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[GIF] Back and Forth (Möbius transformations of the circle)

Posted 3 years ago
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Möbius transformations of the circle

Back and Forth

One fact I've known for a while but never really dived into is that Möbius transformations of the circle can be realized by inverse stereographic projecting to the sphere (here I'm thinking of the circle as the equator of the sphere, so inverse stereographic projection is just the identity in this case), rotating the sphere in space (say, around the south pole), and then stereographically projecting from the new "north pole" back to the circle. The animation shows what happens when you do this to 15 equally-spaced points on the circle, where the sphere is being rotated by an angle of $\pi/3$ around the axis $(\cos \psi, \sin \psi, 0)$ anchored at $(0,0,-1)$, and we let $\psi$ vary from 0 to $2\pi$.

Here's the code. First, some helper functions to tell me the center and pole of the rotated sphere:

RotatedCenter[?_, axis_] := RotationTransform[?, axis, {0, 0, -1}][{0, 0, 0}];
RotatedPole[?_, axis_] := RotatedCenter[?, axis] + {0, 0, 1};

And then a function which performs stereographic projection from any point $q$ to the plane $z=0$:

StereoPointToPlane[p_, q_] := 1/(p[[3]] - q[[3]]) {q[[1]] p[[3]] - p[[1]] q[[3]], q[[2]] p[[3]] - p[[2]] q[[3]]};

Finally, the big Manipulate for the animation:

DynamicModule[{axis, ? = ?/3., n = 15,  viewpoint = {0, 0, 2}, 
  cols = RGBColor /@ {"#FF9F1C", "#2EC4B6", "#E71D36", "#011627"}},
  axis = {Cos[?], Sin[?], 0};
         RotationTransform[?, axis, {0, 0, -1}][Append[#, 0]], 
         RotatedPole[?, axis]], 0], .075] & /@ 
   PlotRange -> 1.2, ViewPoint -> viewpoint, Boxed -> False, 
   Lighting -> 
    Append[Table[{"Point", cols[[i]], 
       Append[2 CirclePoints[3][[i]], 0]}, {i, 1, 3}], {"Ambient", cols[[-1]], viewpoint}], 
   Background -> cols[[-1]], ImageSize -> 540],
  {?, 0, 2 ?}]

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