .. controls (1,-0.3) .. (2,-0.05) .. controls (2.2,-0.2) .. (b);
+    \draw [rounded corners,->, thick, red] (a) .. controls (1,1) .. (2,1) .. controls (2.5,1) .. (b);
+    \draw [rounded corners,->, thick, blue] (a) .. controls (1,-1) .. (2,-0.5) .. controls (2.2,-1) .. (b);
+    \node at (1,1.2) [red] {$\gamma_1$};
+    \node at (0.5,-0.8) [blue] {$\gamma_2$};
+\end{tikzpicture}
+\end{document}

tikz/topology-non-homotop-paths/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-non-homotop-paths
+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-non-homotop-paths/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-non-homotop-paths.png)

tikz/topology-non-homotop-paths/topology-non-homotop-paths.tex

@@ -0,0 +1,18 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+
+\usepackage{pgfplots}
+\usepackage{tikz}
+\usetikzlibrary{patterns}
+
+\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=180:$a$] at (0,0) {};
+    \node (b)[point,label=0:$b$]   at (3, 0) {};
+    \draw[orange,pattern color=orange,pattern=north east lines] (1.5,0) circle (0.3cm);
+    \draw [rounded corners,->, thick, red] (a) .. controls (1,1) .. (2,1) .. controls (2.5,1) .. (b);
+    \draw [rounded corners,->, thick, blue] (a) .. controls (1,-1) .. (2,-0.5) .. controls (2.2,-1) .. (b);
+    \node at (1,1.2) [red] {$\gamma_1$};
+    \node at (0.5,-0.8) [blue] {$\gamma_2$};
+\end{tikzpicture}
+\end{document}

tikz/topology-paths-in-r2/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-paths-in-r2
+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-paths-in-r2/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-paths-in-r2.png)

tikz/topology-paths-in-r2/topology-paths-in-r2.tex

@@ -0,0 +1,34 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+
+\usepackage{pgfplots}
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+    \begin{axis}[
+        legend pos=south west,
+        axis x line=middle,
+        axis y line=middle,
+        %grid = major,
+        width=9cm,
+        height=9cm,
+        grid style={dashed, gray!30},
+        xmin=-2,     % start the diagram at this x-coordinate
+        xmax= 1,    % end   the diagram at this x-coordinate
+        ymin=-1,     % start the diagram at this y-coordinate
+        ymax= 5,   % end   the diagram at this y-coordinate
+        %axis background/.style={fill=white},
+        %xlabel=$x$,
+        %ylabel=$y$,
+        ticks=none,
+        %tick align=outside,
+        %minor tick num=-3,
+        enlargelimits=true,
+        tension=0.08]
+        \addplot[mark=none, orange, smooth cycle, thick, tension=1, dashed] coordinates {%
+   (0,0) (-1,1) (-2,2) (-1,3) (0, 3) (1, 4)};
+        \addplot[mark=none, blue, smooth cycle, thick, tension=3] coordinates {%
+   (0,0) (-1,1) (-2,2) (-1,3) (0, 3) (1, 4)};
+    \end{axis} 
+\end{tikzpicture}
+\end{document}