# [GIF] Rise (Conformal image of parallel lines)

Posted 7 months ago
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 RiseThis 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}] ]