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\usepackage[colorinlistoftodos]{todonotes} \usepackage{pgfplots} \usepackage[arrow, matrix, curve]{xy} +\usepackage{caption} % get newlines within captions \usepackage{tikz} \usepackage{tikz-3dplot} \usepackage{tkz-fct} -\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns} +\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns,fadings,decorations.pathreplacing} \usepackage{cleveref} % has to be after hyperref, ntheorem, amsthm \usepackage{shortcuts} diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index cbf84b8..597eabd 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -459,31 +459,33 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. sodass $\pi$ stetig wird. \begin{beispiel}[Stereographische Projektion] \xindex{Projektion!stereographische} - $\mdr^n$ und $S^n \setminus \Set{P}$ sind homöomorph für - beliebiges $P \in S^n$ + $\mdr^n$ und $S^n \setminus \Set{N}$ sind homöomorph für + beliebiges $N \in S^n$ \begin{align*} S^n &= \Set{x \in \mdr^{n+1} | \|x\| = 1}\\ &= \Set{x \in \mdr^{n+1} | \sum_{i=1}^{n+1} x_i^2} \end{align*} - Sei ohne Einschränkung $P = \begin{pmatrix}0\\ \vdots\\ 1\end{pmatrix}$. + Sei ohne Einschränkung $N = \begin{pmatrix}0\\ \vdots\\ 1\end{pmatrix}$. \begin{align*} - f: &S^n \setminus \Set{P} \rightarrow \mdr^n\\ - Q &\mapsto \overline{L_Q \cap H}^\text{genau ein Punkt} + f: &S^n \setminus \Set{N} \rightarrow \mdr^n\\ + P &\mapsto \overbrace{L_P \cap H}^\text{genau ein Punkt} \end{align*} wobei $\mdr^n = H = \Set{\begin{pmatrix}x_1\\ \vdots \\ x_{n+1}\end{pmatrix} \in \mdr^{n+1} | x_{n+1} = 0}$ - und $L_Q$ die Gerade in $\mdr^{n+1}$ durch $P$ und $Q$ ist. + und $L_P$ die Gerade in $\mdr^{n+1}$ durch $N$ und $P$ ist. - \todo[inline]{Bild einer Kugel einfügen, die von einer Ebene $H$ - geschnitten wird. $P$ ist ganz oben, ein beliebiger Punkt - Q ist mit dabei und die Gerade PQ schneidet die Ebene.} + \begin{figure}[htp] + \centering + \input{figures/stereographic-projection} + \caption{Visualisierung der sphärischen Projektion\\Bildquelle: \href{http://texample.net/tikz/examples/map-projections/}{texample.net/tikz/examples/map-projections}} + \end{figure} - Sei $Q = \begin{pmatrix}x_1\\ \vdots \\ x_{n+1}\end{pmatrix}$, so - ist $x_{n+1} < 1$, also ist $L_Q$ nicht parallel zu $H$. Also - schneiden sich $L_Q$ und $H$ in genau einem Punkt. + Sei $P = \begin{pmatrix}x_1\\ \vdots \\ x_{n+1}\end{pmatrix}$, so + ist $x_{n+1} < 1$, also ist $L_P$ nicht parallel zu $H$. Also + schneiden sich $L_P$ und $H$ in genau einem Punkt $\hat{P}$. Es gilt: $f$ ist bijektiv und die Umkehrabbildung ist ebenfalls stetig. diff --git a/documents/GeoTopo/figures/stereographic-projection.tex b/documents/GeoTopo/figures/stereographic-projection.tex new file mode 100644 index 0000000..9459e21 --- /dev/null +++ b/documents/GeoTopo/figures/stereographic-projection.tex @@ -0,0 +1,97 @@ +%% 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}