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[GIF] Rise (Conformal image of parallel lines)

Posted 7 months ago
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Conformal image of parallel lines Conformal image of parallel lines

Rise

This is basically the same setup as Vanishing Point and :eyes:, conformally mapping an infinite strip into the region between two circles, though in this case the outer circle has infinite radius, and so is a straight line. The conformal map is ( $-i$ times) the Cayley transform:

Cayley[z_] := (z - I)/(z + I);

The circles in the animation are just the images of equally-spaced, parallel lines in the infinite strip. Here's the code for a Manipulate version of the dark animation (for the actual GIF, I changed {x, -5, 5, .01} to {x, -50, 50, .01}, which takes much longer but looks marginally better; see also this comment for the process of exporting to GIF):

With[{d = 1/60.},
 Manipulate[
  Graphics[
   {Table[
     {GrayLevel[y + 1],
      Polygon[Table[ReIm[-I Cayley[x + I y]], {x, -5, 5, .01}]]},
     {y, -1 + d - s, 0, d}]},
   PlotRange -> 1.5 {{-4, 4}, {-1, 5}}, ImageSize -> 540, 
   Background -> Black],
  {s, 0, d}]
 ]

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