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Bilder hinzugefügt und verbessert

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Martin Thoma 2014-01-16 23:21:56 +01:00
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1
documents/GeoTopo/Arbeitszeit.md
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@ -34,3 +34,4 @@ in dem Erstellen dieses Skripts steckt:
|16.01.2014 | 12:15 - 12:30 | Verbesserung an 2 Bildern
|16.01.2014 | 12:45 - 13:30 | TikZ'en eines Bildes
|16.01.2014 | 17:00 - 19:30 | Digitalisieren der Vorlesung von 14.01.2014
|16.01.2014 | 21:30 - | TikZ'en von Bildern

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documents/GeoTopo/GeoTopo.pdf
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67
documents/GeoTopo/Kapitel4.tex
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@ -168,7 +168,7 @@ scheiden.
$g \cap \overline{PQ} \neq \emptyset$\\
$\overset{\mathclap{\ref{axiom:3.2}}}{\Rightarrow} P$ und $Q$ liegen in verschiedenen Halbebenen bzgl. $g$\\
$\Rightarrow$ \obda $R$ und $P$ liegen in verschieden
Halbebenen bzgl. $P$\todo{bzgl. P? Nicht PQ?}\\
Halbebenen bzgl. $g$\\
$\Rightarrow g \cap \overline{RP} \neq \emptyset$
\end{beweis}
@ -364,12 +364,12 @@ schneiden sich.
$h \in G$ mit $P \in h$ und $g \cap h \neq \emptyset$.
\end{proposition}
\begin{figure}[htp]
\centering
\input{figures/geometry-6.tex}
\caption{Situation aus \cref{prop:14.7}}
\label{fig:bild-6}
\end{figure}
\begin{figure}[htp]
\centering
\input{figures/geometry-6.tex}
\caption{Situation aus \cref{prop:14.7}}
\label{fig:bild-6}
\end{figure}
\begin{beweis}
Sei $f \in G$ mit $P \in f$. Ist $f \cap g = \emptyset$, so setze
@ -383,13 +383,6 @@ schneiden sich.
\underline{Z.~Z.:} $h \cap g = \emptyset$.
Andernfalls sei $\Set{R} = h \cap g$.
\begin{figure}[htp]
\centering
\input{figures/geometry-7.tex}
\caption{TODO}
\label{fig:bild-6}
\end{figure}
\end{beweis}
\begin{bemerkung}
@ -400,21 +393,13 @@ schneiden sich.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Mitschrieb vom 16.01.2014 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{prop:14.7}%In Vorlesung: Proposition 14.7
Gelten \cref{axiom:1}~-~\cref{axiom:4}, so gibt es zu $g \in G$
und $P \in X \setminus g$ stets ein $h \in G$ mit $P \in h$ und
$h \cap g = \emptyset$.
\end{proposition}
\todo[inline]{Bild mit paralleler gerade ... das hatte ich doch schon mal}
\begin{beweis}
Sei $\varphi$ die Isometrie, die $Q$ auf $P$ und $P$ auf $P'$
mit $P' \in f, d(P,P') = d(P, Q)$ abbildet und die Halbebenen
bzgl. $f$ erhält.
\end{beweis}
\begin{behauptung}[Herz]
\begin{behauptung}[Herz]\label{beh:herz}
$\varphi(g) \cap g = \emptyset$
\end{behauptung}
@ -422,28 +407,44 @@ schneiden sich.
Ist $\varphi(g) \cap g \neq \emptyset$, so ist $R$ der Schnittpunkt.
\end{beweis}
\todo[inline]{Skizze zu Behauptung Herz}
\begin{figure}[htp]
\centering
\input{figures/geometry-7.tex}
\caption{Skizze zu \cref{beh:herz}}
\label{fig:bild-6}
\end{figure}
\begin{definition}%In Vorlesung: 14.8
\begin{definition}\label{def:14.8}%In Vorlesung: 14.8
\begin{enumerate}[label=\alph*),ref=\alph*]
\item Ein \textbf{Winkel}\xindex{Winkel} ist ein Punkt $P \in X$ \label{def:14.8a}
\item \label{def:14.8a} Ein \textbf{Winkel}\xindex{Winkel} ist ein Punkt $P \in X$
zusammen mit $2$ Halbgeraden mit Anfangspunkt $P$.\\
Man schreibt: $\angle R_1 P R_2$ bzw. $\angle R_2 P R_1$\footnote{Für dieses Skript gilt: $\angle R_1 P R_2 = \angle R_2 P R_1$. Also sind insbesondere alle Winkel $ \leq 180^\circ$.}
\item Zwei Winkel sind \textbf{gleich}, wenn es eine Isometrie gibt,
die den einen Winkel auf den anderen abbildet.
\item \label{def:14.8c}$\angle R_1' P' R_2'$ heißt \textbf{kleiner} als
\item \label{def:14.8c} $\angle R_1' P' R_2'$ heißt \textbf{kleiner} als
$\angle R_1 P R_2$, wenn es eine Isometrie $\varphi$
gibt, mit $\varphi(P) = P'$, $\varphi(PR'_1+) = P' R_1 +$
und $\varphi(R_2')$ liegt in der gleichen Halbebene
bzgl. $PR_1$ wie $R_2$ und in der gleichen Halbebene
bzgl. $PR_2$ wie $R_1$
\item \label{def:14.8d} \ref{def:14.8d} Im Dreieck $\triangle PQR$ gibt es {\color{green} Innenwinkel} und
{\color{red} Außenwinkel}.
\item \label{def:14.8d} Im Dreieck $\triangle PQR$ gibt es Innenwinkel und
Außenwinkel.
\end{enumerate}
\end{definition}
\todo[inline]{$\angle R_1' P' R_2'$ ist kleiner als $\angle R_1 P R_2$, vgl. \cref{def:14.8c} (Bild 4)}
\todo[inline]{{\color{green} Innenwinkel} und {\color{red} Außenwinkel} in $\triangle PQR$, vgl. \cref{def:14.8d} (Bild 5)}
\begin{figure}[ht]
\centering
\subfloat[$\angle R_1' P' R_2'$ ist kleiner als $\angle R_1 P R_2$, vgl. \cref{def:14.8c}]{
\input{figures/smaller-angle.tex}
\label{fig:def.14.8.1}
}%
\subfloat[{\color{green} Innenwinkel} und {\color{blue} Außenwinkel} in $\triangle PQR$, vgl. \cref{def:14.8d} (Bild 5)]{
\input{figures/interiour-exteriour-angles-triangle.tex}
\label{fig:def.14.8.2}
}
\label{fig:def.14.8.0}
\caption{Situation aus \cref{def:14.8}}
\end{figure}
\begin{korollar}\label{kor:14.9}%In Vorlesung: Bemerkung 14.9
In einem Dreieck ist jeder Innenwinkel kleiner als jeder nicht
@ -527,8 +528,8 @@ Sei im Folgenden \enquote{IWS} die \enquote{Innenwinkelsumme}.
\begin{figure}[ht]
\centering
\subfloat[Summe der Winkel $\alpha$, $\beta$ und $\gamma$ (Bild 11)]{
\input{figures/todo.tex}
\subfloat[Summe der Winkel $\alpha$, $\beta$ und $\gamma$]{
\input{figures/three-angles.tex}
\label{fig:prop14.11.1}
}%
\subfloat[Situation aus \cref{prop:14.11} (Bild 12)]{

3
documents/GeoTopo/figures/geometry-6.tex
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@ -10,7 +10,8 @@
\tkzDefLine[parallel=through P](Q,H1) \tkzGetPoint{b}
\tkzMarkAngle[arc=l,size=1cm,color=green,fill=green!20](b,P,Phelper)
\tkzDrawLine[dashed](P,b)
\tkzLabelLine[pos=0.8,below](P,b){$f$}
\tkzLabelLine[pos=0.8,below](P,b){$h$}
\tkzLabelLine[pos=-0.6,left](P,Q){$f$}
\tkzLabelLine[pos=0.8,below](Q,H1){$g$}
\tkzLabelPoint[above left](P){$P$}
\tkzDrawLine[add=0.2 and 0.7](Q,P)

6
documents/GeoTopo/figures/geometry-7.tex
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@ -15,7 +15,7 @@
\tkzDrawLine(Q,R)
\tkzDrawLine(P,R)
\tkzDrawPoints(P,Q,R)
\tkzLabelPoint[below left](P){$P$}
\tkzLabelPoint[above left](Q){$Q$}
\tkzLabelPoint[above=0.2cm](R){$R$}
\node at ($(R) + (0.03,0.4)$) {$R$}; %top
\node at ($(Q) + (-0.3,-0.22)$) {$Q$}; %left
\node at ($(P) + ( 0.3,-0.18)$) {$P$}; %right
\end{tikzpicture}

28
documents/GeoTopo/figures/interiour-exteriour-angles-triangle.tex Normal file
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@ -0,0 +1,28 @@
\begin{tikzpicture}
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
\tkzSetUpLine[line width=1]
\tkzDefPoints{0/1/P, -1/-1/Q, 1/-1/R, -2/-1/links, 2/-1/rechts, -1.5/-2/helperLeft, 1.5/-2/helperRight, -0.25/1.5/helperTopLeft, 0.25/1.5/helperTopRight}
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](R,Q,P)
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](P,R,Q)
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](Q,P,R)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](P,Q,links)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](helperLeft,Q,R)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](helperTopLeft,P,Q)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](R,P,helperTopRight)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](rechts,R,P)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](Q,R,helperRight)
\tkzDrawLine[add=0.35 and 0.35](P,Q)
\tkzDrawLine[add=0.35 and 0.35](P,R)
\tkzDrawLine[add=0.4 and 0.4](Q,R)
\node at ($(P) + (0.03,0.4)$) {$P$};
\node at ($(Q) + (-0.3,-0.22)$) {$Q$};
\node at ($(R) + ( 0.3,-0.18)$) {$R$};
%\tkzLabelPoint[above=0.2cm](P){$P$}
%\tkzLabelPoint[below left](Q){$Q$}
%\tkzLabelPoint[below right](R){$R$}
\tkzDrawPoints(P, Q, R)
\end{tikzpicture}

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documents/GeoTopo/figures/smaller-angle.tex Normal file
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@ -0,0 +1,21 @@
\begin{tikzpicture}
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
\tkzSetUpLine[line width=1]
\tkzDefPoints{0/0/P, 1.5/0/R1S, 3/0/R1, 1/1/G, 1/2/R2}
\tkzMarkAngle[arc=lll,size=1.2cm,color=red,fill=red!20](R1S,P,R2)
\tkzDrawLine[add=0 and 0.3,color=green](P,R1)
\tkzDrawLine[add=0 and 0.6](P,R2)
\tkzLabelPoint[below left](P){$P$}
\tkzLabelPoint[below](R1S){$R_1'$}
\tkzLabelPoint[below](R1){$R_1$}
\tkzInterLC(P,R1)(R1S,P) \tkzGetPoints{D}{E}
\tkzInterLC(P,G)(R1S,P) \tkzGetPoints{F}{R2S}
%\tkzDrawCircle(R1S,D)
\tkzLabelPoint[below](R2S){$R_2'$}
\tkzLabelPoint[above left](R2){$R_2$}
\tkzDrawLine[add=0 and 1,color=green](P,R2S)
\tkzMarkAngle[arc=l,size=0.8cm,color=green,fill=green!20](R1S,P,R2S)
\tkzDrawPoints(P, R1S, R1, R2,R2S)
\end{tikzpicture}

21
documents/GeoTopo/figures/three-angles.tex Normal file
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@ -0,0 +1,21 @@
\begin{tikzpicture}
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
\tkzSetUpLine[line width=1]
\tkzDefPoints{0/0/P, 1/0/helperRight, 1/1/helperTopRight, -1/1/helperTopLeft, -1/0/helperLeft, -1/-0.3/helperBottomLeft}
\tkzMarkAngle[arc=l,size=0.8cm,color=green,fill=green!20](helperRight,P,helperTopRight)
\tkzMarkAngle[arc=ll,size=0.8cm,color=blue,fill=blue!20](helperTopRight,P,helperTopLeft)
\tkzMarkAngle[arc=lll,size=0.8cm,color=red,fill=red!20](helperTopLeft,P,helperBottomLeft)
\path[draw] ++(25:.4) node[rotate=0] {$\alpha$};
\path[draw] ++(90:.4) node[rotate=0] {$\beta$};
\path[draw] ++(160:.4) node[rotate=0] {$\gamma$};
\tkzDrawLine[add=0 and 1.0](P, helperRight)
\tkzDrawLine[add=0 and 0.3](P, helperTopRight)
\tkzDrawLine[add=0 and 0.3](P, helperTopLeft)
\tkzDrawLine[add=0 and 1.0](P, helperLeft)
\tkzDrawLine[add=0 and 0.8](P, helperBottomLeft)
\tkzDrawPoints(P)
\tkzLabelPoint[below](P){$P$}
\end{tikzpicture}

7
documents/GeoTopo/shortcuts.sty
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@ -98,5 +98,8 @@
\crefname{lemma}{Lemma}{Lemmata}
\crefname{korollar}{Korollar}{Korollare}
\crefname{folgerung}{Folgerung}{Folgerungen}
\let\OldAngle\angle
\let\angle\sphericalangle
\crefname{definition}{Definition}{Definitionen}
\crefname{plaindefinition}{Definition}{Definitionen}
\crefname{behauptung}{Behauptung}{Behauptungen}
%\let\OldAngle\angle
%\let\angle\sphericalangle

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6
tikz/geometry-7/geometry-7.tex
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@ -20,8 +20,8 @@
\tkzDrawLine(Q,R)
\tkzDrawLine(P,R)
\tkzDrawPoints(P,Q,R)
\tkzLabelPoint[below left](P){$P$}
\tkzLabelPoint[above left](Q){$Q$}
\tkzLabelPoint[above=0.2cm](R){$R$}
\node at ($(R) + (0.03,0.4)$) {$R$}; %top
\node at ($(Q) + (-0.3,-0.22)$) {$Q$}; %left
\node at ($(P) + ( 0.3,-0.18)$) {$P$}; %right
\end{tikzpicture}
\end{document}

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tikz/interiour-exteriour-angles-triangle/Makefile Normal file
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@ -0,0 +1,31 @@
SOURCE = interiour-exteriour-angles-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

3
tikz/interiour-exteriour-angles-triangle/Readme.md Normal file
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@ -0,0 +1,3 @@
Compiled example
----------------
![Example](interiour-exteriour-angles-triangle.png)

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34
tikz/interiour-exteriour-angles-triangle/interiour-exteriour-angles-triangle.tex Normal file
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@ -0,0 +1,34 @@
\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{tkz-euclide}
\usetkzobj{all}
\begin{document}
\begin{tikzpicture}
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
\tkzSetUpLine[line width=1]
\tkzDefPoints{0/1/P, -1/-1/Q, 1/-1/R, -2/-1/links, 2/-1/rechts, -1.5/-2/helperLeft, 1.5/-2/helperRight, -0.25/1.5/helperTopLeft, 0.25/1.5/helperTopRight}
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](R,Q,P)
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](P,R,Q)
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](Q,P,R)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](P,Q,links)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](helperLeft,Q,R)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](helperTopLeft,P,Q)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](R,P,helperTopRight)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](rechts,R,P)
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](Q,R,helperRight)
\tkzDrawLine[add=0.35 and 0.35](P,Q)
\tkzDrawLine[add=0.35 and 0.35](P,R)
\tkzDrawLine[add=0.4 and 0.4](Q,R)
\node at ($(P) + (0.03,0.4)$) {$P$};
\node at ($(Q) + (-0.3,-0.22)$) {$Q$};
\node at ($(R) + ( 0.3,-0.18)$) {$R$};
%\tkzLabelPoint[above=0.2cm](P){$P$}
%\tkzLabelPoint[below left](Q){$Q$}
%\tkzLabelPoint[below right](R){$R$}
\tkzDrawPoints(P, Q, R)
\end{tikzpicture}
\end{document}

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tikz/smaller-angle/Makefile Normal file
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@ -0,0 +1,31 @@
SOURCE = smaller-angle
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/smaller-angle/Readme.md Normal file
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@ -0,0 +1,3 @@
Compiled example
----------------
![Example](smaller-angle.png)

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27
tikz/smaller-angle/smaller-angle.tex Normal file
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@ -0,0 +1,27 @@
\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{tkz-euclide}
\usetkzobj{all}
\begin{document}
\begin{tikzpicture}
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
\tkzSetUpLine[line width=1]
\tkzDefPoints{0/0/P, 1.5/0/R1S, 3/0/R1, 1/1/G, 1/2/R2}
\tkzMarkAngle[arc=lll,size=1.2cm,color=red,fill=red!20](R1S,P,R2)
\tkzDrawLine[add=0 and 0.3,color=green](P,R1)
\tkzDrawLine[add=0 and 0.6](P,R2)
\tkzLabelPoint[below left](P){$P$}
\tkzLabelPoint[below](R1S){$R_1'$}
\tkzLabelPoint[below](R1){$R_1$}
\tkzInterLC(P,R1)(R1S,P) \tkzGetPoints{D}{E}
\tkzInterLC(P,G)(R1S,P) \tkzGetPoints{F}{R2S}
%\tkzDrawCircle(R1S,D)
\tkzLabelPoint[below](R2S){$R_2'$}
\tkzLabelPoint[above left](R2){$R_2$}
\tkzDrawLine[add=0 and 1,color=green](P,R2S)
\tkzMarkAngle[arc=l,size=0.8cm,color=green,fill=green!20](R1S,P,R2S)
\tkzDrawPoints(P, R1S, R1, R2,R2S)
\end{tikzpicture}
\end{document}

31
tikz/three-angles/Makefile Normal file
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@ -0,0 +1,31 @@
SOURCE = three-angles
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/three-angles/Readme.md Normal file
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@ -0,0 +1,3 @@
Compiled example
----------------
![Example](three-angles.png)

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27
tikz/three-angles/three-angles.tex Normal file
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\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{tkz-euclide}
\begin{document}
\usetkzobj{all}
\begin{tikzpicture}
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
\tkzSetUpLine[line width=1]
\tkzDefPoints{0/0/P, 1/0/helperRight, 1/1/helperTopRight, -1/1/helperTopLeft, -1/0/helperLeft, -1/-0.3/helperBottomLeft}
\tkzMarkAngle[arc=l,size=0.8cm,color=green,fill=green!20](helperRight,P,helperTopRight)
\tkzMarkAngle[arc=ll,size=0.8cm,color=blue,fill=blue!20](helperTopRight,P,helperTopLeft)
\tkzMarkAngle[arc=lll,size=0.8cm,color=red,fill=red!20](helperTopLeft,P,helperBottomLeft)
\path[draw] ++(25:.4) node[rotate=0] {$\alpha$};
\path[draw] ++(90:.4) node[rotate=0] {$\beta$};
\path[draw] ++(160:.4) node[rotate=0] {$\gamma$};
\tkzDrawLine[add=0 and 1.0](P, helperRight)
\tkzDrawLine[add=0 and 0.3](P, helperTopRight)
\tkzDrawLine[add=0 and 0.3](P, helperTopLeft)
\tkzDrawLine[add=0 and 1.0](P, helperLeft)
\tkzDrawLine[add=0 and 0.8](P, helperBottomLeft)
\tkzDrawPoints(P)
\tkzLabelPoint[below](P){$P$}
\end{tikzpicture}
\end{document}