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z_pWLTH#+Y5^Ac`+T;{#HN+cqWWJW_HB41%vCK|k^*`Pw*SEjW}9k-*F*Eniy^tQEH z`ZwlCZ$Y&n^6i9XwNzR_jHX_NF7MLYdSUoD&3Lsmp(fF_UWMYf>(;z#MD5&SY0sP! z6QsvlGY@n|Yje#t!W&+Np!MJ9Z%2)QHjuHbN0jZonc7WrjUNr=m70lCbe3?53kqYk zWd|6wv^c83X#Mn;>jqH&NTk++?l^61V@G ulC??oyXn_9huEOct+;0Aec1lzR+vmd65JdY0X!?`*MA!_GBJ5^xc>!$UO(yp diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 5328ae3..3c9e445 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/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} diff --git a/documents/GeoTopo/figures/topology-oriented-triangle.tex b/documents/GeoTopo/figures/topology-oriented-triangle.tex new file mode 100644 index 0000000..6acfee8 --- /dev/null +++ b/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} diff --git a/documents/GeoTopo/figures/topology-parametric-surface-mapping.tex b/documents/GeoTopo/figures/topology-parametric-surface-mapping.tex new file mode 100644 index 0000000..891a5b6 --- /dev/null +++ b/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} diff --git a/tikz/topology-oriented-triangle/Makefile b/tikz/topology-oriented-triangle/Makefile new file mode 100644 index 0000000..0c1b58a --- /dev/null +++ b/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 diff --git a/tikz/topology-oriented-triangle/Readme.md b/tikz/topology-oriented-triangle/Readme.md new file mode 100644 index 0000000..1de15f1 --- /dev/null +++ b/tikz/topology-oriented-triangle/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-oriented-triangle.png) diff --git a/tikz/topology-oriented-triangle/topology-oriented-triangle.png b/tikz/topology-oriented-triangle/topology-oriented-triangle.png new file mode 100644 index 0000000000000000000000000000000000000000..7e535752f70eadb2515165adc490b5fcd22f8001 GIT binary patch literal 19146 zcmd43hg*|d-!+&*m1aaisS=t|7LF}~w1D(tiJ*cML5j2} z2r2&~y;#p!>B&5J zD2LOQyg71xhU351Uk;d_VLu)J*I$n~PWCIgHE|H15RgdZ^DF1%XlJ-i 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node[label=right:$e_1$] {} (c); + \draw[->, very thick] (c) edge node[label=left:$e_2$] {} (a); +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-parametric-surface-mapping/Makefile b/tikz/topology-parametric-surface-mapping/Makefile new file mode 100644 index 0000000..c3004a1 --- /dev/null +++ b/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 diff --git a/tikz/topology-parametric-surface-mapping/Readme.md b/tikz/topology-parametric-surface-mapping/Readme.md new file mode 100644 index 0000000..a76576a --- /dev/null +++ b/tikz/topology-parametric-surface-mapping/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-parametric-surface-mapping.png) diff --git a/tikz/topology-parametric-surface-mapping/topology-parametric-surface-mapping.png b/tikz/topology-parametric-surface-mapping/topology-parametric-surface-mapping.png new file mode 100644 index 0000000000000000000000000000000000000000..3439c9113cb80f8e48bcd8d5ede0840b5dec0c1d GIT binary patch literal 31576 zcmZU4c{tQx^#6<*yGYry8)S>@yHFUi%Mw{iqe+$!lXVcvZe$sI8Kkmg6tazc>|<#Z zreqDtHul~8-hIE%_phJl@yy$M9_QYB?z!jO^E$8d!pg$<91|ZC2n0F@H@R*D0?`1E 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z`FwlI?Dh@o7c4Z%#&v3(wJR;Mt(Vf?psuNzKb)22Kw5ok$wlZ#H*slE zG=3Dtxg~&)w=YNWL)2 S;R<{O8>FM9kEqwQi~c_ZX?4*6 literal 0 HcmV?d00001 diff --git a/tikz/topology-parametric-surface-mapping/topology-parametric-surface-mapping.tex b/tikz/topology-parametric-surface-mapping/topology-parametric-surface-mapping.tex new file mode 100644 index 0000000..d4389c3 --- /dev/null +++ b/tikz/topology-parametric-surface-mapping/topology-parametric-surface-mapping.tex @@ -0,0 +1,33 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{amsmath,amssymb}% math symbols / fonts +\usepackage{tikz} +\usepackage{tqft} +\usetikzlibrary{patterns} + +\begin{document} + \begin{tikzpicture} + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt] + \draw (0,0) ellipse (2cm and 1cm); + \def\ringa{(-0.3,0) circle (0.5cm)} + \def\ringb{(+0.3,0) circle (0.5cm)} + + \draw \ringa; + \draw[red] \ringb; + + %\node at (-1,0.3) {$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}