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

Posted 2 years ago
3376 Views
|
|
3 Total Likes
|
 Renewable ResourceThis 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}] ]