diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 7d147f3..ad060fc 100644 Binary files a/documents/GeoTopo/GeoTopo.pdf and b/documents/GeoTopo/GeoTopo.pdf differ diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 6259a57..9b12abe 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/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. diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 0d85a70..45279c9 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/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} diff --git a/documents/GeoTopo/figures/topology-3.tex b/documents/GeoTopo/figures/topology-3.tex new file mode 100644 index 0000000..51a47a4 --- /dev/null +++ b/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} diff --git a/documents/GeoTopo/figures/topology-oktaeder.tex b/documents/GeoTopo/figures/topology-oktaeder.tex index 2329aa5..8b1fa71 100644 --- a/documents/GeoTopo/figures/topology-oktaeder.tex +++ b/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); diff --git a/tikz/topology-3/Makefile b/tikz/topology-3/Makefile new file mode 100644 index 0000000..9d82caa --- /dev/null +++ b/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 diff --git a/tikz/topology-3/Readme.md b/tikz/topology-3/Readme.md new file mode 100644 index 0000000..1ac6295 --- /dev/null +++ b/tikz/topology-3/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-3.png) diff --git a/tikz/topology-3/topology-3.png b/tikz/topology-3/topology-3.png new file mode 100644 index 0000000..ebe1779 Binary files /dev/null and b/tikz/topology-3/topology-3.png differ diff --git a/tikz/topology-3/topology-3.tex b/tikz/topology-3/topology-3.tex new file mode 100644 index 0000000..aef0d50 --- /dev/null +++ b/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}