From 8acbd45e8dc835db7da6ca70ef18fe7b8aecc549 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Tue, 14 Jan 2014 17:39:10 +0100 Subject: [PATCH] misc --- documents/GeoTopo/GeoTopo.pdf | Bin 4711573 -> 4711224 bytes documents/GeoTopo/Kapitel4.tex | 33 ++++++++++-------- documents/GeoTopo/figures/geometry-6.pdf | Bin 0 -> 10277 bytes documents/GeoTopo/figures/geometry-7.pdf | Bin 0 -> 10457 bytes documents/GeoTopo/figures/knot-trefoil.pdf | Bin 0 -> 1329 bytes documents/GeoTopo/figures/sin-cos.pdf | Bin 0 -> 46430 bytes .../GeoTopo/figures/spherical-coordinates.pdf | Bin 0 -> 18685 bytes .../figures/topology-r-spiral-covering-s.pdf | Bin 0 -> 309716 bytes .../GeoTopo/figures/torus-three-paths.pdf | Bin 0 -> 38738 bytes .../figures/triangleSimplizialkomplex.pdf | Bin 0 -> 16749 bytes documents/GeoTopo/shortcuts.sty | 3 ++ tikz/geometry-6/geometry-6.png | Bin 10154 -> 10482 bytes tikz/geometry-6/geometry-6.tex | 3 +- 13 files changed, 23 insertions(+), 16 deletions(-) create mode 100644 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zxD_lXq%t7t!j-4ia1!!fUU>Tc7yVzrSWb8AQBUnd)7Ri{mq3#r{^T&=<2D+w?IwmW KBogmr4f_|5>~)3! diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index b7caf56..75e9f5c 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -139,21 +139,21 @@ aufgestellt. Identifizierung von $P \cong P'$ und $Q \cong Q'$ eine weitere Isometrie.) \item \textbf{Parallelenaxiom}: Für jedes $g \in G$ und jedes - $P \in X \setminus g$ gibt es höchstens ein $k \in G$ mit + $P \in X \setminus g$ gibt es höchstens ein $h \in G$ mit $h \cap g = \emptyset$.\footnote{$h$ heißt \enquote{Parallele zu $g$ durch $P$}.} \end{enumerate} \end{definition} -\todo[inline]{Bilder zu Parallelenaxiom, Inzidenzaxiom und Bewegungsaxiom} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mitschrieb vom 14.01.2014 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{satz}[Satz von Rasch]\label{satz:rasch} %In Vorlesung: Bemerkung 14.5 Seien $P$, $Q$, $R$ nicht kollinear, $g \in G$ mit $g \cap \Set{P, Q, R} = \emptyset$ - und $g \cap \overline{PQ} \neq \emptyset$. Dann ist - $g \cap \overline{PR} \neq \emptyset$ oder $g \cap \overline{QR} \neq \emptyset$. + und $g \cap \overline{PQ} \neq \emptyset$. + + Dann ist $g \cap \overline{PR} \neq \emptyset$ oder + $g \cap \overline{QR} \neq \emptyset$. \end{satz} \begin{beweis} @@ -198,7 +198,7 @@ aufgestellt. Sind $P \neq Q$ Fixpunkte einer Isometrie, so ist $\varphi(R) = R$ für jedes $R \in PQ$. \end{behauptung} - \begin{beweis} + \begin{beweis}[zu Beh. 2] Seien $P$, $Q$ und $R$ Fixpunkte von $\varphi$, $R \in PG$ und $A \notin \overline{PQ} \cup \overline{PR} \cup \overline{QR}$. Sei $B \in \overline{PQ} \setminus \Set{P, Q}$. Dann ist @@ -210,7 +210,7 @@ aufgestellt. \begin{figure}[htp] \centering \input{figures/geometry-1.tex} - \caption{$P, Q, R$ sind Fixpunke, $B \in \overline{PQ} \setminus \Set{P,Q}$, $A \notin PQ \cup PR \cup QR$} + \caption{$P, Q, R$ sind Fixpunkte, $B \in \overline{PQ} \setminus \Set{P,Q}$, $A \notin PQ \cup PR \cup QR$} \label{fig:geometry-1} \end{figure} @@ -241,7 +241,7 @@ aufgestellt. \begin{figure}[htp] \centering \input{figures/geometry-2.tex} - \caption{Die beiden roten und die beiden blauen Linien sind gleich lang. Intuitiv weiß man, dass daraus folgt, dass $\varphi_1(R) = \varphi_2(R)$ gilt.}. + \caption{Die beiden roten und die beiden blauen Linien sind gleich lang. Intuitiv weiß man, dass daraus folgt, dass $\varphi_1(R) = \varphi_2(R)$ gilt.} \label{fig:bild-2} \end{figure} \end{beweis} @@ -249,7 +249,9 @@ aufgestellt. \begin{korollar}\label{kor:14.6}%In Vorlesung: Bemerkung 14.6 Seien $P, Q \in X$, $P \neq Q$, $A, B \in X \setminus PQ$ in der selben Halbebene bzgl. $PQ$ mit $d(A, P) = d(B, P)$ - und $d(A, Q) = d(B, Q)$. Dann ist $A = B$. + und $d(A, Q) = d(B, Q)$. + + Dann ist $A = B$. \end{korollar} \begin{beweis} durch Widerspruch\\ \underline{Annahme}: $A \neq B$ @@ -332,17 +334,18 @@ aufgestellt. wie man sie aus der Schule kennt, beweisen. \end{bemerkung} -\begin{proposition}%In Vorlesung: Proposition 14.7 +\begin{proposition}\label{prop:14.7}%In Vorlesung: Proposition 14.7 Sei $(X, d, G)$ eine Geometrie mit den Axiomen \ref{axiom:1}~-~\ref{axiom:4}. - Dannn gibt es zu jedem $g \in G$ und jedem $P \in X \setminus g$ ein - $k \in G$ mit $P \in h$ und $g \cap h \neq \emptyset$. + + Dann gibt es zu jedem $g \in G$ und jedem $P \in X \setminus g$ ein + $h \in G$ mit $P \in h$ und $g \cap h \neq \emptyset$. \end{proposition} \begin{figure}[htp] \centering %\input{figures/geometry-6.tex} \includegraphics[width=0.5\linewidth, keepaspectratio]{figures/geometry-6.pdf} - \caption{TODO} + \caption{Situation aus \cref{prop:14.7}} \label{fig:bild-6} \end{figure} @@ -350,7 +353,7 @@ aufgestellt. Sei $f \in G$ mit $P \in f$. Ist $f \cap g = \emptyset$, so setze $h := f$. Andernfalls sei $\Set{Q} : = f \cap g$. - Sei $\varphi$ \underline{die} Isometrie mit $\varphi(Q) = P$, + Sei $\varphi$ die eindeutige Isometrie mit $\varphi(Q) = P$, $\varphi(P) = P'$, die die Halbebenen bzgl. $f$ nicht vertauscht. Setze $h := \varphi(g)$. @@ -369,6 +372,6 @@ aufgestellt. \end{beweis} \begin{bemerkung} - Jder Innenwinkel eines Dreiecks ist kleiner als alle nicht-anliegenden + Jeder Innenwinkel eines Dreiecks ist kleiner als alle nicht-anliegenden Außenwinkel. \end{bemerkung} diff --git a/documents/GeoTopo/figures/geometry-6.pdf b/documents/GeoTopo/figures/geometry-6.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5d2aa16e1a442664dd23f4fdca225a923dfa4b05 GIT binary patch literal 10277 zcma)?WlUvJvbJ$KxH}x&8h3Yh8X9+R91ia8?v1-c<23Hx(70>k?k<=4?o7U$%uVL* z>>sPvu9fwsQdQYWJ(SAgk_;@2901BcbAw9&P9QVT-uOF^pC7;^Z)Rr!vIO$*Z~_1O z0We8g*?`QPfJ~A$Mj$hBGZTALGk~BVz!~IZW@HO+U#Zq;wVG;SP6Kjcd>dtKjr)Cs{-_uIT{|ESQUeRyP0;yf~pE-a(;7`}}x z&hu#=Lifs!X%%0v#1I3SY7OIpnIMc!LqRM_WPTBt*~_i)$nVgp_e|e#!_ll`C@W( z8f%o+6mAPrgl&mZ5u0RfTaT-xnX1vlpNx*A%kq)ts>7>cTWPR+_4X&wtIAEK!3(%>RpY3GgbA?h zz@yY&ue+HZ_aJ!-{O$*=YzS|C;_c=wG60};i01#zu0nc9Eym;^|GpHTEue9Iim?;aAQ6bX6KmZSgkOUF 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70da043..2d4d3bc 100644 --- a/tikz/geometry-6/geometry-6.tex +++ b/tikz/geometry-6/geometry-6.tex @@ -15,7 +15,8 @@ \tkzDefLine[parallel=through P](Q,H1) \tkzGetPoint{b} \tkzMarkAngle[arc=l,size=1cm,color=green,fill=green!20](b,P,Phelper) \tkzDrawLine[dashed](P,b) - \tkzLabelLine[pos=0.8,below](P,b){$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)