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[GIF] Renewable Resource (More Möbius transformations of the sphere)

Posted 1 year ago
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Möbius transformations of the sphere

Renewable Resource

This animation shows the same family of Möbius transformations of the sphere as Inner Light: the north and south poles are fixed and circles of latitude move from north to south, achieving maximum speed at the equator: I'm just doing inverse stereographic projection of a scaling of the complex plane.

In this case I'm actually only showing the "back" half of the sphere, with one pink spotlight placed at $(0,0,-3/4)$ and pointed up and one yellow spotlight placed at $(0,0,3/4)$ and pointed down.

To produce the GIF I used m=200, but this is much too slow for a Manipulate object since the resulting graphics object has almost 100,000 little spheres. With m=10 as in the below code we only lose small bands around the two poles.

InverseStereo3D[{x_, y_}] := {2 x/(1 + x^2 + y^2), 2 y/(1 + x^2 + y^2), (1 - x^2 - y^2)/(1 + x^2 + y^2)};

With[{n = 60, d = 1/8., m = 10, viewpoint = 5 {1, 0, 0}, 
  cols = RGBColor /@ {"#F3368D", "#FFC468", "#2F2B2B"}},
 Manipulate[
  Graphics3D[
   {Sphere[#, .02] & /@
     Join[
      Flatten[
       Table[
        InverseStereo3D /@ {(r + s) {Cos[θ], Sin[θ]}, 
          1/(r - s) {Cos[θ], Sin[θ]}},
        {θ, π/2., 3 π/2, 2 π/n}, {r, 1 + d, m, d}],
       2],
      Table[
       InverseStereo3D[(1 + s) {Cos[θ], Sin[θ]}], {θ, π/2., 3 π/2, 2 π/n}]]},
   PlotRange -> Sqrt[2], Boxed -> False, ImageSize -> 540, 
   ViewPoint -> viewpoint, Background -> cols[[-1]], 
   Lighting -> {{"Spot", cols[[1]], {{0, 0, -.75}, {0, 0, 1}}, π/2}, 
      {"Spot",  cols[[2]], {{0, 0, .75}, {0, 0, -1}}, π/2}, 
     {"Ambient", cols[[-1]], viewpoint}}],
  {s, 0, d}]
 ]

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