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[?], Sin[?]}, 
          1/(r - s) {Cos[?], Sin[?]}},
        {?, ?/2., 3 ?/2, 2 ?/n}, {r, 1 + d, m, d}],
       2],
      Table[
       InverseStereo3D[(1 + s) {Cos[?], Sin[?]}], {?, ?/2., 3 ?/2, 2 ?/n}]]},
   PlotRange -> Sqrt[2], Boxed -> False, ImageSize -> 540, 
   ViewPoint -> viewpoint, Background -> cols[[-1]], 
   Lighting -> {{"Spot", cols[[1]], {{0, 0, -.75}, {0, 0, 1}}, ?/2}, 
      {"Spot",  cols[[2]], {{0, 0, .75}, {0, 0, -1}}, ?/2}, 
     {"Ambient", cols[[-1]], viewpoint}}],
  {s, 0, d}]
 ]