# [GIF] Canis (Circle transformation)

Posted 3 years ago
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 CanisI've been playing around lately with stereographic projection and Möbius transformations of the circle. This is neither stereographic projection nor a Möbius transformation, but it came out of the same collection of ideas.I start with 32 equally spaced points on the equator of the sphere. Now, rotate these points around the axis $(1,0,1)$ and project back down to the equator along the meridian passing through the point: the result is the above animation (after dropping the last coordinate and treating the resulting points as living in the plane). And here's the full 3-D picture showing the rotations and the projection map; the dark points are the rotating points, the light points are the points on the equator that I'm projecting to, and the thin lines show the meridians:Source code: SphereToCircle[p_] := {Cos[#], Sin[#], 0} &[ToSphericalCoordinates[p][[3]]]; RotateAndProject[p_, ?_, w_] := SphereToCircle[RotationTransform[?, w][p]]; With[{n = 32, cols = RGBColor /@ {"#DDFEE4", "#132F2B"}}, Manipulate[ Graphics[ {PointSize[.02], cols[[1]], Point /@ Table[RotateAndProject[{Cos[?], Sin[?], 0}, t, {1, 0, 1}][[;; 2]], {?, 0, 2 ? - 2 ?/n, 2 ?/n}]}, PlotRange -> Sqrt[2], ImageSize -> 540, Background -> cols[[-1]]], {t, 0, 2 ?}] ]