# [GIF] School’s Out (Stereographic projection of random sphere paths)

Posted 8 months ago
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 School’s OutThis one is very much the same idea as Pathways: basically, a bunch of random points on the sphere undergoing two simultaneous random rotations. There are three main differences: (i) more points; (ii) all points are chosen from the positive orthant; (iii) I'm applying stereographic projection to map the points down to the plane. Of course, that means we need to define stereographic projection: Stereo[p_] := p[[;; -2]]/(1 - p[[-1]]); And then to generate a bunch of random points and two random rotation axes: spherepoints = Abs /@ Normalize /@ RandomVariate[NormalDistribution[], {100, 3}]; n = Normalize /@ RandomVariate[NormalDistribution[], {2, 3}]; And then put everything together: With[{cols = RGBColor /@ {"#fdff01", "#6cf068", "#000018"}}, Manipulate[ Graphics[ {Thickness[.006], CapForm["Round"], Line[Stereo /@ #, VertexColors -> Table[ Directive[Blend[cols[[;; 2]], t/25], Opacity[1 - t/200]], {t, 0, 200}]] & /@ Transpose[ Table[spherepoints.RotationMatrix[2 θ, n[[1]]].RotationMatrix[3 θ, n[[2]]], {θ, -s, π/4 - s, π/800}]]}, ImageSize -> 540, PlotRange -> 2, Background -> cols[[-1]]], {s, 0, 2 π}] ] If you really want to play around with the exact points and axes I used, you can unzip this file and load them from the resulting files using the following commands: Get["spherepoints.ws"]; Get["n.ws"];