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Move GeoTopo to submodule
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[submodule "documents/NeuralNets"]
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[submodule "documents/NeuralNets"]
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path = documents/NeuralNets
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path = documents/NeuralNets
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url = https://github.com/Marvin182/NeuralNets.git
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url = https://github.com/Marvin182/NeuralNets.git
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[submodule "documents/GeoTopo"]
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path = documents/GeoTopo
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url = https://github.com/MartinThoma/GeoTopo.git
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documents/GeoTopo
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documents/GeoTopo
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Subproject commit 32ca0d1ffff10d43c992705f5c4528b007b975dd
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\chapter*{Abkürzungsverzeichnis\markboth{Abkürzungsverzeichnis}{Abkürzungsverzeichnis}}
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\addcontentsline{toc}{chapter}{Abkürzungsverzeichnis}
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\begin{acronym}
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\acro{Beh.}{Behauptung}
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\acro{Bew.}{Beweis}
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\acro{bzgl.}{bezüglich}
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\acro{bzw.}{beziehungsweise}
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\acro{ca.}{circa}
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\acro{d. h.}{das heißt}
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\acro{Def.}{Definition}
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\acro{etc.}{et cetera}
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\acro{ex.}{existieren}
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\acro{Hom.}{Homomorphismus}
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\acro{o. B. d. A.}{ohne Beschränkung der Allgemeinheit}
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\acro{Prop.}{Proposition}
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\acro{sog.}{sogenannte}
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\acro{Vor.}{Voraussetzung}
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\acro{vgl.}{vergleiche}
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\acro{z. B.}{zum Beispiel}
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\acro{zhgd.}{zusammenhängend}
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\acro{z. z.}{zu zeigen}
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\end{acronym}
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\chapter*{Bildquellen\markboth{Bildquellen}{Bildquellen}}
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\addcontentsline{toc}{chapter}{Bildquellen}
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Alle Bilder, die hier nicht aufgeführt sind, wurden von Martin Thoma erstellt.
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Teilweise wurden die im folgenden aufgelisteten Bilder noch leicht
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modifiziert.
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\begin{itemize}
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\item[Abb. \ref{fig:s2}] $S^2$: Tom Bombadil, \href{http://tex.stackexchange.com/a/42865/5645}{tex.stackexchange.com/a/42865}
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\item[Abb. \ref{fig:cube}] Würfel: Jan Hlavacek, \href{http://tex.stackexchange.com/a/12069/5645}{tex.stackexchange.com/a/12069}
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\item[Abb. \ref{fig:torus}] $T^2$: Jake, \href{http://tex.stackexchange.com/a/70979/5645}{tex.stackexchange.com/a/70979/5645}
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\item[Abb. \ref{fig:stereographic-projection}] Stereographische Projektion: \href{http://texample.net/tikz/examples/map-projections/}{texample.net/tikz/examples/map-projections}
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\item[Abb. \ref{fig:Knoten}] Knoten von Jim.belk aus der \enquote{\href{https://commons.wikimedia.org/wiki/Category:Blue_knots}{Blue knots}}-Serie:
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\begin{itemize}
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\item Trivialer Knoten: \href{https://commons.wikimedia.org/wiki/File:Blue_Unknot.png}{\path{commons.wikimedia.org/wiki/File:Blue_Unknot.png}}
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\item Kleeblattknoten: \href{https://commons.wikimedia.org/wiki/File:Blue_Trefoil_Knot.png}{\path{commons.wikimedia.org/wiki/File:Blue_Trefoil_Knot.png}}
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\item Achterknoten: \href{https://commons.wikimedia.org/wiki/File:Blue_Figure-Eight_Knot.png}{\path{commons.wikimedia.org/wiki/File:Blue_Figure-Eight_Knot.png}}
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\item $6_2$-Knoten: \href{https://commons.wikimedia.org/wiki/File:Blue_6_2_Knot.png}{\path{commons.wikimedia.org/wiki/File:Blue_6_2_Knot.png}}
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\end{itemize}
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\item[Abb. \ref{fig:reidemeister-zuege}] Reidemeister-Züge: YAMASHITA Makoto (\href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{1}, \href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{2}, \href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{3})
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\item[Abb. \ref{fig:treefoil-knot-three-colors}] Kleeblattknoten, 3-Färbung: Jim.belk, \href{https://commons.wikimedia.org/wiki/File:Tricoloring.png}{\path{commons.wikimedia.org/wiki/File:Tricoloring.png}}
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\item[Abb. \ref{fig:double-torus}] Doppeltorus: Oleg Alexandrov, \href{https://commons.wikimedia.org/wiki/File:Double_torus_illustration.png}{\path{commons.wikimedia.org/wiki/File:Double\_torus\_illustration.png}}
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\item[Abb. \ref{fig:faltungsdiagramm}] Faltungsdiagramm: Jérôme Urhausen, Email vom 11.02.2014.
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\item[Abb. \ref{fig:torus-three-paths}] 3 Pfade auf Torus: \href{http://tex.stackexchange.com/users/484/charles-staats}{Charles Staats}, \href{http://tex.stackexchange.com/a/149991/5645}{\path{tex.stackexchange.com/a/149991/5645}}
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\item[Abb. \ref{fig:ueberlappung-r1-spirale-s1}] Überlagerung von $S^1$ mit $\mdr$: \href{http://tex.stackexchange.com/users/22467/alex}{Alex}, \href{http://tex.stackexchange.com/a/149706/5645}{\path{tex.stackexchange.com/a/149706/5645}}
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\item[Abb. \ref{fig:bem:14.9}] Sphärisches Dreieck: \href{https://commons.wikimedia.org/wiki/User:DemonDeLuxe}{Dominique Toussaint},\\
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\href{https://commons.wikimedia.org/wiki/File:Spherical_triangle_3d_opti.png}{\path{commons.wikimedia.org/wiki/File:Spherical_triangle_3d_opti.png}}
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\item[Abb. \ref{fig:moebius-strip}] Möbiusband: \href{http://tex.stackexchange.com/users/2552/jake}{Jake},
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\href{http://tex.stackexchange.com/a/118573/5645}{\path{tex.stackexchange.com/a/118573/5645}}
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\item[Abb. \ref{fig:torus-gauss-kruemmung}] Krümmung des Torus: \href{http://tex.stackexchange.com/users/484/charles-staats}{Charles Staats}, \href{http://tex.stackexchange.com/a/149991/5645}{\path{tex.stackexchange.com/a/149991/5645}}
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\end{itemize}
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@ -1,61 +0,0 @@
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%!TEX root = GeoTopo.tex
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\markboth{Ergänzende Definitionen und Sätze}{Ergänzende Definitionen und Sätze}
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\chapter*{Ergänzende Definitionen und Sätze}
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\addcontentsline{toc}{chapter}{Ergänzende Definitionen und Sätze}
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Da dieses Skript in die Geometrie und Topologie einführen soll, sollten soweit
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wie möglich alle benötigten Begriffe definiert und erklärt werden. Die folgenden
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Begriffe wurden zwar verwendet, aber nicht erklärt, da sie Bestandteil der
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Vorlesungen \enquote{Analysis I und II} sowie \enquote{Lineare Algebra und analytische Geometrie I und II}
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sind. Jedoch will ich zumindest die Definitionen bereitstellen.
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\begin{definition}\xindex{Häufungspunkt}%
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Sei $D \subseteq \mdr$ und $x_0 \in \mdr$. $x_0$ heißt ein \textbf{Häufungspunkt}
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von $D :\gdw \exists$ Folge $x_n$ in $D \setminus \Set{x_0}$ mit $x_n \rightarrow x_0$.
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\end{definition}
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Folgende Definition wurde dem Skript von Herrn Prof.~Dr.~Leuzinger für
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Lineare Algebra entnommen:
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\begin{definition}\xindex{Abbildung!affine}%
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Es seien $V$ und $W$ $\mdk$-Vektorräume und $\mda(V)$ und $\mda(W)$ die
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zugehörigen affinen Räume. Eine Abbildung $f:V \rightarrow W$ heißt \textbf{affin},
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falls für alle $a, b \in V$ und alle $\lambda, \mu \in \mdk$ mit $\lambda + \mu = 1$ gilt:
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\[f(\lambda a + \mu b) = \lambda f(a) + \mu f(b)\]
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\end{definition}
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\begin{definition}\xindex{Orthonormalbasis}%
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Sei $V$ ein Vektorraum und $S \subseteq V$ eine Teilmenge.
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$S$ heißt eine \textbf{Orthonormalbasis} von $V$, wenn gilt:
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\begin{defenumprops}
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\item $S$ ist eine Basis von $V$
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\item $\forall v \in S: \|v\| = 1$
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\item $\forall v_1, v_2 \in S: v_1 \neq v_2 \Rightarrow \langle v_1, v_2 \rangle = 0$
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\end{defenumprops}
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\end{definition}
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\begin{satz*}[Zwischenwertsatz]\xindex{Zwischenwertsatz}%
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Sei $a<b$ und $f \in\ C[a, b]:=C([a, b])$, weiter sei $y_0 \in \mdr$ und
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$f(a) < y_0 < f(b)$ oder $f(b) < y_0 < f(a)$. Dann existiert ein
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$x_0 \in [a, b]$ mit $f(x_0) = y_0$.
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\end{satz*}
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\begin{definition}\xindex{Eigenwert}\xindex{Eigenvektor}%
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Sei $V$ ein Vektorraum über einem Körper $\mdk$ und $f: V \rightarrow V$ eine
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lineare Abbildung.
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$v \in V \setminus \Set{0}$ heißt \textbf{Eigenvektor} $:\Leftrightarrow \exists \lambda \in \mdk: f(v) = \lambda v$.
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Wenn ein solches $\lambda \in \mdk$ existiert, heißt es \textbf{Eigenwert} von $f$.
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\end{definition}
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\begin{satz*}[Binomischer Lehrsatz]\xindex{Lehrsatz!Binomischer}%
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Sei $x, y \in \mdr$. Dann gilt:
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\[(x+y)^n = \sum_{k=0}^{n}\binom{n}{k} x^{n-k}y^{k} \;\;\; \forall n \in \mdn_0\]
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\end{satz*}
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\begin{definition}\xindex{Kreuzprodukt}\index{Vektorprodukt|see{Kreuzprodukt}}
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Seien $a, b \in \mdr^3$ Vektoren.
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\[ a \times b := \begin{pmatrix}a_1\\b_3\\a_3 \end{pmatrix} \times \begin{pmatrix}a_1\\b_3\\a_3 \end{pmatrix} = \begin{pmatrix}a_2 b_3 - a_3 b_2\\a_3 b_1 - a_1 b_3\\a_1 b_2 - a_2 b_1 \end{pmatrix}\]
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\end{definition}
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\documentclass[a5paper,oneside]{scrbook}
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\usepackage{etoolbox}
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\usepackage{amsmath,amssymb}% math symbols / fonts
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\usepackage{mathtools} % \xRightarrow
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\usepackage{nicefrac} % \nicefrac
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\usepackage[utf8]{inputenc} % this is needed for umlauts
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\usepackage[ngerman]{babel} % this is needed for umlauts
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\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
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\usepackage[framed,amsmath,thmmarks,hyperref]{ntheorem}
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\usepackage{framed}
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\usepackage{marvosym}
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\usepackage{makeidx} % for automatically generation of an index
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\usepackage{xcolor}
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\usepackage[bookmarks,bookmarksnumbered,hypertexnames=false,pdfpagelayout=OneColumn,colorlinks,hyperindex=false]{hyperref} % has to be after makeidx
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\usepackage{enumitem} % Better than \usepackage{enumerate}, because it allows to set references
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\usepackage{tabto}
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\usepackage{braket} % needed for \Set
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\usepackage{csquotes} % \enquote{}
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\usepackage{subfig} % multiple figures in one
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\usepackage{parskip} % nicer paragraphs
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\usepackage{xifthen} % \isempty
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\usepackage{changepage} % for the adjustwidth environment
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\usepackage{pst-solides3d}
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\usepackage[colorinlistoftodos]{todonotes}
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\usepackage{pgfplots}
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\pgfplotsset{compat=1.7}
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\usepackage[arrow, matrix, curve]{xy}
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\usepackage{caption} % get newlines within captions
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\usepackage{cancel}
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\usepackage{tikz} % draw
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\usepackage{tikz-3dplot} % draw
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\usepackage{tkz-fct} % draw
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\usepackage{tkz-euclide} % draw
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\usetkzobj{all} % tkz-euclide
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\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns,fadings,decorations.pathreplacing}
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\usepackage{tqft}
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\usepackage{xspace} % for new commands; decides weather I want to insert a space after the command
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\usepackage[german,nameinlink]{cleveref} % has to be after hyperref, ntheorem, amsthm
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\usepackage[left=10mm,right=10mm, top=2mm, bottom=10mm]{geometry}
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\usepackage{../shortcuts}
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\hypersetup{
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pdfauthor = {Martin Thoma},
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pdfkeywords = {Geometrie und Topologie},
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pdftitle = {Fragen zu Definitionen}
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}
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\allowdisplaybreaks
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Begin document %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\chapter{Fragen zu Definitionen}
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\section*{17.) Simpliziale Abbildungen}
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Wenn man Simpliziale Abbildungen wie folgt definiert
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\begin{definition}\xindex{Abbildung!simpliziale}%
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Seien $K, L$ Simplizialkomplexe. Eine stetige Abbildung
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\[f:|K| \rightarrow |L|\]
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heißt \textbf{simplizial}, wenn für
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jedes $\Delta \in K$ gilt:
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\begin{defenum}
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\item $f(\Delta) \in L$
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\item $f|_{\Delta} : \Delta \rightarrow f(\Delta)$ ist eine
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affine Abbildung.
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\end{defenum}
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\end{definition}
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\todo[inline]{Dann ist die Forderung \enquote{$f(\Delta) \in L$} doch immer erfüllt, oder?
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Gibt es eine Abbildung
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$f:|K| \rightarrow |L|$
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mit $f(\Delta) \notin L$?}
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\section*{18.) ÜB 1, Aufgabe 2}
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\underline{Vor.:} Es sei $(X, d)$ ein metrischer Raum, $A \subseteq X$.
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Weiter bezeichne $\fT$ die von $d$ auf $X$ erzeugte Topologie $\fT'$, die von
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der auf $A \times A$ eingeschränkten Metrik $d|_{A \times A}$ erzeugte Topologie.
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\underline{Beh.:} Die Topologie $\fT'$ und $\fT|_A$ (Spurtopologie) stimmen überein.
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\underline{Bew.:}
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\enquote{$\fT|_A \subseteq \fT'$}:
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Sei $U \in \fT|_A = \Set{V \cap A | V \in \fT}$.\\
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Dann ex. also $V \in \fT$ mit
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$U = V \cap A$.\\
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Sei $x \in U$.\\
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Da $V \in \fT$, ex. nach Bemerkung~3 ein $r > 0$ mit
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\begin{align*}
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\fB_r(x) := \Set{y \in X | d(x,y) < r} &\subseteq V\\
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\Set{y \in A | d(x,y) < r} &\subseteq V \cap A = U
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\end{align*}
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also ist $U$ offen bzgl. $d|_{A \times A}$.
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\todo[inline]{Wieso ist $U$ offen bzgl. $d|_{A \times A}$?}
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Da $x \in U$ beliebig gewählt war gilt: $\fT|_A \subseteq \fT'$
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\section*{19.) Topologische Gruppe und stetige Gruppenoperation}
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\begin{definition}%
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Sei $G$ eine Mannigfaltigkeit und $(G, \circ)$ eine Gruppe.
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\begin{defenum}
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\item $G$ heißt \textbf{topologische Gruppe}\xindex{Gruppe!topologische},
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wenn die Abbildungen $\circ: G \times G \rightarrow G$
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und $\iota: G \rightarrow G$ definiert durch
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\[g \circ h := g \cdot h \text{ und } \iota(g) := g^{-1}\]
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stetig sind.
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\item Ist $G$ eine differenzierbare Mannigfaltigkeit, so heißt
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$G$ \textbf{Lie-Gruppe}\xindex{Lie-Gruppe}, wenn
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$(G, \circ)$ und $(G, \iota)$ differenzierbar sind.
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\end{defenum}
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\end{definition}
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\begin{definition}
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Sei $G$ eine Gruppe, $X$ ein topologischer Raum und
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$\circ: G \times X \rightarrow X$ eine Gruppenoperation.
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\begin{defenum}
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\item \xindex{Gruppe operiert durch Homöomorphismen}\textbf{$G$ operiert durch Homöomorphismen}, wenn für jedes $g \in G$
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die Abbildung
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\[m_g: X \rightarrow X, x \mapsto g \circ x\]
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ein Homöomorphismus ist.
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\item Ist $G$ eine topologische Gruppe, so heißt die Gruppenoperation $\circ$
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|
||||||
\textbf{stetig}\xindex{Gruppenoperation!stetige}, wenn
|
|
||||||
$\circ: G \times X \rightarrow X$ stetig ist.
|
|
||||||
\end{defenum}
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\todo[inline]{Wenn $G$ eine topologische Gruppe ist, dann ist $\circ$ doch auf jeden Fall stetig! Was soll die Definition? Des Weiteren verstehe ich $g \circ h := g \cdot h$ nicht. Was ist $\cdot$?}
|
|
||||||
|
|
||||||
\section*{22.) MF-Beispiel}
|
|
||||||
$\praum^n(\mdr) = (\mdr^{n+1} \setminus \Set{0})/_\sim = S^n /_\sim$ und $\praum^n(\mdc)$ sind Mannigfaltigkeiten
|
|
||||||
der Dimension $n$ bzw. $2n$, da gilt:
|
|
||||||
|
|
||||||
Sei $U_i := \Set{(x_0: \dots : x_n) \in \praum^n(\mdr) | x_i \neq 0}\;\forall i \in 0, \dots, n$.
|
|
||||||
Dann ist $\praum^n(\mdr) = \bigcup_{i=0}^n U_i$ und die Abbildung
|
|
||||||
\begin{align*}
|
|
||||||
U_i &\rightarrow \mdr^n\\
|
|
||||||
(x_0 : \dots : x_n) &\mapsto \left (\frac{x_0}{x_i}, \dots, \cancel{\frac{x_i}{x_i}}, \dots, \frac{x_n}{x_i} \right )\\
|
|
||||||
(y_1 : \dots : y_{i-1} : 1 : y_i : \dots : y_n) &\mapsfrom (y_1, \dots, y_n)
|
|
||||||
\end{align*}
|
|
||||||
ist bijektiv.
|
|
||||||
\todo[inline]{Was wird im Folgenden gemacht?}
|
|
||||||
Die $U_i$ mit $i = 0, \dots, n$ bilden einen $n$-dimensionalen Atlas:
|
|
||||||
\begin{align*}
|
|
||||||
x &= (1:0:0) \in U_0 \rightarrow \mdr^2 & x &\mapsto (0,0)\\
|
|
||||||
y &= (0:1:1) \in U_2 \rightarrow \mdr^2 & y &\mapsto (0,1)
|
|
||||||
\end{align*}
|
|
||||||
$\text{Umgebung: } \fB_1 (0,1) \rightarrow \Set{(1:u:v) | \|(u,v)\| < 1} = V_1$\\
|
|
||||||
$\text{Umgebung: } \fB_1 (0,1) \rightarrow \Set{(w:z:1) | w^2 + z^2 < 1} = V_2$\\
|
|
||||||
|
|
||||||
$V_1 \cap V_2 = \emptyset$?
|
|
||||||
|
|
||||||
$(a:b:c) \in V_1 \cap V_2$\\
|
|
||||||
$\Rightarrow a \neq 0$ und $(\frac{b}{a})^2 + (\frac{c}{a})^2 < 1 \Rightarrow \frac{c}{a} < 1$\\
|
|
||||||
$\Rightarrow c \neq 0$ und $(\frac{a}{c})^2 + (\frac{b}{c})^2 < 1 \Rightarrow \frac{a}{c} < 1$\\
|
|
||||||
$\Rightarrow$ Widerspruch
|
|
||||||
|
|
||||||
|
|
||||||
\section*{23) Hyperbolische Geraden erfüllen 3.ii}
|
|
||||||
\begin{bemerkung}[Eigenschaften der hyperbolischen Geraden]
|
|
||||||
Die hyperbolischen Geraden erfüllen das Anordnungsaxiom 3 ii
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}\leavevmode
|
|
||||||
Sei $g \in G_1 \dcup G_2$ eine hyperbolische Gerade.\\
|
|
||||||
\underline{Fall 1:} $g = \Set{z \in \mdh | |z-m| = r} \in G_1$\\
|
|
||||||
Dann gilt:
|
|
||||||
\[\mdh = \underbrace{\Set{z \in \mdh | |z-m| < r}}_{=:H_1 \text{ (Kreisinneres)}} \dcup \underbrace{\Set{z \in \mdh | |z-m| < r}}_{=:H_2 \text{ (Kreisäußeres)}}\]
|
|
||||||
Da $r > 0$ ist $H_1$ nicht leer, da $r \in \mdr$ ist $H_2$ nicht leer.
|
|
||||||
|
|
||||||
\underline{Zu zeigen:} $\forall A \in H_i$, $B \in H_j$ mit
|
|
||||||
$i,j \in \Set{1,2}$ gilt:
|
|
||||||
$\overline{AB} \cap g \neq \emptyset \Leftrightarrow i \neq j$\\
|
|
||||||
\enquote{$\Leftarrow$}: Da $d_\mdh$ stetig ist, folgt diese Richtung
|
|
||||||
direkt. Alle Punkte in $H_1$ haben einen Abstand von $m$ der kleiner
|
|
||||||
ist als $r$ und alle Punkte in $H_2$ haben einen Abstand von $m$ der
|
|
||||||
größer ist als $r$. Da man jede Strecke von $A$ nach $B$ insbesondere
|
|
||||||
auch als stetige Abbildung $f: \mdr \rightarrow \mdr_{>0}$ auffassen
|
|
||||||
kann, greift der Zwischenwertsatz $\Rightarrow$ $\overline{AB} \cap g \neq \emptyset$
|
|
||||||
|
|
||||||
\enquote{$\Rightarrow$}:
|
|
||||||
\todo[inline]{TODO}
|
|
||||||
|
|
||||||
\underline{Fall 2:} $g = \Set{z \in \mdh | \Re{z} = x} \in G_2$\\
|
|
||||||
Die disjunkte Zerlegung ist:
|
|
||||||
\[\mdh = \underbrace{\Set{z \in \mdh | \Re(z) < x}}_{=: H_1 \text{ (Links)}} \dcup \underbrace{\Set{z \in \mdh | \Re(z) > x}}_{=: H_2 \text{ (Rechts)}}\]
|
|
||||||
|
|
||||||
\underline{Zu zeigen:} $\forall A \in H_i$, $B \in H_j$ mit
|
|
||||||
$i,j \in \Set{1,2}$ gilt:
|
|
||||||
$\overline{AB} \cap g \neq \emptyset \Leftrightarrow i \neq j$\\
|
|
||||||
\enquote{$\Leftarrow$}: Wie zuvor mit dem Zwischenwertsatz.
|
|
||||||
|
|
||||||
\enquote{$\Rightarrow$}:
|
|
||||||
\todo[inline]{TODO}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\section*{25.) Fragen}
|
|
||||||
\begin{enumerate}
|
|
||||||
\item Kapitel II:
|
|
||||||
\begin{enumerate}
|
|
||||||
\item Frage 7: Anschaulich ist mir klar, warum durch Verkleben gegenüberliegernder Seiten ein Torus entsteht. Was wird hier erwartet?
|
|
||||||
\end{enumerate}
|
|
||||||
\item Kapitel III
|
|
||||||
\begin{enumerate}
|
|
||||||
\item Deformationsretrakt: Das hatten wir nicht in der Vorlesung, oder? Ich meine mich zwar an das Wort zu erinnern (aus einem Übungsblatt? Einem Tutorium?) Könntest du bitte nochmals erklären was das ist?
|
|
||||||
Das ist zwar auf Blatt 7 und 8 vorgekommen, aber sonst nie.
|
|
||||||
\item Damit verbunden: Was genau ist eine "Einbettung"?
|
|
||||||
\item Was bedeutet der Pfeil: $f:S^1 \hookrightarrow \mdr^2\;\;\;$ Einbettung der Kreislinie in die Ebene
|
|
||||||
\item Was ist eine Inklusionsabbildung?
|
|
||||||
\item Was ist ein Homotopietyp? (Ist das eventuell die Anzahl der Homotopieklassen?)
|
|
||||||
\item Frage 4: Was ist eine Rose?
|
|
||||||
\item Frage 5: Wieso ist $\GL(n, \mdr)$ eine Lie-Gruppe?
|
|
||||||
\end{enumerate}
|
|
||||||
\end{enumerate}
|
|
||||||
\end{document}
|
|
||||||
|
|
@ -1,7 +0,0 @@
|
||||||
SOURCE = Fragen
|
|
||||||
make:
|
|
||||||
pdflatex $(SOURCE).tex -output-format=pdf
|
|
||||||
make clean
|
|
||||||
|
|
||||||
clean:
|
|
||||||
rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.thm
|
|
||||||
Binary file not shown.
|
|
@ -1,14 +0,0 @@
|
||||||
{
|
|
||||||
"folders":
|
|
||||||
[
|
|
||||||
{
|
|
||||||
"path": "/home/moose/Downloads/LaTeX-examples/documents/GeoTopo",
|
|
||||||
"file_exclude_patterns": ["*.aux", "*.fdb_latexmk", "*.out", "*.idx", "*.toc", "*.ilg", "*.thm", "*.ind"]
|
|
||||||
}
|
|
||||||
],
|
|
||||||
"settings":
|
|
||||||
{
|
|
||||||
"tab_size": 4,
|
|
||||||
"translate_tabs_to_spaces": false
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
@ -1,119 +0,0 @@
|
||||||
\documentclass[DIV15,BCOR12mm]{scrbook}
|
|
||||||
\pdfoutput=1
|
|
||||||
\newif\ifAFive\AFivefalse
|
|
||||||
\ifAFive
|
|
||||||
\KOMAoptions{paper=a5,twoside=true}
|
|
||||||
\else
|
|
||||||
\KOMAoptions{paper=a4,twoside=false}
|
|
||||||
\fi
|
|
||||||
\usepackage{etoolbox}
|
|
||||||
\usepackage{amsmath,amssymb}% math symbols / fonts
|
|
||||||
\usepackage{mathtools} % \xRightarrow
|
|
||||||
\usepackage{nicefrac} % \nicefrac
|
|
||||||
\usepackage[utf8]{inputenc} % this is needed for umlauts
|
|
||||||
\usepackage[ngerman]{babel} % this is needed for umlauts
|
|
||||||
\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
|
|
||||||
\usepackage[framed,amsmath,thmmarks,hyperref]{ntheorem}
|
|
||||||
\usepackage{framed}
|
|
||||||
\usepackage{marvosym}
|
|
||||||
\usepackage{makeidx} % for automatically generation of an index
|
|
||||||
\usepackage{xcolor}
|
|
||||||
\usepackage[bookmarks,bookmarksnumbered,hypertexnames=false,pdfpagelayout=OneColumn,colorlinks,hyperindex=false]{hyperref} % has to be after makeidx
|
|
||||||
\usepackage{breakurl} % allow line breaks in \href{ ... }
|
|
||||||
\ifAFive
|
|
||||||
\hypersetup{hidelinks=true}
|
|
||||||
% no \else branch needed in this case
|
|
||||||
\fi
|
|
||||||
\usepackage{enumitem} % Better than \usepackage{enumerate}, because it allows to set references
|
|
||||||
\usepackage{tabto}
|
|
||||||
\usepackage{braket} % needed for \Set
|
|
||||||
\usepackage{csquotes} % \enquote{}
|
|
||||||
\usepackage{subfig} % multiple figures in one
|
|
||||||
\usepackage{parskip} % nicer paragraphs
|
|
||||||
\usepackage{xifthen} % \isempty
|
|
||||||
\usepackage{changepage} % for the adjustwidth environment
|
|
||||||
\usepackage{pst-solides3d}
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=1.7}
|
|
||||||
\usepackage[arrow, matrix, curve]{xy}
|
|
||||||
\usepackage{caption} % get newlines within captions
|
|
||||||
\usepackage{tikz} % draw
|
|
||||||
\usepackage{tikz-3dplot} % draw
|
|
||||||
\usepackage{tkz-fct} % draw
|
|
||||||
\usepackage{tkz-euclide} % draw
|
|
||||||
\usetkzobj{all} % tkz-euclide
|
|
||||||
\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns,fadings,decorations.pathreplacing}
|
|
||||||
\usepackage{tqft}
|
|
||||||
\usepackage{xspace} % for new commands; decides weather I want to insert a space after the command
|
|
||||||
\usepackage[german,nameinlink,noabbrev]{cleveref} % has to be after hyperref, ntheorem, amsthm
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\usepackage{array,xtab,ragged2e} % for symbol table
|
|
||||||
\newlength\mylengtha
|
|
||||||
\newlength\mylengthb
|
|
||||||
\newcolumntype{P}[1]{>{\RaggedRight}p{#1}}
|
|
||||||
\tabcolsep=3pt % default: 6pt
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\usepackage{acronym}
|
|
||||||
\usepackage{cancel}
|
|
||||||
\usepackage{shortcuts}
|
|
||||||
|
|
||||||
\usepackage{fancyhdr}
|
|
||||||
\pagestyle{fancy}
|
|
||||||
\renewcommand{\chaptermark}[1]%
|
|
||||||
{\markboth{\MakeUppercase{\thechapter.\ #1}}{}}
|
|
||||||
\renewcommand{\sectionmark}[1]%
|
|
||||||
{\markright{\MakeUppercase{\thesection.\ #1}}}
|
|
||||||
\renewcommand{\headrulewidth}{0.5pt}
|
|
||||||
\renewcommand{\footrulewidth}{0pt}
|
|
||||||
\newcommand{\helv}{%
|
|
||||||
\fontfamily{phv}\fontseries{b}\fontsize{9}{11}\selectfont}
|
|
||||||
\fancyhf{}
|
|
||||||
\fancyhead[LO,RE]{\helv \thepage}
|
|
||||||
\fancyhead[LE]{\helv \leftmark}
|
|
||||||
\fancyhead[RO]{\helv \rightmark}
|
|
||||||
\fancypagestyle{plain}{%
|
|
||||||
\fancyhead{}
|
|
||||||
\renewcommand{\headrulewidth}{0pt}
|
|
||||||
}
|
|
||||||
|
|
||||||
\hypersetup{
|
|
||||||
pdfauthor = {Martin Thoma},
|
|
||||||
pdfkeywords = {Geometrie, Topologie},
|
|
||||||
pdftitle = {Geometrie und Topologie}
|
|
||||||
}
|
|
||||||
|
|
||||||
\makeindex
|
|
||||||
\allowdisplaybreaks
|
|
||||||
\usepackage{microtype}
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
\pagenumbering{roman}
|
|
||||||
\setcounter{page}{1}
|
|
||||||
\input{titlepage}
|
|
||||||
\input{Vorwort}
|
|
||||||
\tableofcontents
|
|
||||||
\ifAFive
|
|
||||||
\cleardoublepage
|
|
||||||
\fi
|
|
||||||
\pagenumbering{arabic}
|
|
||||||
\setcounter{page}{1}
|
|
||||||
\input{Kapitel1}
|
|
||||||
\input{Kapitel2}
|
|
||||||
\input{Kapitel3}
|
|
||||||
\input{Kapitel4}
|
|
||||||
\input{Kapitel5}
|
|
||||||
\input{Loesungen}
|
|
||||||
|
|
||||||
\appendix
|
|
||||||
\input{Bildquellen}
|
|
||||||
\clearpage
|
|
||||||
\input{Abkuerzungen}
|
|
||||||
\clearpage
|
|
||||||
\input{Definitionen}
|
|
||||||
\clearpage
|
|
||||||
\input{Symbolverzeichnis}
|
|
||||||
\clearpage
|
|
||||||
\addcontentsline{toc}{chapter}{Stichwortverzeichnis}
|
|
||||||
\renewcommand{\indexname}{Stichwortverzeichnis}
|
|
||||||
\printindex
|
|
||||||
\end{document}
|
|
||||||
|
|
@ -1,61 +0,0 @@
|
||||||
\clearpage
|
|
||||||
\section*{Übungsaufgaben}
|
|
||||||
\addcontentsline{toc}{section}{Übungsaufgaben}
|
|
||||||
|
|
||||||
\begin{aufgabe}[Sierpińskiraum]\label{ub1:aufg1}\xindex{Sierpińskiraum}%
|
|
||||||
Es sei $X := \Set{0,1}$ und $\fT_X := \Set{\emptyset, \Set{0}, X}$.
|
|
||||||
Dies ist der sogenannte Sierpińskiraum.
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item Beweisen Sie, dass $(X, \fT_X)$ ein topologischer Raum ist.
|
|
||||||
\item Ist $(X, \fT_X)$ hausdorffsch?
|
|
||||||
\item Ist $\fT_X$ von einer Metrik erzeugt?
|
|
||||||
\end{enumerate}
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}\label{ub1:aufg4}
|
|
||||||
Es sei $\mdz$ mit der von den Mengen $U_{a,b} := a + b \mdz (a \in \mdz, b \in \mdz \setminus \Set{0})$
|
|
||||||
erzeugten Topologie versehen.
|
|
||||||
|
|
||||||
Zeigen Sie:
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item Jedes $U_{a,b}$ und jede einelementige Teilmenge von $\mdz$ ist abgeschlossen.
|
|
||||||
\item $\Set{-1, 1}$ ist nicht offen.
|
|
||||||
\item Es gibt unendlich viele Primzahlen.
|
|
||||||
\end{enumerate}
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}[Cantorsches Diskontinuum]\label{ub2:aufg4}\xindex{Cantorsches Diskontinuum}%
|
|
||||||
Für jedes $i \in \mdn$ sei $P_i := \Set{0,1}$ mit der diskreten
|
|
||||||
Topologie. Weiter Sei $P := \prod_{i \in \mdn} P_i$.
|
|
||||||
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item Wie sehen die offenen Mengen von $P$ aus?
|
|
||||||
\item Was können Sie über den Zusammenhang von $P$ sagen?
|
|
||||||
\end{enumerate}
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}[Kompaktheit]\label{ub3:aufg1}
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item Ist $\GL_n(\mdr) = \Set{A \in \mdr^{n \times n} | \det(A) \neq 0}$ kompakt?\xindex{Gruppe!allgemeine lineare}
|
|
||||||
\item Ist $\SL_n(\mdr) = \Set{A \in \mdr^{n \times n} | \det(A) = 1}$ kompakt?\xindex{Gruppe!spezielle lineare}
|
|
||||||
\item Ist $\praum(\mdr)$ kompakt?\xindex{Raum!projektiver}
|
|
||||||
\end{enumerate}
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}[Begriffe]\label{ub3:meinsExtra}
|
|
||||||
Definieren sie die Begriffe \enquote{Homomorphismus} und
|
|
||||||
\enquote{Homöomorphismus}.
|
|
||||||
|
|
||||||
Geben Sie, falls möglich, ein Beispiel für folgende Fälle an.
|
|
||||||
Falls es nicht möglich ist, begründen Sie warum.
|
|
||||||
\begin{bspenum}
|
|
||||||
\item Ein Homomorphismus, der zugleich ein Homöomorphismus ist,
|
|
||||||
\item ein Homomorphismus, der kein Homöomorphismus ist,
|
|
||||||
\item ein Homöomorphismus, der kein Homomorphismus ist
|
|
||||||
\end{bspenum}
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}[Begriffe]\label{ub3:meinsExtra2}
|
|
||||||
Definieren sie die Begriffe \enquote{Isomorphismus},
|
|
||||||
\enquote{Isotopie} und \enquote{Isometrie}.
|
|
||||||
\end{aufgabe}
|
|
||||||
File diff suppressed because it is too large
Load diff
|
|
@ -1,15 +0,0 @@
|
||||||
\clearpage
|
|
||||||
\section*{Übungsaufgaben}
|
|
||||||
\addcontentsline{toc}{section}{Übungsaufgaben}
|
|
||||||
|
|
||||||
\begin{aufgabe}[Zusammenhang]\label{ub4:aufg1}
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item Beweisen Sie, dass eine topologische Mannigfaltigkeit
|
|
||||||
genau dann wegzusammenhängend ist, wenn sie zusammenhängend
|
|
||||||
ist
|
|
||||||
\item Betrachten Sie nun wie in \cref{bsp:mannigfaltigkeit8}
|
|
||||||
den Raum $X:= (\mdr \setminus \Set{0}) \cup \Set{0_1, 0_2}$
|
|
||||||
versehen mit der dort definierten Topologie. Ist $X$
|
|
||||||
wegzusammenhängend?
|
|
||||||
\end{enumerate}
|
|
||||||
\end{aufgabe}
|
|
||||||
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|
|
@ -1,16 +0,0 @@
|
||||||
%\clearpage
|
|
||||||
%\section*{Übungsaufgaben}
|
|
||||||
%\addcontentsline{toc}{section}{Übungsaufgaben}
|
|
||||||
|
|
||||||
%Die Lösung ist zu lang (vgl. Loesungen.tex)
|
|
||||||
%\begin{aufgabe}\label{ub7:aufg1}
|
|
||||||
% Berechnen Sie die Homotogiegruppen von $S^1$ und $S^2$, indem Sie
|
|
||||||
% zu $S^1$ bzw. $S^2$ homöomorphe Simplizialkomplexe betrachten.
|
|
||||||
%\end{aufgabe}
|
|
||||||
|
|
||||||
|
|
||||||
% Auch diese Aufgabe ist zu lang...
|
|
||||||
%\begin{aufgabe}\label{ub7:aufg3}
|
|
||||||
% Es sei $G$ eine topologische Gruppe und $e$ ihr neutrales
|
|
||||||
% Element. Man beweise, dass $\pi_1(G,e)$ abelsch ist.
|
|
||||||
%\end{aufgabe}
|
|
||||||
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|
|
@ -1,49 +0,0 @@
|
||||||
\clearpage
|
|
||||||
\section*{Übungsaufgaben}
|
|
||||||
\addcontentsline{toc}{section}{Übungsaufgaben}
|
|
||||||
|
|
||||||
\begin{aufgabe}\label{ub11:aufg1}
|
|
||||||
Seien $(X, d)$ eine absolute Ebene und $P, Q, R \in X$ Punkte.
|
|
||||||
Der \textit{Scheitelwinkel}\xindex{Scheitelwinkel} des Winkels $\angle PQR$ ist
|
|
||||||
der Winkel, der aus den Halbgeraden $QP^-$ und $QR^-$ gebildet
|
|
||||||
wird. Die \textit{Nebenwinkel}\xindex{Nebenwinkel} von $\angle PQR$
|
|
||||||
sind die von $QP^+$ und $QR^-$ bzw. $QP^-$ und $QR^+$ gebildeten
|
|
||||||
Winkel.
|
|
||||||
|
|
||||||
Zeigen Sie:
|
|
||||||
\begin{aufgabeenum}
|
|
||||||
\item Die beiden Nebenwinkel von $\angle PQR$ sind gleich.
|
|
||||||
\item Der Winkel $\angle PQR$ ist gleich seinem Scheitelwinkel.
|
|
||||||
\end{aufgabeenum}
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}\label{ub11:aufg3}
|
|
||||||
Sei $(X, d)$ eine absolute Ebene. Der \textit{Abstand}\xindex{Abstand} eines
|
|
||||||
Punktes $P$ zu einer Menge $Y \subseteq X$ von Punkten ist
|
|
||||||
definiert durch $d(P, Y) := \inf{d(P, y) | y \in Y}$.
|
|
||||||
|
|
||||||
Zeigen Sie:
|
|
||||||
\begin{aufgabeenum}
|
|
||||||
\item \label{ub11:aufg3.a} Ist $\triangle ABC$ ein Dreieck, in dem die Seiten
|
|
||||||
$\overline{AB}$ und $\overline{AC}$ kongruent sind, so
|
|
||||||
sind die Winkel $\angle ABC$ und $\angle BCA$ gleich.
|
|
||||||
\item \label{ub11:aufg3.b} Ist $\triangle ABC$ ein beliebiges Dreieck, so liegt
|
|
||||||
der längeren Seite der größere Winkel gegenüber und
|
|
||||||
umgekehrt.
|
|
||||||
\item \label{ub11:aufg3.c} Sind $g$ eine Gerade und $P \notin g$ ein Punkt, so gibt
|
|
||||||
es eine eindeutige Gerade $h$ mit $P \in h$ und die
|
|
||||||
$g$ im rechten Winkel schneidet. Diese Grade heißt
|
|
||||||
\textit{Lot}\xindex{Lot} von $P$ auf $g$ und der
|
|
||||||
Schnittpunkt des Lots mit $g$ heißt \textit{Lotfußpunkt}\xindex{Lotfußpunkt}.
|
|
||||||
\end{aufgabeenum}
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}\label{ub-tut-24:a1}
|
|
||||||
Seien $f, g, h \in G$ und paarweise verschieden.
|
|
||||||
|
|
||||||
Zeigen Sie: $f \parallel g \land g \parallel h \Rightarrow f \parallel h$
|
|
||||||
\end{aufgabe}
|
|
||||||
|
|
||||||
\begin{aufgabe}\label{ub-tut-24:a3}%
|
|
||||||
Beweise den Kongruenzsatz $SSS$.
|
|
||||||
\end{aufgabe}
|
|
||||||
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|
|
@ -1,655 +0,0 @@
|
||||||
%!TEX root = GeoTopo.tex
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Mitschrieb vom 30.01.2014 %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\chapter{Krümmung}
|
|
||||||
|
|
||||||
\begin{definition}\xindex{Kurve}%
|
|
||||||
Sei $f: [a, b] \rightarrow \mdr^n$ eine eine Funktion aus $C^\infty$.
|
|
||||||
Dann heißt $f$ \textbf{Kurve}.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\section{Krümmung von Kurven}\label{sec:Kurvenkrümmung}
|
|
||||||
\begin{definition}%In Vorlesung: Def.+Bem. 16.1
|
|
||||||
Sei $\gamma: I = [a, b] \rightarrow \mdr^n$ eine Kurve.
|
|
||||||
|
|
||||||
\begin{defenum}
|
|
||||||
\item Die Kurve $\gamma$ heißt
|
|
||||||
\textbf{durch Bogenlänge parametrisiert}\xindex{parametrisiert!durch Bogenlänge},
|
|
||||||
wenn gilt:
|
|
||||||
\[\|\gamma'(t)\|_2 = 1 \;\;\; \forall t \in I\]
|
|
||||||
Dabei ist $\gamma'(t) = \left (\gamma_1'(t), \gamma_2'(t), \dots, \gamma_n'(t) \right)$.
|
|
||||||
\item $l(\gamma) = \int_a^b \|\gamma'(t)\| \mathrm{d} t$ heißt
|
|
||||||
\textbf{Länge von $\gamma$}\xindex{Kurve!Länge einer}.
|
|
||||||
\end{defenum}
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}[Eigenschaften von Kurven I]%In Vorlesung: Def.+Bem. 16.1
|
|
||||||
Sei $\gamma: I = [a, b] \rightarrow \mdr^n$ eine $C^\infty$-Funktion.
|
|
||||||
|
|
||||||
\begin{bemenum}
|
|
||||||
\item Ist $\gamma$ durch Bogenlänge parametrisiert, so ist $l(\gamma) = b-a$.
|
|
||||||
\item \label{bem:16.1d} Ist $\gamma$ durch Bogenlänge parametrisiert, so ist
|
|
||||||
$\gamma'(t)$ orthogonal zu $\gamma''(t)$ für alle $t \in I$.
|
|
||||||
\end{bemenum}
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}\leavevmode
|
|
||||||
\begin{enumerate}[label=\alph*)]
|
|
||||||
\item $l(\gamma) = \int_a^b \|\gamma'(t)\| \mathrm{d} t = \int_a^b 1 \mathrm{d} t = b - a$.
|
|
||||||
\item Im Folgenden wird die Aussage nur für $\gamma: [a, b] \rightarrow \mdr^2$ bewiesen.
|
|
||||||
Allerdings funktioniert der Beweis im $\mdr^n$ analog. Es muss nur
|
|
||||||
die Ableitung angepasst werden.
|
|
||||||
\begin{align*}
|
|
||||||
1 &= \|\gamma'(t)\| = \|\gamma'(t)\|^2 = \langle \gamma'(t), \gamma'(t) \rangle\\
|
|
||||||
\Rightarrow 0 &= \frac{\mathrm{d}}{\mathrm{d}t} \langle \gamma'(t), \gamma'(t) \rangle\\
|
|
||||||
&= \frac{\mathrm{d}}{\mathrm{d}t} (\gamma_1'(t)\gamma_1'(t) + \gamma_2'(t)\gamma_2'(t))\\
|
|
||||||
&= 2 \cdot (\gamma_1''(t) \cdot \gamma_1'(t) + \gamma_2''(t) \cdot \gamma_2'(t))\\
|
|
||||||
&= 2 \cdot \langle \gamma''(t), \gamma'(t) \rangle
|
|
||||||
\end{align*}
|
|
||||||
\end{enumerate}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{definition}%In Vorlesung: Definition 16.2
|
|
||||||
Sei $\gamma: I \rightarrow \mdr^2$ eine durch Bogenlänge
|
|
||||||
parametrisierte Kurve.
|
|
||||||
|
|
||||||
\begin{defenum}
|
|
||||||
\item Für $t \in I$ sei $n(t)$ \textbf{Normalenvektor}\xindex{Normalenvektor}
|
|
||||||
an $\gamma$ in $t$ wenn gilt:
|
|
||||||
\[\langle n(t), \gamma'(t) \rangle = 0 \text{, } \|n(t)\|=1 \text{ und } \det((\gamma'(t), n(t))) = +1\]
|
|
||||||
\item Seit $\kappa: I \rightarrow \mdr$ so, dass gilt:
|
|
||||||
\[\gamma''(t) = \kappa(t) \cdot n(t)\]
|
|
||||||
Dann heißt $\kappa(t)$ \textbf{Krümmung}\xindex{Krümmung}
|
|
||||||
von $\gamma$ in $t$.
|
|
||||||
\end{defenum}
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
Da $n(t)$ und $\gamma''(t)$ nach \cref{bem:16.1d} linear
|
|
||||||
abhängig sind, existiert $\kappa(t)$.
|
|
||||||
|
|
||||||
\begin{beispiel}%In Vorlesung: Beispiel 16.3
|
|
||||||
Gegeben sei ein Kreis mit Radius $r$, d.~h. mit Umfang $2\pi r$.
|
|
||||||
Es gilt:
|
|
||||||
|
|
||||||
\[\gamma(t) = \left (r \cdot \cos \frac{t}{r}, r \cdot \sin \frac{t}{r} \right ) \text{ für } t \in [0, 2\pi r]\]
|
|
||||||
ist parametrisiert durch Bogenlänge, da gilt:
|
|
||||||
|
|
||||||
\begin{align*}
|
|
||||||
\gamma'(t) &= \left ((r \cdot \frac{1}{r}) (- \sin \frac{t}{r}), r \frac{1}{r} \cos \frac{t}{r} \right )\\
|
|
||||||
&= \left (- \sin \frac{t}{r}, \cos \frac{t}{r} \right )
|
|
||||||
\end{align*}
|
|
||||||
|
|
||||||
Der Normalenvektor von $\gamma$ in $t$ ist
|
|
||||||
\[n(t) = \left (- \cos \frac{t}{r}, - \sin \frac{t}{r} \right )\]
|
|
||||||
da gilt:
|
|
||||||
|
|
||||||
\begin{align*}
|
|
||||||
\langle n(t), \gamma'(t) \rangle &=
|
|
||||||
\left \langle
|
|
||||||
\begin{pmatrix}- \cos \frac{t}{r}\\ - \sin \frac{t}{r}\end{pmatrix},
|
|
||||||
\begin{pmatrix}- \sin \frac{t}{r}\\ \cos \frac{t}{r}\end{pmatrix}
|
|
||||||
\right \rangle\\
|
|
||||||
&= (- \cos \frac{t}{r}) \cdot (- \sin \frac{t}{r}) + (- \sin \frac{t}{r}) \cdot (\cos \frac{t}{r})\\
|
|
||||||
&= 0\\
|
|
||||||
\|n(t)\| &= \left \| (- \cos \frac{t}{r}, - \sin \frac{t}{r}) \right \|\\
|
|
||||||
&=(- \cos \frac{t}{r})^2 + (- \sin \frac{t}{r})^2\\
|
|
||||||
&= 1\\
|
|
||||||
\det(\gamma_1'(t), n(t)) &= \left \|
|
|
||||||
\begin{pmatrix}
|
|
||||||
- \sin \frac{t}{r} & - \cos \frac{t}{r}\\
|
|
||||||
\cos \frac{t}{r} & - \sin \frac{t}{r}
|
|
||||||
\end{pmatrix}
|
|
||||||
\right \|\\
|
|
||||||
&= (- \sin \frac{t}{r})^2 - (- \cos \frac{t}{r}) \cdot \cos \frac{t}{r}\\
|
|
||||||
&= 1
|
|
||||||
\end{align*}
|
|
||||||
|
|
||||||
Die Krümmung ist für jedes $t$ konstant $\frac{1}{r}$, da gilt:
|
|
||||||
\begin{align*}
|
|
||||||
\gamma''(t) &= \left (- \frac{1}{r} \cos \frac{t}{r}, - \frac{1}{r} \sin \frac{t}{r} \right )\\
|
|
||||||
&= \frac{1}{r} \cdot \left (- \cos \frac{t}{r}, - \sin \frac{t}{r} \right )\\
|
|
||||||
\Rightarrow \kappa(t) &= \frac{1}{r}
|
|
||||||
\end{align*}
|
|
||||||
\end{beispiel}
|
|
||||||
|
|
||||||
\begin{definition}%In Vorlesung: Def.+Bem. 16.4
|
|
||||||
Sei $\gamma: I \rightarrow \mdr^3$ eine durch Bogenlänge parametrisierte
|
|
||||||
Kurve.
|
|
||||||
|
|
||||||
\begin{defenum}
|
|
||||||
\item Für $t \in I$ heißt $\kappa(t) := \|\gamma''(t)\|$ die
|
|
||||||
\textbf{Krümmung}\xindex{Krümmung} von $\gamma$ in $t$.
|
|
||||||
\item Ist für $t \in I$ die Ableitung $\gamma''(t) \neq 0$,
|
|
||||||
so heißt $\frac{\gamma''(t)}{\|\gamma''(t)\|}$ \textbf{Normalenvektor}\xindex{Normalenvektor}
|
|
||||||
an $\gamma$ in $t$.
|
|
||||||
\item \label{def:16.4c} $b(t)$ sei ein Vektor, der $\gamma'(t), n(t)$
|
|
||||||
zu einer orientierten Orthonormalbasis von $\mdr^3$ ergänzt.
|
|
||||||
Also gilt:
|
|
||||||
\[\det(\gamma'(t), n(t), b(t)) = 1\]
|
|
||||||
$b(t)$ heißt \textbf{Binormalenvektor}\xindex{Binormalenvektor},
|
|
||||||
die Orthonormalbasis
|
|
||||||
\[\Set{\gamma'(t), n(t), b(t)}\]
|
|
||||||
heißt \textbf{begleitendes Dreibein}\xindex{Dreibein!begleitendes}.
|
|
||||||
\end{defenum}
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}[Eigenschaften von Kurven II]%In Vorlesung: Def.+Bem 16.4
|
|
||||||
Sei $\gamma: I \rightarrow \mdr^3$ durch Bogenlänge parametrisierte
|
|
||||||
Kurve.
|
|
||||||
|
|
||||||
\begin{bemenum}
|
|
||||||
\item $n(t)$ ist orthogonal zu $\gamma'(t)$.
|
|
||||||
\item $b(t)$ aus \cref{def:16.4c} ist eindeutig.
|
|
||||||
\end{bemenum}
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\section{Tangentialebene}\index{Tangentialebene|(}
|
|
||||||
Erinnerung Sie sich an \cref{def:8.5} \enquote{reguläre Fläche}.
|
|
||||||
|
|
||||||
Äquivalent dazu ist: $S$ ist lokal von der Form
|
|
||||||
\[V(f) = \Set{x \in \mdr^3 | f(x) = 0 }\]
|
|
||||||
für eine $C^\infty$-Funktion $f: \mdr^3 \rightarrow \mdr$.
|
|
||||||
|
|
||||||
\begin{definition}\label{def:Tangentialebene}%In Vorlesung: 17.1
|
|
||||||
Sei $S \subseteq \mdr^3$ eine reguläre Fläche, $s \in S$,
|
|
||||||
$F: U \rightarrow V \cap S$ eine lokale Parametrisierung um $s \in V$:
|
|
||||||
\[(u,v) \mapsto (x(u,v), y(u,v), z(u,v))\]
|
|
||||||
Für $p=F^{-1}(s) \in U$ sei
|
|
||||||
\[ J_F(p) = \begin{pmatrix}
|
|
||||||
\frac{\partial x}{\partial u} (p) & \frac{\partial x}{\partial v} (p)\\
|
|
||||||
\frac{\partial y}{\partial u} (p) & \frac{\partial y}{\partial v} (p)\\
|
|
||||||
\frac{\partial z}{\partial u} (p) & \frac{\partial z}{\partial v} (p)
|
|
||||||
\end{pmatrix}\]
|
|
||||||
und $D_p F: \mdr^2 \rightarrow \mdr^3$ die durch $J_F (p)$
|
|
||||||
definierte lineare Abbildung.
|
|
||||||
|
|
||||||
Dann heißt $T_s S := \Bild(D_p F)$ die \textbf{Tangentialebene}\xindex{Tangentialebene}
|
|
||||||
an $s \in S$.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}[Eigenschaften der Tangentialebene]%
|
|
||||||
\begin{bemenum}
|
|
||||||
\item $T_s S$ ist $2$-dimensionaler Untervektorraum von $\mdr^3$.%In Vorlesung: 17.2
|
|
||||||
\item $T_s S = \langle \tilde{u}, \tilde{v} \rangle$, wobei $\tilde{u}, \tilde{v}$
|
|
||||||
die Spaltenvektoren der Jacobi-Matrix $J_F(p)$ sind.
|
|
||||||
\item $T_s S$ hängt nicht von der gewählten Parametrisierung ab.%In Vorlesung: 17.3
|
|
||||||
\item Sei $S=V(f)$ eine reguläre Fläche in $\mdr^3$, also %In Vorlesung: Bemerkung 17.4
|
|
||||||
$f:V \rightarrow \mdr$ eine $C^\infty$-Funktion, $V \subseteq \mdr^3$
|
|
||||||
offen, $\grad(f)(x) \neq 0$ für alle $x \in S$.
|
|
||||||
|
|
||||||
Dann ist $T_s S = (\grad(f)(s))^\perp$ für jedes $s \in S$.
|
|
||||||
\end{bemenum}
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}\leavevmode
|
|
||||||
\begin{enumerate}[label=\alph*)]
|
|
||||||
\item \label{bew:tangentialebene.a} $J_F$ ist eine $3 \times 2$-Matrix, die mit einem $2 \times 1$-Vektor
|
|
||||||
multipliziert wird. Das ist eine lineare Abbildung und aus der
|
|
||||||
linearen Algebra ist bekannt, das das Bild ein Vektorraum ist.
|
|
||||||
Da $\rang(J_F) = 2$, ist auch $\dim (T_s S) = 2$.
|
|
||||||
\item Hier kann man wie in \cref{bew:tangentialebene.a} argumentieren
|
|
||||||
\item $T_s S = \{x \in \mdr^3 | \exists \text{parametrisierte Kurve }
|
|
||||||
\gamma:[- \varepsilon, + \varepsilon] \rightarrow S
|
|
||||||
\text{ für ein } \varepsilon > 0
|
|
||||||
\text{ mit } \gamma(0) = s \text{ und } \gamma'(0) = x
|
|
||||||
\}$\\
|
|
||||||
Wenn jemand diesen Beweis führt, bitte an info@martin-thoma.de
|
|
||||||
schicken.%TODO
|
|
||||||
\item Sei $x \in T_s S, \gamma:[-\varepsilon, +\varepsilon] \rightarrow S$
|
|
||||||
eine parametrisierte Kurve mit $\varepsilon > 0$ und $\gamma'(0) = s$,
|
|
||||||
sodass $\gamma'(0) = x$ gilt. Da $\gamma(t) \in S$ für alle
|
|
||||||
$t \in [-\varepsilon, \varepsilon]$, ist $f \circ \gamma = 0$\\
|
|
||||||
$\Rightarrow 0 = (f \circ \gamma)'(0) = \langle \grad(f)(\gamma(0)), \gamma'(0) \rangle$\\
|
|
||||||
$\Rightarrow T_s S \subseteq \grad (f)(s)^\perp$\\
|
|
||||||
$\xRightarrow{\dim = 2} T_s S = (\grad(f)(s))^\perp$
|
|
||||||
\end{enumerate}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Mitschrieb vom 04.02.2014 %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\begin{definition}%In Vorlesung: Def.+Bem 17.5
|
|
||||||
\begin{defenum}
|
|
||||||
\item Ein \textbf{Normalenfeld}\xindex{Normalenfeld} auf der regulären
|
|
||||||
Fläche $S \subseteq \mdr^3$ ist eine Abbildung $n: S \rightarrow S^2 \subseteq \mdr^3$
|
|
||||||
mit $n(s) \in T_s S^\perp$ für jedes $s \in S$.
|
|
||||||
\item $S$ heißt \textbf{orientierbar}\xindex{Fläche!orientierbare},
|
|
||||||
wenn es ein stetiges Normalenfeld auf $S$ gibt.
|
|
||||||
\end{defenum}
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
Manchmal wird zwischen einem \textit{Normalenfeld} und einem
|
|
||||||
\textit{Einheitsnormalenfeld}\xindex{Einheitsnormalenfeld} unterschieden.
|
|
||||||
Im Folgenden werden diese Begriffe jedoch synonym benutzt.
|
|
||||||
|
|
||||||
\begin{bemerkung}[Eigenschaften von Normalenfeldern]%In Vorlesung: Def.+Bem 17.5
|
|
||||||
\begin{bemenum}
|
|
||||||
\item Ein Normalenfeld auf $S$ ist genau dann stetig, wenn es
|
|
||||||
glatt ist (also $C^\infty$).
|
|
||||||
\item Zu jedem $s \in S$ gibt es eine Umgebung $V \subseteq \mdr^3$
|
|
||||||
von $s$ und eine lokale Parametrisierung $F: U \rightarrow V$
|
|
||||||
von $S$ um $s$, sodass auf $F(U) = V \cap S$
|
|
||||||
ein stetiges Normalenfeld existiert.
|
|
||||||
\item $S$ ist genau dann orientierbar, wenn es einen
|
|
||||||
differenzierbaren Atlas von $S$ aus lokalen Parametrisierungen
|
|
||||||
$F_i: U_i \rightarrow V_i,\;i \in I$ gibt, sodass
|
|
||||||
für alle $i, j \in F$ und alle $s \in V_i \cap V_j \cap S$
|
|
||||||
gilt:
|
|
||||||
\[\det(\underbrace{D_s \overbrace{F_j \circ F_i^{-1}}^{V_i \rightarrow V_j}}_{\in \mdr^{3 \times 3}}) > 0\]
|
|
||||||
\end{bemenum}
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}
|
|
||||||
Wird hier nicht geführt.%TODO: Übung? Übungsblatt?
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{beispiel}[Normalenfelder]
|
|
||||||
\begin{bspenum}
|
|
||||||
\item $S = S^2$, $n_1 = \id_{S^2}$ ist ein stetiges Normalenfeld.\\
|
|
||||||
Auch $n_2 = - \id_{S^2}$ ist ein stetiges Normalenfeld.
|
|
||||||
\item $S = \text{Möbiusband}$ (vgl. \cref{fig:moebius-strip})
|
|
||||||
ist nicht orientierbar. Es existiert ein Normalenfeld,
|
|
||||||
aber kein stetiges Normalenfeld.
|
|
||||||
\end{bspenum}
|
|
||||||
\end{beispiel}
|
|
||||||
|
|
||||||
\begin{figure}[htp]\xindex{Möbiusband}
|
|
||||||
\centering
|
|
||||||
\includegraphics[width=0.5\linewidth, keepaspectratio]{figures/moebius-strip.pdf}
|
|
||||||
\caption{Möbiusband}
|
|
||||||
\label{fig:moebius-strip}
|
|
||||||
\end{figure}
|
|
||||||
\index{Tangentialebene|)}
|
|
||||||
\section{Gauß-Krümmung}\index{Gauß-Krümmung|(}
|
|
||||||
\begin{bemerkung}\label{bem:18.1}%In Vorlesung: Bemerkung 18.1
|
|
||||||
Sei $S$ eine reguläre Fläche, $s \in S$, $n(s)$ ist ein Normalenvektor
|
|
||||||
in $s$, $x \in T_s S$, $\|x\| = 1$.
|
|
||||||
|
|
||||||
Sei $E$ der von $x$ und $n(s)$ aufgespannte 2-dimensionale
|
|
||||||
Untervektorraum von $\mdr^3$.
|
|
||||||
|
|
||||||
Dann gibt es eine Umgebung $V \subseteq \mdr^3$ von $s$, sodass
|
|
||||||
\[C := (s + E) \cap S \cap V\]
|
|
||||||
das Bild einer durch Bogenlänge parametrisierten Kurve
|
|
||||||
$\gamma:[-\varepsilon, \varepsilon] \rightarrow S$ enthält mit
|
|
||||||
$\gamma(0) = s$ und $\gamma'(0) = x$.
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}
|
|
||||||
\enquote{Satz über implizite Funktionen}\footnote{Siehe z.~B.
|
|
||||||
\url{https://github.com/MartinThoma/LaTeX-examples/tree/master/documents/Analysis\%20II}}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{definition}\xindex{Normalkrümmung}%In Vorlesung: Definition 18.2
|
|
||||||
In der Situation aus \cref{bem:18.1} heißt die Krümmung $\kappa_\gamma(0)$
|
|
||||||
der Kurve $\gamma$ in der Ebene $(s+ E)$ im Punkt $s$ die
|
|
||||||
\textbf{Normalkrümmung} von $S$ in $s$ in Richtung
|
|
||||||
$x = \gamma'(0)$.
|
|
||||||
|
|
||||||
Man schreibt: $\kappanor(s, x) := \kappa_\gamma(0)$
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\underline{Hinweis}: Die Krümmung ist nur bis auf das Vorzeichen bestimmt.
|
|
||||||
|
|
||||||
\begin{beispiel}[Gauß-Krümmung]%In Vorlesung: Beispiel 18.3
|
|
||||||
\begin{bspenum}
|
|
||||||
\item $S = S^2 = V(X^2 + Y^2 + Z^2 - 1)$ ist die Kugel um den Ursprung mit Radius~1,
|
|
||||||
$n = \id$, $s=(0,0,1)$, $x=(1,0,0)$\\
|
|
||||||
$\Rightarrow E = \mdr \cdot x + \mdr \cdot n(s)$ ($x,z\text{-Ebene}$)
|
|
||||||
|
|
||||||
$C = E \cap S$ ist Kreislinie\\
|
|
||||||
$\kappanor(s, x) = \frac{1}{r} = 1$
|
|
||||||
\item $S = V(X^2 + Z^2 - 1) \subseteq \mdr^3$ ist ein Zylinder (siehe \cref{fig:regular-zylinder}).
|
|
||||||
$s = (1,0,0)$\\
|
|
||||||
$x_1 = (0,1,0) \Rightarrow E_1 = \mdr \cdot e_1 + \mdr \cdot e_2$ ($x,y\text{-Ebene}$)\\
|
|
||||||
$S \cap E_1 = V(X^2 + Y^2 - 1) \cap E$, Kreislinie in $E$\\
|
|
||||||
$\Rightarrow \kappanor(s, x_1) = \pm 1$\\
|
|
||||||
$x_2 = (0, 0, 1), E_2 = \mdr \cdot e_1 + \mdr \cdot e_3$ ($x,z\text{-Ebene}$)\\
|
|
||||||
$V \cap E_2 \cap S = \Set{(1, 0, z) \in \mdr^3 | z \in \mdr}$ ist eine Gerade\\
|
|
||||||
$\Rightarrow \kappanor(s, x_2) = 0$
|
|
||||||
\item $S = V(X^2 - Y^2 - Z)$, $s = (0,0,0)$ (Hyperbolisches Paraboloid\xindex{Paraboloid!hyperbolisches}, siehe \cref{fig:hyperbolic-paraboloid})\\
|
|
||||||
$x_1 = (1,0,0)$, $n(s) = (0,0,1)$\\
|
|
||||||
$x_2 = (0, 1, 0)$\\
|
|
||||||
$\kappanor(s, x_1) = \hphantom{-}2$\\
|
|
||||||
$\kappanor(s, x_2) = -2$
|
|
||||||
\end{bspenum}
|
|
||||||
\end{beispiel}
|
|
||||||
|
|
||||||
\begin{figure}[ht]
|
|
||||||
\centering
|
|
||||||
\subfloat[$S = V(X^2 + Z^2 - 1)$]{
|
|
||||||
\resizebox{0.4\linewidth}{!}{\input{figures/cylinder.tex}}
|
|
||||||
\label{fig:regular-zylinder}
|
|
||||||
}%
|
|
||||||
\subfloat[$S = V(X^2 - Y^2 - Z)$]{
|
|
||||||
\resizebox{0.4\linewidth}{!}{\input{figures/hyperbolic-paraboloid.tex}}
|
|
||||||
\label{fig:hyperbolic-paraboloid}
|
|
||||||
}%
|
|
||||||
\label{fig:regular-surfaces}
|
|
||||||
\caption{Beispiele für reguläre Flächen}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Mitschrieb vom 06.02.2014 %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\begin{definition}\label{def:18.4}\xindex{Normalkrümmung}%In Vorlesung: Def. 18.4
|
|
||||||
Sei $S \subseteq \mdr^3$ eine reguläre Fläche, $s \in S$ und $n$ ein
|
|
||||||
stetiges Normalenfeld auf $S$.
|
|
||||||
|
|
||||||
$\gamma:[-\varepsilon, \varepsilon] \rightarrow S$ eine nach
|
|
||||||
Bogenlänge parametrisierte Kurve ($\varepsilon > 0$) mit
|
|
||||||
$\gamma(0) = s$ und $\gamma''(0) \neq 0$.
|
|
||||||
|
|
||||||
Sei $n(0) := \frac{\gamma''(0)}{\|\gamma''(0)\|}$. Zerlege
|
|
||||||
\[n(0) = n(0)^t + n(0)^\perp \text{ mit } n(0)^t \in T_s S \text{ und } n(0)^\perp \in (T_s S)^\perp\]
|
|
||||||
|
|
||||||
Dann ist $n(0)^\perp = \langle n(0), n(s) \rangle \cdot n(s)$\\
|
|
||||||
$\kappanor(s, \gamma) := \langle \gamma''(0), n(s) \rangle$
|
|
||||||
die \textbf{Normalkrümmung}.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}
|
|
||||||
Sei $\overline{\gamma}(t) = \gamma(-t)$, $t \in [- \varepsilon, \varepsilon]$.
|
|
||||||
Dann ist $\kappanor(s, \overline{\gamma}) = \kappanor(s, \gamma)$.
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}
|
|
||||||
$\overline{\gamma}''(0) = \gamma''(0)$, da $\overline{\gamma}'(0) = - \gamma'(0)$.
|
|
||||||
|
|
||||||
Es gilt: $\kappanor(s,\gamma)$ hängt nur von $|\gamma'(0)|$ ab
|
|
||||||
und ist gleich $\kappanor(s, \gamma'(0))$.
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{bemerkung}%In Vorlesung: Bem.+Def. 18.6
|
|
||||||
Sei $S$ eine reguläre Fläche und $n=n(s)$ ein Normalenvektor an
|
|
||||||
$S$ in $s$.
|
|
||||||
|
|
||||||
Sei $T_{s}^{1} S = \Set{x \in T_s S | \|x\| = 1} \cong S^1$.
|
|
||||||
Dann ist
|
|
||||||
\[ \kappanor^n(s): T^1_s S \rightarrow \mdr, \;\;\; x \mapsto \kappanor(s,x)\]
|
|
||||||
eine glatte Funktion und
|
|
||||||
$\Bild \kappanor^n(s)$ ist ein abgeschlossenes Intervall.
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{definition}\xindex{Hauptkrümmung}\xindex{Gauß-Krümmung}%In Vorlesung: Bem.+Def. 18.6
|
|
||||||
Sei $S$ eine reguläre Fläche und $n=n(s)$ ein Normalenvektor an
|
|
||||||
$S$ in $s$.
|
|
||||||
|
|
||||||
\begin{defenum}
|
|
||||||
\item $\begin{aligned}[t]
|
|
||||||
\kappa^n_1(s) :&= \min \Set{\kappanor^n(s,x) | x \in T_s^1 S} \text{ und }\\
|
|
||||||
\kappa^n_2(s) :&= \max \Set{\kappanor^n(s,x) | x \in T_s^1 S}
|
|
||||||
\end{aligned}$
|
|
||||||
heißen \textbf{Hauptkrümmungen} von $S$ in $s$.
|
|
||||||
\item $K(s) := \kappa_1^n(s) \cdot \kappa_2^n(s)$ heißt
|
|
||||||
\textbf{Gauß-Krümmung} von $S$ in $s$.
|
|
||||||
\end{defenum}
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}%In Vorlesung: Bem.+Def. 18.6
|
|
||||||
Ersetzt man $n$ durch $-n$, so gilt:
|
|
||||||
|
|
||||||
\begin{align*}
|
|
||||||
\kappanor^{-n}(s, x) &= - \kappanor^n(x)\; \forall x \in T_s^1 S\\
|
|
||||||
\Rightarrow \kappa_1^{-n}(s) &= - \kappa_2^n(s)\\
|
|
||||||
\kappa_2^{-n}(s) &= - \kappa_1^n (s)\\
|
|
||||||
\text{ und } K^{-n}(s) &= K^n(s) =: K(s)
|
|
||||||
\end{align*}
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beispiel}
|
|
||||||
\begin{bspenum}
|
|
||||||
\item $S = S^2$. Dann ist $\kappa_1(s) = \kappa_2(s) = \pm 1\;\forall s \in S^2$\\
|
|
||||||
$\Rightarrow K(s) = 1$
|
|
||||||
\item Zylinder:\\
|
|
||||||
$\kappa_1(s) = 0, \kappa_2(s) = 1 \Rightarrow K(s) = 0$
|
|
||||||
\item Sattelpunkt auf hyperbolischem Paraboloid:\\
|
|
||||||
$\kappa_1(s) < 0, \kappa_2(s) = 0 \rightarrow K(s) < 0$
|
|
||||||
\item $S = \text{Torus}$. Siehe \cref{fig:torus-gauss-kruemmung}\\
|
|
||||||
\begin{figure}[htp]\xindex{Torus}
|
|
||||||
\centering
|
|
||||||
\includegraphics[width=0.95\linewidth, keepaspectratio]{figures/torus-gauss-kruemmung.pdf}
|
|
||||||
\caption{$K(s_1) > 0$, $K(s_2) = 0$, $K(s_3) < 0$}
|
|
||||||
\label{fig:torus-gauss-kruemmung}
|
|
||||||
\end{figure}
|
|
||||||
\end{bspenum}
|
|
||||||
\end{beispiel}
|
|
||||||
|
|
||||||
\begin{bemerkung}%In Vorlesung: Bem. 18.7
|
|
||||||
Sei $S$ eine reguläre Fläche, $s \in S$ ein Punkt.
|
|
||||||
\begin{bemenum}
|
|
||||||
\item Ist $K(s) > 0$, so liegt $S$ in einer Umgebung von $s$
|
|
||||||
ganz auf einer Seite von $T_s S + s$.
|
|
||||||
\item Ist $K(s) < 0$, so schneidet jede Umgebung von $s$ in $S$
|
|
||||||
beide Seiten von $T_s S + s$.
|
|
||||||
\end{bemenum}
|
|
||||||
\end{bemerkung}
|
|
||||||
\index{Gauß-Krümmung|)}
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Mitschrieb vom 11.02.2014 %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Erste und zweite Fundamentalform}%In Vorlesung: §19
|
|
||||||
Sei $S \subseteq \mdr^3$ eine reguläre Fläche, $s \in S$, $T_s S$ die Tangentialebene
|
|
||||||
an $S$ in $s$ und $F: U \rightarrow V$ eine lokale Parametrisierung von $S$ um
|
|
||||||
$s$. Weiter sei $p := F^{-1}(s)$.
|
|
||||||
|
|
||||||
\begin{definition}\xindex{Fundamentalform!erste}%In Vorlesung: Bem.+Def. 19.1
|
|
||||||
Sei $I_S \in \mdr^{2 \times 2}$ definiert als
|
|
||||||
\begin{align*}
|
|
||||||
I_S :&= \begin{pmatrix}
|
|
||||||
g_{1,1}(s) & g_{1,2}(s)\\
|
|
||||||
g_{1,2}(s) & g_{2,2}(s)
|
|
||||||
\end{pmatrix} =
|
|
||||||
\begin{pmatrix}
|
|
||||||
E(s) & F(s) \\
|
|
||||||
F(s) & G(s)
|
|
||||||
\end{pmatrix}\\
|
|
||||||
\text{mit } g_{i,j} &= g_s(D_p F(e_i), D_p F(e_j))\\
|
|
||||||
&= \langle \frac{\partial F}{\partial u_i} (p), \frac{\partial F}{\partial u_j} (p) \rangle \;\;\; i,j \in \Set{1,2}
|
|
||||||
\end{align*}
|
|
||||||
Die Matrix $I_S$ heißt \textbf{erste Fundamentalform}
|
|
||||||
von $S$ bzgl. der Parametrisierung $F$.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}%In Vorlesung: Bem.+Def. 19.1
|
|
||||||
\begin{bemenum}
|
|
||||||
\item \label{bem:19.1a} Die Einschränkung des Standardskalarproduktes des $\mdr^3$ auf
|
|
||||||
$T_s S$ macht $T_s S$ zu einem euklidischen Vektorraum.
|
|
||||||
\item $\Set{D_p F(e_1), D_p F(e_2)}$ ist eine Basis von $T_s S$.
|
|
||||||
\item Bzgl. der Basis $\Set{D_p F(e_1), D_p F(e_2)}$ hat das
|
|
||||||
Standardskalarprodukt aus \cref{bem:19.1a} die Darstellungsmatrix
|
|
||||||
$I_S$.
|
|
||||||
\item $g_{i,j}(s)$ ist eine differenzierbare Funktion von $s$.
|
|
||||||
\end{bemenum}
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{bemerkung}
|
|
||||||
\[\det(I_S) = \left \| \frac{\partial F}{\partial u_1}(p) \times \frac{\partial F}{\partial u_2}(p) \right \|^2\]
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}\leavevmode
|
|
||||||
Sei $\frac{\partial F}{\partial u_1}(p) = \begin{pmatrix}
|
|
||||||
x_1\\ x_2 \\ x_3
|
|
||||||
\end{pmatrix}, \;\;\; \frac{\partial F}{\partial u_2}(p) = \begin{pmatrix}
|
|
||||||
y_1\\ y_2 \\ y3
|
|
||||||
\end{pmatrix}$
|
|
||||||
|
|
||||||
Dann ist $\frac{\partial F}{\partial u_1}(p) \times \frac{\partial F}{\partial u_2}(p) = \begin{pmatrix}
|
|
||||||
z_1 \\ z_2 \\ z_3
|
|
||||||
\end{pmatrix}$ mit
|
|
||||||
\begin{align*}
|
|
||||||
z_1 &= x_2 y_3 - x_3 y_2\\
|
|
||||||
z_2 &= x_3 y_1 - x_1 y_3\\
|
|
||||||
z_3 &= x_1 y_2 - x_2 y_1\\
|
|
||||||
\Rightarrow \|\frac{\partial F}{\partial u_1} (p) \times \frac{\partial F}{\partial u_2} (p)\| &= z_1^2 + z_2^2 + z_3^2\\
|
|
||||||
\end{align*}
|
|
||||||
\begin{align*}
|
|
||||||
\det(I_S) &= g_{1,1} g_{2,2} - g_{1,2}^2\\
|
|
||||||
&= \left \langle \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \right \rangle \left \langle \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix} \right \rangle - \left \langle \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix} \right \rangle^2\\
|
|
||||||
&= (x_1^2 + x_2^2 + x_3^2) (y_1^2 + y_2^2 + y_3^2) - (x_1 y_1 + x_2 y_2 + x_3 y_3)^2
|
|
||||||
\end{align*}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{definition}\xindex{Flächenelement}%In Vorlesung: Def.+Bem. 19.3 / Erinnerung
|
|
||||||
\begin{defenum}
|
|
||||||
\item Das Differential $\mathrm{d} A = \sqrt{\det (I)} \mathrm{d} u_1 \mathrm{d} u_2$
|
|
||||||
heißt \textbf{Flächenelement} von $S$ bzgl. der Parametrisierung $F$.
|
|
||||||
\item \label{def:berechenbares-integral}Für eine Funktion $f: V \rightarrow \mdr$ heißt
|
|
||||||
\[\int_V f \mathrm{d} A := \int_U f(\underbrace{F(u_1, u_2)}_{=: s}) \sqrt{\det I(s)} \mathrm{d} u_1 \mathrm{d} u_2\]
|
|
||||||
der \textbf{Wert des Integrals} von $f$ über $V$, falls das Integral rechts
|
|
||||||
existiert.
|
|
||||||
\end{defenum}
|
|
||||||
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}
|
|
||||||
\begin{bemenum}
|
|
||||||
\item $\int_V f \mathrm{d} A$ ist unabhängig von der gewählten Parametrisierung.
|
|
||||||
\item Sei $f: S \rightarrow \mdr$ eine Funktion, die im Sinne von
|
|
||||||
\cref{def:berechenbares-integral} lokal integrierbar ist.
|
|
||||||
|
|
||||||
Dann ist $\int_S f \mathrm{d} A$ wohldefiniert, falls (z.~B.) $S$
|
|
||||||
kompakt ist.
|
|
||||||
|
|
||||||
Etwa:
|
|
||||||
\begin{align*}
|
|
||||||
\int_S f \mathrm{d} A &= \sum_{i=1}^n \int_{\mathrlap{V_i}} f \mathrm{d} A \\
|
|
||||||
&- \sum_{i \neq j} \int_{\mathrlap{V_i \cap V_j}} f \mathrm{d} A \\
|
|
||||||
&+ \sum_{i,j,k} \int_{\mathrlap{V_i \cap V_j \cap V_k}} f \mathrm{d} A\\
|
|
||||||
&- \dots
|
|
||||||
\end{align*}
|
|
||||||
\end{bemenum}
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{beweis}\leavevmode
|
|
||||||
\begin{enumerate}[label=\alph*)]
|
|
||||||
\item Mit Transformationsformel.%TODO
|
|
||||||
\item Ist dem Leser überlassen.%TODO
|
|
||||||
\end{enumerate}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{proposition}\xindex{Weingarten-Abbildung}\label{prop:5.1}%
|
|
||||||
Sei $S \subseteq \mdr^3$ eine reguläre, orientierbare Fläche mit glatten
|
|
||||||
Normalenfeld $n: S \rightarrow S^2$. Dann gilt:
|
|
||||||
|
|
||||||
\begin{propenum}
|
|
||||||
\item \label{prop:5.1a} $n$ induziert für jedes $s \in S$ eine lineare Abbildung $d_s n: T_s S \rightarrow T_{n(s)} S^2$
|
|
||||||
durch
|
|
||||||
\[d_s n(x) = \frac{\mathrm{d}}{\mathrm{d} t} n (\underbrace{s \text{\enquote{+}} tx}_{\mathclap{\text{Soll auf Fläche $S$ bleiben}}}) \Bigr |_{t=0}\]
|
|
||||||
Die Abbildung $d_s n$ heißt \textbf{Weingarten-Abbildung}
|
|
||||||
\item $T_{n(s)} S^2 = T_s S$.
|
|
||||||
\item $d_s n$ ist ein Endomorphismus von $T_s S$.
|
|
||||||
\item $d_s n$ ist selbstadjungiert bzgl. des Skalarproduktes $I_S$.
|
|
||||||
\end{propenum}
|
|
||||||
\end{proposition}
|
|
||||||
|
|
||||||
\underline{Hinweis:} Die Weingarten-Abbildung wird auch \textit{Formoperator}\index{Formoperator|see{Weingarten-Abbildung}} genannt.
|
|
||||||
\clearpage
|
|
||||||
\begin{beweis}\leavevmode
|
|
||||||
\begin{enumerate}[label=\alph*)]
|
|
||||||
\item Wenn jemand diesen Beweis führt, bitte an info@martin-thoma.de
|
|
||||||
schicken.
|
|
||||||
\item $T_{n(S)} S^2 = \langle n(s) \rangle^\perp = T_s S$
|
|
||||||
\item Wegen \cref{prop:5.1a} ist $d_s n$ ein Homomorphismus.%\\
|
|
||||||
%TODO: Warum sollte das ein Endomorphismus sein?
|
|
||||||
\item Zu zeigen: $\forall x,y \in I_s S: \langle x, d_s n (y) \rangle = \langle d_s n(x), y \rangle$
|
|
||||||
|
|
||||||
Aufgrund der Bilinearität des Skalarproduktes genügt es diese Eigenschaft
|
|
||||||
für die Basisvektoren zu zeigen.
|
|
||||||
|
|
||||||
Sei $x_i = D_p F(e_i) = \frac{\partial F}{\partial u_i} (p)\;\;\; i = 1,2$
|
|
||||||
|
|
||||||
\underline{Beh.:}
|
|
||||||
$\langle x_i, d_s n(x_j) \rangle = \langle \frac{\partial^2 F}{\partial u_i \partial u_j} (p), d_s n (x_i) \rangle$
|
|
||||||
|
|
||||||
$\Rightarrow \langle \frac{\partial^2 F}{\partial u_i \partial u_j} (p), d_s n (x_i) \rangle = \langle x_j, d_s n (x_i) \rangle$
|
|
||||||
|
|
||||||
\underline{Bew.:} $
|
|
||||||
\begin{aligned}[t]
|
|
||||||
0 &= \hphantom{\frac{\mathrm{d}}{\mathrm{d}t} \left (\right.} \langle \frac{\partial F}{\partial u} (p + t e_j), n(p + t e_j) \rangle\\
|
|
||||||
\Rightarrow 0 &= \frac{\mathrm{d}}{\mathrm{d}t} \left (\langle \frac{\partial F}{\partial u} (p + t e_j), n(p + t e_j) \rangle \right) \Bigr |_{t=0}\\
|
|
||||||
&= \langle \underbrace{\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial F}{\partial u_i} (p + t e_j)}_{\frac{\partial^2 F}{\partial u_j \partial u_i} (p)} \Bigr |_{t=0}, n(s) \rangle + \langle x_i, d_s n \underbrace{D_p F (e_j)}_{x_j}\rangle
|
|
||||||
\end{aligned}$
|
|
||||||
\end{enumerate}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Mitschrieb vom 13.02.2014 %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\begin{definition}\xindex{Fundamentalform!zweite}%In Vorlesung: Def. + Bem. 19.5 a)
|
|
||||||
Die durch $-d_s n$ definierte symmetrische Bilinearform auf $T_s S$ heißt
|
|
||||||
\textbf{zweite Fundamentalform} von $S$ in $s$ bzgl. $F$.
|
|
||||||
|
|
||||||
Man schreibt: $II_s(x,y) = \langle - d_s n(x), y \rangle = I_s (-d_s n(x), y)$
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{bemerkung}%%In Vorlesung: Def. + Bem. 19.5 b)
|
|
||||||
Bezüglich der Basis $\Set{x_1, x_2}$ von $T_s S$ hat $II_s$ die Darstellungsmatrix
|
|
||||||
\[(h^{(s)}_{i,j})_{i,j=1,2} \text{ mit } h_{i,j}(s) = \langle \frac{\partial^2 F}{\partial u_i \partial u_j} (p), n(s) \rangle \]
|
|
||||||
\end{bemerkung}
|
|
||||||
|
|
||||||
\begin{proposition}\label{prop:19.6}%In Vorlesung: Proposition 19.6
|
|
||||||
Sei $\gamma:[- \varepsilon, \varepsilon] \rightarrow S$ eine nach Bogenlänge
|
|
||||||
parametrisierte Kurve mit $\gamma(0) = s$. Dann gilt:
|
|
||||||
\[\kappanor(s, \gamma) = II_s(\gamma'(0), \gamma'(0))\]
|
|
||||||
\end{proposition}
|
|
||||||
|
|
||||||
\begin{beweis}
|
|
||||||
Nach \cref{def:18.4} ist $\kappanor(s, \gamma) = \langle \gamma''(0), n(s) \rangle$.
|
|
||||||
Nach Voraussetzung gilt
|
|
||||||
\[n(\gamma(t)) \perp \gamma'(t) \Leftrightarrow \langle \gamma''(0), n(s) \rangle = 0\]
|
|
||||||
Die Ableitung nach $t$ ergibt
|
|
||||||
\begin{align*}
|
|
||||||
0 &= \frac{\mathrm{d}}{\mathrm{d}t}(\langle n (\gamma(t)), \gamma'(t))\\
|
|
||||||
&= \left \langle \frac{\mathrm{d}}{\mathrm{d}t} n(\gamma(t)) \Bigr |_{t=0}, \gamma'(0) \right \rangle + \langle n(s), \gamma''(0) \rangle\\
|
|
||||||
&= \langle d_s n (\gamma'(0)), \gamma'(0) \rangle + \kappanor(s,\gamma)\\
|
|
||||||
&= - II_s(\gamma'(0), \gamma'(0)) + \kappanor(s, \gamma)
|
|
||||||
\end{align*}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{folgerung}\xindex{Normalkrümmung}%In Vorlesung: Folgerung 19.7
|
|
||||||
Die beiden Definitionen von Normalkrümmung in \cref{sec:Kurvenkrümmung} stimmen
|
|
||||||
überein:
|
|
||||||
\[\kappanor(s, \gamma) = \kappanor(s, \gamma'(0))\]
|
|
||||||
\end{folgerung}
|
|
||||||
|
|
||||||
\begin{satz}%In Vorlesung: Satz 19.8
|
|
||||||
Sei $S \subseteq \mdr^3$ eine reguläre, orientierbare Fläche und $s \in S$.
|
|
||||||
\begin{satzenum}
|
|
||||||
\item Die Hauptkrümmungen $\kappa_1(s), \kappa_2(s)$ sind die Eigenwerte
|
|
||||||
von $II_s$.
|
|
||||||
\item Für die Gauß-Krümmung gilt: $K(s) = \det(II_s)$
|
|
||||||
\end{satzenum}
|
|
||||||
\end{satz}
|
|
||||||
|
|
||||||
\begin{beweis}\leavevmode
|
|
||||||
\begin{enumerate}[label=\alph*)]
|
|
||||||
\item $II_s$ ist symmetrisch, $I_s S$ hat also eine Orthonormalbasis aus
|
|
||||||
Eigenvektoren $y_1, y_2$ von $II_s$. Ist $x \in T_s S$, $\|x\| = 1$,
|
|
||||||
so gibt es $\varphi \in [0,2\pi)$ mit $x = \cos \varphi \cdot y_1 + \sin \varphi \cdot y_2$.
|
|
||||||
|
|
||||||
Seien $\lambda_1, \lambda_2$ die Eigenwerte von $II_s$, also
|
|
||||||
$II_s(y_i, y_i) = \lambda_i$. Dann gilt:
|
|
||||||
\begin{align*}
|
|
||||||
II_s (x,x) &= \cos^2 \varphi \lambda_1 + \sin^2 \varphi \lambda_2\\
|
|
||||||
&= (1- \sin^2 \varphi) \lambda_1 + \sin^2 \varphi \lambda_2\\
|
|
||||||
&= \lambda_1 + \sin^2 \varphi (\lambda_2 - \lambda_1) \geq \lambda_1\\
|
|
||||||
&= \cos^2 \varphi + (1 - \cos^2 \varphi) \lambda_2\\
|
|
||||||
&= \lambda_2 - \cos^2 \varphi (\lambda_2 - \lambda_1) \leq \lambda_2\\
|
|
||||||
\xRightarrow{\crefabbr{prop:19.6}} \lambda_1 &= \min \Set{\kappanor (s,x) | x \in T^1_s S}\\
|
|
||||||
\lambda_2 &= \max \Set{\kappanor (s,x) | x \in T^1_s S}
|
|
||||||
\end{align*}
|
|
||||||
\end{enumerate}
|
|
||||||
\end{beweis}
|
|
||||||
|
|
||||||
\begin{satz}[Satz von Gauß-Bonnet]\xindex{Satz von!Gauß-Bonnet}%
|
|
||||||
Sei $S \subseteq \mdr^3$ eine kompakte orientierbare reguläre Fläche. Dann gilt:
|
|
||||||
\[\int_S K(s) \mathrm{d}A = 2 \pi \chi(S)\]
|
|
||||||
Dabei ist $\chi(S)$ die Euler-Charakteristik von $S$.
|
|
||||||
\end{satz}
|
|
||||||
|
|
||||||
\begin{beweis}
|
|
||||||
Der Beweis wird hier nicht geführt. Er kann in \enquote{Elementare Differentialgeometrie}
|
|
||||||
von Christian Bär (2. Auflage), ISBN 978-3-11-022458-0, ab Seite 281 nachgelesen werden.
|
|
||||||
\end{beweis}
|
|
||||||
|
|
@ -1,355 +0,0 @@
|
||||||
%!TEX root = GeoTopo.tex
|
|
||||||
\chapter*{Lösungen der Übungsaufgaben\markboth{Lösungen der Übungsaufgaben}{Lösungen der Übungsaufgaben}}
|
|
||||||
\addcontentsline{toc}{chapter}{Lösungen der Übungsaufgaben}
|
|
||||||
\begin{solution}[\ref{ub1:aufg1}]
|
|
||||||
\textbf{Teilaufgabe a)} Es gilt:
|
|
||||||
\begin{enumerate}[label=(\roman*)]
|
|
||||||
\item $\emptyset, X \in \fT_X$.
|
|
||||||
\item $\fT_X$ ist offensichtlich unter Durchschnitten abgeschlossen,
|
|
||||||
d.~h. es gilt für alle $U_1, U_2 \in \fT_X: U_1 \cap U_2 \in \fT_X$.
|
|
||||||
\item Auch unter beliebigen Vereinigungen ist $\fT_X$ abgeschlossen,
|
|
||||||
d.~h. es gilt für eine beliebige Indexmenge $I$ und alle
|
|
||||||
$U_i \in \fT_X$ für alle $i \in I: \bigcup_{i \in I} U_i \in \fT_X$
|
|
||||||
\end{enumerate}
|
|
||||||
|
|
||||||
Also ist $(X, \fT_X)$ ein topologischer Raum.
|
|
||||||
|
|
||||||
\textbf{Teilaufgabe b)} Wähle $x=1, y=0$. Dann gilt $x \neq y$
|
|
||||||
und die einzige Umgebung von $x$ ist $X$. Da $y=0 \in X$ können
|
|
||||||
also $x$ und $y$ nicht durch offene Mengen getrennt werden.
|
|
||||||
$(X, \fT_X)$ ist also nicht hausdorffsch.
|
|
||||||
|
|
||||||
\textbf{Teilaufgabe c)} Nach Bemerkung \ref{Trennungseigenschaft}
|
|
||||||
sind metrische Räume hausdorffsch. Da $(X, \fT_X)$ nach (b) nicht
|
|
||||||
hausdorffsch ist, liefert die Kontraposition der Trennungseigenschaft,
|
|
||||||
dass $(X, \fT_X)$ kein metrischer Raum sein kann.
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub1:aufg4}]
|
|
||||||
\textbf{Teilaufgabe a)}
|
|
||||||
|
|
||||||
\textbf{Beh.:} $\forall a \in \mdz: \Set{a}$ ist abgeschlossen.
|
|
||||||
|
|
||||||
Sei $a \in \mdz$ beliebig. Dann gilt:
|
|
||||||
|
|
||||||
Wenn jemand diese Aufgabe gemacht hat, bitte die Lösung an info@martin-thoma.de
|
|
||||||
schicken.%TODO
|
|
||||||
|
|
||||||
\textbf{Teilaufgabe b)}
|
|
||||||
|
|
||||||
\textbf{Beh.:} $\Set{-1, 1}$ ist nicht offen
|
|
||||||
|
|
||||||
\textbf{Bew.:} durch Widerspruch
|
|
||||||
|
|
||||||
Annahme: $\Set{-1, 1}$ ist offen.
|
|
||||||
|
|
||||||
Dann gibt es $T \subseteq \fB$, sodass $\bigcup_{M \in T} M = \Set{-1, 1}$.
|
|
||||||
Aber alle $U \in \fB$ haben unendlich viele Elemente. Auch endlich
|
|
||||||
viele Schnitte von Elementen in $\fB$ haben unendlich viele
|
|
||||||
Elemente $\Rightarrow$ keine endliche nicht-leere Menge kann
|
|
||||||
in dieser Topologie offen sein $\Rightarrow \Set{-1,1}$ ist
|
|
||||||
nicht offen. $\qed$
|
|
||||||
|
|
||||||
\textbf{Teilaufgabe c)}
|
|
||||||
|
|
||||||
\textbf{Beh.:} Es gibt unendlich viele Primzahlen.
|
|
||||||
|
|
||||||
\textbf{Bew.:} durch Widerspruch
|
|
||||||
|
|
||||||
Annahme: Es gibt nur endlich viele Primzahlen $p \in \mdp$
|
|
||||||
|
|
||||||
Dann ist
|
|
||||||
\[\mdz \setminus \Set{-1, +1} \overset{\text{FS d. Arithmetik}}= \bigcup_{p \in \mdp} U_{0,p}\]
|
|
||||||
endlich. Das ist ein Widerspruch zu $|\mdz|$ ist unendlich und
|
|
||||||
$|\Set{-1,1}|$ ist endlich. $\qed$
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub2:aufg4}]
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item \textbf{Beh.:} Die offenen Mengen von $P$ sind
|
|
||||||
Vereinigungen von Mengen der Form
|
|
||||||
\[\prod_{j \in J} U_j \times \prod_{i \in \mdn, i \neq j} P_i\]
|
|
||||||
wobei $J \subseteq \mdn$ endlich und $U_j \subseteq P_j$
|
|
||||||
offen ist.
|
|
||||||
\begin{beweis}
|
|
||||||
Nach Definition der Produkttopologie bilden Mengen
|
|
||||||
der Form
|
|
||||||
\[\prod_{i \in J} U_j \times \prod_{i \in \mdn \setminus J} P_i\]
|
|
||||||
wobei $J \subseteq \mdn$ endlich und $U_j \subseteq P_j$ offen
|
|
||||||
$\forall{j \in J}$
|
|
||||||
eine Basis der Topologie.
|
|
||||||
|
|
||||||
Damit sind die offenen
|
|
||||||
Mengen von $P$ Vereinigungen von Mengen der obigen
|
|
||||||
Form. $\qed$
|
|
||||||
\end{beweis}
|
|
||||||
\item \textbf{Beh.:} Die Zusammenhangskomponenten von $P$
|
|
||||||
sind alle einpunktig.\xindex{Total Unzusammenhängend}
|
|
||||||
\begin{beweis}
|
|
||||||
Es seinen $x,y \in P$ und $x$ sowie $y$ liegen in der
|
|
||||||
gleichen Zusammenhangskomponente $Z \subseteq P$.
|
|
||||||
Da $Z$ zusammenhängend ist und $\forall{i \in I}: p_i : P \rightarrow P_i$
|
|
||||||
ist stetig, ist $p_i(Z) \subseteq P_i$ zusammenhängend
|
|
||||||
für alle $i \in \mdn$. Die zusammenhängenden Mengen
|
|
||||||
von $P_i$ sind genau $\Set{0}$ und $\Set{1}$, d.~h.
|
|
||||||
für alle $i \in \mdn$ gilt entweder $p_i(Z) \subseteq \Set{0}$
|
|
||||||
oder $p_i(Z) \subseteq \Set{1}$. Es sei $z_i \in \Set{0,1}$
|
|
||||||
so, dass $p_i(Z) \subseteq \Set{z_i}$ für alle $i \in \mdn$.
|
|
||||||
Dann gilt also:
|
|
||||||
\[\underbrace{p_i(x)}_{= x_i} = z_i = \underbrace{p_i(y)}_{= y_i} \forall i \in \mdn\]
|
|
||||||
Somit folgt: $x = y \qed$
|
|
||||||
|
|
||||||
\end{beweis}
|
|
||||||
\end{enumerate}
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub3:aufg1}]
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item \textbf{Beh.:} $\GL_n(\mdr)$ ist nicht kompakt.\\
|
|
||||||
\textbf{Bew.:} $\det: \GL_n(\mdr) \rightarrow \mdr \setminus \Set{0}$
|
|
||||||
ist stetig. Außerdem ist
|
|
||||||
$\det(\GL_n(\mdr)) = \mdr \setminus \Set{0}$ nicht
|
|
||||||
kompakt. $\overset{\ref{kor:5.6}}{\Rightarrow}$
|
|
||||||
$\GL_n(\mdr)$ ist nicht kompakt. $\qed$
|
|
||||||
\item \textbf{Beh.:} $\SL_1(\mdr)$ ist nicht kompakt, für $n > 1$ ist $\SL_n(\mdr)$ kompakt.\\
|
|
||||||
\textbf{Bew.:} Für $\SL_1(\mdr)$ gilt:
|
|
||||||
$\SL_1(\mdr) = \Set{A \in \mdr^{1 \times 1} | \det A = 1} = \begin{pmatrix}1\end{pmatrix} \cong \Set{1}$.
|
|
||||||
$\overset{\ref{kor:5.6}}{\Rightarrow} \SL_1(\mdr)$ ist
|
|
||||||
kompakt.\\
|
|
||||||
|
|
||||||
$\SL_n(\mdr) \subseteq \GL_n(\mdr)$ lässt sich mit einer
|
|
||||||
Teilmenge des $\mdr^{n^2}$ identifizieren. Nach \cref{satz:heine-borel}
|
|
||||||
sind diese genau dann kompakt, wenn sie beschränkt und
|
|
||||||
abgeschlossen sind. Definiere nun für für $n \in \mdn_{\geq 2}, m \in \mdn$:
|
|
||||||
\[A_m = \text{diag}_n(m, \frac{1}{m}, \dots, 1)\]
|
|
||||||
Dann gilt: $\det A_m = 1$, d.~h. $A_m \in \SL_n(\mdr)$,
|
|
||||||
und $A_m$ ist unbeschränkt, da $\|A_m\|_\infty =m \xrightarrow[m \rightarrow \infty]{} \infty$.$\qed$
|
|
||||||
\item \textbf{Beh.:} $\praum(\mdr)$ ist kompakt.\\
|
|
||||||
\textbf{Bew.:} $\praum(\mdr) \cong S^n/_{x \sim -x}$.
|
|
||||||
Per Definition der Quotiententopologie ist die Klassenabbildung stetig.
|
|
||||||
Da $S^n$ als abgeschlossene und beschränkte Teilmenge
|
|
||||||
des $\mdr^{n+1}$ kompakt ist $\overset{\ref{kor:5.6}}{\Rightarrow}$
|
|
||||||
$\praum(\mdr)$ ist kompakt. $\qed$
|
|
||||||
\end{enumerate}
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub3:meinsExtra}]
|
|
||||||
Die Definition von Homöomorphismus kann auf \cpageref{def:homoeomorphismus}
|
|
||||||
nachgelesen werden.
|
|
||||||
|
|
||||||
\begin{definition}\xindex{Homomorphismus}%
|
|
||||||
Seien $(G, *)$ und $(H, \circ)$ Gruppen und
|
|
||||||
$\varphi:G \rightarrow H$ eine Abbildung.
|
|
||||||
|
|
||||||
$\varphi$ heißt \textbf{Homomorphismus}, wenn
|
|
||||||
\[\forall g_1, g_2 \in G: \varphi(g_1 * g_2) = \varphi(g_1) \circ \varphi(g_2)\]
|
|
||||||
gilt.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
Es folgt direkt:
|
|
||||||
\begin{bspenum}
|
|
||||||
\item Sei $X = \mdr$ mit der Standarttopologie und $\varphi_1: \id_\mdr$ und $\mdr = (\mdr,+)$. Dann ist $\varphi_1$ ein Gruppenhomomorphismus und ein Homöomorphismus.
|
|
||||||
\item Sei $G = (\mdz, +)$ und $H = (\mdz / 3 \mdz, +)$. Dann ist $\varphi_2 : G \rightarrow H, x \mapsto x \mod 3$ ein Gruppenhomomorphismus.
|
|
||||||
Jedoch ist $\varphi_2$ nicht injektiv, also sicher kein Homöomorphismus.
|
|
||||||
\item Sei $X$ ein topologischer Raum. Dann ist $\id_X$ ein Homöomorphismus. Da keine Verknüpfung auf $X$ definiert wurde, ist $X$ keine Gruppe und daher auch kein Gruppenhomomorphismus.
|
|
||||||
\end{bspenum}
|
|
||||||
|
|
||||||
Also: Obwohl die Begriffe ähnlich klingen, werden sie in ganz unterschiedlichen
|
|
||||||
Kontexten verwendet.
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub3:meinsExtra2}]
|
|
||||||
Die Definition einer Isotopie kann auf \cpageref{def:Isotopie} nachgelesen
|
|
||||||
werden, die einer Isometrie auf \cpageref{def:Isometrie}.
|
|
||||||
|
|
||||||
\begin{definition}\xindex{Isomorphismus}%
|
|
||||||
Seien $(G, *)$ und $(H, \circ)$ Gruppen und
|
|
||||||
$\varphi:G \rightarrow H$ eine Abbildung.
|
|
||||||
|
|
||||||
$\varphi$ heißt \textbf{Isomorphismus}, wenn $\varphi$ ein bijektiver
|
|
||||||
Homomorphismus ist.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
Eine Isotopie ist also für Knoten definiert, Isometrien machen nur in
|
|
||||||
metrischen Räumen Sinn und ein Isomorphismus benötigt eine Gruppenstruktur.
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub4:aufg1}]
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item \textbf{Vor.:} Sei $M$ eine topologische Mannigfaltigkeit.\\
|
|
||||||
\textbf{Beh.:} $M$ ist wegzusammehängend $\gdw M$ ist zusammenhängend
|
|
||||||
\begin{beweis}
|
|
||||||
\enquote{$\Rightarrow$}: Da $M$ insbesondere ein
|
|
||||||
topologischer Raum ist folgt diese Richtung direkt
|
|
||||||
aus \cref{kor:wegzusammehang-impliziert-zusammenhang}.
|
|
||||||
|
|
||||||
\enquote{$\Leftarrow$}: Seien $x,y \in M$ und
|
|
||||||
\[Z := \Set{z \in M | \exists \text{Weg von } x \text{ nach } z}\]
|
|
||||||
Es gilt:
|
|
||||||
\begin{enumerate}[label=(\roman*)]
|
|
||||||
\item $Z \neq \emptyset$, da $M$ lokal wegzusammenhängend ist
|
|
||||||
\item $Z$ ist offen, da $M$ lokal wegzusammenhängend ist
|
|
||||||
\item $Z^C := \Set{\tilde{z} \in M | \nexists \text{Weg von } x \text{ nach } \tilde{z}}$ ist offen
|
|
||||||
|
|
||||||
Da $M$ eine Mannigfaltigkeit ist, existiert zu jedem
|
|
||||||
$\tilde{z} \in Z^C$ eine offene und wegzusammenhängende Umgebung
|
|
||||||
$U_{\tilde{z}} \subseteq M$.
|
|
||||||
|
|
||||||
Es gilt sogar $U_{\tilde{z}} \subseteq Z^C$, denn
|
|
||||||
gäbe es ein $U_{\tilde{z}} \ni \overline{z} \in Z$,
|
|
||||||
so gäbe es Wege $\gamma_2:[0,1] \rightarrow M, \gamma_2(0) = \overline{z}, \gamma_2(1) = x$
|
|
||||||
und $\gamma_1:[0,1] \rightarrow M, \gamma_1(0) = \tilde{z}, \gamma_1(1) = \overline{z}$.
|
|
||||||
Dann wäre aber
|
|
||||||
\begin{align*}
|
|
||||||
\gamma:[0,1] &\rightarrow M,\\
|
|
||||||
\gamma(x) &= \begin{cases}
|
|
||||||
\gamma_1(2x) &\text{falls } 0 \leq x \leq \frac{1}{2}\\
|
|
||||||
\gamma_2(2x-1) &\text{falls } \frac{1}{2} < x \leq 1
|
|
||||||
\end{cases}
|
|
||||||
\end{align*}
|
|
||||||
ein stetiger Weg von $\tilde{z}$ nach $x$
|
|
||||||
$\Rightarrow$ Widerspruch.
|
|
||||||
|
|
||||||
Da $M$ zusammenhängend ist und $M = \underbrace{Z}_{\mathclap{\text{offen}}} \cup \underbrace{Z^C}_{\mathclap{\text{offen}}}$,
|
|
||||||
sowie $Z \neq \emptyset$ folgt $Z^C = \emptyset$.
|
|
||||||
Also ist $M=Z$ wegzusammenhängend.$\qed$
|
|
||||||
\end{enumerate}
|
|
||||||
\end{beweis}
|
|
||||||
\item \textbf{Beh.:} $X$ ist wegzusammenhängend.\\
|
|
||||||
\begin{beweis}
|
|
||||||
$X:= (\mdr \setminus \Set{0}) \cup \Set{0_1, 0_2}$
|
|
||||||
und $(\mdr \setminus \Set{0}) \cup \Set{0_2}$ sind
|
|
||||||
homöomorph zu $\mdr$. Also sind die einzigen kritischen
|
|
||||||
Punkte, die man nicht verbinden können könnte
|
|
||||||
$0_1$ und $0_2$.
|
|
||||||
|
|
||||||
Da $(\mdr \setminus \Set{0}) \cup \Set{0_1}$ homöomorph
|
|
||||||
zu $\mdr$ ist, exisitert ein Weg $\gamma_1$ von $0_1$
|
|
||||||
zu einem beliebigen Punkt $a \in \mdr \setminus \Set{0}$.
|
|
||||||
|
|
||||||
Da $(\mdr \setminus \Set{0}) \cup \Set{0_2}$ ebenfalls
|
|
||||||
homöomorph zu $\mdr$ ist, existiert außerdem ein Weg
|
|
||||||
$\gamma_2$ von $a$ nach $0_2$. Damit existiert ein
|
|
||||||
(nicht einfacher)
|
|
||||||
Weg $\gamma$ von $0_1$ nach $0_2$. $\qed$
|
|
||||||
\end{beweis}
|
|
||||||
\end{enumerate}
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
%Das scheint mir etwas zu lang zu sein...
|
|
||||||
%\begin{solution}[\ref{ub7:aufg1}]
|
|
||||||
% \textbf{Beh.:} $H_k = \begin{cases}\mdr &\text{für } k\in \Set{0,1}\\
|
|
||||||
% 0 &\text{für } k \geq 2$
|
|
||||||
% \newcommand{\triangleSimplizialkomplex}{\mathord{\includegraphics[height=5ex]{figures/triangleSimplizialkomplex.pdf}}}
|
|
||||||
% \textbf{Bew.:} $S^1$ ist homöomorph zum Simplizialkomplex
|
|
||||||
% $X = \triangleSimplizialkomplex$, d.~h. dem Rand
|
|
||||||
% von $\Delta^2$. Es gilt:
|
|
||||||
% \[X = \Set{\underbrace{v_0, v_1, v_2}_{A_0(X)}, \underbrace{\Delta (v_1, v_2)}_{=: a_0}, \underbrace{\underbrace{\Delta (v_0, v_2)}_{=: a_1}, \underbrace{\Delta(v_0, v_1)}_{=: a_2}}_{A_1(X)}}\]
|
|
||||||
% Damit folgt:
|
|
||||||
% \begin{enumerate}
|
|
||||||
% \item Für $k \geq 2$ ist $C_k(X) \cong 0$, da es in diesen
|
|
||||||
% Dimensionen keine Simplizes gibt, d.~h. $A_k(X) = \emptyset$ gilt.\\
|
|
||||||
% Also: $H_k(X) \cong 0 \; \forall k \geq 2$
|
|
||||||
% \item $C_0(X) = \Set{\sum_{i=0}^2 c_i v_i | c_i \in \mdr}$, da
|
|
||||||
% $A_0(x)$ Basis von $C_0(X)$ ist;\\
|
|
||||||
% $C_1(X) = \Set{\sum_{i=0}^2 c_i a_i | c_i \in \mdr}$, da
|
|
||||||
% $A_1(X)$ Basis von $C_1(X)$ ist.
|
|
||||||
% \item Für die Randabbildungen $d_i: C_i(X) \rightarrow C_{i-1}(X)$ gilt:
|
|
||||||
% $d_0 \equiv 0$, $d_1: C_1(X) \rightarrow C_0(X)$ ist definiert durch
|
|
||||||
% $d_1(a_k) = \sum_{i=0}^1 (-1)^i \partial_i(a_k) = \partial_0 (a_k) - \partial_1(a_k) \; \forall k \in \Set{0,1,2}$
|
|
||||||
% \end{enumerate}
|
|
||||||
%\end{solution}
|
|
||||||
|
|
||||||
%Auch diese Aufgabe ist zu lang
|
|
||||||
%\begin{solution}[\ref{ub7:aufg3}]
|
|
||||||
%
|
|
||||||
%\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub11:aufg3}]
|
|
||||||
\textbf{Vor.:} Sei $(X, d)$ eine absolute Ebene, $A, B, C \in X$
|
|
||||||
und $\triangle ABC$ ein Dreieck.
|
|
||||||
|
|
||||||
\begin{enumerate}[label=(\alph*)]
|
|
||||||
\item \textbf{Beh.:} $\overline{AB} \cong \overline{AC} \Rightarrow \angle ABC \cong \angle ACB$\\
|
|
||||||
\textbf{Bew.:} Sei $\overline{AB} \cong \overline{AC}$.\\
|
|
||||||
$\Rightarrow \exists$ Isometrie $\varphi$ mit $\varphi(B) = C$ und
|
|
||||||
$\varphi(C) = B$ und $\varphi(A) = A$.\\
|
|
||||||
$\Rightarrow \varphi(\angle ABC) = \angle ACB$\\
|
|
||||||
$\Rightarrow \angle ABC \cong \angle ACB \qed$
|
|
||||||
\item \textbf{Beh.:} Der längeren Seite von $\triangle ABC$ liegt der größere Winkel gegenüber und
|
|
||||||
umgekehrt.\\
|
|
||||||
\textbf{Bew.:} Sei $d(A,C) > d(A,B)$. Nach \ref{axiom:3.1}
|
|
||||||
gibt es $C' \in AC^+$ mit $d(A, C') = d(A,B)$\\
|
|
||||||
$\Rightarrow C'$ liegt zwischen $A$ und $C$.\\
|
|
||||||
Es gilt $\measuredangle ABC' < \measuredangle ABC$ und
|
|
||||||
aus \cref{ub11:aufg3.a} folgt: $\measuredangle ABC' = \measuredangle AC' B$.\\
|
|
||||||
$\angle BC' A$ ist ein nicht anliegender Außenwinkel zu
|
|
||||||
$\angle BCA \xRightarrow{\crefabbr{bem:14.9}} \measuredangle BC' A > \measuredangle BCA$\\
|
|
||||||
$\Rightarrow \measuredangle BCA < \measuredangle BC' A = \measuredangle ABC' < \measuredangle ABC $
|
|
||||||
Sei umgekehrt $\measuredangle ABC > \measuredangle BCA$,
|
|
||||||
kann wegen 1. Teil von \cref{ub11:aufg3.b} nicht
|
|
||||||
$d(A,B) > d(A,C)$ gelten.\\
|
|
||||||
Wegen \cref{ub11:aufg3.a} kann nicht $d(A,B) = d(A,C)$
|
|
||||||
gelten.\\
|
|
||||||
$\Rightarrow d(A,B) < d(A, C) \qed$
|
|
||||||
\item \textbf{Vor.:} Sei $g$ eine Gerade, $P \in X$ und $P \notin g$\\
|
|
||||||
\textbf{Beh.:} $\exists!$ Lot\\
|
|
||||||
\textbf{Bew.:} ÜB10 A4(a): Es gibt Geradenspiegelung $\varphi$
|
|
||||||
an $g$. $\varphi$ vertauscht die beiden Halbebenen bzgl.
|
|
||||||
$g$.\\
|
|
||||||
$\Rightarrow \varphi(P)P$ schneidet $g$ in $F$.
|
|
||||||
|
|
||||||
%Nach ÜB 10 A4(a):
|
|
||||||
Es gibt eine Geradenspiegelung $\varphi$ an $g$.
|
|
||||||
$\varphi$ vertauscht die beiden Halbebenen bzgl. $g$
|
|
||||||
$\Rightarrow \varphi(P)P$ schneidet $g$ in $F$.
|
|
||||||
|
|
||||||
Sei $A \in g \setminus \Set{F}$. Dann gilt $\varphi(\angle AFP) = \angle AF \varphi(P) = \pi$
|
|
||||||
$\Rightarrow \angle AFP$ ist rechter Winkel.
|
|
||||||
|
|
||||||
Gäbe es nun $G \in g \setminus \Set{F}$, so dass $PG$ weiteres Lot von $P$ auf $g$ ist,
|
|
||||||
wäre $\triangle PFG$ ein Dreieck mit zwei rechten Innenwinkeln (vgl. \cref{fig:two-perpendiculars}).
|
|
||||||
|
|
||||||
\begin{figure}[htp]
|
|
||||||
\centering
|
|
||||||
\input{figures/two-perpendiculars.tex}
|
|
||||||
\caption{Zwei Lote zu einer Geraden $g$ durch einen Punkt $P$}
|
|
||||||
\label{fig:two-perpendiculars}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
Nach \cref{folgerung:14.10} ist die Summe von zwei Innenwinkeln immer $< \pi$\\
|
|
||||||
$\Rightarrow G$ gibt es nicht. $\qed$
|
|
||||||
\end{enumerate}
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub-tut-24:a1}]
|
|
||||||
Sei $f \parallel h$ und \obda $f \parallel g$.
|
|
||||||
|
|
||||||
$f \nparallel h \Rightarrow f \cap h \neq \emptyset$, sei also $x \in f \cap h$.
|
|
||||||
Mit Axiom \ref{axiom:5} folgt: Es gibt höchstens eine Parallele
|
|
||||||
zu $g$ durch $x$, da $x \notin g$. Diese ist $f$, da $x \in f$
|
|
||||||
und $f \parallel g$. Da aber $x \in h$, kann $h$ nicht parallel
|
|
||||||
zu $g$ sein, denn ansonsten gäbe es zwei Parallelen zu $g$ durch
|
|
||||||
$x$ ($f \neq h$).
|
|
||||||
$\Rightarrow g \nparallel h$ $\qed$
|
|
||||||
\end{solution}
|
|
||||||
|
|
||||||
\begin{solution}[\ref{ub-tut-24:a3}]\xindex{Kongruenzsatz!SSS}%
|
|
||||||
Sei $(X,d,G)$ eine Geometrie, die \ref{axiom:1}-\ref{axiom:4} erfüllt.
|
|
||||||
Seien außerdem $\triangle ABC$ und $\triangle A'B' C'$ Dreiecke, für die gilt:
|
|
||||||
\begin{align*}
|
|
||||||
d(A, B) &= d(A', B')\\
|
|
||||||
d(A, C) &= d(A', C')\\
|
|
||||||
d(B, C) &= d(B', C')
|
|
||||||
\end{align*}
|
|
||||||
|
|
||||||
Sei $\varphi$ die Isometrie mit $\varphi(A) = A'$, $\varphi(B) = B'$ und
|
|
||||||
$\varphi(C')$ liegt in der selben Halbebene bzgl. $AB$ wie $C$. Diese
|
|
||||||
Isometrie existiert wegen \ref{axiom:4}.
|
|
||||||
|
|
||||||
Es gilt $d(A,C) = d(A', C') = d(\varphi(A'), \varphi(C')) = d(A, \varphi(C'))$
|
|
||||||
und $d(B,C) = d(B', C') = d(\varphi(B'), \varphi(C')) = d(B, \varphi(C'))$.\\
|
|
||||||
$\xRightarrow{\crefabbr{kor:14.6}} C = \varphi(C)$.
|
|
||||||
|
|
||||||
Es gilt also $\varphi(\triangle A'B'C') = \triangle ABC$. $\qed$
|
|
||||||
\end{solution}
|
|
||||||
|
|
@ -1,26 +0,0 @@
|
||||||
SOURCE = GeoTopo
|
|
||||||
|
|
||||||
make:
|
|
||||||
sketch figures/torus.sketch > figures/torus.tex
|
|
||||||
pdflatex $(SOURCE).tex -interaction=batchmode -output-format=pdf # aux-files for makeindex / makeglossaries
|
|
||||||
makeindex $(SOURCE)
|
|
||||||
pdflatex $(SOURCE).tex -interaction=batchmode -output-format=pdf # include index
|
|
||||||
pdflatex $(SOURCE).tex -interaction=batchmode -output-format=pdf # include symbol table
|
|
||||||
make clean # remove intermediate files like *.log and *.aux
|
|
||||||
|
|
||||||
ebook:
|
|
||||||
latexml --dest=$(SOURCE).xml $(SOURCE).tex
|
|
||||||
latexmlpost -dest=$(SOURCE).html $(SOURCE).xml
|
|
||||||
ebook-convert $(SOURCE).html $(SOURCE).epub --language de --no-default-epub-cover
|
|
||||||
|
|
||||||
all:
|
|
||||||
cd definitions;make
|
|
||||||
sed -i 's/\\newif\\ifAFive\\AFivefalse/\\newif\\ifAFive\\AFivetrue/' GeoTopo.tex
|
|
||||||
make
|
|
||||||
mv GeoTopo.pdf other-formats/GeoTopo-A5.pdf
|
|
||||||
sed -i 's/\\newif\\ifAFive\\AFivetrue/\\newif\\ifAFive\\AFivefalse/' GeoTopo.tex
|
|
||||||
make
|
|
||||||
|
|
||||||
|
|
||||||
clean:
|
|
||||||
rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.thm *.idx *.toc *.ind *.ilg figures/torus.tex *.glg *.glo *.gls *.ist *.xdy *.fdb_latexmk *.bak
|
|
||||||
|
|
@ -1,69 +0,0 @@
|
||||||
Dies ist ein **inoffizielles, von Studenten erstelltes Skript**
|
|
||||||
zur Vorlesung "Einführung in Geometrie und Topologie" am KIT bei
|
|
||||||
Herrn Prof. Dr. Herrlich (WS 2013/2014). Da es von Studenten erstellt
|
|
||||||
wird, die die Inhalte noch lernen, sind sehr wahrscheinlich einige
|
|
||||||
Fehler im Skript. Das können Übertragungsfehler, Tippfehler oder
|
|
||||||
Verständnisprobleme sein.
|
|
||||||
|
|
||||||
Verbesserungsvorschläge (auch wenn es nur einzelne Textsetzungsprobleme oder
|
|
||||||
Rechtschreibfehler sind) bitte immer direkt melden oder verbessern!
|
|
||||||
|
|
||||||
Den Verbesserungsvorschlag kann man
|
|
||||||
* entweder direkt selbst umsetzen und einen pull request machen oder
|
|
||||||
* mir per E-Mail (info@martin-thoma.de) schicken.
|
|
||||||
|
|
||||||
Ich werde dann versuchen die Verbesserungsvorschläge zeitnah einzuarbeiten.
|
|
||||||
|
|
||||||
Zeichnungen
|
|
||||||
===========
|
|
||||||
Das erstellen der Zeichnungen ist sehr zeitaufwendig. Das ist der
|
|
||||||
Grund, warum manchmal nur ein "TODO" im Dokument steht.
|
|
||||||
|
|
||||||
Ihr könnt mir gerne Zeichnungen schicken (entweder schön auf Papier
|
|
||||||
Zeichnen und abfotographieren / einscannen oder schon mit Inscape /
|
|
||||||
Gimp / ... oder sogar mit TikZ erstellen).
|
|
||||||
|
|
||||||
Akzeptable Formate sind: .jpg, .pdf, .svg, .png, .gif, .tex, .sketch
|
|
||||||
Alles andere kann ich vermutlich nicht einbinden.
|
|
||||||
|
|
||||||
|
|
||||||
Dokument erzeugen
|
|
||||||
=================
|
|
||||||
Zum erzeugen des Dokuments wird `sketch` und LaTeX benötigt.
|
|
||||||
|
|
||||||
LaTeX installiert man so: [Link](http://martin-thoma.com/how-to-install-the-latest-latex-version/)
|
|
||||||
|
|
||||||
Rechtliches
|
|
||||||
===========
|
|
||||||
Die Autoren kann man über Git ermitteln. Ich schreibe meist nur den
|
|
||||||
Tafelanschrieb der Vorlesung ab; eventuell noch mit ein paar
|
|
||||||
Notizen meinerseits. Wenn mir Verbesserungsvorschläge per E-Mail
|
|
||||||
geschickt werden, ist der Autor sowie das Datum der E-Mail in der
|
|
||||||
Commit-Nachricht von Git zu sehen.
|
|
||||||
|
|
||||||
Bilder habe ich entweder selbst erstellt oder von tex.stackexchange.com.
|
|
||||||
Bei Bildern von tex.stackexchange.com steht der Link auf die Quelle
|
|
||||||
im Quelltext des Bildes (siehe Ordner `figures`).
|
|
||||||
|
|
||||||
Was noch kommen soll
|
|
||||||
====================
|
|
||||||
|
|
||||||
1. Alle `TODOS` auflösen
|
|
||||||
* "Punkt" suchen
|
|
||||||
* Checken, ob alle Seitenumbrüche / Bildgrößen stimmen
|
|
||||||
2. Reviews (Mathematik, LaTeX und Bilder)
|
|
||||||
3. A5-Version drucken
|
|
||||||
* In `GeoTopo.tex`: `\AFivefalse` → `\AFivetrue`
|
|
||||||
* Momentan sind es ca. 100 Seiten in A4. In A5 sind es ca. 159 Seiten.
|
|
||||||
* Druckereien
|
|
||||||
* An der Uni (ca. 8.50 Euro, SW, Spiralbindung)
|
|
||||||
* http://www.epubli.de/ (ca. 9.23 Euro SW + 2.95 Euro Versand, 26.99 Euro farbig)
|
|
||||||
* https://www.viaprinto.de/ (ca. 15 Euro SW, 35 Euro farbig)
|
|
||||||
* http://shop.kopie.de/article/show/diplomarbeit
|
|
||||||
* http://www.drucksofa.com/
|
|
||||||
* http://www.mein-druck.de/category.htm?c=15510
|
|
||||||
* http://www.1buch.de/preisuebersicht/
|
|
||||||
4. Version für Sehgeschädigte:
|
|
||||||
* min `12pt`, besser `14pt`
|
|
||||||
* nicht `article`, `book`, `report` sondern `extarticle`
|
|
||||||
* Sans serif: Arial, Helvetica (`\usepackage{cmbright}`)
|
|
||||||
|
|
@ -1,179 +0,0 @@
|
||||||
%!TEX root = GeoTopo.tex
|
|
||||||
\markboth{Symbolverzeichnis}{Symbolverzeichnis}
|
|
||||||
\twocolumn
|
|
||||||
\chapter*{Symbolverzeichnis}
|
|
||||||
\addcontentsline{toc}{chapter}{Symbolverzeichnis}
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Mengenoperationen %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section*{Mengenoperationen}
|
|
||||||
|
|
||||||
Seien $A, B$ und $M$ Mengen.
|
|
||||||
|
|
||||||
% Set \mylengtha to widest element in first column; adjust
|
|
||||||
% \mylengthb so that the width of the table is \columnwidth
|
|
||||||
\settowidth\mylengtha{$A \subsetneq B$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$A^C $ & Komplement von $A$\\
|
|
||||||
$\mathcal{P}(M)$ & Potenzmenge von $M$\\
|
|
||||||
$\overline{M}$ & Abschluss von $M$\\
|
|
||||||
$\partial M$ & Rand der Menge $M$\\
|
|
||||||
$M^\circ$ & Inneres der Menge $M$\\
|
|
||||||
$A \times B$ & Kreuzprodukt\\
|
|
||||||
$A \subseteq B$ & Teilmengenbeziehung\\
|
|
||||||
$A \subsetneq B$ & echte Teilmengenbeziehung\\
|
|
||||||
$A \setminus B$ & Differenzmenge\\
|
|
||||||
$A \cup B$ & Vereinigung\\
|
|
||||||
$A \dcup B$ & Disjunkte Vereinigung\\
|
|
||||||
$A \cap B$ & Schnitt\\
|
|
||||||
\end{xtabular}
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Geometrie %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section*{Geometrie}
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$\overline{AB} \cong \overline{CD}$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$AB$ & Gerade durch die Punkte $A$ und $B$\\
|
|
||||||
$\overline{AB}$ & Strecke mit Endpunkten $A$ und $B$\\
|
|
||||||
$\triangle ABC$ & Dreieck mit Eckpunkten $A, B, C$\\
|
|
||||||
$\overline{AB} \cong \overline{CD}$& Die Strecken $\overline{AB}$ und $\overline{CD}$ sind isometrisch\\
|
|
||||||
$|K|$ & Geometrische Realisierung des Simplizialkomplexes~$K$\\
|
|
||||||
\end{xtabular}
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Gruppen %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section*{Gruppen}
|
|
||||||
|
|
||||||
Sei $X$ ein topologischer Raum und $K$ ein Körper.
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$\Homoo(X)$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$\Homoo(X)$ & Homöomorphis\-men\-gruppe\\
|
|
||||||
$\Iso(X)$ & Isometrien\-gruppe\\
|
|
||||||
$\GL_n(K)$ & Allgemeine lineare Gruppe (von \textit{\textbf{G}eneral \textbf{L}inear Group})\\
|
|
||||||
$\SL_n(K)$ & Spezielle lineare Gruppe\\
|
|
||||||
$\PSL_n(K)$ & Projektive lineare Gruppe\\
|
|
||||||
$\Perm(X)$ & Permutations\-gruppe\\
|
|
||||||
$\Sym(X)$ & Symmetrische Gruppe\\
|
|
||||||
\end{xtabular}
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Wege %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section*{Wege}
|
|
||||||
|
|
||||||
Sei $\gamma: I \rightarrow X$ ein Weg.
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$\gamma_1 \sim \gamma_2$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$[\gamma]$ & Homotopieklasse von $\gamma$\\
|
|
||||||
$\gamma_1 * \gamma_2$ & Zusammenhängen von Wegen\\
|
|
||||||
$\gamma_1 \sim \gamma_2$ & Homotopie von Wegen\\
|
|
||||||
$\overline{\gamma}(x)$ & Inverser Weg, also $\overline{\gamma}(x) := \gamma(1-x)$\\
|
|
||||||
$C$ & Bild eines Weges $\gamma$, also $C := \gamma([0,1])$
|
|
||||||
\end{xtabular}
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Weiteres %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section*{Weiteres}
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$\fB_\delta(x)$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$\fB$ & Basis einer Topologie\\
|
|
||||||
$\fB_\delta(x)$& $\delta$-Kugel um $x$\\
|
|
||||||
$\calS$ & Subbasis einer Topologie\\
|
|
||||||
$\fT$ & Topologie\\
|
|
||||||
\end{xtabular}
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$X /_\sim$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$\atlas$ & Atlas\\
|
|
||||||
$\praum$ & Projektiver Raum\\
|
|
||||||
$\langle \cdot , \cdot \rangle$ & Skalarprodukt\\
|
|
||||||
$X /_\sim$ & $X$ modulo $\sim$\\
|
|
||||||
$[x]_\sim$ & Äquivalenzklassen von $x$ bzgl. $\sim$\\
|
|
||||||
$\| x \|$ & Norm von $x$\\
|
|
||||||
$| x |$ & Betrag von $x$\\
|
|
||||||
$\langle a \rangle$ & Erzeugnis von $a$\\
|
|
||||||
\end{xtabular}
|
|
||||||
|
|
||||||
$S^n\;\;\;$ Sphäre\\
|
|
||||||
$T^n\;\;\;$ Torus\\
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$f^{-1}(M)$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$f \circ g$&Verkettung von $f$ und $g$\\
|
|
||||||
$\pi_X$ &Projektion auf $X$\\
|
|
||||||
$f|_U$ $f$ &eingeschränkt auf $U$\\
|
|
||||||
$f^{-1}(M)$&Urbild von $M$\\
|
|
||||||
$\rang(M)$ & Rang von $M$\\
|
|
||||||
$\chi(K)$ & Euler-Charakteristik von $K$\\
|
|
||||||
$\Delta^k$ & Standard-Simplex\\
|
|
||||||
$X \# Y$ & Verklebung von $X$ und $Y$\\
|
|
||||||
$d_n$ & Lineare Abbildung aus \cref{kor:9.11}\\
|
|
||||||
$A \cong B$& $A$ ist isometrisch zu $B$\\
|
|
||||||
$f_*$ & Abbildung zwischen Fundamentalgruppen (vgl. \cpageref{korr:11.5})
|
|
||||||
\end{xtabular}
|
|
||||||
|
|
||||||
\onecolumn
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Zahlenmengen %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section*{Zahlenmengen}
|
|
||||||
$\mdn = \Set{1, 2, 3, \dots} \;\;\;$ Natürliche Zahlen\\
|
|
||||||
$\mdz = \mdn \cup \Set{0, -1, -2, \dots} \;\;\;$ Ganze Zahlen\\
|
|
||||||
$\mdq = \mdz \cup \Set{\frac{1}{2}, \frac{1}{3}, \frac{2}{3}} = \Set{\frac{z}{n} \text{ mit } z \in \mdz \text{ und } n \in \mdz \setminus \Set{0}} \;\;\;$ Rationale Zahlen\\
|
|
||||||
$\mdr = \mdq \cup \Set{\sqrt{2}, -\sqrt[3]{3}, \dots}\;\;\;$ Reele Zahlen\\
|
|
||||||
$\mdr_+\;$ Echt positive reele Zahlen\\
|
|
||||||
$\mdr_{+,0}^n := \Set{(x_1, \dots, x_n) \in \mdr^n | x_n \geq 0}\;\;\;$ Halbraum\\
|
|
||||||
$\mdr^\times = \mdr \setminus \Set{0} \;$ Einheitengruppe von $\mdr$\\
|
|
||||||
$\mdc = \Set{a+ib|a,b \in \mdr}\;\;\;$ Komplexe Zahlen\\
|
|
||||||
$\mdp = \Set{2, 3, 5, 7, \dots}\;\;\;$ Primzahlen\\
|
|
||||||
$\mdh = \Set{z \in \mdc | \Im{z} > 0}\;\;\;$ obere Halbebene\\
|
|
||||||
$I = [0,1] \subsetneq \mdr\;\;\;$ Einheitsintervall\\
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$f:S^1 \hookrightarrow \mdr^2$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$f:S^1 \hookrightarrow \mdr^2$& Einbettung der Kreislinie in die Ebene\\
|
|
||||||
$\pi_1(X,x)$ & Fundamentalgruppe im topologischen Raum $X$ um $x \in X$\\
|
|
||||||
$\Fix(f)$ & Menge der Fixpunkte der Abbildung $f$\\
|
|
||||||
$\|\cdot\|_2$ & 2-Norm; Euklidische Norm\\
|
|
||||||
$\kappa$ & Krümmung\\
|
|
||||||
$\kappa_{\ts{Nor}}$ & Normalenkrümmung\\
|
|
||||||
$V(f)$ & Nullstellenmenge von $f$\footnotemark
|
|
||||||
\end{xtabular}
|
|
||||||
\footnotetext{von \textit{\textbf{V}anishing Set}}
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Krümmung %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section*{Krümmung}
|
|
||||||
|
|
||||||
\settowidth\mylengtha{$D_p F: \mdr^2 \rightarrow \mdr^3$}
|
|
||||||
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
|
|
||||||
|
|
||||||
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
|
|
||||||
$D_p F: \mdr^2 \rightarrow \mdr^3$& Lineare Abbildung mit Jacobi-Matrix in $p$ (siehe \cpageref{def:Tangentialebene})\\
|
|
||||||
$T_s S$ & Tangentialebene an $S \subseteq \mdr^3$ durch $s \in S$\\
|
|
||||||
$d_s n(x)$ & Weingarten-Abbildung\\
|
|
||||||
\end{xtabular}
|
|
||||||
|
|
||||||
\index{Faser|see{Urbild}}
|
|
||||||
\index{kongruent|see{isometrisch}}
|
|
||||||
\index{Kongruenz|see{Isometrie}}
|
|
||||||
|
|
@ -1,80 +0,0 @@
|
||||||
%!TEX root = GeoTopo.tex
|
|
||||||
\chapter*{Vorwort}
|
|
||||||
Dieses Skript wurde im Wintersemester 2013/2014
|
|
||||||
von Martin Thoma geschrieben. Es beinhaltet die Mitschriften aus
|
|
||||||
der Vorlesung von Prof.~Dr.~Herrlich sowie die Mitschriften einiger
|
|
||||||
Übungen und Tutorien.
|
|
||||||
|
|
||||||
Das Skript ist kostenlos über \href{http://martin-thoma.com/geotopo/}{martin-thoma.com/geotopo}
|
|
||||||
verfügbar. Wer es gerne in A5 (Schwarz-Weiß, Ringbindung) für 10~Euro hätte,
|
|
||||||
kann mir eine E-Mail schicken (info@martin-thoma.de).
|
|
||||||
|
|
||||||
\section*{Danksagungen}
|
|
||||||
An dieser Stelle möchte ich Herrn~Prof.~Dr.~Herrlich für einige
|
|
||||||
Korrekturvorschläge und einen gut strukturierten Tafelanschrieb
|
|
||||||
danken, der als Vorlage für dieses Skript diente. Tatsächlich basiert
|
|
||||||
die Struktur dieses Skripts auf der Vorlesung von Herrn~Prof.~Dr.~Herrlich
|
|
||||||
und ganze Abschnitte konnten direkt mit \LaTeX{} umgesetzt werden.
|
|
||||||
Vielen Dank für die Erlaubnis, Ihre Inhalte in diesem Skript einbauen
|
|
||||||
zu dürfen!
|
|
||||||
|
|
||||||
Vielen Dank auch an Frau Lenz und Frau Randecker, die es mir erlaubt
|
|
||||||
haben, ihre Übungsaufgaben und Lösungen zu benutzen.
|
|
||||||
|
|
||||||
Jérôme Urhausen hat durch viele Verbesserungsvorschläge und Beweise zu einer erheblichen
|
|
||||||
Qualitätssteigerung am Skript beigetragen und meine Tutorin Sarah hat mir
|
|
||||||
viele Fragen per E-Mail und nach dem Tutorium beantwortet. Danke!
|
|
||||||
|
|
||||||
|
|
||||||
\section*{Was ist Topologie?}
|
|
||||||
|
|
||||||
Die Kugeloberfläche $S^2$ lässt sich durch strecken, stauchen
|
|
||||||
und umformen zur Würfeloberfläche oder
|
|
||||||
der Oberfläche einer Pyramide verformen, aber nicht zum $\mdr^2$
|
|
||||||
oder zu einem Torus $T^2$. Für den $\mdr^2$ müsste man die Oberfläche
|
|
||||||
unendlich ausdehnen und für einen Torus müsste man ein Loch machen.
|
|
||||||
|
|
||||||
\begin{figure}[ht]
|
|
||||||
\centering
|
|
||||||
\subfloat[$S^2$]{
|
|
||||||
\input{figures/s2.tex}
|
|
||||||
\label{fig:s2}
|
|
||||||
}%
|
|
||||||
\subfloat[Würfel]{
|
|
||||||
\input{figures/cube.tex}
|
|
||||||
\label{fig:cube}
|
|
||||||
}%
|
|
||||||
\subfloat[Pyramide]{
|
|
||||||
\input{figures/pyramid.tex}
|
|
||||||
\label{fig:pyramide}
|
|
||||||
}
|
|
||||||
|
|
||||||
\subfloat[$\mdr^2$]{
|
|
||||||
\input{figures/plane-r2.tex}
|
|
||||||
\label{fig:plane-r2}
|
|
||||||
}%
|
|
||||||
\subfloat[$T^2$]{
|
|
||||||
\input{figures/torus.tex} \xindex{Torus}
|
|
||||||
\label{fig:torus}
|
|
||||||
}
|
|
||||||
\label{fig:formen}
|
|
||||||
\caption{Beispiele für verschiedene Formen}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
\section*{Erforderliche Vorkenntnisse}
|
|
||||||
Es wird ein sicherer Umgang mit den Quantoren ($\forall, \exists$),
|
|
||||||
Mengenschreibweisen ($\cup, \cap, \setminus, \emptyset, \mdr, \powerset{M}$)
|
|
||||||
und ganz allgemein formaler Schreibweise vorausgesetzt. Auch die
|
|
||||||
Beweisführung mittels Widerspruchsbeweisen sollte bekannt sein und
|
|
||||||
der Umgang mit komplexen Zahlen $\mdc$, deren Betrag, Folgen und
|
|
||||||
Häufungspunkten nicht weiter schwer fallen.
|
|
||||||
Diese Vorkenntnisse werden vor allem in \enquote{Analysis I} vermittelt.
|
|
||||||
|
|
||||||
Außerdem wird vorausgesetzt, dass (affine) Vektorräume, Faktorräume,
|
|
||||||
lineare Unabhängigkeit, der Spektralsatz und der projektive Raum $\praum(\mdr)$ aus
|
|
||||||
\enquote{Lineare Algebra I} bekannt sind. In \enquote{Lineare Algebra II}
|
|
||||||
wird der Begriff der Orthonormalbasis eingeführt.
|
|
||||||
|
|
||||||
Obwohl es nicht vorausgesetzt wird, könnte es von Vorteil sein
|
|
||||||
\enquote{Einführung in die Algebra und Zahlentheorie} gehört zu
|
|
||||||
haben.
|
|
||||||
|
|
@ -1,9 +0,0 @@
|
||||||
SOURCE = definitionen
|
|
||||||
make:
|
|
||||||
./generateDefinitions.py
|
|
||||||
pdflatex $(SOURCE).tex -interaction=batchmode -output-format=pdf
|
|
||||||
pdflatex $(SOURCE).tex -interaction=batchmode -output-format=pdf
|
|
||||||
make clean
|
|
||||||
|
|
||||||
clean:
|
|
||||||
rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.thm definitionen.tex
|
|
||||||
|
|
@ -1,12 +0,0 @@
|
||||||
\NeedsTeXFormat{LaTeX2e}[1996/12/01]
|
|
||||||
\ProvidesFile{avery5388.cfg}
|
|
||||||
\newcommand{\cardpaper}{a4paper}
|
|
||||||
\newcommand{\cardpapermode}{portrait}
|
|
||||||
\newcommand{\cardrows}{4}
|
|
||||||
\newcommand{\cardcolumns}{2}
|
|
||||||
\setlength{\cardheight}{70mm}
|
|
||||||
\setlength{\cardwidth}{100mm}
|
|
||||||
\setlength{\topoffset}{2mm}
|
|
||||||
\setlength{\oddoffset}{5mm}
|
|
||||||
\setlength{\evenoffset}{5mm}
|
|
||||||
\endinput
|
|
||||||
Binary file not shown.
|
|
@ -1,67 +0,0 @@
|
||||||
\documentclass[a7cards,frame]{flashcards}
|
|
||||||
\usepackage{etoolbox}
|
|
||||||
\usepackage{amsmath,amssymb}% math symbols / fonts
|
|
||||||
\usepackage{mathtools} % \xRightarrow
|
|
||||||
\usepackage{nicefrac} % \nicefrac
|
|
||||||
\usepackage[utf8]{inputenc} % this is needed for umlauts
|
|
||||||
\usepackage[ngerman]{babel} % this is needed for umlauts
|
|
||||||
\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
|
|
||||||
\usepackage[framed,amsmath,thmmarks,hyperref]{ntheorem}
|
|
||||||
\usepackage{framed}
|
|
||||||
\usepackage{marvosym}
|
|
||||||
\usepackage{makeidx} % for automatically generation of an index
|
|
||||||
\usepackage{xcolor}
|
|
||||||
\usepackage[bookmarks,bookmarksnumbered,hypertexnames=false,pdfpagelayout=OneColumn,colorlinks,hyperindex=false]{hyperref} % has to be after makeidx
|
|
||||||
\usepackage{enumitem} % Better than \usepackage{enumerate}, because it allows to set references
|
|
||||||
\usepackage{tabto}
|
|
||||||
\usepackage{braket} % needed for \Set
|
|
||||||
\usepackage{csquotes} % \enquote{}
|
|
||||||
\usepackage{subfig} % multiple figures in one
|
|
||||||
\usepackage{parskip} % nicer paragraphs
|
|
||||||
\usepackage{xifthen} % \isempty
|
|
||||||
\usepackage{changepage} % for the adjustwidth environment
|
|
||||||
\usepackage{pst-solides3d}
|
|
||||||
\usepackage[colorinlistoftodos]{todonotes}
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=1.7}
|
|
||||||
\usepackage[arrow, matrix, curve]{xy}
|
|
||||||
\usepackage{caption} % get newlines within captions
|
|
||||||
\usepackage{tikz} % draw
|
|
||||||
\usepackage{tikz-3dplot} % draw
|
|
||||||
\usepackage{tkz-fct} % draw
|
|
||||||
\usepackage{tkz-euclide} % draw
|
|
||||||
\usetkzobj{all} % tkz-euclide
|
|
||||||
\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns,fadings,decorations.pathreplacing}
|
|
||||||
\usepackage{tqft}
|
|
||||||
\usepackage{xspace} % for new commands; decides weather I want to insert a space after the command
|
|
||||||
\usepackage[german,nameinlink]{cleveref} % has to be after hyperref, ntheorem, amsthm
|
|
||||||
|
|
||||||
%%
|
|
||||||
\newcounter{chapter}
|
|
||||||
\newcommand\chapter{\if@openright\cleardoublepage\else\clearpage\fi
|
|
||||||
\thispagestyle{plain}%
|
|
||||||
\global\@topnum\z@
|
|
||||||
\@afterindentfalse
|
|
||||||
\secdef\@chapter\@schapter}
|
|
||||||
%%
|
|
||||||
\usepackage{../shortcuts}
|
|
||||||
|
|
||||||
\hypersetup{
|
|
||||||
pdfauthor = {Martin Thoma},
|
|
||||||
pdfkeywords = {Geometrie und Topologie},
|
|
||||||
pdftitle = {Geometrie und Topologie - Definitionen}
|
|
||||||
}
|
|
||||||
\allowdisplaybreaks
|
|
||||||
|
|
||||||
\makeatletter
|
|
||||||
\renewcommand{\flashcards@ps@back@begin@plain}
|
|
||||||
% {\vspace*{\fill}\center\flashcards@format@back}% REMOVED
|
|
||||||
{\vspace*{\fill}\flashcards@format@back}% ADDED
|
|
||||||
\makeatother
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Begin document %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\begin{document}
|
|
||||||
%CONTENT%
|
|
||||||
\end{document}
|
|
||||||
|
|
@ -1,36 +0,0 @@
|
||||||
#!/usr/bin/env python3
|
|
||||||
# -*- coding: utf-8 -*-
|
|
||||||
|
|
||||||
import re, glob
|
|
||||||
|
|
||||||
def get_definitions(filename):
|
|
||||||
with open(filename) as f:
|
|
||||||
content = f.read()
|
|
||||||
|
|
||||||
pattern = re.compile(r"^\\begin{definition}(.*?)\\end{definition}", re.DOTALL | re.UNICODE | re.MULTILINE)
|
|
||||||
index_pattern = re.compile(r"\\xindex{(?:.*?@)?(.*?)(?:\|.*?)?}", re.UNICODE)
|
|
||||||
todo_pattern = re.compile(r"\\todo{.*?}", re.UNICODE)
|
|
||||||
definitions = re.findall(pattern, content)
|
|
||||||
def_dict_list = []
|
|
||||||
for definition in definitions:
|
|
||||||
names = re.findall(index_pattern, definition)
|
|
||||||
names = map(lambda s: s.replace("!", ", "), names)
|
|
||||||
name = "\\\\".join(names)
|
|
||||||
definition = re.sub(todo_pattern, "", definition)
|
|
||||||
def_dict_list.append({"name":name, "definition":definition})
|
|
||||||
#return "\n\n".join('\\vspace*{{\\fill}}\n{0}\n\\vspace*{{\\fill}}\\clearpage'.format(definition["definition"]) for definition in def_dict_list)
|
|
||||||
return "\n\n".join('\\begin{{flashcard}}{{ {1} }}\n{{ {0} }}\n\\end{{flashcard}}'.format(definition["definition"], definition["name"]) for definition in def_dict_list)
|
|
||||||
|
|
||||||
def write_definitions_to_template(definitions, template="mathe-vorlage.tex", target="definitionen.tex"):
|
|
||||||
with open(template) as f:
|
|
||||||
content = f.read()
|
|
||||||
content = content.replace('%CONTENT%', definitions)
|
|
||||||
|
|
||||||
with open(target, 'w') as f:
|
|
||||||
f.write(content)
|
|
||||||
|
|
||||||
if __name__ == "__main__":
|
|
||||||
definitions = []
|
|
||||||
for texsource in sorted(glob.glob("../Kapitel*.tex")):
|
|
||||||
definitions.append(get_definitions(texsource))
|
|
||||||
write_definitions_to_template("\n\n\n".join(definitions), "flashcards-try.tex")
|
|
||||||
|
|
@ -1,53 +0,0 @@
|
||||||
\documentclass[a7paper,9pt,landscape]{scrbook}
|
|
||||||
\usepackage{etoolbox}
|
|
||||||
\usepackage{amsmath,amssymb}% math symbols / fonts
|
|
||||||
\usepackage{mathtools} % \xRightarrow
|
|
||||||
\usepackage{nicefrac} % \nicefrac
|
|
||||||
\usepackage[utf8]{inputenc} % this is needed for umlauts
|
|
||||||
\usepackage[ngerman]{babel} % this is needed for umlauts
|
|
||||||
\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
|
|
||||||
\usepackage[framed,amsmath,thmmarks,hyperref]{ntheorem}
|
|
||||||
\usepackage{framed}
|
|
||||||
\usepackage{marvosym}
|
|
||||||
\usepackage{makeidx} % for automatically generation of an index
|
|
||||||
\usepackage{xcolor}
|
|
||||||
\usepackage[bookmarks,bookmarksnumbered,hypertexnames=false,pdfpagelayout=OneColumn,colorlinks,hyperindex=false]{hyperref} % has to be after makeidx
|
|
||||||
\usepackage{enumitem} % Better than \usepackage{enumerate}, because it allows to set references
|
|
||||||
\usepackage{tabto}
|
|
||||||
\usepackage{braket} % needed for \Set
|
|
||||||
\usepackage{csquotes} % \enquote{}
|
|
||||||
\usepackage{subfig} % multiple figures in one
|
|
||||||
\usepackage{parskip} % nicer paragraphs
|
|
||||||
\usepackage{xifthen} % \isempty
|
|
||||||
\usepackage{changepage} % for the adjustwidth environment
|
|
||||||
\usepackage{pst-solides3d}
|
|
||||||
\usepackage[colorinlistoftodos]{todonotes}
|
|
||||||
\usepackage{pgfplots}
|
|
||||||
\pgfplotsset{compat=1.7}
|
|
||||||
\usepackage[arrow, matrix, curve]{xy}
|
|
||||||
\usepackage{caption} % get newlines within captions
|
|
||||||
\usepackage{tikz} % draw
|
|
||||||
\usepackage{tikz-3dplot} % draw
|
|
||||||
\usepackage{tkz-fct} % draw
|
|
||||||
\usepackage{tkz-euclide} % draw
|
|
||||||
\usetkzobj{all} % tkz-euclide
|
|
||||||
\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns,fadings,decorations.pathreplacing}
|
|
||||||
\usepackage{tqft}
|
|
||||||
\usepackage{xspace} % for new commands; decides weather I want to insert a space after the command
|
|
||||||
\usepackage[german,nameinlink]{cleveref} % has to be after hyperref, ntheorem, amsthm
|
|
||||||
\usepackage[left=10mm,right=10mm, top=2mm, bottom=10mm]{geometry}
|
|
||||||
\usepackage{../shortcuts}
|
|
||||||
|
|
||||||
\hypersetup{
|
|
||||||
pdfauthor = {Martin Thoma},
|
|
||||||
pdfkeywords = {Geometrie und Topologie},
|
|
||||||
pdftitle = {Geometrie und Topologie - Definitionen}
|
|
||||||
}
|
|
||||||
\allowdisplaybreaks
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
% Begin document %
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\begin{document}
|
|
||||||
%CONTENT%
|
|
||||||
\end{document}
|
|
||||||
|
|
@ -1,32 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
legend pos=south east,
|
|
||||||
axis x line=middle,
|
|
||||||
axis y line=middle,
|
|
||||||
grid = major,
|
|
||||||
%width=9cm,
|
|
||||||
%height=4.5cm,
|
|
||||||
grid style={dashed, gray!30},
|
|
||||||
xmin= 0, % start the diagram at this x-coordinate
|
|
||||||
xmax= 12, % end the diagram at this x-coordinate
|
|
||||||
ymin=-10, % start the diagram at this y-coordinate
|
|
||||||
ymax= 10, % end the diagram at this y-coordinate
|
|
||||||
%axis background/.style={fill=white},
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$y$,
|
|
||||||
%xticklabels={-2,-1.6,...,7},
|
|
||||||
tick align=outside,
|
|
||||||
%minor tick num=-3,
|
|
||||||
enlargelimits=true]
|
|
||||||
\addplot[domain=0:12, red, thick,samples=500] {1/3*x^1.5};
|
|
||||||
\addplot[domain=0:12, dotted, orange, thick,samples=500] {1*x^1.5};
|
|
||||||
\addplot[domain=0:12, dashed, blue, thick,samples=500] {2*x^1.5};
|
|
||||||
|
|
||||||
\addplot[domain=0:12, red, thick,samples=500] {-1/3*x^1.5};
|
|
||||||
\addplot[domain=0:12, dotted, orange, thick,samples=500] {-1*x^1.5};
|
|
||||||
\addplot[domain=0:12, dashed, blue, thick,samples=500] {-2*x^1.5};
|
|
||||||
\addlegendentry{$a=\frac{1}{3}$}
|
|
||||||
\addlegendentry{$a=1$}
|
|
||||||
\addlegendentry{$a=2$}
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,31 +0,0 @@
|
||||||
\pgfplotsset{
|
|
||||||
colormap={whitered}{
|
|
||||||
color(0cm)=(white);
|
|
||||||
color(1cm)=(orange!75!red)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
\begin{tikzpicture}[scale=0.5]
|
|
||||||
\begin{axis}[
|
|
||||||
colormap name=whitered,
|
|
||||||
width=15cm,
|
|
||||||
view={155}{45},
|
|
||||||
enlargelimits=false,
|
|
||||||
grid=major,
|
|
||||||
domain=-5:5,
|
|
||||||
y domain=-5:5,
|
|
||||||
samples=56, %57 : TeX capacity exceeded, sorry [main memory size=3000000].
|
|
||||||
% see also http://tex.stackexchange.com/a/7954/5645
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$y$,
|
|
||||||
zlabel={$z$},
|
|
||||||
colorbar,
|
|
||||||
colorbar style={
|
|
||||||
at={(-0.1,0)},
|
|
||||||
anchor=south west,
|
|
||||||
height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
|
|
||||||
title={$f(x,y)$}
|
|
||||||
}
|
|
||||||
]
|
|
||||||
\addplot3[surf] {y*y-x*x*x};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
Binary file not shown.
|
Before Width: | Height: | Size: 260 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 401 KiB |
Binary file not shown.
Binary file not shown.
|
Before Width: | Height: | Size: 636 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 343 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 200 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 298 KiB |
|
|
@ -1,14 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\node (Z) at (0,0) {$Z$};
|
|
||||||
\node (Y) at (3,0) {$Y$};
|
|
||||||
\node (X) at (1.5,-1.5) {$X$};
|
|
||||||
\draw[->, above, dashed] (Z) to node {$\tilde{f}$} (Y);
|
|
||||||
\draw[->, below] (Z) to node {$f$} (X);
|
|
||||||
\draw[->, right] (Y) to node {$p$} (X);
|
|
||||||
|
|
||||||
\begin{scope}[xshift=1.3cm,yshift=-0.6cm]
|
|
||||||
\draw (0,0) -- (0.3,0.3);
|
|
||||||
\draw (0.1,0) -- (0.4,0.3);
|
|
||||||
\draw (0.2,0) -- (0.5,0.3);
|
|
||||||
\end{scope}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,17 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/O, 1/0/X, 0/1/Y, 2/1/P}
|
|
||||||
|
|
||||||
\tkzMarkAngle[fill=green!20,size=0.3cm,opacity=.5](X,O,Y)
|
|
||||||
\tkzLabelAngle[pos=0.15](X,O,Y){$\cdot$}
|
|
||||||
|
|
||||||
\tkzDrawLine[add=3 and 2](O,X)
|
|
||||||
\tkzLabelLine[below,pos=3](O,X){$g_1$}
|
|
||||||
\tkzLabelLine[right,pos=3](O,Y){$g_2$}
|
|
||||||
\tkzDrawLine[add=3 and 2](O,Y)
|
|
||||||
|
|
||||||
\tkzLabelPoint(P){$P$}
|
|
||||||
\node at ($(-2,2)$){$X$};
|
|
||||||
\tkzDrawPoints(P)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,34 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/O, 1/0/X, 0/1/Y, 2/1/P}
|
|
||||||
|
|
||||||
\tkzMarkAngle[fill=green!20,size=0.3cm,opacity=.5](X,O,Y)
|
|
||||||
\tkzLabelAngle[pos=0.15](X,O,Y){$\cdot$}
|
|
||||||
|
|
||||||
\tkzDrawLine[add=3 and 2](O,X)
|
|
||||||
\tkzLabelLine[below,pos=3](O,X){$g_1$}
|
|
||||||
\tkzLabelLine[right,pos=3](O,Y){$g_2$}
|
|
||||||
\tkzDrawLine[add=3 and 2](O,Y)
|
|
||||||
|
|
||||||
\tkzDefLine[orthogonal=through P,/tikz/overlay](O,X) \tkzGetPoint{helper}
|
|
||||||
\tkzInterLL(O,X)(P,helper) \tkzGetPoint{xp}
|
|
||||||
\draw [decorate,decoration={brace,amplitude=4pt,mirror}]
|
|
||||||
(O) -- (xp) node [black,midway,xshift=0cm, yshift=-0.3cm]
|
|
||||||
{\footnotesize $x_P$};
|
|
||||||
|
|
||||||
\tkzDefLine[orthogonal=through P,/tikz/overlay](O,Y) \tkzGetPoint{helper}
|
|
||||||
\tkzInterLL(O,Y)(P,helper) \tkzGetPoint{yp}
|
|
||||||
\draw [decorate,decoration={brace,amplitude=4pt}]
|
|
||||||
(O) -- (yp) node [black,midway,xshift=-0.4cm]
|
|
||||||
{\footnotesize $y_P$};
|
|
||||||
|
|
||||||
\tkzDrawPolygon(O,xp,P,yp)
|
|
||||||
|
|
||||||
\tkzLabelPoint[above right](P){$P$}
|
|
||||||
\tkzLabelPoint[below left](O){$0$}
|
|
||||||
\tkzLabelPoint[below](xp){$P_X$}
|
|
||||||
\tkzLabelPoint[left](Y){$P_Y$}
|
|
||||||
\node at ($(-2,2)$){$X$};
|
|
||||||
\tkzDrawPoints(P,Y,xp)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,43 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/O, 1/0/X, 0/1/Y, 2/1/P, 3/3/Q}
|
|
||||||
\tkzDrawLine[add=3 and 2.2](O,X)
|
|
||||||
\tkzLabelLine[below,pos=3](O,X){$g_1$}
|
|
||||||
\tkzLabelLine[left,pos=3](O,Y){$g_2$}
|
|
||||||
\tkzDrawLine[add=3 and 2.2](O,Y)
|
|
||||||
|
|
||||||
\tkzDefLine[orthogonal=through P,/tikz/overlay](O,X) \tkzGetPoint{helper}
|
|
||||||
\tkzInterLL(O,X)(P,helper) \tkzGetPoint{xp}
|
|
||||||
\draw [decorate,decoration={brace,amplitude=4pt,mirror}]
|
|
||||||
(O) -- (xp) node [black,midway,xshift=0cm, yshift=-0.3cm]
|
|
||||||
{\footnotesize $x_P$};
|
|
||||||
|
|
||||||
\tkzDefLine[orthogonal=through P,/tikz/overlay](O,Y) \tkzGetPoint{helper}
|
|
||||||
\tkzInterLL(O,Y)(P,helper) \tkzGetPoint{yp}
|
|
||||||
\draw [decorate,decoration={brace,amplitude=4pt}]
|
|
||||||
(O) -- (yp) node [black,midway,xshift=-0.4cm]
|
|
||||||
{\footnotesize $y_P$};
|
|
||||||
|
|
||||||
\tkzDrawPolygon(O,xp,P,yp)
|
|
||||||
|
|
||||||
\tkzDefLine[orthogonal=through Q,/tikz/overlay](O,X) \tkzGetPoint{helper}
|
|
||||||
\tkzInterLL(O,X)(Q,helper) \tkzGetPoint{xq}
|
|
||||||
\tkzDefLine[orthogonal=through Q,/tikz/overlay](O,Y) \tkzGetPoint{helper}
|
|
||||||
\tkzInterLL(O,Y)(Q,helper) \tkzGetPoint{yq}
|
|
||||||
|
|
||||||
\tkzInterLL(yp,P)(Q,xq) \tkzGetPoint{qxp}
|
|
||||||
\tkzInterLL(xp,P)(Q,yq) \tkzGetPoint{R}
|
|
||||||
|
|
||||||
\tkzDrawPolygon(O,xq,Q,yq)
|
|
||||||
|
|
||||||
\tkzDrawSegments[green](xp,xq R,Q)
|
|
||||||
\tkzDrawSegments[very thick,orange](yp,yq P,R)
|
|
||||||
|
|
||||||
\tkzLabelPoint[above right](P){$P$}
|
|
||||||
\tkzLabelPoint[above right](Q){$Q$}
|
|
||||||
\tkzLabelPoint[below left](O){$0$}
|
|
||||||
\tkzLabelPoint[above](R){$R$}
|
|
||||||
\node at ($(-2,2)$){$X$};
|
|
||||||
\tkzDrawPoints(P,Q,R)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,51 +0,0 @@
|
||||||
% Source: http://tex.stackexchange.com/a/12069/5645
|
|
||||||
\begin{tikzpicture}[scale=0.5]
|
|
||||||
\clip (-3,-3) rectangle (3,3);
|
|
||||||
\coordinate (tf) at (0,0);
|
|
||||||
\coordinate (bf) at (0,-3);
|
|
||||||
\coordinate (tr) at (15:2.5cm);
|
|
||||||
\coordinate (tl) at (165:2.5cm);
|
|
||||||
|
|
||||||
% You can change the perspective by playing with the 5, 5, 15:
|
|
||||||
\coordinate (fr) at ($ (tf)!5!(tr) $);
|
|
||||||
\coordinate (fl) at ($ (tf)!5!(tl) $);
|
|
||||||
\coordinate (fb) at ($ (tf)!15!(bf) $);
|
|
||||||
|
|
||||||
\path[name path=brpath] (bf) -- (fr);
|
|
||||||
\path[name path=rbpath] (tr) -- (fb);
|
|
||||||
\path[name path=blpath] (bf) -- (fl);
|
|
||||||
\path[name path=lbpath] (tl) -- (fb);
|
|
||||||
\path[name path=trpath] (tl) -- (fr);
|
|
||||||
\path[name path=tlpath] (tr) -- (fl);
|
|
||||||
|
|
||||||
\draw[name intersections={of=brpath and rbpath}] (intersection-1)coordinate (br){};
|
|
||||||
\draw[name intersections={of=blpath and lbpath}] (intersection-1)coordinate (bl){};
|
|
||||||
\draw[name intersections={of=trpath and tlpath}] (intersection-1)coordinate (tb){};
|
|
||||||
|
|
||||||
\shade[right color=gray!10, left color=black!50, shading angle=105] (tf) -- (bf) -- (bl) -- (tl) -- cycle;
|
|
||||||
\shade[left color=gray!10, right color=black!50, shading angle=75] (tf) -- (bf) -- (br) -- (tr) -- cycle;
|
|
||||||
|
|
||||||
\begin{scope}
|
|
||||||
\clip (tf) -- (tr) -- (tb) -- (tl) -- cycle;
|
|
||||||
\shade[inner color = gray!5, outer color=black!50, shading=radial] (tf) ellipse (3cm and 1.5cm);
|
|
||||||
\end{scope}
|
|
||||||
|
|
||||||
\draw (tf) -- (bf);
|
|
||||||
\draw (tf) -- (tr);
|
|
||||||
\draw (tf) -- (tl);
|
|
||||||
\draw (tr) -- (br);
|
|
||||||
\draw (bf) -- (br);
|
|
||||||
\draw (tl) -- (bl);
|
|
||||||
\draw (bf) -- (bl);
|
|
||||||
\draw (tb) -- (tr);
|
|
||||||
\draw (tb) -- (tl);
|
|
||||||
|
|
||||||
%set the sizes of the little cubes:
|
|
||||||
\def\tone{.4}\def\ttwo{.75}\def\fone{.36}\def\ftwo{.70}
|
|
||||||
\draw ($ (bf)!\tone!(br) $) -- ($ (tf)!\tone!(tr) $) -- ($ (tl)!\tone!(tb) $);
|
|
||||||
\draw ($ (bf)!\ttwo!(br) $) -- ($ (tf)!\ttwo!(tr) $) -- ($ (tl)!\ttwo!(tb) $);
|
|
||||||
\draw ($ (bf)!\tone!(bl) $) -- ($ (tf)!\tone!(tl) $) -- ($ (tr)!\tone!(tb) $);
|
|
||||||
\draw ($ (bf)!\ttwo!(bl) $) -- ($ (tf)!\ttwo!(tl) $) -- ($ (tr)!\ttwo!(tb) $);
|
|
||||||
\draw ($ (tl)!\fone!(bl) $) -- ($ (tf)!\fone!(bf) $) -- ($ (tr)!\fone!(br) $);
|
|
||||||
\draw ($ (tl)!\ftwo!(bl) $) -- ($ (tf)!\ftwo!(bf) $) -- ($ (tr)!\ftwo!(br) $);
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,33 +0,0 @@
|
||||||
\pgfplotsset{
|
|
||||||
colormap={whitered}{
|
|
||||||
color(0cm)=(white);
|
|
||||||
color(1cm)=(orange!75!red)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
colormap name=whitered,
|
|
||||||
width=15cm,
|
|
||||||
view={340}{25},
|
|
||||||
enlargelimits=false,
|
|
||||||
grid=major,
|
|
||||||
domain=0:5,
|
|
||||||
y domain=0:2*pi,
|
|
||||||
xmin=-1.5, xmax=1.5,
|
|
||||||
ymin=-1.5, ymax=1.5, zmin=0.0,
|
|
||||||
samples=30, %57 : TeX capacity exceeded, sorry [main memory size=3000000].
|
|
||||||
% see also http://tex.stackexchange.com/a/7954/5645
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$y$,
|
|
||||||
zlabel={$z$},
|
|
||||||
%colorbar,
|
|
||||||
colorbar style={
|
|
||||||
at={(-0.1,0)},
|
|
||||||
anchor=south west,
|
|
||||||
height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
|
|
||||||
title={$f(x,y)$}
|
|
||||||
}
|
|
||||||
]
|
|
||||||
\addplot3 [surf,z buffer=sort] ({cos(deg(y))},{sin(deg(y))},{x});
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
Binary file not shown.
|
|
@ -1,24 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
|
|
||||||
\node (P)[point,label={[label distance=0cm]210:$P$}] at (0,0) {};
|
|
||||||
\node (B)[point,label={[label distance=0cm]-90:$B$}] at (2.5,0) {};
|
|
||||||
\node (Q)[point,label={[label distance=0cm]-90:$Q$}] at (4,0) {};
|
|
||||||
\node (C)[point,label={[label distance=0cm]90:$C$}] at (1.5,1.5) {};
|
|
||||||
\node (R)[point,label={[label distance=0cm]90:$R$}] at (2.5,2.5) {};
|
|
||||||
|
|
||||||
\node (A)[point,label={[label distance=0cm]0:$A$}] at (0.5,3) {};
|
|
||||||
|
|
||||||
\draw[very thick] (P) edge node {} (B);
|
|
||||||
\draw[very thick] (B) edge node {} (Q);
|
|
||||||
\draw[very thick] (P) edge node {} (C);
|
|
||||||
\draw[very thick] (C) edge node {} (R);
|
|
||||||
|
|
||||||
\draw[very thick] (B) edge node {} (C);
|
|
||||||
\draw[very thick] (C) edge node {} (A);
|
|
||||||
|
|
||||||
\draw[very thick] (Q) edge node {} (R);
|
|
||||||
|
|
||||||
\draw[very thick] ($(P)!-1cm!(Q)$) -- ($(Q)!-1cm!(P)$);
|
|
||||||
\draw[very thick] ($(A)!-0.3cm!(B)$) -- ($(B)!-1cm!(A)$);
|
|
||||||
\draw[very thick] ($(R)!-1cm!(Q)$) -- ($(Q)!-1cm!(R)$);
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,15 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
|
|
||||||
\node (P)[point,label={[label distance=0cm]-90:$P$}] at (0,0) {};
|
|
||||||
\node (Q)[point,label={[label distance=0cm]-90:$Q$}] at (5,1) {};
|
|
||||||
\node (A)[point,label={[label distance=0cm]180:$A$}] at (2,2) {};
|
|
||||||
\node (B)[point,label={[label distance=0cm]190:$B$}] at (1,3) {};
|
|
||||||
|
|
||||||
\draw[very thick] (P) edge node {} (Q);
|
|
||||||
\draw[very thick, red] (P) edge node {} (A);
|
|
||||||
\draw[very thick, red] (P) edge node {} (B);
|
|
||||||
\draw[very thick, blue] (Q) edge node {} (A);
|
|
||||||
\draw[very thick, blue] (Q) edge node {} (B);
|
|
||||||
|
|
||||||
\draw[very thick] ($(P)!-1cm!(Q)$) -- ($(Q)!-1cm!(P)$);
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,21 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/P, 4/0/Q, 1/0.5/B, 1/2/A}
|
|
||||||
\tkzInterLL(P,B)(Q,A) \tkzGetPoint{C}
|
|
||||||
|
|
||||||
\tkzDrawLine[color=red](P,A)
|
|
||||||
\tkzDrawLine(Q,A)
|
|
||||||
\tkzDrawLine(P,Q)
|
|
||||||
\tkzDrawLine[add=0 and 0.5](P,C)
|
|
||||||
\tkzDrawSegments(B,Q)
|
|
||||||
\tkzDrawLine(P,Q)
|
|
||||||
|
|
||||||
\tkzDrawPoints(P,C,A)
|
|
||||||
\tkzDrawPoints[fill=red,color=red](Q,B)
|
|
||||||
\tkzLabelPoint[below](P){$P$}
|
|
||||||
\tkzLabelPoint[below](Q){$Q$}
|
|
||||||
\tkzLabelPoint[below](B){$B$}
|
|
||||||
\tkzLabelPoint[below](C){$C$}
|
|
||||||
\tkzLabelPoint[below](A){$A$}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,17 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/P, 4/1/Q, 2/3/A, 0/3/B}
|
|
||||||
|
|
||||||
\tkzDrawSegments(P,Q Q,A A,P)
|
|
||||||
\tkzDrawSegments[dashed](P,B B,Q)
|
|
||||||
\tkzDrawLine(P,Q)
|
|
||||||
\tkzDrawLine[color=red](P,A)
|
|
||||||
|
|
||||||
\tkzDrawPoints(P,C,A)
|
|
||||||
\tkzDrawPoints[fill=red,color=red](Q,B)
|
|
||||||
\tkzLabelPoint[below](P){$P$}
|
|
||||||
\tkzLabelPoint[below](Q){$Q$}
|
|
||||||
\tkzLabelPoint[above](A){$A$}
|
|
||||||
\tkzLabelPoint[above](B){$B$}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,20 +0,0 @@
|
||||||
\usetkzobj{all}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/P, 4/1/Q, 2/3/A, 5/3/B, -0.6/-0.15/Pstrich}
|
|
||||||
\tkzInterLL(P,B)(A,Q) \tkzGetPoint{C}
|
|
||||||
\tkzDrawPoints(P,Q,A,B,C,Pstrich)
|
|
||||||
|
|
||||||
\tkzDrawLine(P,Q)
|
|
||||||
\tkzDrawLine(P,A)
|
|
||||||
\tkzDrawLine(A,Q)
|
|
||||||
\tkzDrawLine(P,B)
|
|
||||||
|
|
||||||
\tkzLabelPoint[below](P){$P$}
|
|
||||||
\tkzLabelPoint[above](Pstrich){$P'$}
|
|
||||||
\tkzLabelPoint[below](Q){$Q$}
|
|
||||||
\tkzLabelPoint[below](A){$A$}
|
|
||||||
\tkzLabelPoint[below](B){$B$}
|
|
||||||
\tkzLabelPoint[above](C){$C$}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,20 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/Q, 4/1/H1, 1/2/P}
|
|
||||||
\tkzDefPoint(1.5,3){Phelper}
|
|
||||||
\tkzMarkAngle[arc=l,size=1cm,color=green,fill=green!20](H1,Q,P)
|
|
||||||
\tkzDrawLine(Q,H1)
|
|
||||||
|
|
||||||
\tkzLabelPoint[above left](Q){$Q$}
|
|
||||||
\tkzDefLine[parallel=through P](Q,H1) \tkzGetPoint{b}
|
|
||||||
\tkzMarkAngle[arc=l,size=1cm,color=green,fill=green!20](b,P,Phelper)
|
|
||||||
\tkzDrawLine[dashed](P,b)
|
|
||||||
\tkzLabelLine[pos=0.8,below](P,b){$h$}
|
|
||||||
\tkzLabelLine[pos=-0.6,left](P,Q){$f$}
|
|
||||||
\tkzLabelLine[pos=0.8,below](Q,H1){$g$}
|
|
||||||
\tkzLabelPoint[above left](P){$P$}
|
|
||||||
\tkzDrawLine[add=0.2 and 0.7](Q,P)
|
|
||||||
\tkzDrawPoints(P,Q)
|
|
||||||
\tkzMarkSegments[mark=||](Q,H1 P,b)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,21 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/Q, 2/0/P, 1/2/R}
|
|
||||||
|
|
||||||
|
|
||||||
\pgfmathsetmacro{\firstAngle}{0}
|
|
||||||
\pgfmathsetmacro{\secondAngle}{-120}
|
|
||||||
\path[draw,red, fill=red!40] (Q) -- ++(\firstAngle:.4) arc[start angle=\firstAngle, delta angle=\secondAngle,radius=.4];
|
|
||||||
|
|
||||||
\tkzMarkAngle[arc=l,size=0.8cm,color=green,fill=green!20](Q,R,P)
|
|
||||||
\path[draw] ++(-50:.2) node[rotate=-50] {$\alpha$};
|
|
||||||
\node at (1,1.5) {$\beta$};
|
|
||||||
\tkzDrawLine(Q,P)
|
|
||||||
\tkzDrawLine(Q,R)
|
|
||||||
\tkzDrawLine(P,R)
|
|
||||||
\tkzDrawPoints(P,Q,R)
|
|
||||||
\node at ($(R) + (0.03,0.4)$) {$R$}; %top
|
|
||||||
\node at ($(Q) + (-0.3,-0.22)$) {$Q$}; %left
|
|
||||||
\node at ($(P) + ( 0.3,-0.18)$) {$P$}; %right
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,32 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/A, 4/0/B, 2/2/C, 6/2/A'}
|
|
||||||
|
|
||||||
\tkzMarkAngle[arc=l,size=1.15cm,color=green,fill=green!20,opacity=.8](A',A,C)
|
|
||||||
\tkzLabelAngle[pos=0.9](A',A,C){$\alpha_1$}
|
|
||||||
\tkzMarkAngle[arc=lll,size=1.15cm,color=green,fill=green!20,opacity=.8](B,A,A')
|
|
||||||
\tkzLabelAngle[pos=0.9](B,A,A'){$\alpha_2$}
|
|
||||||
|
|
||||||
\tkzMarkAngle[arc=l,size=0.7cm,color=red,fill=red!20,opacity=.8](C,B,A)
|
|
||||||
\tkzLabelAngle[pos=0.5](C,B,A){$\beta$}
|
|
||||||
|
|
||||||
\tkzMarkAngle[arc=l,size=0.6cm,color=blue,fill=blue!20,opacity=.8](A,C,B)
|
|
||||||
\tkzLabelAngle[pos=0.4](A,C,B){$\gamma$}
|
|
||||||
|
|
||||||
\tkzDrawSegments(A,B A,C A,A' B,C C,A')
|
|
||||||
\tkzInterLL(A,A')(B,C)\tkzGetPoint{M}
|
|
||||||
\node at ($(M) + (0.1,0.25)$) {$M$};
|
|
||||||
|
|
||||||
\tkzLabelPoint[below left](A){$A$}
|
|
||||||
\tkzLabelPoint[below right](B){$B$}
|
|
||||||
\tkzLabelPoint[above left](C){$C$}
|
|
||||||
\tkzLabelPoint[above right](A'){$A'$}
|
|
||||||
|
|
||||||
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20,opacity=.8](B,A,C)
|
|
||||||
\path[draw] ++(25:.35) node[rotate=0] {$\alpha$};
|
|
||||||
|
|
||||||
|
|
||||||
\tkzDrawSegments(A,B A,C)
|
|
||||||
\tkzDrawPoints(A,B,C,A',M)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,19 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{-1/0/Q, 0/0/M, 0/-2/A, 0/2/P, 1/0/R, -1.2/-0.5/helper}
|
|
||||||
\tkzFillPolygon[color = green!10](Q,M,A)
|
|
||||||
\tkzFillPolygon[color = green!10](M,P,R)
|
|
||||||
\tkzMarkAngle[arc=l,size=0.4cm,color=red,fill=red!20](P,R,M)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.4cm,color=green,fill=green!20](helper,Q,M)
|
|
||||||
%\path[draw] ++(25:.3) node[rotate=0] {$\alpha$};
|
|
||||||
%\node at (1,1.5) {$\beta$};
|
|
||||||
\tkzDrawSegments(Q,A)
|
|
||||||
\tkzDrawLines(Q,R A,P Q,P P,R)
|
|
||||||
\tkzDrawPoints(Q,M,A,P,R)
|
|
||||||
\tkzLabelPoint[below left](Q){$Q$}
|
|
||||||
\tkzLabelPoint[below right](M){$M$}
|
|
||||||
\tkzLabelPoint[above left](A){$A$}
|
|
||||||
\tkzLabelPoint[above right](P){$P$}
|
|
||||||
\tkzLabelPoint[above right](R){$R$}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,56 +0,0 @@
|
||||||
% Author: Marc van Dongen
|
|
||||||
% Source: http://www.texample.net/tikz/examples/hilbert-curve/
|
|
||||||
\newdimen\HilbertLastX
|
|
||||||
\newdimen\HilbertLastY
|
|
||||||
\newcounter{HilbertOrder}
|
|
||||||
|
|
||||||
\def\DrawToNext#1#2{%
|
|
||||||
\advance \HilbertLastX by #1
|
|
||||||
\advance \HilbertLastY by #2
|
|
||||||
\pgfpathlineto{\pgfqpoint{\HilbertLastX}{\HilbertLastY}}
|
|
||||||
% Alternative implementation using plot streams:
|
|
||||||
% \pgfplotstreampoint{\pgfqpoint{\HilbertLastX}{\HilbertLastY}}
|
|
||||||
}
|
|
||||||
|
|
||||||
% \Hilbert[right_x,right_y,left_x,left_x,up_x,up_y,down_x,down_y]
|
|
||||||
\def\Hilbert[#1,#2,#3,#4,#5,#6,#7,#8] {
|
|
||||||
\ifnum\value{HilbertOrder} > 0%
|
|
||||||
\addtocounter{HilbertOrder}{-1}
|
|
||||||
\Hilbert[#5,#6,#7,#8,#1,#2,#3,#4]
|
|
||||||
\DrawToNext {#1} {#2}
|
|
||||||
\Hilbert[#1,#2,#3,#4,#5,#6,#7,#8]
|
|
||||||
\DrawToNext {#5} {#6}
|
|
||||||
\Hilbert[#1,#2,#3,#4,#5,#6,#7,#8]
|
|
||||||
\DrawToNext {#3} {#4}
|
|
||||||
\Hilbert[#7,#8,#5,#6,#3,#4,#1,#2]
|
|
||||||
\addtocounter{HilbertOrder}{1}
|
|
||||||
\fi
|
|
||||||
}
|
|
||||||
|
|
||||||
|
|
||||||
% \hilbert((x,y),order)
|
|
||||||
\def\hilbert((#1,#2),#3){%
|
|
||||||
\advance \HilbertLastX by #1
|
|
||||||
\advance \HilbertLastY by #2
|
|
||||||
\pgfpathmoveto{\pgfqpoint{\HilbertLastX}{\HilbertLastY}}
|
|
||||||
% Alternative implementation using plot streams:
|
|
||||||
% \pgfplothandlerlineto
|
|
||||||
% \pgfplotstreamstart
|
|
||||||
% \pgfplotstreampoint{\pgfqpoint{\HilbertLastX}{\HilbertLastY}}
|
|
||||||
\setcounter{HilbertOrder}{#3}
|
|
||||||
\Hilbert[1mm,0mm,-1mm,0mm,0mm,1mm,0mm,-1mm]
|
|
||||||
\pgfusepath{stroke}%
|
|
||||||
}
|
|
||||||
|
|
||||||
\begin{figure}[htp]%
|
|
||||||
\centering
|
|
||||||
% draw Hilbert curves of order n=1,...,5
|
|
||||||
% Warning! Curves with order > 6 may crash TeX
|
|
||||||
\subfloat[$n=1$]{\tikz[scale=18] \hilbert((0mm,0mm),1);}~~
|
|
||||||
\subfloat[$n=2$]{\tikz[scale=6] \hilbert((0mm,0mm),2);}~~
|
|
||||||
\subfloat[$n=3$]{\tikz[scale=2.6] \hilbert((0mm,0mm),3);}~~
|
|
||||||
\subfloat[$n=4$]{\tikz[scale=1.2] \hilbert((0mm,0mm),4);}~~
|
|
||||||
\subfloat[$n=5$]{\tikz[scale=0.58] \hilbert((0mm,0mm),5);}%
|
|
||||||
\caption{Hilbert-Kurve}\xindex{Hilbert-Kurve}
|
|
||||||
\label{fig:hilbert-curve}
|
|
||||||
\end{figure}%
|
|
||||||
|
|
@ -1,17 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=3,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzInit[xmax=7,ymax=3,xmin=-1,ymin=0]
|
|
||||||
\tkzDefPoints{2/0/m1,4/0/m2,1/1.1/a1,1/0/a1x, 2.5/2.0/a2,2.5/0/a2x}
|
|
||||||
\tkzDrawSegments(a1x,a1 a2x,a2)
|
|
||||||
\tkzAxeXY[ticks=false]
|
|
||||||
|
|
||||||
\tkzDrawArc[R,line width=1pt,color=red](m1,1.5 cm)(0,180)
|
|
||||||
\tkzDrawArc[R,line width=1pt](m2,2.5 cm)(0,180)
|
|
||||||
\tkzDrawPoints(m1,m2,a1,a2)
|
|
||||||
\tkzLabelPoint[above](m1) {$m$}
|
|
||||||
\tkzLabelPoint[above](m2) {$\lambda^2 m$}
|
|
||||||
\tkzLabelPoint[above](a1) {$m+\iu r$}
|
|
||||||
\tkzLabelPoint[above](a2) {$\lambda^2 m+\iu \lambda^2 r$}
|
|
||||||
\node[red] at ($(m1)+(1.5,-0.2)$) {$m+1$};
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,18 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=3,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=0.5]
|
|
||||||
\tkzInit[xmax=4.5,ymax=3,xmin=-1,ymin=0]
|
|
||||||
\tkzDefPoints{0/0/O, 3/3/lz, 2/2/z, 3/0/lzx, 2/0/zx}
|
|
||||||
\tkzDrawLines(O,lz zx,z)
|
|
||||||
\tkzDrawLine[add=0 and 0.2](lzx,lz)
|
|
||||||
\tkzAxeXY[ticks=false]
|
|
||||||
|
|
||||||
%\tkzDrawArc[R,line width=1pt,color=red](m1,1.5 cm)(0,180)
|
|
||||||
%\tkzDrawArc[R,line width=1pt](m2,2.5 cm)(0,180)
|
|
||||||
\tkzDrawPoints(z,lz)
|
|
||||||
\tkzLabelPoint[left](z) {$z$}
|
|
||||||
\tkzLabelPoint[above right](zx) {$x$}
|
|
||||||
\tkzLabelPoint[right](lz) {$\lambda^2 z$}
|
|
||||||
\tkzLabelPoint[above right](lzx) {$\lambda^2 x$}
|
|
||||||
%\node[red] at ($(m1)+(1.5,-0.2)$) {$m+1$};
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,15 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=3,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzInit[xmax=6,ymax=5,xmin=-5,ymin=0]
|
|
||||||
\tkzDefPoints{-2/0/A,3.5/0/B,-0.85/0/C,2/0/D,2/2/P}
|
|
||||||
\tkzDefPoints{-1/0/X, -1/5/Y}
|
|
||||||
\tkzDrawLine[add=0.1 and 0.1](X,Y)
|
|
||||||
\tkzAxeXY
|
|
||||||
|
|
||||||
\tkzDrawArc[R,line width=1pt,color=red](A,0.5 cm)(0,180)
|
|
||||||
\tkzDrawArc[R,line width=1pt](B,2.5 cm)(0,180)
|
|
||||||
\tkzDrawArc[R,line width=1pt](C,3.5 cm)(0,180)
|
|
||||||
\tkzDrawArc[R,line width=1pt](D,2.0 cm)(0,180)
|
|
||||||
\tkzDrawPoints(P)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,31 +0,0 @@
|
||||||
\pgfplotsset{
|
|
||||||
colormap={whitered}{
|
|
||||||
color(0cm)=(white);
|
|
||||||
color(1cm)=(orange!75!red)
|
|
||||||
}
|
|
||||||
}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
colormap name=whitered,
|
|
||||||
width=15cm,
|
|
||||||
view={340}{25},
|
|
||||||
enlargelimits=false,
|
|
||||||
grid=major,
|
|
||||||
domain=-2:2,
|
|
||||||
y domain=-2:2,
|
|
||||||
samples=40, %57 : TeX capacity exceeded, sorry [main memory size=3000000].
|
|
||||||
% see also http://tex.stackexchange.com/a/7954/5645
|
|
||||||
xlabel=$x$,
|
|
||||||
ylabel=$y$,
|
|
||||||
zlabel={$z$},
|
|
||||||
colorbar,
|
|
||||||
colorbar style={
|
|
||||||
at={(-0.1,0)},
|
|
||||||
anchor=south west,
|
|
||||||
height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
|
|
||||||
title={$f(x,y)$}
|
|
||||||
}
|
|
||||||
]
|
|
||||||
\addplot3[surf,draw=black] {x^2-y^2};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,13 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=3,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzInit[xmax=5,ymax=4,xmin=-1,ymin=0]
|
|
||||||
\tkzDefPoints{2/2/Z1,2/3/Z2,2/0/A}
|
|
||||||
\tkzAxeXY
|
|
||||||
|
|
||||||
\tkzDrawLine[add=2 and 1, color=orange](Z1,Z2)
|
|
||||||
\tkzDrawPoints(Z1, Z2)
|
|
||||||
\tkzLabelPoint[right](Z1){$Z_1$}
|
|
||||||
\tkzLabelPoint[right](Z2){$Z_2$}
|
|
||||||
\node[orange] at ($(A)+(0.5,0.3)$) {$\Re(Z_1)$};
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,17 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=3,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzInit[xmax=5,ymax=4,xmin=-1,ymin=0]
|
|
||||||
\tkzDefPoints{1/1/Z1,2/2/Z2,3/0/A}
|
|
||||||
\tkzAxeXY
|
|
||||||
|
|
||||||
\tkzDrawPoints(Z1, Z2)
|
|
||||||
\tkzLabelPoint[right](Z1){$Z_1$}
|
|
||||||
\tkzLabelPoint[below](Z2){$Z_2$}
|
|
||||||
\node (m) at ($(Z1)!0.5!(Z2)$) {};
|
|
||||||
\tkzDrawSegments[dashed](Z1,Z2 A,Z1 A,Z2)
|
|
||||||
|
|
||||||
\tkzDefLine[perpendicular=through m](Z1,Z2)\tkzGetPoint{c}
|
|
||||||
\tkzDrawLine[add=2 and 1,dashed,thick](m, c)
|
|
||||||
\tkzDrawArc[R,line width=1pt,color=orange](A,2.24 cm)(0,180)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,28 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/1/P, -1/-1/Q, 1/-1/R, -2/-1/links, 2/-1/rechts, -1.5/-2/helperLeft, 1.5/-2/helperRight, -0.25/1.5/helperTopLeft, 0.25/1.5/helperTopRight}
|
|
||||||
|
|
||||||
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](R,Q,P)
|
|
||||||
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](P,R,Q)
|
|
||||||
\tkzMarkAngle[arc=l,size=0.6cm,color=green,fill=green!20](Q,P,R)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](P,Q,links)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](helperLeft,Q,R)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](helperTopLeft,P,Q)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](R,P,helperTopRight)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](rechts,R,P)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.6cm,color=blue,fill=blue!20](Q,R,helperRight)
|
|
||||||
\tkzDrawLine[add=0.35 and 0.35](P,Q)
|
|
||||||
\tkzDrawLine[add=0.35 and 0.35](P,R)
|
|
||||||
\tkzDrawLine[add=0.4 and 0.4](Q,R)
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
\node at ($(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$}
|
|
||||||
\tkzDrawPoints(P, Q, R)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,24 +0,0 @@
|
||||||
\begin{tikzpicture}[scale=3]
|
|
||||||
\tkzSetUpPoint[shape=circle,size=3,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzInit[xmax=1.2,ymax=1,xmin=-1.2,ymin=0]
|
|
||||||
\pgfmathsetmacro{\Radius}{1}
|
|
||||||
\tkzDefPoints{2.0/1.5/Z, 0/0/O, 0/1/i}
|
|
||||||
|
|
||||||
%% Konstruktion von 1/ \overline{z} und -1/ \overline{z}
|
|
||||||
\tkzTangent[from with R = Z,/tikz/overlay](O,\Radius cm) \tkzGetPoints{T1}{T2}
|
|
||||||
\tkzInterLL(T1,T2)(O,Z) \tkzGetPoint{dZ}
|
|
||||||
%%
|
|
||||||
|
|
||||||
\tkzDrawArc[R,line width=1pt,color=orange](O,\Radius cm)(0,180)
|
|
||||||
\tkzMarkAngle[size=1mm](Z,dZ,T1)
|
|
||||||
\tkzLabelAngle[pos=0.06](Z,dZ,T1){$\cdot$}
|
|
||||||
\tkzAxeXY
|
|
||||||
|
|
||||||
\tkzDrawPoints(Z, dZ, T1)
|
|
||||||
\tkzLabelPoint[above left](Z){$z = r \cdot e^{\iu \varphi}$}
|
|
||||||
\tkzLabelPoint[below right](dZ){$\frac{1}{\overline{z}} = \frac{1}{r} \cdot e^{\iu \varphi}$}
|
|
||||||
\tkzDrawSegments[dashed](O,Z)
|
|
||||||
\tkzDrawLine[dashed, add=0 and 0.5](Z,T1)
|
|
||||||
\tkzDrawSegments[dashed](T1,dZ)
|
|
||||||
\end{tikzpicture}
|
|
||||||
Binary file not shown.
|
|
@ -1,42 +0,0 @@
|
||||||
% The following answers were used to create this image:
|
|
||||||
% - http://tex.stackexchange.com/a/45824/5645 - Grid
|
|
||||||
% - http://tex.stackexchange.com/a/373/5645 - Torus
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
|
|
||||||
\newcommand*{\xMin}{0}%
|
|
||||||
\newcommand*{\xMax}{6}%
|
|
||||||
\newcommand*{\yMin}{0}%
|
|
||||||
\newcommand*{\yMax}{6}%
|
|
||||||
|
|
||||||
\draw (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
|
|
||||||
\draw[xscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
|
|
||||||
\draw[rotate=180] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
|
|
||||||
\draw[yscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
|
|
||||||
|
|
||||||
\draw (-2,.2) .. controls (-1.5,-0.3) and (-1,-0.5) .. (0,-.5) .. controls (1,-0.5) and (1.5,-0.3) .. (2,0.2);
|
|
||||||
|
|
||||||
\draw (-1.75,0) .. controls (-1.5,0.3) and (-1,0.5) .. (0,.5) .. controls (1,0.5) and (1.5,0.3) .. (1.75,0);
|
|
||||||
|
|
||||||
\begin{scope}[shift={(5,-3)}]
|
|
||||||
\foreach \i in {\xMin,...,\xMax} {
|
|
||||||
\draw [very thin,gray] (\i,\yMin) -- (\i,\yMax) node [below] at (\i,\yMin) {$\i$};
|
|
||||||
}
|
|
||||||
\foreach \i in {\yMin,...,\yMax} {
|
|
||||||
\draw [very thin,gray] (\xMin,\i) -- (\xMax,\i) node [left] at (\xMin,\i) {$\i$};
|
|
||||||
}
|
|
||||||
|
|
||||||
\node (P)[point,red] at (1.2,2.2) {};
|
|
||||||
\node (Q)[point,red] at (1.2,1.6) {};
|
|
||||||
\draw[ultra thick, red] (P) -- (Q);
|
|
||||||
|
|
||||||
\begin{scope}[shift={(2,1)}]
|
|
||||||
\node (P)[point,red] at (1.2,2.2) {};
|
|
||||||
\node (Q)[point,red] at (1.2,1.6) {};
|
|
||||||
\draw[ultra thick, red] (P) -- (Q);
|
|
||||||
\end{scope}
|
|
||||||
\draw (-1, -0.5) node[below] {$T \xrightarrow{\text{Liften}} \mathbb{R}^2 / \mathbb{Z}^2$};
|
|
||||||
\draw[red,dashed] (-5,1.5) ellipse (0.5cm and 1cm);
|
|
||||||
|
|
||||||
\draw[red] (-5,2.5) arc (-270:-90:0.5 and 1) ;
|
|
||||||
\end{scope}
|
|
||||||
\end{tikzpicture}
|
|
||||||
Binary file not shown.
|
|
@ -1,67 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
axis x line=middle,
|
|
||||||
axis y line=middle,
|
|
||||||
%width=9cm,
|
|
||||||
%height=4.5cm,
|
|
||||||
xmin=-1, % start the diagram at this x-coordinate
|
|
||||||
xmax= 5, % end the diagram at this x-coordinate
|
|
||||||
ymin=-1, % start the diagram at this y-coordinate
|
|
||||||
ymax= 5, % end the diagram at this y-coordinate
|
|
||||||
xlabel=$Y$,
|
|
||||||
ylabel=$X$,
|
|
||||||
ticks=none,
|
|
||||||
enlargelimits=true,
|
|
||||||
after end axis/.code={
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=15pt}, orange] (axis cs:0,3.6) -- (axis cs:0,2.5) node [midway,left=20pt,orange] {$V_{x,y}$};
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=12pt}, green] (axis cs:1.5,0) -- (axis cs:2.5,0) node [midway,below=16pt,green] {$U_{x,y}$};
|
|
||||||
}]
|
|
||||||
|
|
||||||
\addplot[mark=none, red, smooth cycle, thick, fill=red!30] coordinates {(1,1) (2,0.5) (3,1.5) (3,2) (3.5,3) (3.2, 5) (2.2, 4.7) (1.5, 4.2) (1.1, 3.9) (0.9, 2.5)};
|
|
||||||
\node[red] at (axis cs:4,4) [anchor=south] {$W_i$};
|
|
||||||
|
|
||||||
% Draw help lines
|
|
||||||
%\addplot[dashed] coordinates {(1.5,0) (1.5,3.6)};
|
|
||||||
%\addplot[dashed] coordinates {(2.5,0) (2.5,3.6)};
|
|
||||||
%\addplot[dashed] coordinates {(0,2.5) (2.5,2.5)};
|
|
||||||
%\addplot[dashed] coordinates {(0,3.6) (2.5,3.6)};
|
|
||||||
|
|
||||||
% Draw solid square
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(2.5,2.5) (2.5,3.6)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(2.5,3.6) (1.5,3.6)};
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(1.5,3.6) (1.5,2.5)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(1.5,2.5) (2.5,2.5)};
|
|
||||||
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(3.0,1.5) (3.0,2.6)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(3.0,2.6) (1.1,2.6)};
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(1.1,1.5) (1.1,2.6)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(1.1,1.5) (3.0,1.5)};
|
|
||||||
|
|
||||||
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(3.0,1.5) (3.0,2.6)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(3.0,2.6) (1.1,2.6)};
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(1.1,1.5) (1.1,2.6)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(1.1,1.5) (3.0,1.5)};
|
|
||||||
|
|
||||||
\addplot[mark=none, blue, thick, dashed] coordinates {((1.8,0) (1.8,5)};
|
|
||||||
\addplot[mark=none, blue, thick, dashed] coordinates {((2.2,0) (2.2,5)};
|
|
||||||
|
|
||||||
% Draw x and annotation
|
|
||||||
\node at (axis cs:2,3) [anchor=-90] {$x$};
|
|
||||||
\addplot[mark=*] coordinates {(2,3)};
|
|
||||||
|
|
||||||
% Draw ticks of help lines
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(1.5, -0.1) (1.5,0.1)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(2.5, -0.1) (2.5,0.1)};
|
|
||||||
\addplot[mark=none, green, thick] coordinates {(1.5, 0) (2.5,0)};
|
|
||||||
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(-0.1, 2.5) (0.1,2.5)};
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(-0.1, 3.6) (0.1,3.6)};
|
|
||||||
\addplot[mark=none, orange, thick] coordinates {(0, 2.5) (0,3.6)};
|
|
||||||
|
|
||||||
% Draw axis text
|
|
||||||
\node at (axis cs:0,3) [anchor=east] {$y$};
|
|
||||||
\node at (axis cs:2,0) [anchor=north] {$x$};
|
|
||||||
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,47 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
axis x line=middle,
|
|
||||||
axis y line=middle,
|
|
||||||
%width=9cm,
|
|
||||||
%height=4.5cm,
|
|
||||||
xmin=-1, % start the diagram at this x-coordinate
|
|
||||||
xmax= 5, % end the diagram at this x-coordinate
|
|
||||||
ymin=-1, % start the diagram at this y-coordinate
|
|
||||||
ymax= 5, % end the diagram at this y-coordinate
|
|
||||||
xlabel=$X_1$,
|
|
||||||
ylabel=$X_2$,
|
|
||||||
ticks=none,
|
|
||||||
enlargelimits=true,
|
|
||||||
after end axis/.code={
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=15pt}] (axis cs:0,3.6) -- (axis cs:0,2.5) node [midway,left=20pt] {$U_2$};
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=12pt}] (axis cs:1.5,0) -- (axis cs:2.5,0) node [midway,below=16pt] {$U_1$};
|
|
||||||
}]
|
|
||||||
|
|
||||||
\addplot[mark=none, orange, smooth cycle, thick, fill=orange!30] coordinates {(1,1) (2,0.5) (3,1.5) (3,2) (3.5,3) (3.2, 5) (2.2, 4.7) (1.5, 4.2) (1.1, 3.9) (0.9, 2.5)};
|
|
||||||
\node[orange] at (axis cs:4,4) [anchor=south] {$U$};
|
|
||||||
|
|
||||||
% Draw help lines
|
|
||||||
\addplot[dashed] coordinates {(1.5,0) (1.5,3.6)};
|
|
||||||
\addplot[dashed] coordinates {(2.5,0) (2.5,3.6)};
|
|
||||||
\addplot[dashed] coordinates {(0,2.5) (2.5,2.5)};
|
|
||||||
\addplot[dashed] coordinates {(0,3.6) (2.5,3.6)};
|
|
||||||
|
|
||||||
% Draw solid square
|
|
||||||
\addplot[mark=none, red, thick, fill=red!30] coordinates {(2.5,2.5) (2.5,3.6) (1.5,3.6) (1.5,2.5) (2.5,2.5)};
|
|
||||||
|
|
||||||
% Draw x and annotation
|
|
||||||
\node[blue] at (axis cs:2,3) [anchor=south west] {$x$};
|
|
||||||
\addplot[mark=*, blue] coordinates {(2,3)};
|
|
||||||
|
|
||||||
% Draw ticks of help lines
|
|
||||||
\addplot[mark=none, red, thick] coordinates {(1.5, -0.1) (1.5,0.1)};
|
|
||||||
\addplot[mark=none, red, thick] coordinates {(2.5, -0.1) (2.5,0.1)};
|
|
||||||
\addplot[mark=none, red, thick] coordinates {(-0.1, 2.5) (0.1,2.5)};
|
|
||||||
\addplot[mark=none, red, thick] coordinates {(-0.1, 3.6) (0.1,3.6)};
|
|
||||||
|
|
||||||
% Draw axis text
|
|
||||||
\node[blue] at (axis cs:0,3) [anchor=east] {$x_2$};
|
|
||||||
\node[blue] at (axis cs:2,0) [anchor=north] {$x_1$};
|
|
||||||
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,37 +0,0 @@
|
||||||
\tikzset{
|
|
||||||
point/.style={
|
|
||||||
thick,
|
|
||||||
draw=gray,
|
|
||||||
cross out,
|
|
||||||
inner sep=0pt,
|
|
||||||
minimum width=4pt,
|
|
||||||
minimum height=4pt,
|
|
||||||
},
|
|
||||||
}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
|
|
||||||
\draw[->] (-1.5,0) -- (5.5,0) node [below] {$\mathbb{R}$};
|
|
||||||
|
|
||||||
\foreach \x in {-1,...,5}
|
|
||||||
\draw (\x,0.1) -- (\x,-0.1) node [below] {\x};
|
|
||||||
|
|
||||||
\foreach \x in {-1,...,4} {
|
|
||||||
\draw[red] (\x+0.6,0.01) -- (\x+0.6,-0.14) node [below] {};
|
|
||||||
\draw[red] (\x+1.2,0.01) -- (\x+1.2,-0.14) node [below] {};
|
|
||||||
\draw[red] (\x+0.6,-0.07) -- (\x+1.2,-0.07) node [below] {};
|
|
||||||
}
|
|
||||||
|
|
||||||
\begin{scope}[shift={(0,-2)}]
|
|
||||||
\draw[thick] (0cm,0cm) circle(1cm);
|
|
||||||
\draw[thick, red] ([shift={(216:1cm)}]-0.0,0) arc (216:-72:1cm);
|
|
||||||
\draw (0:1cm) node[point, label=right:{$0$}] {};
|
|
||||||
\path node[point, blue, label={[blue,above]{$\overline{a}$}}] (posU) at (-252:1cm) {};
|
|
||||||
\path node[label={[red,left]{$U$}}] at (30:1cm) {};
|
|
||||||
\end{scope}
|
|
||||||
\draw (3.7cm,0cm) node[point, blue, label={[blue,above]{$a$}}] (posA) {};
|
|
||||||
\draw (0.7cm,0cm) node[point, blue, label={[blue,above]{$\pi^{-1}(u)$}}] {};
|
|
||||||
\draw[dashed, blue, thick] plot [smooth] coordinates{(posU) (0.2,-0.8) (2.5,-1) (posA)};
|
|
||||||
|
|
||||||
\draw[blue, dashed, thick] (3.7cm,0cm) arc (0:180:1.5 and 0.5);
|
|
||||||
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,27 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
axis x line=middle,
|
|
||||||
axis y line=middle,
|
|
||||||
xmin=-1.5, % start the diagram at this x-coordinate
|
|
||||||
xmax= 1.5, % end the diagram at this x-coordinate
|
|
||||||
ymin=-1.5, % start the diagram at this y-coordinate
|
|
||||||
ymax= 1.5, % end the diagram at this y-coordinate
|
|
||||||
ticks=none,
|
|
||||||
enlargelimits=true,
|
|
||||||
after end axis/.code={
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=2pt}] (axis cs:0,1) -- (axis cs:-1,1) node [midway,above=5pt] {$r$};
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=2pt}] (axis cs:1,1) -- (axis cs:0,1) node [midway,above=5pt] {$r$};
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=2pt}] (axis cs:1,0) -- (axis cs:1,1) node [midway,right=5pt] {$r$};
|
|
||||||
\draw [decorate,decoration={brace,mirror,raise=2pt}] (axis cs:1,-1) -- (axis cs:1,0) node [midway,right=5pt] {$r$};
|
|
||||||
}]
|
|
||||||
|
|
||||||
|
|
||||||
% Draw solid square
|
|
||||||
\addplot[mark=none, thick] coordinates {(-1,-1) (1,-1) (1,1) (-1,1) (-1,-1)};
|
|
||||||
\addplot[mark=*] coordinates {(0,0)};
|
|
||||||
|
|
||||||
% Draw axis text
|
|
||||||
\node at (axis cs:-1,0.5) [anchor=east] {$\mathfrak{B}_r(0) = $};
|
|
||||||
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,21 +0,0 @@
|
||||||
\tdplotsetmaincoords{110}{50}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
[tdplot_main_coords,
|
|
||||||
cube/.style={very thick,black},
|
|
||||||
grid/.style={very thin,gray},
|
|
||||||
axis/.style={->,blue,thick}]
|
|
||||||
|
|
||||||
%draw a grid in the x-y plane
|
|
||||||
\foreach \x in {-0.5,0,...,2.5}
|
|
||||||
\foreach \y in {-0.5,0,...,2.5}
|
|
||||||
{
|
|
||||||
\draw[grid] (\x,-0.5) -- (\x,2.5);
|
|
||||||
\draw[grid] (-0.5,\y) -- (2.5,\y);
|
|
||||||
}
|
|
||||||
|
|
||||||
|
|
||||||
%draw the axes
|
|
||||||
\draw[axis] (-1,0,0) -- (3,0,0) node[anchor=west]{$y$};
|
|
||||||
\draw[axis] (0,-1,0) -- (0,3,0) node[anchor=west]{$x$};
|
|
||||||
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,7 +0,0 @@
|
||||||
\begin{tikzpicture}[scale=.5, z={(.707,.3)}]
|
|
||||||
\draw (2,3,2) -- (0,0,0) -- (4,0,0) -- (4,0,4) -- (2,3,2)
|
|
||||||
-- (4,0,0);
|
|
||||||
\draw[color=gray, style=dashed] (2,3,2) -- (0,0,4)
|
|
||||||
-- (0,0,0);
|
|
||||||
\draw[color=gray, style=dashed] (0,0,4) -- (4,0,4);
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,25 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/A, 4/0/B, 4/4/C, 0/4/D, 1/0/W, 4/1/X, 3/4/Y, 0/3/Z}
|
|
||||||
\tkzDrawPolygon(A,B,C,D)
|
|
||||||
\tkzDrawPolygon(W,X,Y,Z)
|
|
||||||
\tkzLabelSegment[below](A,W){$b$}
|
|
||||||
\tkzLabelSegment[below](W,B){$a$}
|
|
||||||
\tkzLabelSegment[right](B,X){$b$}
|
|
||||||
\tkzLabelSegment[right](X,C){$a$}
|
|
||||||
\tkzLabelSegment[above](C,Y){$b$}
|
|
||||||
\tkzLabelSegment[above](Y,D){$a$}
|
|
||||||
\tkzLabelSegment[left](D,Z){$b$}
|
|
||||||
\tkzLabelSegment[left](Z,A){$a$}
|
|
||||||
\tkzLabelAngle[pos=-0.24](D,C,B){$\cdot$}
|
|
||||||
\tkzMarkAngle[arc=l,size=0.4cm](D,C,B)
|
|
||||||
\tkzLabelAngle[pos=0.24](C,B,A){$\cdot$}
|
|
||||||
\tkzMarkAngle[arc=l,size=0.4cm](C,B,A)
|
|
||||||
\tkzLabelAngle[pos=0.24](B,A,D){$\cdot$}
|
|
||||||
\tkzMarkAngle[arc=l,size=0.4cm](B,A,D)
|
|
||||||
\tkzLabelAngle[pos=0.24](A,D,C){$\cdot$}
|
|
||||||
\tkzMarkAngle[arc=l,size=0.4cm](A,D,C)
|
|
||||||
\tkzLabelAngle[pos=0.24](W,Z,Y){$\gamma$}
|
|
||||||
\tkzMarkAngle[arc=l,size=0.4cm](W,Z,Y)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,16 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/A, 5/0/B}
|
|
||||||
\tkzInterCC[R,/tikz/overlay](A,4cm)(B,3cm) \tkzGetPoints{C}{D}
|
|
||||||
\tkzDrawPolygon(A,B,C)
|
|
||||||
\tkzDrawPoints(A,B,C)
|
|
||||||
\tkzLabelSegment[below](A,B){$c$}
|
|
||||||
\tkzLabelSegment[above left](A,C){$b$}
|
|
||||||
\tkzLabelSegment[above right](B,C){$a$}
|
|
||||||
\tkzLabelPoint[below](A){$A$}
|
|
||||||
\tkzLabelPoint[below](B){$B$}
|
|
||||||
\tkzLabelPoint[above](C){$C$}
|
|
||||||
\tkzLabelAngle[pos=0.24](A,C,B){$\cdot$}
|
|
||||||
\tkzMarkAngle[arc=l,size=0.4cm](A,C,B)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,9 +0,0 @@
|
||||||
\begin{tikzpicture}[thick]
|
|
||||||
\draw (-1,-1) -- (1,-1) -- (1,1) -- (-1,1) -- cycle;
|
|
||||||
\draw (0cm,0cm) circle(0.9cm);
|
|
||||||
|
|
||||||
\begin{scope}[scale=1.7]
|
|
||||||
\draw (-1,-1) -- (1,-1) -- (1,1) -- (-1,1) -- cycle;
|
|
||||||
\draw (0cm,0cm) circle(0.9cm);
|
|
||||||
\end{scope}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,9 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/A, 4/0/B, 4/2/C, 0/2/D, 3/2/E, 4/1/F, 1/0/G, 1/-1/H}
|
|
||||||
\tkzDrawPolygon[fill=black!20](A,B,C,D)
|
|
||||||
\tkzDrawPolygon[orange,fill=orange!20](E,C,F)
|
|
||||||
\tkzDrawPolygon[orange,fill=orange!20](A,G,H)
|
|
||||||
\tkzDrawPoints(A,B,C,D)
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,9 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/A, 4/0/B, 4/2/C, 0/2/D}
|
|
||||||
\tkzDrawPolygon(A,B,C,D)
|
|
||||||
\tkzDrawPolygon[pattern=north east lines](A,B,C)
|
|
||||||
\tkzDrawPolygon[pattern=north west lines](C,D,A)
|
|
||||||
\tkzDrawPoints(A,B,C,D)
|
|
||||||
\end{tikzpicture}
|
|
||||||
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|
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|
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|
Before Width: | Height: | Size: 72 KiB |
|
|
@ -1,9 +0,0 @@
|
||||||
% Source: http://tex.stackexchange.com/a/42865/5645
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\draw (-1,0) arc (180:360:1cm and 0.5cm);
|
|
||||||
\draw[dashed] (-1,0) arc (180:0:1cm and 0.5cm);
|
|
||||||
\draw (0,1) arc (90:270:0.5cm and 1cm);
|
|
||||||
\draw[dashed] (0,1) arc (90:-90:0.5cm and 1cm);
|
|
||||||
\draw (0,0) circle (1cm);
|
|
||||||
\shade[ball color=blue!10!white,opacity=0.20] (0,0) circle (1cm);
|
|
||||||
\end{tikzpicture}
|
|
||||||
Binary file not shown.
|
|
@ -1,21 +0,0 @@
|
||||||
\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}
|
|
||||||
|
|
@ -1,41 +0,0 @@
|
||||||
\tikzset{
|
|
||||||
point/.style={
|
|
||||||
thick,
|
|
||||||
draw=gray,
|
|
||||||
cross out,
|
|
||||||
inner sep=0pt,
|
|
||||||
minimum width=4pt,
|
|
||||||
minimum height=4pt,
|
|
||||||
},
|
|
||||||
}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
legend pos=south east,
|
|
||||||
axis x line=middle,
|
|
||||||
axis y line=middle,
|
|
||||||
%grid = major,
|
|
||||||
width=12cm,
|
|
||||||
height=8cm,
|
|
||||||
%grid style={dashed, gray!30},
|
|
||||||
xmin=-4, % start the diagram at this x-coordinate
|
|
||||||
xmax= 8, % end the diagram at this x-coordinate
|
|
||||||
ymin=-4, % start the diagram at this y-coordinate
|
|
||||||
ymax= 4, % end the diagram at this y-coordinate
|
|
||||||
axis background/.style={fill=white},
|
|
||||||
%xticklabels={-2,-1.6,...,2},
|
|
||||||
%yticklabels={-8,-7,...,8},
|
|
||||||
%tick align=outside,
|
|
||||||
enlargelimits=true,
|
|
||||||
tension=0.08]
|
|
||||||
% plot the stirling-formulae
|
|
||||||
\addplot[domain=-4:8, red, thick,samples=500] {0.5*x};
|
|
||||||
\addplot[domain=-2:2, red, thick,samples=500] {2*x};
|
|
||||||
\addplot[domain=-4:4, red, thick,samples=500] {x};
|
|
||||||
\addplot[domain=-4:8, red, thick,samples=500] {-0.5*x};
|
|
||||||
\addplot[color=red,only marks,mark=o]
|
|
||||||
plot coordinates {
|
|
||||||
(1.5,3)
|
|
||||||
(1.5,1.5)
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,17 +0,0 @@
|
||||||
\pgfplotsset{
|
|
||||||
colormap={whitered}{
|
|
||||||
color(0cm)=(white);
|
|
||||||
color(1cm)=(orange!75!red)
|
|
||||||
}
|
|
||||||
%colormap={color}{color(0cm)=(white); color(1cm)=(blue)}
|
|
||||||
}
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[view={60}{30}]
|
|
||||||
\addplot3[surf,
|
|
||||||
samples=50,
|
|
||||||
domain=1:2,y domain=0:2*pi,
|
|
||||||
z buffer=sort]
|
|
||||||
%({(2 + tan(deg(y)))*cos((deg(x)))}, {(2 + cos(x)) * sin(x)}, {x});
|
|
||||||
({x * cos(deg(y))}, {x * sin(deg(y))}, {1/x});
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
Binary file not shown.
|
|
@ -1,5 +0,0 @@
|
||||||
\begin{tikzpicture}[thick]
|
|
||||||
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
|
|
||||||
\draw[fill=orange!20] (-2,0) -- (-1,0.5) -- (0,2) -- (1,0.5) -- (2,0) -- (1,-0.5) -- (0,-2) -- (-1,-0.5) -- cycle;
|
|
||||||
\node (a)[point,label=$x$] at (0,0) {};
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,97 +0,0 @@
|
||||||
%% helper macros
|
|
||||||
\begin{tikzpicture} % CENT
|
|
||||||
\newcommand\pgfmathsinandcos[3]{%
|
|
||||||
\pgfmathsetmacro#1{sin(#3)}%
|
|
||||||
\pgfmathsetmacro#2{cos(#3)}%
|
|
||||||
}
|
|
||||||
\newcommand\LongitudePlane[3][current plane]{%
|
|
||||||
\pgfmathsinandcos\sinEl\cosEl{#2} % elevation
|
|
||||||
\pgfmathsinandcos\sint\cost{#3} % azimuth
|
|
||||||
\tikzset{#1/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
|
|
||||||
}
|
|
||||||
\newcommand\LatitudePlane[3][current plane]{%
|
|
||||||
\pgfmathsinandcos\sinEl\cosEl{#2} % elevation
|
|
||||||
\pgfmathsinandcos\sint\cost{#3} % latitude
|
|
||||||
\pgfmathsetmacro\yshift{\cosEl*\sint}
|
|
||||||
\tikzset{#1/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
|
|
||||||
}
|
|
||||||
\newcommand\DrawLongitudeCircle[2][1]{
|
|
||||||
\LongitudePlane{\angEl}{#2}
|
|
||||||
\tikzset{current plane/.prefix style={scale=#1}}
|
|
||||||
% angle of "visibility"
|
|
||||||
\pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
|
|
||||||
\draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
|
|
||||||
\draw[current plane,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
|
|
||||||
}
|
|
||||||
\newcommand\DrawLatitudeCircle[2][1]{
|
|
||||||
\LatitudePlane{\angEl}{#2}
|
|
||||||
\tikzset{current plane/.prefix style={scale=#1}}
|
|
||||||
\pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
|
|
||||||
% angle of "visibility"
|
|
||||||
\pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
|
|
||||||
\draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
|
|
||||||
\draw[current plane,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
|
|
||||||
}
|
|
||||||
|
|
||||||
\tikzset{%
|
|
||||||
>=latex, % option for nice arrows
|
|
||||||
inner sep=0pt,%
|
|
||||||
outer sep=2pt,%
|
|
||||||
mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
|
|
||||||
fill=black,circle}%
|
|
||||||
}
|
|
||||||
%% some definitions
|
|
||||||
|
|
||||||
\def\R{2.5} % sphere radius
|
|
||||||
\def\angEl{35} % elevation angle
|
|
||||||
\def\angAz{-105} % azimuth angle
|
|
||||||
\def\angPhi{-40} % longitude of point P
|
|
||||||
\def\angBeta{19} % latitude of point P
|
|
||||||
|
|
||||||
%% working planes
|
|
||||||
|
|
||||||
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
|
|
||||||
\tikzset{xyplane/.estyle={cm={cos(\angAz),sin(\angAz)*sin(\angEl),-sin(\angAz),
|
|
||||||
cos(\angAz)*sin(\angEl),(0,-\H)}}}
|
|
||||||
\LongitudePlane[xzplane]{\angEl}{\angAz}
|
|
||||||
\LongitudePlane[pzplane]{\angEl}{\angPhi}
|
|
||||||
\LatitudePlane[equator]{\angEl}{0}
|
|
||||||
|
|
||||||
%% draw xyplane and sphere
|
|
||||||
|
|
||||||
\draw[xyplane] (-2*\R,-2*\R) rectangle (2.2*\R,2.8*\R);
|
|
||||||
\fill[ball color=white] (0,0) circle (\R); % 3D lighting effect
|
|
||||||
\draw (0,0) circle (\R);
|
|
||||||
|
|
||||||
%% characteristic points
|
|
||||||
|
|
||||||
\coordinate (O) at (0,0);
|
|
||||||
\coordinate[mark coordinate] (N) at (0,\H);
|
|
||||||
\coordinate[mark coordinate] (S) at (0,-\H);
|
|
||||||
\path[pzplane] (\angBeta:\R) coordinate[mark coordinate] (P);
|
|
||||||
\path[pzplane] (\R,0) coordinate (PE);
|
|
||||||
\path[xzplane] (\R,0) coordinate (XE);
|
|
||||||
\path (PE) ++(0,-\H) coordinate (Paux); % to aid Phat calculation
|
|
||||||
\coordinate[mark coordinate] (Phat) at (intersection cs: first line={(N)--(P)},
|
|
||||||
second line={(S)--(Paux)});
|
|
||||||
|
|
||||||
%% draw meridians and latitude circles
|
|
||||||
|
|
||||||
\DrawLatitudeCircle[\R]{0} % equator
|
|
||||||
\DrawLongitudeCircle[\R]{\angAz} % xzplane
|
|
||||||
\DrawLongitudeCircle[\R]{\angAz+90} % yzplane
|
|
||||||
\DrawLongitudeCircle[\R]{\angPhi} % pzplane
|
|
||||||
|
|
||||||
%% draw xyz coordinate system
|
|
||||||
|
|
||||||
\draw[xyplane,<->] (1.8*\R,0) node[below] {$x$} -- (0,0) -- (0,2.4*\R)
|
|
||||||
node[right] {$y$};
|
|
||||||
\draw[->] (0,-\H) -- (0,1.6*\R) node[above] {$z$};
|
|
||||||
|
|
||||||
%% draw lines and put labels
|
|
||||||
|
|
||||||
\draw[blue,dashed] (P) -- (N) +(0.3ex,0.6ex) node[above left,black] {$\mathbf{N}$};
|
|
||||||
\draw[blue] (P) -- (Phat) node[above right,black] {$\mathbf{\hat{P}}$};
|
|
||||||
\path (S) +(0.4ex,-0.4ex) node[below] {$\mathbf{0}$};
|
|
||||||
\draw (P) node[above right] {$\mathbf{P}$};
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,21 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]
|
|
||||||
\tkzSetUpLine[line width=1]
|
|
||||||
\tkzDefPoints{0/0/P, 1/0/helperRight, 1/1/helperTopRight, -1/1/helperTopLeft, -1/0/helperLeft, -1/-0.3/helperBottomLeft}
|
|
||||||
|
|
||||||
\tkzMarkAngle[arc=l,size=0.8cm,color=green,fill=green!20](helperRight,P,helperTopRight)
|
|
||||||
\tkzMarkAngle[arc=ll,size=0.8cm,color=blue,fill=blue!20](helperTopRight,P,helperTopLeft)
|
|
||||||
\tkzMarkAngle[arc=lll,size=0.8cm,color=red,fill=red!20](helperTopLeft,P,helperBottomLeft)
|
|
||||||
\path[draw] ++(25:.4) node[rotate=0] {$\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$}
|
|
||||||
\end{tikzpicture}
|
|
||||||
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|
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|
Before Width: | Height: | Size: 171 KiB |
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|
Before Width: | Height: | Size: 65 KiB |
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|
Before Width: | Height: | Size: 89 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 64 KiB |
|
|
@ -1,12 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
|
|
||||||
\node (Pleft) at (0,0) {};
|
|
||||||
\node (P)[point,label=90:$P$] at (2,0) {};
|
|
||||||
\node (R)[point,label=90:$R$] at (4,0) {};
|
|
||||||
\node (Rright) at (6,0) {};
|
|
||||||
\draw[dashed,very thick] (Pleft) -- (P);
|
|
||||||
\draw[dotted,very thick] (P) -- (R) -- (Rright);
|
|
||||||
\draw [thick,decoration={brace,mirror,raise=0.2cm},decorate] (Pleft) -- (P) node [pos=0.5,anchor=north,yshift=-0.25cm] {$PR^-$};
|
|
||||||
\draw [thick,decoration={brace,mirror,raise=0.2cm},decorate] (P) -- (R) node [pos=0.5,anchor=north,yshift=-0.25cm] {$\overline{PR}$};
|
|
||||||
\draw [thick,decoration={brace,mirror,raise=0.8cm},decorate] (P) -- (Rright) node [pos=0.5,anchor=north,yshift=-0.85cm] {$PR^+$};
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,10 +0,0 @@
|
||||||
\begin{tikzpicture}[thick]
|
|
||||||
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
|
|
||||||
\node[blue] at (4.5, 1.7) {$a$};
|
|
||||||
\node[purple] at (7.5, 1.7) {$b$};
|
|
||||||
\begin{scope}[xshift=5cm, yshift=1cm]
|
|
||||||
\draw[blue,->] ( 0,0)+(150:0.7cm) arc (150:510:0.7cm);
|
|
||||||
\draw[purple,<-] (1.4,0)+( 40:0.7cm) arc (40:400:0.7cm);
|
|
||||||
\node (z)[point,label={[label distance=0.2cm]-90:$x$}] at (0.7,0) {};
|
|
||||||
\end{scope}
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,10 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
|
|
||||||
\node (a)[point] at (0.4,0) {};
|
|
||||||
\node (b)[point] at (1,1) {};
|
|
||||||
\node (c)[point] at (2,1) {};
|
|
||||||
\node (d)[point] at (2.6,0) {};
|
|
||||||
\node (e)[point] at (2,-1) {};
|
|
||||||
\node (f)[point] at (1,-1) {};
|
|
||||||
\draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- (e.center) -- (f.center) -- cycle;
|
|
||||||
\end{tikzpicture}
|
|
||||||
|
|
@ -1,33 +0,0 @@
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
axis x line=none,
|
|
||||||
axis y line=none,
|
|
||||||
%width=9cm,
|
|
||||||
%height=4.5cm,
|
|
||||||
xmin= 0, % start the diagram at this x-coordinate
|
|
||||||
xmax= 5, % end the diagram at this x-coordinate
|
|
||||||
ymin= 0, % start the diagram at this y-coordinate
|
|
||||||
ymax= 5, % end the diagram at this y-coordinate
|
|
||||||
xlabel=$Y$,
|
|
||||||
ylabel=$X$,
|
|
||||||
ticks=none,
|
|
||||||
enlargelimits=true]
|
|
||||||
|
|
||||||
\addplot[mark=none, red, smooth cycle, thick, fill=red!30] coordinates {(0,0) (2,0.2) (3,1.5) (3,2) (3.5,3) (3.2, 5) (2.2, 4.7) (1.5, 4.2) (1.1, 3.9) (0.2, 2.5)};
|
|
||||||
\node[red] at (axis cs:4,4) [anchor=south] {$X_i$};
|
|
||||||
|
|
||||||
% Draw solid square
|
|
||||||
\addplot[mark=none, thick] coordinates {(1.5,2.0) (2.5,2.0) (2.5,3.6) (1.5,3.6) (1.5,2.0)};
|
|
||||||
\node at (axis cs:2.7,3.2) [anchor=90] {$K$};
|
|
||||||
|
|
||||||
|
|
||||||
% Draw x and annotation
|
|
||||||
\node at (axis cs:1.8,3.2) [anchor=-90] {$x$};
|
|
||||||
\draw (axis cs:1.8,3.2) circle[radius=0.6];
|
|
||||||
\addplot[mark=*] coordinates {(1.8,3.2)};
|
|
||||||
|
|
||||||
\node at (axis cs:0.8,1.2) [anchor=-90] {$y$};
|
|
||||||
\draw (axis cs:0.8,1.2) circle[radius=0.6];
|
|
||||||
\addplot[mark=*] coordinates {(0.8,1.2)};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
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Loading…
Add table
Add a link
Reference in a new issue