Viele kleine Verbesserungen

documents/GeoTopo/Kapitel1.tex

@@ -587,7 +587,7 @@ sodass $\pi$ stetig wird.
     \end{enumerate}
 \end{korollar}
 
-\begin{beweis}
+\begin{beweis}\leavevmode
     \begin{enumerate}[label=\alph*)]
         \item Sei $Z(x) = A_1 \cup A_2$ mit $A_i \neq \emptyset$ abgeschlossen,
               disjunkt.

documents/GeoTopo/Kapitel2.tex

@@ -162,8 +162,7 @@ U_i = \Set{(x_0: \dots : x_n) \in \mdp^n(\mdr) | x_i \neq 0} &\rightarrow \mdr^n
     \end{enumerate}
 \end{korollar}
 
-\begin{beweis}
-    von a und b:
+\begin{beweis}\leavevmode
     \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*]
         \item Sei $y \in \mdr^n \setminus V(F)$. Weil $F$ stetig ist,
               gibt es $\delta > 0$, sodass $F(\fB_\delta(y)) \subseteq \fB_\varepsilon(F(y))$
@@ -321,7 +320,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
         \item $f$ heißt \textbf{differenzierbar}
               (von Klasse $C^k$), wenn $f$ in jedem $x \in X$ 
               differenzierbar ist.
-        \item $f$ heißt \textbf{Diffieomorphismus}\xindex{Diffieomorphismus},
+        \item $f$ heißt \textbf{Diffeomorphismus}\xindex{Diffeomorphismus},
               wenn $f$ differenzierbar von Klasse $C^\infty$ ist und
               es eine differenzierbare Abbildung $g: Y \rightarrow X$
               von Klasse $C^\infty$ gibt mit $g \circ f = \text{id}_X$
@@ -347,7 +346,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
 
 \begin{beispiel}
     $f: \mdr \rightarrow \mdr, \;\;\; x \mapsto x^3$ ist kein
-    Diffieomorphismis, aber Homöomorphismus, da mit $g(x) := \sqrt[3]{x}$
+    Diffeomorphismis, aber Homöomorphismus, da mit $g(x) := \sqrt[3]{x}$
     gilt: $f \circ g = \text{id}_\mdr, \;\;\; g \circ f = \text{id}_\text{\mdr}$
 \end{beispiel}
 
@@ -535,7 +534,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
         h  &\mapsto g \cdot h
     \end{align*}
 
-    ein Diffieomorphismus.
+    ein Diffeomorphismus.
 \end{bemerkung}
 
 \section{Simplizialkomplex}
@@ -618,33 +617,33 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
 \begin{figure}[ht]
     \centering
     \subfloat[1D Simplizialkomplex]{
-        \parbox{4cm}{\centering\input{figures/topology-1-d-simplizialkomplex}}
+        \parbox[c][4cm]{4cm}{\centering\input{figures/topology-1-d-simplizialkomplex}}
         \label{fig:simplizialkomplex-1-d}
     }%
     \subfloat[2D Simplizialkomplex (ohne untere Fläche!)]{
-        \parbox{4cm}{\centering\input{figures/topology-pyramid.tex}}
+        \parbox[c][4cm]{4cm}{\centering\input{figures/topology-pyramid.tex}}
         \label{fig:simplizialkomplex-2-d}
     }%
     \subfloat[2D Simplizialkomplex]{
-        \parbox{5cm}{\centering\input{figures/topology-oktaeder.tex}}
+        \parbox[c][4cm]{5cm}{\centering\input{figures/topology-oktaeder.tex}}
         \label{fig:simplizialkomplex-2-d-okateder}
     }%
 
     \subfloat[1D Simplizialkomplex]{
-        \parbox{5cm}{\centering\input{figures/topology-cube.tex}}
+        \parbox[c][4cm]{5cm}{\centering\input{figures/topology-cube.tex}}
         \label{fig:simplizialkomplex-cube}
     }%
     \subfloat[2D Simplizialkomplex]{
-        \parbox{5cm}{\centering\input{figures/topology-cube-divided.tex}}
+        \parbox[c][4cm]{5cm}{\centering\input{figures/topology-cube-divided.tex}}
         \label{fig:simplizialkomplex-cube-divided}
     }
 
     \subfloat[$P$ ist kein Teilsimplex, da Eigenschaft (ii) verletzt ist]{
-        \parbox{5cm}{\centering\input{figures/topology-triangle-no-simplicial-complex.tex}}
+        \parbox[c][4cm]{5cm}{\centering\input{figures/topology-triangle-no-simplicial-complex.tex}}
         \label{fig:no-simplizialkomplex-triangles}
     }%
     \subfloat[Simplizialkomplex]{
-        \parbox{5cm}{\centering\input{figures/topology-triangle-simplicial-complex.tex}}
+        \parbox[c][4cm]{5cm}{\centering\input{figures/topology-triangle-simplicial-complex.tex}}
         \label{fig:simplizialkomplex-triangles}
     }%
     \label{fig:simplizialkomplexe}
@@ -733,20 +732,20 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
 \begin{figure}[ht]
     \centering
     \subfloat[Dies wird häufig auch als Multigraph bezeichnet.]{
-        \parbox{4cm}{\centering\input{figures/topology-graph-simple.tex}}
+        \parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-simple.tex}}
         \label{fig:topology-graph-simple}
     }%
     \subfloat[Planare Einbettung des Tetraeders]{
-        \parbox{4cm}{\centering\input{figures/topology-graph-tetraeder.tex}}
+        \parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-tetraeder.tex}}
         \label{fig:topology-graph-tetraeder}
     }
 
     \subfloat[$K_5$]{
-        \parbox{4cm}{\centering\input{figures/topology-graph-k-5.tex}}
+        \parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-k-5.tex}}
         \label{fig:k-5}
     }%
     \subfloat[$K_{3,3}$]{
-        \parbox{4cm}{\centering\input{figures/topology-graph-k-3-3.tex}}
+        \parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-k-3-3.tex}}
         \label{fig:k-3-3}
     }%
     \label{fig:graphen-beispiele}
@@ -827,11 +826,13 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
               eine eine Triangulierungen $T$, die sowohl um $T_1$ als
               auch um $T_2$ Verfeinerung ist.
 
-              \todo[inline]{Komische Zeichung}
+              \begin{center}
+                  \input{figures/topology-3.tex}\todo{Was bedeutet diese Zeichnung?}
+              \end{center}
 
               Nach Korollar~\ref{kor:simplex-unterteilung} ist
-              $\chi(\partial P_1) = \chi(T_1) = \chi(T) = \chi(T_2) = \chi(\partial P_2) = 2$.
-              Weil \obda{} $P_2$ ein Tetraeder ist.
+              $\chi(\partial P_1) = \chi(T_1) = \chi(T) = \chi(T_2) = \chi(\partial P_2) = 2$,
+              weil \obda{} $P_2$ ein Tetraeder ist.
     \end{enumerate}
 \end{beweis}
 

documents/GeoTopo/figures/topology-3.tex

@@ -0,0 +1,75 @@
+\newenvironment{customlegend}[1][]{%
+    \begingroup
+    % inits/clears the lists (which might be populated from previous
+    % axes):
+    \csname pgfplots@init@cleared@structures\endcsname
+    \pgfplotsset{#1}%
+}{%
+    % draws the legend:
+    \csname pgfplots@createlegend\endcsname
+    \endgroup
+}%
+
+% makes \addlegendimage available (typically only available within an
+% axis environment):
+\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
+
+%%--------------------------------
+
+% definition to insert numbers
+\pgfkeys{/pgfplots/number in legend/.style={%
+        /pgfplots/legend image code/.code={%
+            \node at (0.295,-0.0225){#1};
+        },%
+    },
+}
+
+\pgfdeclarelayer{background}
+\pgfdeclarelayer{foreground}
+\pgfsetlayers{background,main,foreground}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \tikzstyle{smallpoint}=[circle,thick,draw=red,fill=red,inner sep=0pt,minimum width=3pt,minimum height=3pt]
+    \begin{pgfonlayer}{foreground}
+        \node (a)[point] at (0,0) {};
+        \node (b)[point] at (2,0) {};
+        \node (c)[point] at (3,0.5) {};
+        \node (d)[point] at (0,3) {};
+        \node (e)[point] at (2,3) {};
+        \node (f)[point] at (3,1.5) {};
+        \node (g)[point] at (2,2.5) {};
+        \node (h)[point] at (1,1.5) {};
+    \end{pgfonlayer}
+
+    \draw (h.center) -- (e.center) -- (f.center) -- (b.center) -- cycle;
+    \draw (h.center) -- (f.center);
+
+    \draw[green, densely dashed] (a.center) -- (d.center) -- (g.center) -- (c.center) -- cycle;
+    \draw[green, densely dashed] (a.center) -- (g.center);
+
+    \begin{pgfonlayer}{foreground}
+        \node (x)[point, red,fill=red] at (1.79,0.31) {};
+        \node (x1)[smallpoint] at (1.2,1.5) {};
+        \node (x2)[smallpoint] at (1.71,2.56) {};
+        \node (x3)[smallpoint] at (2.5,1.5) {};
+        \node (x4)[smallpoint] at (2.72,1.06) {};
+    \end{pgfonlayer}
+    \draw[blue, densely dotted] (x.center) -- (x1.center);
+    \draw[blue, densely dotted] (x.center) -- (x2.center);
+    \draw[blue, densely dotted] (x.center) -- (x3.center);
+    \draw[blue, densely dotted] (x.center) -- (x4.center);
+
+
+    \begin{customlegend}[
+    legend entries={
+        $T_1$,
+        $T_2$,
+        $?$
+    },
+    legend style={at={(4.5,3.5)},font=\footnotesize}] % <= to define position and font legend
+    % the following are the "images" and numbers in the legend
+        \addlegendimage{black}
+        \addlegendimage{green,densely dashed}
+        \addlegendimage{blue, densely dotted}
+    \end{customlegend}
+\end{tikzpicture}

documents/GeoTopo/figures/topology-oktaeder.tex

@@ -5,7 +5,7 @@
     \node (c)[point] at (3,1) {};
     \node (d)[point] at (1,1) {};
     \node (e)[point] at (1.5,3) {};
-    \node (f)[point] at (1.5,-2) {};
+    \node (f)[point] at (1.5,-1) {};
     \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center);
     \draw (a.center) -- (e.center);
     \draw[dashed] (a.center) -- (d.center) -- (c.center);

tikz/topology-3/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-3
+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-3/Readme.md

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

tikz/topology-3/topology-3.tex

@@ -0,0 +1,83 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+\usetikzlibrary{calc,shadings}
+\usepackage{pgfplots}
+
+\begin{document}
+
+\newenvironment{customlegend}[1][]{%
+    \begingroup
+    % inits/clears the lists (which might be populated from previous
+    % axes):
+    \csname pgfplots@init@cleared@structures\endcsname
+    \pgfplotsset{#1}%
+}{%
+    % draws the legend:
+    \csname pgfplots@createlegend\endcsname
+    \endgroup
+}%
+
+% makes \addlegendimage available (typically only available within an
+% axis environment):
+\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
+
+%%--------------------------------
+
+% definition to insert numbers
+\pgfkeys{/pgfplots/number in legend/.style={%
+        /pgfplots/legend image code/.code={%
+            \node at (0.295,-0.0225){#1};
+        },%
+    },
+}
+
+\pgfdeclarelayer{background}
+\pgfdeclarelayer{foreground}
+\pgfsetlayers{background,main,foreground}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \tikzstyle{smallpoint}=[circle,thick,draw=red,fill=red,inner sep=0pt,minimum width=3pt,minimum height=3pt]
+    \begin{pgfonlayer}{foreground}
+        \node (a)[point] at (0,0) {};
+        \node (b)[point] at (2,0) {};
+        \node (c)[point] at (3,0.5) {};
+        \node (d)[point] at (0,3) {};
+        \node (e)[point] at (2,3) {};
+        \node (f)[point] at (3,1.5) {};
+        \node (g)[point] at (2,2.5) {};
+        \node (h)[point] at (1,1.5) {};
+    \end{pgfonlayer}
+
+    \draw (h.center) -- (e.center) -- (f.center) -- (b.center) -- cycle;
+    \draw (h.center) -- (f.center);
+
+    \draw[green, densely dashed] (a.center) -- (d.center) -- (g.center) -- (c.center) -- cycle;
+    \draw[green, densely dashed] (a.center) -- (g.center);
+
+    \begin{pgfonlayer}{foreground}
+        \node (x)[point, red,fill=red] at (1.79,0.31) {};
+        \node (x1)[smallpoint] at (1.2,1.5) {};
+        \node (x2)[smallpoint] at (1.71,2.56) {};
+        \node (x3)[smallpoint] at (2.5,1.5) {};
+        \node (x4)[smallpoint] at (2.72,1.06) {};
+    \end{pgfonlayer}
+    \draw[blue, densely dotted] (x.center) -- (x1.center);
+    \draw[blue, densely dotted] (x.center) -- (x2.center);
+    \draw[blue, densely dotted] (x.center) -- (x3.center);
+    \draw[blue, densely dotted] (x.center) -- (x4.center);
+
+
+    \begin{customlegend}[
+    legend entries={
+        $T_1$,
+        $T_2$,
+        $?$
+    },
+    legend style={at={(4.5,3.5)},font=\footnotesize}] % <= to define position and font legend
+    % the following are the "images" and numbers in the legend
+        \addlegendimage{black}
+        \addlegendimage{green,densely dashed}
+        \addlegendimage{blue, densely dotted}
+    \end{customlegend}
+\end{tikzpicture}
+\end{document}