File:Belt trick 1.gif

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[][][s, t]]);
F[s_, t_] = FullSimplify[][][s, 0].F2[s, t];
i = {{I, 0}, {0, -I}};
j = {{0, -1}, {1, 0}};
k = i.j;
Components[A_] := (1/2) {Tr[][i], 
   Tr[][j], Tr[][k]};
R[s_][t_] = 
  Transpose[][][][][s, t].i.ConjugateTranspose[][s, t]]]],
    FullSimplify[][][][s, t].j.ConjugateTranspose[][s, t]]]],
    FullSimplify[][][][s, t].k.ConjugateTranspose[][s, t]]]]}];
PlotRotation[R_] := 
 ParametricPlot3D[][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[][][s], {s, 0, 1, 0.02}];
Export["a.gif", a];