tex-projects/LaTeX-examples
2
0
Fork 0
mirror of https://github.com/MartinThoma/LaTeX-examples.git synced 2025-04-26 06:48:04 +02:00
Code Activity

added many images

Browse source
This commit is contained in:
Martin Thoma 2013-12-03 14:59:22 +01:00
parent 8654f2d45e
commit a8e6ccd140
24 changed files with 357 additions and 7 deletions
Download patch file Download diff file Expand all files Collapse all files

2
documents/GeoTopo/Arbeitszeit.txt
View file

@ -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

BIN
documents/GeoTopo/GeoTopo.pdf
View file

Binary file not shown.

6
documents/GeoTopo/Kapitel2.tex
View file

@ -867,7 +867,9 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
\end{korollar}
\begin{beispiel}
\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 $Z_n \rightarrow H_n = Z_n / B_n: \dim Z_n = b_n + \dim B_n$

62
documents/GeoTopo/Kapitel3.tex
View file

@ -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}

10
documents/GeoTopo/figures/topology-circle-two-paths.tex Normal file
View file

@ -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}

16
documents/GeoTopo/figures/topology-homotop-paths.tex Normal file
View file

@ -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}

10
documents/GeoTopo/figures/topology-non-homotop-paths.tex Normal file
View file

@ -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}

27
documents/GeoTopo/figures/topology-paths-in-r2.tex Normal file
View file

@ -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}

31
tikz/topology-circle-two-paths/Makefile Normal file
View file

@ -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

3
tikz/topology-circle-two-paths/Readme.md Normal file
View file

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

BIN
tikz/topology-circle-two-paths/topology-circle-two-paths.png Normal file
View file

Binary file not shown.
Side by side

After

Width:  |  Height:  |  Size: 12 KiB

18
tikz/topology-circle-two-paths/topology-circle-two-paths.tex Normal file
View file

@ -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}

31
tikz/topology-homotop-paths/Makefile Normal file
View file

@ -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

3
tikz/topology-homotop-paths/Readme.md Normal file
View file

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

BIN
tikz/topology-homotop-paths/topology-homotop-paths.png Normal file
View file

Binary file not shown.
Side by side

After

Width:  |  Height:  |  Size: 27 KiB

23
tikz/topology-homotop-paths/topology-homotop-paths.tex Normal file
View file

@ -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}

31
tikz/topology-non-homotop-paths/Makefile Normal file
View file

@ -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

3
tikz/topology-non-homotop-paths/Readme.md Normal file
View file

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

BIN
tikz/topology-non-homotop-paths/topology-non-homotop-paths.png Normal file
View file

Binary file not shown.
Side by side

After

Width:  |  Height:  |  Size: 18 KiB

18
tikz/topology-non-homotop-paths/topology-non-homotop-paths.tex Normal file
View file

@ -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}

31
tikz/topology-paths-in-r2/Makefile Normal file
View file

@ -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

3
tikz/topology-paths-in-r2/Readme.md Normal file
View file

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

BIN
tikz/topology-paths-in-r2/topology-paths-in-r2.png Normal file
View file

Binary file not shown.
Side by side

After

Width:  |  Height:  |  Size: 24 KiB

34
tikz/topology-paths-in-r2/topology-paths-in-r2.tex Normal file
View file

@ -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}