Example: Homotopy of paths

Published 2011-12-31 | Author: Alain Matthes

Jpgfdraw example rewritten in TikZ

Following an illustration in Singer/Thorpe: Lecture Notes in Elementary Topology and Geometry, the example has been drawn by Stefan Kottwitz using jpgfdraw, and programmed by Alain Matthes on TeX.SX .

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Homotopy of paths

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% Homotopy of paths
% Author: Alain Matthes
  \node at (0,0) {$F : I \times I \rightarrow X$};
  \node[label=below:$x_1$]  (x1) at (6,0)  {$\bullet$};
  \node[label=above:$x_0$]  (x0) at (9,4)  {$\bullet$};  
  \node  at (9.5,2)  {$\subset X$}; 
  \draw (x1.center) to [out=5,in=-90]++(2.8,1.8) to[out=90,in=-95](x0.center);
  \draw (x1.center) to [out=10,in=-110]++(2.6,2) to[out=70,in=-103](x0.center); 
  \draw (x1.center) to [out=15,in=-105](x0.center);
  \draw (x1.center) to [out=30,in=-150](x0.center);
  \draw (x1.center) to [out=45,in=-170](x0.center); 
  \draw (x1.center) to [out=50,in=-105]++(1.2,3)to [out=75,in=-172](x0.center); 
  \draw (x1.center) to [out=55,in=-100]++(1.0,3) to[out=80,in=-175](x0.center); 
  \draw (x1.center) to [out=60,in=-90]++(0.8,3) to[out=90,in=-180] (x0.center);
  \begin{scope}[every node/.style={draw, anchor=text, rectangle split,
    rectangle split parts=7,minimum width=2cm}]
    \node (R) at (2,4){ \nodepart{two} \nodepart{three}
    \nodepart{four}$I\times I$\nodepart{five}\nodepart{six}\nodepart{seven}};
  \draw[decorate,decoration={brace,mirror,raise=6pt,amplitude=10pt}, thick]
    (R.north west)--(R.south west) ;
  \draw[decorate,decoration={brace,raise=6pt,amplitude=10pt}, thick]
    (R.north east)--(R.south east); 
  \draw[->] ($(R.west)+(-20pt,0)$) to[out=-180,in=240] ++(0,2)
    to [out=60,in=120]node[above,midway]{$F(0,t_2)$}(x0) ; 
  \draw[->] ($(R.north)+(0,10pt)$) to [out=60,in=120]
    node[above,midway]{$\beta \simeq \alpha$} ++(4.5,-1) ; 
  \draw[->] ($(R.east)+(20pt,0)$)  to [out=0,in=140]
    node[right,midway]{$F(1,t_2)$}(x1) ; 
  \draw[->] ($(R.south)+(0,-20pt)$)  to [out=-85,in=-30]
    node[below,midway]{$\alpha$}++(7,0) ;    


  • #1 Traian Surtea, April 13, 2013 at 7:31 a.m.

    To improve the "learning power" of this example it would be better to use different colours for the paths $\alpha (t) = F(t,0)$ (let's say red), $\beta (t) = F(t,1)$ (let's say blue) and to draw an intermediate path let's say $\gamma (t) = F(t,1/2)$ with magenta. Congrats to both authors.

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