From 6536ec2326bf598bdb3ed41c9872c88784d4544c Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sat, 25 Jan 2014 00:22:50 +0100 Subject: [PATCH] Vorlesung vom 23.01.2014 teilweise geTeXt --- documents/GeoTopo/Arbeitszeit.md | 1 + documents/GeoTopo/GeoTopo.pdf | Bin 4919360 -> 4929464 bytes documents/GeoTopo/Kapitel2.tex | 2 +- documents/GeoTopo/Kapitel4.tex | 103 ++++++++++++++++-- documents/GeoTopo/Symbolverzeichnis.tex | 5 + .../hyperbolic-geometry-not-parallel.tex | 15 +++ documents/GeoTopo/shortcuts.sty | 4 + .../hyperbolic-geometry-not-parallel/Makefile | 31 ++++++ .../Readme.md | 3 + .../hyperbolic-geometry-not-parallel.png | Bin 0 -> 16393 bytes .../hyperbolic-geometry-not-parallel.tex | 23 ++++ 11 files changed, 179 insertions(+), 8 deletions(-) create mode 100644 documents/GeoTopo/figures/hyperbolic-geometry-not-parallel.tex create mode 100644 tikz/hyperbolic-geometry-not-parallel/Makefile create mode 100644 tikz/hyperbolic-geometry-not-parallel/Readme.md create mode 100644 tikz/hyperbolic-geometry-not-parallel/hyperbolic-geometry-not-parallel.png create mode 100644 tikz/hyperbolic-geometry-not-parallel/hyperbolic-geometry-not-parallel.tex diff --git a/documents/GeoTopo/Arbeitszeit.md b/documents/GeoTopo/Arbeitszeit.md index 940cfdc..4c7879a 100644 --- a/documents/GeoTopo/Arbeitszeit.md +++ b/documents/GeoTopo/Arbeitszeit.md @@ -43,3 +43,4 @@ in dem Erstellen dieses Skripts steckt: |23.01.2014 | 09:00 - 10:00 | TikZ'en eines Bildes und Bemerkungen |23.01.2014 | 10:30 - 12:15 | TikZ'en von Bildern |24.01.2014 | 15:00 - 15:15 | Flag um Dokument in A4 (für den Bildschirm) bzw. 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z$#Y+H>fhrlZa;jQOAz4(rha;8QI$wcIluh$dfp5B!dBV#jDv;T>D3K8*y*m>Oa76W z((59G>Gkuf^?H%-Z(}&_(@ZGG{b_f-Lfb3dflBDQYCZJcCD@g#N`KWWzZyCk$M8HL z`e=h>39Qcv;0w>9bf3k5f3)|&-}`%-yr8wYzvcN~)j3ar!#{-imH$~j{-Mcf?O5Fr z&Vxdwh$_LlR?2a(mueuMh=h(f0n7aV@cqBJJWK{gX5+JxQ1$3N!v= 0} = \Set{(x,y) \in \mdr^2 | y > 0}\] + die obere Halbebene bzw. Poincaré-Halbebene und $G = G_1 \cup G_2$ + mit + \begin{align*} + G_1 &= \Set{g_1 \subseteq \mdh | \exists m \in \mdr, r \in \mdr_{>0}: g_1 \in \mdc: g_1 = \Set{|z-m|=r}}\\ + G_2 &= \Set{g_2 \subseteq \mdh | \exists x \in \mdr: g_2 = \Set{z \in \mdc: \Re{z} = x} \cap \mdh} + \end{align*} + + Die Elemente von $\mdh$ heißen \textbf{hyperbolische Geraden}\xindex{Gerade!hyperbolische} +\end{definition} + +\begin{bemerkung}[Eigenschaften der hyperbolischen Geraden] + Die hyperbolischen Geraden erfüllen\dots + \begin{enumerate}[label=\alph*)] + \item \dots die Inzidenzaxiome \ref{axiom:1} + \item \dots das Anordnungsaxiom \ref{axiom:3.2} + \item \dots nicht das Parallelenaxiom \ref{axiom:5} + \end{enumerate} +\end{bemerkung} + +\begin{beweis}\leavevmode + \begin{enumerate}[label=\alph*)] + \item Offensichtlich sind \ref{axiom:1.3} und \ref{axiom:1.2} + erfüllt. Für \ref{axiom:1.1} gilt:\\ + Gegeben $z_1, z_2 \in \mdh$\\ + \textbf{Existenz:} $\Re(z_1) = \Re(z_2)$ + $\Rightarrow z_1$ und $z_2$ liegen auf + \[g = \Set{z \in \mdc | \Re(z) = \Re{z_1} \land \mdh}\] + \item TODO + \item Siehe \cref{fig:hyperbolische-halbebene-axiom-5}. + \begin{figure}[htp] + \centering + \input{figures/hyperbolic-geometry-not-parallel.tex} + \caption{Hyperbolische Geraden erfüllen \ref{axiom:5} nicht.} + \label{fig:hyperbolische-halbebene-axiom-5} + \end{figure} + \end{enumerate} +\end{beweis} + +\begin{proposition}%In Vorlesung: Proposition 15.2 + \begin{enumerate}[label=\alph*)] + \item Die Gruppe $\SL_2(\mdr)$ operiert auf $\mdh$ durch + \[\sigma(z):= \begin{pmatrix}a & b\\c & d\end{pmatrix} \circ z := \frac{az + b}{cz + d}\] + \item Es ist $\sigma(z) = (-\sigma)(z)$ für alle $\sigma \in \SL_2(\mdr)$ + und $z \in \mdh$. Daher operiert $\PSL_2(\mdr) = \SL_2(\mdr) /_{(\pm I)}$ + auf $\mdh$. + \item $\PSL_2(\mdr)$ operiert auf $\mdr \cup \Set{\infty}$. + Diese Gruppenoperation ist 3-fach transitiv, d.~h. zu + $x_0 < x_1 < x_\infty \in \mdr$ gibt es genau ein + $\sigma \in \PSL_2(\mdr)$ mit $\sigma(x_0) = 0$, + $\sigma(x_1) = 1$, $\sigma(x_\infty) = \infty$ + \item \label{prop:15.2d} $\SL_2(\mdr)$ wird von den Matrizen + \[\begin{pmatrix}\lambda & 0\\ 0 & \lambda^-1\end{pmatrix}, \lambda \in \mdr\] + \[\begin{pmatrix}1 & a\\ 0 & 1\end{pmatrix}, a \in \mdr\] + \[\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}\] + erzeugt + \item $\PSL_2(\mdr)$ operiert auf $G$ + \end{enumerate} +\end{proposition} + +\begin{beweis}\leavevmode + \begin{enumerate}[label=\alph*)] + \item Sei $z = x + iy \in \mdh$, d.~h. $y>0$ und + $\sigma=\begin{pmatrix}a&b\\c&d\end{pmatrix} \in \SL_2(\mdr)$ + \begin{align} + \Rightarrow \sigma(z) &= \frac{ax + aiy + b}{cx + ciy +d}\\ + &= \frac{ax + aiy + b}{cx + ciy +d} \cdot \frac{cx+d-iy}{cx+d-iy}\\ + &= \frac{\Re(...) + i(aycx + ayd - ciyax - cyb)}{(cx+d)^2 + (cy)^2}\\ + &= \frac{\Re(...) + i(ad-bc)y}{(cx+d)^2 + (cy)^2} + &\overset{\mathclap{\SL_2(\mdr)}}{=} \frac{\Re(...) + iy}{(cx+d)^2 + (cy)^2} + \end{align} + $\Rightarrow \Im(\sigma(z)) = \frac{y}{(cx+d)^2 + (cy)^2} > 0$ + \item TODO b) + \item TODO c) + \item TODO d) + \item Es genügt die Aussage für Matrizen aus \cref{prop:15.2d} + zu zeigen. + \begin{itemize} + \item $\sigma = \begin{pmatrix}\lambda & 0\\ 0 & \lambda^{-1}\end{pmatrix}$, also $\sigma(z) = \lambda^2 z$ + \end{itemize} + \end{enumerate} +\end{beweis} + % Die Übungsaufgaben sollen ganz am Ende des Kapitels sein. \input{Kapitel4-UB} diff --git a/documents/GeoTopo/Symbolverzeichnis.tex b/documents/GeoTopo/Symbolverzeichnis.tex index d1c390b..2669d6e 100644 --- a/documents/GeoTopo/Symbolverzeichnis.tex +++ b/documents/GeoTopo/Symbolverzeichnis.tex @@ -29,6 +29,7 @@ $\mdr^+\;$ Echt positive reele Zahlen\\ $\mdr^\times\;$ Einheitengruppe von $\mdr$ ($\mdr \setminus \Set{0}$)\\ $\mdc\;\;\;$ Komplexe Zahlen ($\Set{a+ib|a,b \in \mdr}$)\\ $\mdp\;\;\;$ Primzahlen ($2, 3, 5, 7, \dots$)\\ +$\mdh\;\;\;$ obere Halbebene ($\Set{z \in \mdc | \Im{z} > 0}$)\\ \section*{Geometrie} $AB\;\;\;$ Gerade durch die Punkte $A$ und $B$\\ @@ -43,6 +44,8 @@ $\triangle ABC\;\;\;$ Dreieck mit Eckpunkten $A, B, C$\\ $\Homoo(X)\;\;\;$ Homöomorphismengruppe\\ $\Iso(X)\;\;\;$ Isometriengruppe\\ $\GL_n(K)\;\;\;$ Allgemeine lineare Gruppe (general linear group)\\ +$\SL_n(K)\;\;\;$ Spezielle lineare Gruppe\\ +$\PSL_n(K)\;\;\;$ Projektive lineare Gruppe\\ $\Perm(X)\;\;\;$ Permutationsgruppe\\ $\Sym(X)\;\;\;$ Symmetrische Gruppe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -79,3 +82,5 @@ $\pi_1(X,x)\;\;\;$ Fundamentalgruppe im topologischen Raum $X$ um $x \in X$\\ $\Fix(f)\;\;\;$ Menge der Fixpunkte der Abbildung $f$ \index{Faser|see{Urbild}} +\index{kongruent|see{isometrisch}} +\index{Kongruenz|see{Isometrie}} diff --git a/documents/GeoTopo/figures/hyperbolic-geometry-not-parallel.tex b/documents/GeoTopo/figures/hyperbolic-geometry-not-parallel.tex new file mode 100644 index 0000000..8841a7b --- /dev/null +++ b/documents/GeoTopo/figures/hyperbolic-geometry-not-parallel.tex @@ -0,0 +1,15 @@ +\begin{tikzpicture} + \tkzSetUpPoint[shape=circle,size=3,color=black,fill=black] + \tkzSetUpLine[line width=1] + \tkzInit[xmax=6,ymax=5,xmin=-5,ymin=0] + \tkzDefPoints{-2/0/A,3.5/0/B,-0.85/0/C,2/0/D,2/2/P} + \tkzDefPoints{-1/0/X, -1/5/Y} + \tkzDrawLine[add=0.1 and 0.1](X,Y) + \tkzAxeXY + + \tkzDrawArc[R,line width=1pt,color=red](A,0.5 cm)(0,180) + \tkzDrawArc[R,line width=1pt](B,2.5 cm)(0,180) + \tkzDrawArc[R,line width=1pt](C,3.5 cm)(0,180) + \tkzDrawArc[R,line width=1pt](D,2.0 cm)(0,180) + \tkzDrawPoints(P) +\end{tikzpicture} diff --git a/documents/GeoTopo/shortcuts.sty b/documents/GeoTopo/shortcuts.sty index a0e6d1c..5d18fbd 100644 --- a/documents/GeoTopo/shortcuts.sty +++ b/documents/GeoTopo/shortcuts.sty @@ -64,6 +64,7 @@ \def\mdq{\ensuremath{\mathbb{Q}}} \def\mdz{\ensuremath{\mathbb{Z}}} \def\mdn{\ensuremath{\mathbb{N}}} +\def\mdh{\ensuremath{\mathbb{H}}} \def\gdw{\ensuremath{\Leftrightarrow}} \def\atlas{\ensuremath{\mathcal{A}}} @@ -71,6 +72,7 @@ \def\GL{\ensuremath{\mathrm{GL}}} \def\SL{\ensuremath{\mathrm{SL}}} +\def\PSL{\ensuremath{\mathrm{PSL}}} \newcommand\mapsfrom{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} \newcommand\dcup{\mathbin{\dot{\cup}}} \newcommand{\id}{\textnormal{id}} @@ -84,6 +86,8 @@ \DeclareMathOperator{\Diffeo}{Diffeo} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\IWS}{IWS} +%\DeclareMathOperator{\Re}{Re} +%\DeclareMathOperator{\Im}{Im} %%%Text %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand\obda{o.~B.~d.~A.\xspace} diff --git a/tikz/hyperbolic-geometry-not-parallel/Makefile b/tikz/hyperbolic-geometry-not-parallel/Makefile new file mode 100644 index 0000000..9da0a96 --- /dev/null +++ b/tikz/hyperbolic-geometry-not-parallel/Makefile @@ -0,0 +1,31 @@ +SOURCE = hyperbolic-geometry-not-parallel +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/hyperbolic-geometry-not-parallel/Readme.md b/tikz/hyperbolic-geometry-not-parallel/Readme.md new file mode 100644 index 0000000..bc8fa7a --- /dev/null +++ b/tikz/hyperbolic-geometry-not-parallel/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](hyperbolic-geometry-not-parallel.png) diff --git a/tikz/hyperbolic-geometry-not-parallel/hyperbolic-geometry-not-parallel.png b/tikz/hyperbolic-geometry-not-parallel/hyperbolic-geometry-not-parallel.png new file mode 100644 index 0000000000000000000000000000000000000000..d9f4880cf880ecc25252b6214b85c1ceb7188bbe GIT binary patch literal 16393 zcmcJ$c{tQ>6gT{x8Dn3|z9%aCGG)nLgoKJ@Uo&Kv?Azd%EFna8Qb^VaW1q5xu_Ymc zK`8saFYoPnp7*c!pZ9sM=bh`CYu4}mz3+3Mb3W(X=X0X&>uS(YvQt73M5Cpt_7H;L z;4O@UoD{s2geoY57ZT4~S_b6c5lH?d5`3m`*EIHoAUfjt6K1L1#SOk>d#-NuT;C1- z+{gNv9pvNVBlgtQ$HynSI8p0y_lpLRxC72ELgq6F#X1a|az8YoX@pyylD# zB*?=aTym`7gG=)Yb%e1$h;pzI8T|M;iL9X`J?Dm_+I4mYruWx`th6Gdtu*`@NEqH& zFsDpx9nC6IQ7wN zmkHwIWhS5Z^r4>f?mPyDYbAqIv7l7}!}KE6a1AxKS4?Sdu}11^I9zqu&!jw<~! z4~goUX!A&i9y>#V4jA@>7Cp&JL^Z}UjCb_vp#=+(q5P|w+I<*rqApFV#4-6O&%V;( z^aI 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