From 84416324f51da7a247deee185fbff3c418aa38a5 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Fri, 21 Feb 2014 12:48:06 +0100 Subject: [PATCH] Triangulierung des Torus --- documents/GeoTopo/GeoTopo.pdf | Bin 5227666 -> 5229545 bytes documents/GeoTopo/Kapitel2.tex | 34 ++++++++++ .../GeoTopo/definitions/definitionen.pdf | Bin 361595 -> 361595 bytes .../figures/torus-invalid-triangulation-1.tex | 32 ++++++++++ .../figures/torus-invalid-triangulation-2.tex | 39 ++++++++++++ .../figures/torus-triangulation-minimal.tex | 53 ++++++++++++++++ .../GeoTopo/figures/torus-triangulation.tex | 51 +++++++++++++++ .../GeoTopo/other-formats/GeoTopo-A5.pdf | Bin 5280302 -> 5282698 bytes tikz/torus-invalid-triangulation-1/Makefile | 31 +++++++++ tikz/torus-invalid-triangulation-1/Readme.md | 3 + .../torus-invalid-triangulation-1.png | Bin 0 -> 13226 bytes .../torus-invalid-triangulation-1.tex | 42 +++++++++++++ tikz/torus-invalid-triangulation-2/Makefile | 31 +++++++++ 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b/documents/GeoTopo/Kapitel2.tex index 21bacf6..955ac1e 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -871,6 +871,40 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch Dann heißt $h$ eine \textbf{Triangulierung} von $X$. \end{definition} +\begin{beispiel}[Triangulierung des Torus]\xindex{Torus}% + Für eine Triangulierung des Torus werden mindestens 14 Dreiecke benötigt. + Beispiele für fehlerhafte \enquote{Triangulierungen} sind in \cref{fig:torus-triangulierung-fails} + zu sehen. Korrekte Triangulierungen sind in \cref{fig:torus-triangulierung}. + + \begin{figure}[htp] + \centering + \subfloat[Die beiden markierten Dreiecke schneiden sich im Mittelpunkt und in einer Seite.]{ + \resizebox{0.45\linewidth}{!}{\input{figures/torus-invalid-triangulation-1.tex}} + \label{fig:torus-triangulierung-fail-1} + }% + \subfloat[Die beiden markierten Dreiecke schneiden sich im Mittelpunkt und außen.]{ + \resizebox{0.45\linewidth}{!}{\input{figures/torus-invalid-triangulation-2.tex}} + \label{fig:torus-triangulierung-fail-2} + }% + \label{fig:torus-triangulierung-fails} + \caption{Fehlerhafte Triangulierungen} + \end{figure} + + \begin{figure}[htp] + \centering + \subfloat[Einfache Triangulierung]{ + \resizebox{0.45\linewidth}{!}{\input{figures/torus-triangulation.tex}} + \label{fig:torus-triangulierung-simple} + }% + \subfloat[Minimale Triangulierung]{ + \resizebox{0.45\linewidth}{!}{\input{figures/torus-triangulation-minimal.tex}} + \label{fig:torus-triangulierung-minimal} + }% + \label{fig:torus-triangulierung} + \caption{Triangulierungen des Torus} + \end{figure} +\end{beispiel} + \begin{satz}[Eulersche Polyederformel]\xindex{Eulersche Polyederformel}% Sei $P$ ein konvexes Polyeder in $\mdr^3$, d.~h. $\partial P$ ist ein 2-dimensionaler Simplizialkomplex, sodass gilt: diff --git a/documents/GeoTopo/definitions/definitionen.pdf b/documents/GeoTopo/definitions/definitionen.pdf index 6f4690c8e61a88e39e282db88637cb393527e482..8ae548ded27b2583faad1b7eca9d8b5b2dae47b0 100644 GIT binary patch delta 116 zcmey}A@;jNtf7Umg{g(Pg=Gutk#bfeV>2U*=_kusMYzQgYo>k8($O-`IZXaX- delta 116 zcmey}A@;jNtf7Umg{g(Pg=Gutk#bf;V?$%3=_kusk8($O-`7ZXZ|x diff --git a/documents/GeoTopo/figures/torus-invalid-triangulation-1.tex b/documents/GeoTopo/figures/torus-invalid-triangulation-1.tex new file mode 100644 index 0000000..c0ec25b --- /dev/null +++ b/documents/GeoTopo/figures/torus-invalid-triangulation-1.tex @@ -0,0 +1,32 @@ +\begin{tikzpicture} + \node (a) at (0,0) {}; + \node (b) at (1,0) {}; + \node (c) at (1,1) {}; + \node (d) at (0,1) {}; + \coordinate (m) at ($(a)!0.5!(c)$); + \coordinate (ab) at ($(a)!0.5!(b)$); + \coordinate (bc) at ($(b)!0.5!(c)$); + \coordinate (cd) at ($(c)!0.5!(d)$); + \coordinate (ad) at ($(a)!0.5!(d)$); + \draw[pattern=north west lines] (d.center) -- (ad.center) -- (m.center); + \draw[pattern=dots] (m.center) -- (bc.center) -- (c.center); + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- cycle; + \begin{scope}[decoration={ + markings, + mark=at position 0.6 with {\arrow{>}}} + ] + \draw[postaction={decorate}] (a.center) -- (b.center); + \draw[postaction={decorate}] (d.center) -- (c.center); + \end{scope} + + \begin{scope}[decoration={ + markings, + mark=at position 0.55 with {\arrow{>>}}} + ] + \draw[postaction={decorate}] (b.center) -- (c.center); + \draw[postaction={decorate}] (a.center) -- (d.center); + \end{scope} + \draw (ab.center) -- (cd.center); + \draw (a.center) -- (c.center); + \draw (b.center) -- (d.center); +\end{tikzpicture} \ No newline at end of file diff --git a/documents/GeoTopo/figures/torus-invalid-triangulation-2.tex b/documents/GeoTopo/figures/torus-invalid-triangulation-2.tex new file mode 100644 index 0000000..c63ff05 --- /dev/null +++ b/documents/GeoTopo/figures/torus-invalid-triangulation-2.tex @@ -0,0 +1,39 @@ +\begin{tikzpicture} + \node (a) at (0,0) {}; + \node (b) at (1,0) {}; + \node (c) at (1,1) {}; + \node (d) at (0,1) {}; + \coordinate (m) at ($(a)!0.5!(c)$); + \coordinate (ab) at ($(a)!0.5!(b)$); + \coordinate (bc) at ($(b)!0.5!(c)$); + \coordinate (cd) at ($(c)!0.5!(d)$); + \coordinate (ad) at ($(a)!0.5!(d)$); + \coordinate (left-intersection) at ($(m)!0.5!(d)$); + \coordinate (right-intersection) at ($(m)!0.5!(c)$); + \draw[pattern=north west lines] (ad.center) -- (left-intersection.center) -- (m.center); + \draw[pattern=dots] (m.center) -- (right-intersection.center) -- (bc.center); + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- cycle; + \draw (bc.center) -- (ad.center); + \draw (cd.center) -- (ad.center); + \draw (cd.center) -- (bc.center); + \draw (ad.center) -- (ab.center); + \draw (ab.center) -- (bc.center); + \begin{scope}[decoration={ + markings, + mark=at position 0.6 with {\arrow{>}}} + ] + \draw[postaction={decorate}] (a.center) -- (b.center); + \draw[postaction={decorate}] (d.center) -- (c.center); + \end{scope} + + \begin{scope}[decoration={ + markings, + mark=at position 0.55 with {\arrow{>>}}} + ] + \draw[postaction={decorate}] (b.center) -- (c.center); + \draw[postaction={decorate}] (a.center) -- (d.center); + \end{scope} + \draw (ab.center) -- (cd.center); + \draw (a.center) -- (c.center); + \draw (b.center) -- (d.center); +\end{tikzpicture} \ No newline at end of file diff --git a/documents/GeoTopo/figures/torus-triangulation-minimal.tex b/documents/GeoTopo/figures/torus-triangulation-minimal.tex new file mode 100644 index 0000000..bccb0a5 --- /dev/null +++ b/documents/GeoTopo/figures/torus-triangulation-minimal.tex @@ -0,0 +1,53 @@ +\begin{tikzpicture} + \node (a) at (0,0) {}; + \node (b) at (1,0) {}; + \node (c) at (1,1) {}; + \node (d) at (0,1) {}; + \coordinate (m) at ($(a)!0.5!(c)$); + \coordinate (ab2) at ($(a)!0.5!(b)$); + \coordinate (ab1) at ($(a)!0.5!(ab2)$); + \coordinate (ab3) at ($(b)!0.5!(ab2)$); + \coordinate (bc2) at ($(b)!0.5!(c)$); + \coordinate (bc1) at ($(b)!0.5!(bc2)$); + \coordinate (bc3) at ($(c)!0.5!(bc2)$); + \coordinate (cd2) at ($(c)!0.5!(d)$); + \coordinate (cd1) at ($(c)!0.5!(cd2)$); + \coordinate (cd3) at ($(d)!0.5!(cd2)$); + \coordinate (ad2) at ($(a)!0.5!(d)$); + \coordinate (ad1) at ($(a)!0.5!(ad2)$); + \coordinate (ad3) at ($(d)!0.5!(ad2)$); + \coordinate (corner1) at ($(a)!0.25!(bc3)$); + \coordinate (corner2) at ($(c)!0.25!(ad1)$); + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- cycle; + %horizontal + \draw (a.center) -- (bc3.center); + \draw (cd3.center) -- (ad3.center); + \draw (ab3.center) -- (bc1.center); + \draw (cd3.center) -- (ad1.center); + \draw (ab3.center) -- (bc3.center); + \draw (ab1.center) -- (corner1.center); + \draw (ad1.center) -- (corner1.center); + \draw (ab3.center) -- (corner1.center); + \draw (c.center) -- (corner2.center); + \draw (cd1.center) -- (corner2.center); + \draw (bc3.center) -- (corner2.center); + \draw (cd3.center) -- (corner2.center); + \draw (corner1.center) -- (corner2.center); + \draw (ad1.center) -- (corner2.center); + + \begin{scope}[decoration={ + markings, + mark=at position 0.6 with {\arrow{>}}} + ] + \draw[postaction={decorate}] (a.center) -- (b.center); + \draw[postaction={decorate}] (d.center) -- (c.center); + \end{scope} + + \begin{scope}[decoration={ + markings, + mark=at position 0.55 with {\arrow{>>}}} + ] + \draw[postaction={decorate}] (b.center) -- (c.center); + \draw[postaction={decorate}] (a.center) -- (d.center); + \end{scope} +\end{tikzpicture} \ No newline at end of file diff --git a/documents/GeoTopo/figures/torus-triangulation.tex b/documents/GeoTopo/figures/torus-triangulation.tex new file mode 100644 index 0000000..62db400 --- /dev/null +++ b/documents/GeoTopo/figures/torus-triangulation.tex @@ -0,0 +1,51 @@ +\begin{tikzpicture} + \node (a) at (0,0) {}; + \node (b) at (1,0) {}; + \node (c) at (1,1) {}; + \node (d) at (0,1) {}; + \coordinate (m) at ($(a)!0.5!(c)$); + \coordinate (ab2) at ($(a)!0.5!(b)$); + \coordinate (ab1) at ($(a)!0.5!(ab2)$); + \coordinate (ab3) at ($(b)!0.5!(ab2)$); + \coordinate (bc2) at ($(b)!0.5!(c)$); + \coordinate (bc1) at ($(b)!0.5!(bc2)$); + \coordinate (bc3) at ($(c)!0.5!(bc2)$); + \coordinate (cd2) at ($(c)!0.5!(d)$); + \coordinate (cd1) at ($(c)!0.5!(cd2)$); + \coordinate (cd3) at ($(d)!0.5!(cd2)$); + \coordinate (ad2) at ($(a)!0.5!(d)$); + \coordinate (ad1) at ($(a)!0.5!(ad2)$); + \coordinate (ad3) at ($(d)!0.5!(ad2)$); + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- cycle; + %horizontal + \draw (bc1.center) -- (ad1.center); + \draw (bc2.center) -- (ad2.center); + \draw (bc3.center) -- (ad3.center); + %vertical + \draw (ab1.center) -- (cd3.center); + \draw (ab2.center) -- (cd2.center); + \draw (ab3.center) -- (cd1.center); + %diagonal + \draw (ad3.center) -- (cd3.center); + \draw (ad2.center) -- (cd2.center); + \draw (ad1.center) -- (cd1.center); + \draw (a.center) -- (c.center); + \draw (ab1.center) -- (bc3.center); + \draw (ab2.center) -- (bc2.center); + \draw (ab3.center) -- (bc1.center); + \begin{scope}[decoration={ + markings, + mark=at position 0.6 with {\arrow{>}}} + ] + \draw[postaction={decorate}] (a.center) -- (b.center); + \draw[postaction={decorate}] (d.center) -- (c.center); + \end{scope} + + \begin{scope}[decoration={ + markings, + mark=at position 0.55 with {\arrow{>>}}} + ] + \draw[postaction={decorate}] (b.center) -- (c.center); + \draw[postaction={decorate}] (a.center) -- (d.center); + \end{scope} +\end{tikzpicture} \ No newline at end of file diff --git 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{}; + \node (c) at (1,1) {}; + \node (d) at (0,1) {}; + \coordinate (m) at ($(a)!0.5!(c)$); + \coordinate (ab) at ($(a)!0.5!(b)$); + \coordinate (bc) at ($(b)!0.5!(c)$); + \coordinate (cd) at ($(c)!0.5!(d)$); + \coordinate (ad) at ($(a)!0.5!(d)$); + \draw[pattern=north west lines] (d.center) -- (ad.center) -- (m.center); + \draw[pattern=dots] (m.center) -- (bc.center) -- (c.center); + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- cycle; + \begin{scope}[decoration={ + markings, + mark=at position 0.6 with {\arrow{>}}} + ] + \draw[postaction={decorate}] (a.center) -- (b.center); + \draw[postaction={decorate}] (d.center) -- (c.center); + \end{scope} + + \begin{scope}[decoration={ + markings, + mark=at position 0.55 with {\arrow{>>}}} + ] + \draw[postaction={decorate}] (b.center) -- (c.center); + \draw[postaction={decorate}] (a.center) -- (d.center); + \end{scope} + \draw (ab.center) -- (cd.center); + \draw (a.center) -- (c.center); + \draw (b.center) -- (d.center); + + %\draw[pattern=north west lines] (a.center) -- (b.center) -- (z.center) --cycle; + %\draw[pattern=dots] (b.center) -- (c.center) -- (z.center) --cycle; + %\draw[pattern=crosshatch] (a.center) -- (c.center) -- (z.center) --cycle; +\end{tikzpicture} +\end{document} diff --git a/tikz/torus-invalid-triangulation-2/Makefile b/tikz/torus-invalid-triangulation-2/Makefile new file mode 100644 index 0000000..4d96bd9 --- /dev/null +++ b/tikz/torus-invalid-triangulation-2/Makefile @@ -0,0 +1,31 @@ +SOURCE = torus-invalid-triangulation-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/torus-invalid-triangulation-2/Readme.md b/tikz/torus-invalid-triangulation-2/Readme.md new file mode 100644 index 0000000..161b3ac --- /dev/null +++ b/tikz/torus-invalid-triangulation-2/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](torus-invalid-triangulation-2.png) diff --git a/tikz/torus-invalid-triangulation-2/torus-invalid-triangulation-2.png b/tikz/torus-invalid-triangulation-2/torus-invalid-triangulation-2.png new file mode 100644 index 0000000000000000000000000000000000000000..64ea452b0462e51bed9b422bc954bb37a7504778 GIT binary patch literal 12951 zcmdVBd00|i7dN~)QqrVSmS#DjXjwUAT4^d3rZ}XPmQ6}!Sa(k4Sf*g30}hd6gIdv@ z*-Qtw zf4#?ysRmO405d3_yDOWdUH$UP*~!If<2#?Ol`Oy%mV(gfVP)^JTqRL7XehEh*C 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E0F?a|S^xk5 literal 0 HcmV?d00001 diff --git a/tikz/torus-invalid-triangulation-2/torus-invalid-triangulation-2.tex b/tikz/torus-invalid-triangulation-2/torus-invalid-triangulation-2.tex new file mode 100644 index 0000000..ef0b2a9 --- /dev/null +++ b/tikz/torus-invalid-triangulation-2/torus-invalid-triangulation-2.tex @@ -0,0 +1,45 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{tikz} +\usetikzlibrary{patterns,calc,decorations.markings} + +\begin{document} +\begin{tikzpicture} + \node (a) at (0,0) {}; + \node (b) at (1,0) {}; + \node (c) at (1,1) {}; + \node (d) at (0,1) {}; + \coordinate (m) at ($(a)!0.5!(c)$); + \coordinate (ab) at ($(a)!0.5!(b)$); + \coordinate (bc) at ($(b)!0.5!(c)$); + \coordinate (cd) at ($(c)!0.5!(d)$); + \coordinate (ad) at ($(a)!0.5!(d)$); + \coordinate (left-intersection) at ($(m)!0.5!(d)$); + \coordinate (right-intersection) at ($(m)!0.5!(c)$); + \draw[pattern=north west lines] (ad.center) -- (left-intersection.center) -- (m.center); + \draw[pattern=dots] (m.center) -- (right-intersection.center) -- (bc.center); + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- cycle; + \draw (bc.center) -- (ad.center); + \draw (cd.center) -- (ad.center); + \draw (cd.center) -- (bc.center); + \draw (ad.center) -- (ab.center); + \draw (ab.center) -- (bc.center); + \begin{scope}[decoration={ + markings, + mark=at position 0.6 with {\arrow{>}}} + ] + \draw[postaction={decorate}] (a.center) -- (b.center); + \draw[postaction={decorate}] (d.center) -- (c.center); + \end{scope} + + \begin{scope}[decoration={ + markings, + mark=at position 0.55 with {\arrow{>>}}} + ] + \draw[postaction={decorate}] (b.center) -- (c.center); + \draw[postaction={decorate}] (a.center) -- (d.center); + \end{scope} + \draw (ab.center) -- (cd.center); + \draw (a.center) -- (c.center); + \draw (b.center) -- (d.center); +\end{tikzpicture} +\end{document} diff --git a/tikz/torus-triangulation-minimal/Makefile b/tikz/torus-triangulation-minimal/Makefile new file mode 100644 index 0000000..c51b548 --- /dev/null +++ b/tikz/torus-triangulation-minimal/Makefile @@ -0,0 +1,31 @@ +SOURCE = torus-triangulation-minimal +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/torus-triangulation-minimal/Readme.md b/tikz/torus-triangulation-minimal/Readme.md new file mode 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+\usetikzlibrary{patterns,calc,decorations.markings} + +\begin{document} +\begin{tikzpicture} + \node (a) at (0,0) {}; + \node (b) at (1,0) {}; + \node (c) at (1,1) {}; + \node (d) at (0,1) {}; + \coordinate (m) at ($(a)!0.5!(c)$); + \coordinate (ab2) at ($(a)!0.5!(b)$); + \coordinate (ab1) at ($(a)!0.5!(ab2)$); + \coordinate (ab3) at ($(b)!0.5!(ab2)$); + \coordinate (bc2) at ($(b)!0.5!(c)$); + \coordinate (bc1) at ($(b)!0.5!(bc2)$); + \coordinate (bc3) at ($(c)!0.5!(bc2)$); + \coordinate (cd2) at ($(c)!0.5!(d)$); + \coordinate (cd1) at ($(c)!0.5!(cd2)$); + \coordinate (cd3) at ($(d)!0.5!(cd2)$); + \coordinate (ad2) at ($(a)!0.5!(d)$); + \coordinate (ad1) at ($(a)!0.5!(ad2)$); + \coordinate (ad3) at ($(d)!0.5!(ad2)$); + \coordinate (corner1) at ($(a)!0.25!(bc3)$); + \coordinate (corner2) at ($(c)!0.25!(ad1)$); + \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- cycle; + %horizontal + \draw (a.center) -- (bc3.center); + \draw (cd3.center) -- (ad3.center); + \draw (ab3.center) -- (bc1.center); + \draw (cd3.center) -- (ad1.center); + \draw (ab3.center) -- (bc3.center); + \draw (ab1.center) -- (corner1.center); + \draw (ad1.center) -- (corner1.center); + \draw (ab3.center) -- (corner1.center); + \draw (c.center) -- (corner2.center); + \draw (cd1.center) -- (corner2.center); + \draw (bc3.center) -- (corner2.center); + \draw (cd3.center) -- (corner2.center); + \draw (corner1.center) -- (corner2.center); + \draw (ad1.center) -- (corner2.center); + + \begin{scope}[decoration={ + markings, + mark=at position 0.6 with {\arrow{>}}} + ] + \draw[postaction={decorate}] (a.center) -- (b.center); + \draw[postaction={decorate}] (d.center) -- (c.center); + \end{scope} + + \begin{scope}[decoration={ + markings, + mark=at position 0.55 with {\arrow{>>}}} + ] + \draw[postaction={decorate}] (b.center) -- (c.center); + \draw[postaction={decorate}] (a.center) -- (d.center); + \end{scope} +\end{tikzpicture} +\end{document} diff --git a/tikz/torus-triangulation/Makefile b/tikz/torus-triangulation/Makefile new file mode 100644 index 0000000..24090c3 --- /dev/null +++ b/tikz/torus-triangulation/Makefile @@ -0,0 +1,31 @@ +SOURCE = torus-triangulation +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/torus-triangulation/Readme.md 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