File:Belt trick 1.gif

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Belt_trick_1.gif ‎(514 × 402 pixels, file size: 636 KB, MIME type: image/gif, looped, 102 frames, 1.0 s)
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Lua error in package.lua at line 80: module 'strict' not found.

Summary

Description

An illustration of the belt trick, showing a homotopy between a one-parameter rotation through an angle of 4π in three dimensions and the identity rotation.
Source

Created in Mathematica 9.0 (source below)
Date

2014-09-06
Author

Slawekb
Permission
(Reusing this file)

See below.
Source
F1[s_, t_] = {{E^(2 \[Pi] I t), Tan[\[Pi] s/2]}, {-Tan[\[Pi] s/2], 
   E^(-2 \[Pi] I t)}};
F2[s_, t_] = (F1[s, t]/Sqrt[Det[F1[s, t]]]);
F[s_, t_] = FullSimplify[Inverse[F2[s, 0]].F2[s, t]];
i = {{I, 0}, {0, -I}};
j = {{0, -1}, {1, 0}};
k = i.j;
Components[A_] := (1/2) {Tr[A.ConjugateTranspose[i]], 
   Tr[A.ConjugateTranspose[j]], Tr[A.ConjugateTranspose[k]]};
R[s_][t_] = 
  Transpose[{FullSimplify[
     ComplexExpand[Components[F[s, t].i.ConjugateTranspose[F[s, t]]]]],
    FullSimplify[
     ComplexExpand[Components[F[s, t].j.ConjugateTranspose[F[s, t]]]]],
    FullSimplify[
     ComplexExpand[
      Components[F[s, t].k.ConjugateTranspose[F[s, t]]]]]}];
PlotRotation[R_] := 
 ParametricPlot3D[R[t][[1]] t + R[t][[2]] s, {t, 0, 1}, {s, 0, 1/7.5}, 
  Boxed -> False, Axes -> False, PlotPoints -> 20, PlotRange -> All];
a = Table[PlotRotation[R[s]], {s, 0, 1, 0.02}];
Export["a.gif", a];

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
Lua error in package.lua at line 80: module 'strict' not found. Lua error in package.lua at line 80: module 'strict' not found.
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File history

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Date/TimeThumbnailDimensionsUserComment
current08:31, 17 January 2017Thumbnail for version as of 08:31, 17 January 2017514 × 402 (636 KB)127.0.0.1 (talk)<p>An illustration of the <a href="//commons.wikimedia.org/w/index.php?title=Belt_trick&action=edit&redlink=1" class="new" title="Belt trick (page does not exist)">belt trick</a>, showing a homotopy between a one-parameter rotation through an angle of 4π in three dimensions and the identity rotation. </p>
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