diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 1023546..0ff91ed 100644 Binary files a/documents/GeoTopo/GeoTopo.pdf and b/documents/GeoTopo/GeoTopo.pdf differ diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 525dc88..4669a82 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -600,5 +600,135 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \caption{Beispiele für $k$-Simplexe} \end{figure} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Mitschrieb vom 21.11.2013 % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{definition} + \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \item Eine endliche Menge $K$ von Simplizes im $\mdr^n$ + heißt (endlicher) \textbf{Simplizialkomplex}\xindex{Simplizialkomplex}, + wenn gilt: + \begin{enumerate}[label=(\roman*)] + \item Für $\Delta \in K$ und $S \subseteq \Delta$ Teilsimplex + ist $S \in K$ + \item Für $\Delta_1, \Delta_2 \in K$ ist + $\Delta_1 \cap \Delta_2$ leer + oder ein Teilsimplex von $\Delta_1$ und von + $\Delta_2$ + \end{enumerate} + \item $|K| := \bigcup_{\Delta \in K} \Delta$ (mit Spurtoplogie) + heißt \textbf{geometrische Realisierung}\xindex{Realisierung!geometrische} + von $K$. + \item Ist $d = \max \Set{ k | K \text{ enthält } k-\text{Simplex}}$, + so heißt $d$ \textbf{Dimension}\xindex{Dimension} von + $K$. + \end{enumerate} +\end{definition} + +\xindex{Oktaeder}\xindex{Würfel} +\begin{figure}[ht] + \centering + \subfloat[1D Simplizialkomplex]{ + \parbox{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}} + \label{fig:simplizialkomplex-2-d} + }% + \subfloat[2D Simplizialkomplex]{ + \parbox{5cm}{\centering\input{figures/topology-oktaeder.tex}} + \label{fig:simplizialkomplex-2-d-okateder} + }% + + \subfloat[1D Simplizialkomplex]{ + \parbox{5cm}{\centering\input{figures/topology-cube.tex}} + \label{fig:simplizialkomplex-cube} + }% + \subfloat[2D Simplizialkomplex]{ + \parbox{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}} + \label{fig:no-simplizialkomplex-triangles} + }% + \subfloat[Simplizialkomplex]{ + \parbox{5cm}{\centering\input{figures/topology-triangle-simplicial-complex.tex}} + \label{fig:simplizialkomplex-triangles} + }% + \label{fig:simplizialkomplexe} + \caption{Beispiele für Simplizialkomplexe} +\end{figure} + +\begin{definition} + Seien $K, L$ Simplizialkomplexe. Eine stetige Abbildung + \[f:|K| \rightarrow |L|\] + heißt \textbf{simplizial}\xindex{Abbildung!simpliziale}, wenn für + jedes $\Delta \in K$ gilt: + \begin{enumerate}[label=(\roman*)] + \item $f(\Delta) \in L$ + \item $f|_{\Delta} : \Delta \rightarrow f(\Delta)$ ist eine + affine Abbildung. + \end{enumerate} +\end{definition} + +\begin{beispiel} + \begin{enumerate}[label=\arabic*)] + \item $\varphi(e_1) := b_1$, $\varphi(e_2) := b_2$\\ + $\varphi$ ist eine eindeutig bestimmte lineare Abbildung + + \input{figures/topology-linear-mapping.tex} + + \item Folgende Abbildung $\Delta^n \rightarrow \Delta^{n-1}$ + ist simplizial: + + \input{figures/topology-triangle-to-line.tex} + \item \todo[inline]{wozu dient das Beispiel?} + + \input{figures/topology-2.tex} + \end{enumerate} +\end{beispiel} + +\begin{definition} + Sei $K$ ein endlicher Simplizialkomplex. Für $n \geq 0$ sei + $a_n(K)$ die Anzahl der $n$-Simplizes in $K$. + + Dann heißt + \[\chi(K) := \sum_{k=0}^{\dim K} (-1)^n a_n(K)\] + \textbf{Eulerzahl}\xindex{Eulerzahl} (oder Euler-Charakteristik\index{Euler-Charakteristik|see{Eulerzahl}}) + von $K$. +\end{definition} + +\begin{beispiel} + \begin{enumerate}[label=\arabic*)] + \item $\chi(\Delta^1) = 2 - 1 = 1$\\ + $\chi(\Delta^2) = 3 - 3 + 1 = 1$\\ + $\chi(\Delta^3) = 4 - 6 + 4 - 1 = 1$ + \item $\chi(\text{Oktaeder-Oberfläche}) = 6 - 12 + 8 = 2$\\ + $\chi(\text{Rand des Tetraeders}) = 2$\\ + $\chi(\text{Ikosaeder}) = 12 - 30 + 20 = 2$ + \item $\chi(\text{Würfel}) = 8 - 12 + 6 = 2$\\ + $\chi(\text{Würfel, unterteilt in Dreiecksflächen}) = 8 - (12 + 6) + (6 \cdot 2) = 2$ + \end{enumerate} +\end{beispiel} + +\begin{korollar} + $\chi(\Delta^n) = 1$ für jedes $n \in \mdn_0$ +\end{korollar} + +\begin{beweis} + $\Delta^n$ ist die konvexe Hülle von $(e_0, \dots, e_n)$ in $\mdr^{n+1}$. + Jede $(k+1)$-elementige Teilmenge von $\Set{e_0, \dots, e_n}$ + definiert ein $k$-Simplex.\\ + $\Rightarrow a_k(\Delta^n) = \binom{n+1}{k+1}, \;\;\; k = 0, \dots, n$\\ + $\Rightarrow \chi(\Delta^n) = \sum_{k=0}^n (-1)^k \binom{n+1}{k+1}$\\ + $f(x) = (x+1)^{n+1} \stackrel{\substack{\tiny\text{Binomischer}\\\text{Lehrsatz}}}{=} \sum_{k=0}^{n+1} \binom{n+1}{k} x^k$\\ + $\Rightarrow 0 = \sum_{k=0}^{n+1} \binom{n+1}{k} (-1)^k = \chi(\Delta^n) -1$\\ + $\Rightarrow \chi(\Delta^n) = 1 \qed$ +\end{beweis} + % Die Übungsaufgaben sollen ganz am Ende des Kapitels sein. \input{Kapitel2-UB} diff --git a/documents/GeoTopo/Symbolverzeichnis.tex b/documents/GeoTopo/Symbolverzeichnis.tex index 8609354..ef7a236 100644 --- a/documents/GeoTopo/Symbolverzeichnis.tex +++ b/documents/GeoTopo/Symbolverzeichnis.tex @@ -257,6 +257,13 @@ sort=ZZZRang } +\newglossaryentry{Einschraeunkung} +{ + name={$f|_U$}, + description={$f$ eingeschränkt auf U}, + sort=ZZZEinschraenkung +} + % Setze den richtigen Namen für das Glossar \renewcommand*{\glossaryname}{\glossarName} \deftranslation{Glossary}{\glossarName} diff --git a/documents/GeoTopo/figures/topology-1-d-simplizialkomplex.tex b/documents/GeoTopo/figures/topology-1-d-simplizialkomplex.tex new file mode 100644 index 0000000..2ae6154 --- /dev/null +++ b/documents/GeoTopo/figures/topology-1-d-simplizialkomplex.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] at (0.4,0) {}; + \node (b)[point] at (1,1) {}; + \node (c)[point] at (2,1) {}; + \node (d)[point] at (2.6,0) {}; + \node (e)[point] at (2,-1) {}; + \node (f)[point] at (1,-1) {}; + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- (e.center) -- (f.center) -- cycle; +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-2.tex b/documents/GeoTopo/figures/topology-2.tex new file mode 100644 index 0000000..357a53f --- /dev/null +++ b/documents/GeoTopo/figures/topology-2.tex @@ -0,0 +1,78 @@ +\begin{tikzpicture}[scale=1.5] + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + \node (a)[point] at (0,0) {}; + \node (b)[point] at (1,0) {}; + \node (c)[point] at (2,0) {}; + + \node (d)[point] at (0,1) {}; + \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {}; + \node (f)[point] at (2,1) {}; + + \node (g)[point] at (0,2) {}; + \node (h)[point] at (1,2) {}; + \node (i)[point] at (2,2) {}; + + \draw (a.center) -- node[below] {$a$} (b.center); + \draw (b.center) -- node[below] {$b$} (c.center); + \draw (g.center) -- node[above] {$a$} (h.center); + \draw (h.center) -- node[above] {$b$} (i.center); + \draw (d.center) -- node[left] {$c$} (g.center); + \draw (d.center) -- node[left] {$d$} (a.center); + \draw (f.center) -- node[right] {$d$} (c.center); + \draw (i.center) -- node[right] {$c$} (f.center); + + \draw (a.center) -- (b.center) -- (M.center) -- (d.center) -- cycle; + \draw (b.center) -- (c.center) -- (f.center) -- (M.center) -- cycle; + \draw (d.center) -- (M.center) -- (h.center) -- (g.center) -- cycle; + \draw (M.center) -- (f.center) -- (i.center) -- (h.center) -- cycle; + + \draw[thick, blue] (d.center) -- (M.center) -- (f.center); + + \draw[thick, red] (a.center) -- (b.center); + \draw[thick, red] (h.center) -- (i.center); + + % Draw again so that lines are below nodes + \draw (a.center) -- (i.center); + \draw (c.center) -- (g.center); + + \node (a)[point] at (0,0) {}; + \node (b)[point] at (1,0) {}; + \node (c)[point] at (2,0) {}; + + \node (d)[point] at (0,1) {}; + \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {}; + \node (f)[point] at (2,1) {}; + + \node (g)[point] at (0,2) {}; + \node (h)[point] at (1,2) {}; + \node (i)[point] at (2,2) {}; + + \draw[->] (3,1) -- node[above] {Quotient nach} node[below] {Punktspiegelung} (5,1); + + \begin{scope}[xshift=6.5cm,yshift=1cm] + \node (w)[point] at (-1,0) {}; + \node (x)[point] at (0,-1) {}; + \node (y)[point] at (1,0) {}; + \node (z)[point] at (0,1) {}; + \node (m)[point] at (0,0) {}; + + \node (1) at (-0.3,+0.35) {1}; + \node (2) at (+0.3,+0.35) {2}; + \node (3) at (+0.3,-0.35) {3}; + \node (4) at (-0.3,-0.35) {4}; + + \draw (w.center) -- (x.center) -- (y.center) -- (z.center) -- cycle; + + \draw[thick, blue] (w.center) -- (m.center); + \draw (m.center) -- (y.center); + \draw (x.center) -- (m.center) -- (z.center); + \draw[thick, red] (z.center) -- (y.center) -- (x.center); + + % Draw again, as lines should be below points + \node (w)[point] at (-1,0) {}; + \node (x)[point] at (0,-1) {}; + \node (y)[point] at (1,0) {}; + \node (z)[point] at (0,1) {}; + \node (m)[point] at (0,0) {}; + \end{scope} +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-cube-divided.tex b/documents/GeoTopo/figures/topology-cube-divided.tex new file mode 100644 index 0000000..f298b41 --- /dev/null +++ b/documents/GeoTopo/figures/topology-cube-divided.tex @@ -0,0 +1,30 @@ +\begin{tikzpicture} + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + \node (a)[point] at (0,0) {}; + \node (b)[point] at (4,0) {}; + \node (c)[point] at (5,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \node (g)[point] at (5,3) {}; + \node (h)[point] at (1,3) {}; + \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle; + \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle; + \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle; + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (h.center); + + \draw[orange] (b.center) -- (e.center) -- (g.center); + \draw[orange,dashed] (a.center) -- (c.center) -- (h.center); + \draw[orange,dashed] (d.center) -- (e.center); + \draw[orange] (f.center) -- (c.center); + + \node (a)[point] at (0,0) {}; + \node (b)[point] at (4,0) {}; + \node (c)[point] at (5,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \node (g)[point] at (5,3) {}; + \node (h)[point] at (1,3) {}; +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-cube.tex b/documents/GeoTopo/figures/topology-cube.tex new file mode 100644 index 0000000..add1842 --- /dev/null +++ b/documents/GeoTopo/figures/topology-cube.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] at (0,0) {}; + \node (b)[point] at (4,0) {}; + \node (c)[point] at (5,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \node (g)[point] at (5,3) {}; + \node (h)[point] at (1,3) {}; + \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle; + \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle; + \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle; + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (h.center); +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-linear-mapping.tex b/documents/GeoTopo/figures/topology-linear-mapping.tex new file mode 100644 index 0000000..2ffbfc1 --- /dev/null +++ b/documents/GeoTopo/figures/topology-linear-mapping.tex @@ -0,0 +1,18 @@ +\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]180:$0$}] at (0,0) {}; + \node (b)[point,label={[label distance=0cm]0:$e_2$}] at (2.5,0) {}; + \node (c)[point,label={[label distance=0cm]90:$e_1$}] at (2,1) {}; + + \begin{scope}[xshift=5cm] + \node (d)[point,label={[label distance=0cm]180:$0$}] at (0,0) {}; + \node (e)[point,label={[label distance=0cm]0:$b_1$}] at (4,0) {}; + \node (f)[point,label={[label distance=0cm]90:$b_2$}] at (2,2) {}; + \end{scope} + + \draw (a.center) -- (b.center) -- (c.center) -- cycle; + \draw (d.center) -- (e.center) -- (f.center) -- cycle; + + \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (5,0.5); +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-oktaeder.tex b/documents/GeoTopo/figures/topology-oktaeder.tex new file mode 100644 index 0000000..2329aa5 --- /dev/null +++ b/documents/GeoTopo/figures/topology-oktaeder.tex @@ -0,0 +1,17 @@ +\begin{tikzpicture} + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + \node (a)[point] at (0,0) {}; + \node (b)[point] at (2,0) {}; + \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) {}; + \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center); + \draw (a.center) -- (e.center); + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (e.center); + + \draw (a.center) -- (f.center) -- (b.center); + \draw (f.center) -- (c.center); + \draw[dashed] (f.center) -- (d.center); +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-pyramid.tex b/documents/GeoTopo/figures/topology-pyramid.tex new file mode 100644 index 0000000..e9226ca --- /dev/null +++ b/documents/GeoTopo/figures/topology-pyramid.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] at (0,0) {}; + \node (b)[point] at (2,0) {}; + \node (c)[point] at (3,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (1.5,3) {}; + \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center); + \draw (a.center) -- (e.center); + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (e.center); +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-triangle-no-simplicial-complex.tex b/documents/GeoTopo/figures/topology-triangle-no-simplicial-complex.tex new file mode 100644 index 0000000..cef2b5f --- /dev/null +++ b/documents/GeoTopo/figures/topology-triangle-no-simplicial-complex.tex @@ -0,0 +1,18 @@ +\begin{tikzpicture} + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + \node (a)[point] at (0,0) {}; + \node (b)[point] at (3,0) {}; + \node (c)[point] at (2,2) {}; + + \begin{scope}[yshift=2cm] + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \end{scope} + + \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {}; + + \draw[pattern=north east lines] (a.center) -- (b.center) -- (c.center) -- cycle; + \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle; + \draw (p.center) -- (d.center); +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-triangle-simplicial-complex.tex b/documents/GeoTopo/figures/topology-triangle-simplicial-complex.tex new file mode 100644 index 0000000..d26d1f1 --- /dev/null +++ b/documents/GeoTopo/figures/topology-triangle-simplicial-complex.tex @@ -0,0 +1,20 @@ +\begin{tikzpicture} + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + \node (a)[point] at (0,0) {}; + \node (b)[point] at (3,0) {}; + \node (c)[point] at (2,2) {}; + + \begin{scope}[yshift=2cm] + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \end{scope} + + \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {}; + + \draw[pattern=north east lines] (a.center) -- (p.center) -- (b.center) -- cycle; + \draw[pattern=north west lines] (a.center) -- (p.center) -- (c.center) -- cycle; + \draw[pattern=vertical lines] (b.center) -- (p.center) -- (c.center) -- cycle; + \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle; + \draw (p.center) -- (d.center); +\end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-triangle-to-line.tex b/documents/GeoTopo/figures/topology-triangle-to-line.tex new file mode 100644 index 0000000..9b3986f --- /dev/null +++ b/documents/GeoTopo/figures/topology-triangle-to-line.tex @@ -0,0 +1,23 @@ +\begin{tikzpicture} + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + + \node (a)[point] at (0,0) {}; + \node (b)[point] at (2.5,0) {}; + \node (c)[point] at (2,1) {}; + + \begin{scope}[xshift=4.5cm, yshift=0.5cm] + \node (d)[point] at (0,0) {}; + \node (e)[point] at (2,0) {}; + \end{scope} + + \begin{scope}[xshift=1.5cm,yshift=0.6cm,rotate=-55] + \draw[->] (0,-0.1) -- (0.4,-0.1); + \draw[->] (0, 0.0) -- (0.5, 0.0); + \draw[->] (0, 0.1) -- (0.6, 0.1); + \end{scope} + + \draw (a.center) -- (b.center) -- (c.center) -- cycle; + \draw (d.center) -- (e.center); + + \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (4,0.5); +\end{tikzpicture} diff --git a/tikz/topology-1-d-simplizialkomplex/Makefile b/tikz/topology-1-d-simplizialkomplex/Makefile new file mode 100644 index 0000000..21e0374 --- /dev/null +++ b/tikz/topology-1-d-simplizialkomplex/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-1-d-simplizialkomplex +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-1-d-simplizialkomplex/Readme.md b/tikz/topology-1-d-simplizialkomplex/Readme.md new file mode 100644 index 0000000..8919feb --- /dev/null +++ b/tikz/topology-1-d-simplizialkomplex/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-1-d-simplizialkomplex.png) diff --git a/tikz/topology-1-d-simplizialkomplex/topology-1-d-simplizialkomplex.png b/tikz/topology-1-d-simplizialkomplex/topology-1-d-simplizialkomplex.png new file mode 100644 index 0000000..c50055e Binary files /dev/null and b/tikz/topology-1-d-simplizialkomplex/topology-1-d-simplizialkomplex.png differ diff --git a/tikz/topology-1-d-simplizialkomplex/topology-1-d-simplizialkomplex.tex b/tikz/topology-1-d-simplizialkomplex/topology-1-d-simplizialkomplex.tex new file mode 100644 index 0000000..e500a81 --- /dev/null +++ b/tikz/topology-1-d-simplizialkomplex/topology-1-d-simplizialkomplex.tex @@ -0,0 +1,15 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0.4,0) {}; + \node (b)[point] at (1,1) {}; + \node (c)[point] at (2,1) {}; + \node (d)[point] at (2.6,0) {}; + \node (e)[point] at (2,-1) {}; + \node (f)[point] at (1,-1) {}; + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- (e.center) -- (f.center) -- cycle; +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-2/Makefile b/tikz/topology-2/Makefile new file mode 100644 index 0000000..86d3228 --- /dev/null +++ b/tikz/topology-2/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-2 +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-2/Readme.md b/tikz/topology-2/Readme.md new file mode 100644 index 0000000..6f45766 --- /dev/null +++ b/tikz/topology-2/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-2.png) diff --git a/tikz/topology-2/topology-2.png b/tikz/topology-2/topology-2.png new file mode 100644 index 0000000..41a85ff Binary files /dev/null and b/tikz/topology-2/topology-2.png differ diff --git a/tikz/topology-2/topology-2.tex b/tikz/topology-2/topology-2.tex new file mode 100644 index 0000000..31df709 --- /dev/null +++ b/tikz/topology-2/topology-2.tex @@ -0,0 +1,83 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{tikz} + +\begin{document} +\begin{tikzpicture}[scale=1.5] + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + \node (a)[point] at (0,0) {}; + \node (b)[point] at (1,0) {}; + \node (c)[point] at (2,0) {}; + + \node (d)[point] at (0,1) {}; + \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {}; + \node (f)[point] at (2,1) {}; + + \node (g)[point] at (0,2) {}; + \node (h)[point] at (1,2) {}; + \node (i)[point] at (2,2) {}; + + \draw (a.center) -- node[below] {$a$} (b.center); + \draw (b.center) -- node[below] {$b$} (c.center); + \draw (g.center) -- node[above] {$a$} (h.center); + \draw (h.center) -- node[above] {$b$} (i.center); + \draw (d.center) -- node[left] {$c$} (g.center); + \draw (d.center) -- node[left] {$d$} (a.center); + \draw (f.center) -- node[right] {$d$} (c.center); + \draw (i.center) -- node[right] {$c$} (f.center); + + \draw (a.center) -- (b.center) -- (M.center) -- (d.center) -- cycle; + \draw (b.center) -- (c.center) -- (f.center) -- (M.center) -- cycle; + \draw (d.center) -- (M.center) -- (h.center) -- (g.center) -- cycle; + \draw (M.center) -- (f.center) -- (i.center) -- (h.center) -- cycle; + + \draw[thick, blue] (d.center) -- (M.center) -- (f.center); + + \draw[thick, red] (a.center) -- (b.center); + \draw[thick, red] (h.center) -- (i.center); + + % Draw again so that lines are below nodes + \draw (a.center) -- (i.center); + \draw (c.center) -- (g.center); + + \node (a)[point] at (0,0) {}; + \node (b)[point] at (1,0) {}; + \node (c)[point] at (2,0) {}; + + \node (d)[point] at (0,1) {}; + \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {}; + \node (f)[point] at (2,1) {}; + + \node (g)[point] at (0,2) {}; + \node (h)[point] at (1,2) {}; + \node (i)[point] at (2,2) {}; + + \draw[->] (3,1) -- node[above] {Quotient nach} node[below] {Punktspiegelung} (5,1); + + \begin{scope}[xshift=6.5cm,yshift=1cm] + \node (w)[point] at (-1,0) {}; + \node (x)[point] at (0,-1) {}; + \node (y)[point] at (1,0) {}; + \node (z)[point] at (0,1) {}; + \node (m)[point] at (0,0) {}; + + \node (1) at (-0.3,+0.35) {1}; + \node (2) at (+0.3,+0.35) {2}; + \node (3) at (+0.3,-0.35) {3}; + \node (4) at (-0.3,-0.35) {4}; + + \draw (w.center) -- (x.center) -- (y.center) -- (z.center) -- cycle; + + \draw[thick, blue] (w.center) -- (m.center); + \draw (m.center) -- (y.center); + \draw (x.center) -- (m.center) -- (z.center); + \draw[thick, red] (z.center) -- (y.center) -- (x.center); + + % Draw again, as lines should be below points + \node (w)[point] at (-1,0) {}; + \node (x)[point] at (0,-1) {}; + \node (y)[point] at (1,0) {}; + \node (z)[point] at (0,1) {}; + \node (m)[point] at (0,0) {}; + \end{scope} +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-cube-divided/Makefile b/tikz/topology-cube-divided/Makefile new file mode 100644 index 0000000..412efd2 --- /dev/null +++ b/tikz/topology-cube-divided/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-cube-divided +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-cube-divided/Readme.md b/tikz/topology-cube-divided/Readme.md new file mode 100644 index 0000000..09150f4 --- /dev/null +++ b/tikz/topology-cube-divided/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-cube-divided.png) diff --git a/tikz/topology-cube-divided/topology-cube-divided.png b/tikz/topology-cube-divided/topology-cube-divided.png new file mode 100644 index 0000000..e1bddce Binary files /dev/null and b/tikz/topology-cube-divided/topology-cube-divided.png differ diff --git a/tikz/topology-cube-divided/topology-cube-divided.tex b/tikz/topology-cube-divided/topology-cube-divided.tex new file mode 100644 index 0000000..bd15712 --- /dev/null +++ b/tikz/topology-cube-divided/topology-cube-divided.tex @@ -0,0 +1,35 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0,0) {}; + \node (b)[point] at (4,0) {}; + \node (c)[point] at (5,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \node (g)[point] at (5,3) {}; + \node (h)[point] at (1,3) {}; + \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle; + \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle; + \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle; + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (h.center); + + \draw[orange] (b.center) -- (e.center) -- (g.center); + \draw[orange,dashed] (a.center) -- (c.center) -- (h.center); + \draw[orange,dashed] (d.center) -- (e.center); + \draw[orange] (f.center) -- (c.center); + + \node (a)[point] at (0,0) {}; + \node (b)[point] at (4,0) {}; + \node (c)[point] at (5,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \node (g)[point] at (5,3) {}; + \node (h)[point] at (1,3) {}; +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-cube/Makefile b/tikz/topology-cube/Makefile new file mode 100644 index 0000000..5488254 --- /dev/null +++ b/tikz/topology-cube/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-cube +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-cube/Readme.md b/tikz/topology-cube/Readme.md new file mode 100644 index 0000000..3a0fa4a --- /dev/null +++ b/tikz/topology-cube/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-cube.png) diff --git a/tikz/topology-cube/topology-cube.png b/tikz/topology-cube/topology-cube.png new file mode 100644 index 0000000..ceb3aaa Binary files /dev/null and b/tikz/topology-cube/topology-cube.png differ diff --git a/tikz/topology-cube/topology-cube.tex b/tikz/topology-cube/topology-cube.tex new file mode 100644 index 0000000..f638fa0 --- /dev/null +++ b/tikz/topology-cube/topology-cube.tex @@ -0,0 +1,21 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0,0) {}; + \node (b)[point] at (4,0) {}; + \node (c)[point] at (5,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \node (g)[point] at (5,3) {}; + \node (h)[point] at (1,3) {}; + \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle; + \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle; + \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle; + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (h.center); +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-linear-mapping/Makefile b/tikz/topology-linear-mapping/Makefile new file mode 100644 index 0000000..f893acd --- /dev/null +++ b/tikz/topology-linear-mapping/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-linear-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-linear-mapping/Readme.md b/tikz/topology-linear-mapping/Readme.md new file mode 100644 index 0000000..0e87736 --- /dev/null +++ b/tikz/topology-linear-mapping/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-linear-mapping.png) diff --git a/tikz/topology-linear-mapping/topology-linear-mapping.png b/tikz/topology-linear-mapping/topology-linear-mapping.png new file mode 100644 index 0000000..5be43da Binary files /dev/null and b/tikz/topology-linear-mapping/topology-linear-mapping.png differ diff --git a/tikz/topology-linear-mapping/topology-linear-mapping.tex b/tikz/topology-linear-mapping/topology-linear-mapping.tex new file mode 100644 index 0000000..6cd8550 --- /dev/null +++ b/tikz/topology-linear-mapping/topology-linear-mapping.tex @@ -0,0 +1,23 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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={[label distance=0cm]180:$0$}] at (0,0) {}; + \node (b)[point,label={[label distance=0cm]0:$e_2$}] at (2.5,0) {}; + \node (c)[point,label={[label distance=0cm]90:$e_1$}] at (2,1) {}; + + \begin{scope}[xshift=5cm] + \node (d)[point,label={[label distance=0cm]180:$0$}] at (0,0) {}; + \node (e)[point,label={[label distance=0cm]0:$b_1$}] at (4,0) {}; + \node (f)[point,label={[label distance=0cm]90:$b_2$}] at (2,2) {}; + \end{scope} + + \draw (a.center) -- (b.center) -- (c.center) -- cycle; + \draw (d.center) -- (e.center) -- (f.center) -- cycle; + + \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (5,0.5); +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-oktaeder/Makefile b/tikz/topology-oktaeder/Makefile new file mode 100644 index 0000000..9ec0fa5 --- /dev/null +++ b/tikz/topology-oktaeder/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-oktaeder +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-oktaeder/Readme.md b/tikz/topology-oktaeder/Readme.md new file mode 100644 index 0000000..aa86fb9 --- /dev/null +++ b/tikz/topology-oktaeder/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-oktaeder.png) diff --git a/tikz/topology-oktaeder/topology-oktaeder.png b/tikz/topology-oktaeder/topology-oktaeder.png new file mode 100644 index 0000000..ae22293 Binary files /dev/null and b/tikz/topology-oktaeder/topology-oktaeder.png differ diff --git a/tikz/topology-oktaeder/topology-oktaeder.tex b/tikz/topology-oktaeder/topology-oktaeder.tex new file mode 100644 index 0000000..7dfe710 --- /dev/null +++ b/tikz/topology-oktaeder/topology-oktaeder.tex @@ -0,0 +1,22 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0,0) {}; + \node (b)[point] at (2,0) {}; + \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) {}; + \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center); + \draw (a.center) -- (e.center); + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (e.center); + + \draw (a.center) -- (f.center) -- (b.center); + \draw (f.center) -- (c.center); + \draw[dashed] (f.center) -- (d.center); +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-pyramid/Makefile b/tikz/topology-pyramid/Makefile new file mode 100644 index 0000000..53e0038 --- /dev/null +++ b/tikz/topology-pyramid/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-pyramid +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-pyramid/Readme.md b/tikz/topology-pyramid/Readme.md new file mode 100644 index 0000000..72fe584 --- /dev/null +++ b/tikz/topology-pyramid/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-pyramid.png) diff --git a/tikz/topology-pyramid/topology-pyramid.png b/tikz/topology-pyramid/topology-pyramid.png new file mode 100644 index 0000000..81653fa Binary files /dev/null and b/tikz/topology-pyramid/topology-pyramid.png differ diff --git a/tikz/topology-pyramid/topology-pyramid.tex b/tikz/topology-pyramid/topology-pyramid.tex new file mode 100644 index 0000000..eacf864 --- /dev/null +++ b/tikz/topology-pyramid/topology-pyramid.tex @@ -0,0 +1,17 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0,0) {}; + \node (b)[point] at (2,0) {}; + \node (c)[point] at (3,1) {}; + \node (d)[point] at (1,1) {}; + \node (e)[point] at (1.5,3) {}; + \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center); + \draw (a.center) -- (e.center); + \draw[dashed] (a.center) -- (d.center) -- (c.center); + \draw[dashed] (d.center) -- (e.center); +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-triangle-no-simplicial-complex/Makefile b/tikz/topology-triangle-no-simplicial-complex/Makefile new file mode 100644 index 0000000..578a130 --- /dev/null +++ b/tikz/topology-triangle-no-simplicial-complex/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-triangle-no-simplicial-complex +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-triangle-no-simplicial-complex/Readme.md b/tikz/topology-triangle-no-simplicial-complex/Readme.md new file mode 100644 index 0000000..1fe6034 --- /dev/null +++ b/tikz/topology-triangle-no-simplicial-complex/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-triangle-no-simplicial-complex.png) diff --git a/tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.png b/tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.png new file mode 100644 index 0000000..0dfcada Binary files /dev/null and b/tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.png differ diff --git a/tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.tex b/tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.tex new file mode 100644 index 0000000..30614ea --- /dev/null +++ b/tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.tex @@ -0,0 +1,24 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0,0) {}; + \node (b)[point] at (3,0) {}; + \node (c)[point] at (2,2) {}; + + \begin{scope}[yshift=2cm] + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \end{scope} + + \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {}; + + \draw[pattern=north east lines] (a.center) -- (b.center) -- (c.center) -- cycle; + \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle; + \draw (p.center) -- (d.center); +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-triangle-on-line/Makefile b/tikz/topology-triangle-on-line/Makefile new file mode 100644 index 0000000..082d019 --- /dev/null +++ b/tikz/topology-triangle-on-line/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-triangle-on-line +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-triangle-on-line/Readme.md b/tikz/topology-triangle-on-line/Readme.md new file mode 100644 index 0000000..b5028d7 --- /dev/null +++ b/tikz/topology-triangle-on-line/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-triangle-on-line.png) diff --git a/tikz/topology-triangle-on-line/topology-triangle-on-line.png b/tikz/topology-triangle-on-line/topology-triangle-on-line.png new file mode 100644 index 0000000..c170553 Binary files /dev/null and b/tikz/topology-triangle-on-line/topology-triangle-on-line.png differ diff --git a/tikz/topology-triangle-on-line/topology-triangle-on-line.tex b/tikz/topology-triangle-on-line/topology-triangle-on-line.tex new file mode 100644 index 0000000..3c5ea19 --- /dev/null +++ b/tikz/topology-triangle-on-line/topology-triangle-on-line.tex @@ -0,0 +1,28 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0,0) {}; + \node (b)[point] at (2.5,0) {}; + \node (c)[point] at (2,1) {}; + + \begin{scope}[xshift=4.5cm, yshift=0.5cm] + \node (d)[point] at (0,0) {}; + \node (e)[point] at (2,0) {}; + \end{scope} + + \begin{scope}[xshift=1.5cm,yshift=0.6cm,rotate=-55] + \draw[->] (0,-0.1) -- (0.4,-0.1); + \draw[->] (0, 0.0) -- (0.5, 0.0); + \draw[->] (0, 0.1) -- (0.6, 0.1); + \end{scope} + + \draw (a.center) -- (b.center) -- (c.center) -- cycle; + \draw (d.center) -- (e.center); + + \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (4,0.5); +\end{tikzpicture} +\end{document} diff --git a/tikz/topology-triangle-simplicial-complex/Makefile b/tikz/topology-triangle-simplicial-complex/Makefile new file mode 100644 index 0000000..232b136 --- /dev/null +++ b/tikz/topology-triangle-simplicial-complex/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-triangle-simplicial-complex +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-triangle-simplicial-complex/Readme.md b/tikz/topology-triangle-simplicial-complex/Readme.md new file mode 100644 index 0000000..d9d0364 --- /dev/null +++ b/tikz/topology-triangle-simplicial-complex/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-triangle-simplicial-complex.png) diff --git a/tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.png b/tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.png new file mode 100644 index 0000000..febea0b Binary files /dev/null and b/tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.png differ diff --git a/tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.tex b/tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.tex new file mode 100644 index 0000000..a9a08f9 --- /dev/null +++ b/tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.tex @@ -0,0 +1,26 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\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] at (0,0) {}; + \node (b)[point] at (3,0) {}; + \node (c)[point] at (2,2) {}; + + \begin{scope}[yshift=2cm] + \node (d)[point] at (1,1) {}; + \node (e)[point] at (0,2) {}; + \node (f)[point] at (4,2) {}; + \end{scope} + + \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {}; + + \draw[pattern=north east lines] (a.center) -- (p.center) -- (b.center) -- cycle; + \draw[pattern=north west lines] (a.center) -- (p.center) -- (c.center) -- cycle; + \draw[pattern=vertical lines] (b.center) -- (p.center) -- (c.center) -- cycle; + \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle; + \draw (p.center) -- (d.center); +\end{tikzpicture} +\end{document}