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[GIF] Dawn (Image of square grid under cosine)

Posted 4 months ago
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Image of square grid under cosine

Dawn

The image of a square grid in the complex plane under the map $z \mapsto \cos z$. To get motion I'm moving the grid diagonally up and to the right.

A few quirks in the code below:

  1. I want to show the way that the map is distorting area, so I've made the grid lines long thin polygons (initially rectangles with redundant vertices on the long sides) rather than lines, so that their images have non-uniform thickness.

  2. On my first attempt, the "vertical" gridlines in the final animation suffered from rendering artifacts, which I think basically boil down to bad choices of automatic triangulations (effectively the issue described in this StackExchange thread). I tried to fix this using @J. M.'s proposed solution from the StackExchange thread, but couldn't get it to work reliably, so I just did the obvious thing and triangulated manually: this is why you see the GraphicsComplex objects and the obnoxious Partition[Flatten[Riffle[Partition junk.

Anyway, here's a Manipulate version of the code (as usual, replace Manipulate with Table or ParallelTable and Export to a GIF file with "AnimationRepetitions" -> Infinity and "DisplayDurations" -> 1/50 to generate the final image):

With[{r = \[Pi], width = \[Pi]/160, m = 100, n = 40, 
  cols = RGBColor /@ {"#120136", "#035aa6", "#40bad5", "#fcbf1e"}},
 Manipulate[
  Graphics[{
    Table[
     {Blend[cols, 1 - 5/4 (y + s)/r],
      Polygon[Join @@ Transpose[
         Table[
            ReIm[Cos[#]] & /@ {x + I (y + s - width), -x + I (y + s + width)},
          {x, -r, r, 2 (r)/m}]]]},
     {y, 2 r/n, r - 2 r/n, 2 r/n}],
    Table[{Blend[cols, 1 - 5/4 (y + s)/r],
      Polygon[Join @@ Transpose[
         Table[
            ReIm[Cos[#]] & /@ {x + I (y + s - width), r - x + I (y + s + width)},
          {x, -r, 0, 2 (r)/m}]]]},
     {y, -2 r/n, 0, 2 r/n}],
    Table[
     GraphicsComplex[
      Join @@ Transpose[
        Table[
         ReIm[-Cos[#]] & /@ {x - width + s + I y, x + width + s + I (r - y)},
         {y, -width, r + width, (r + 2 width)/m}]],
      Polygon[
       Join[
        Partition[Flatten[Riffle[Partition[Range[m + 1], 2, 1],
           Reverse@Range[m + 3, 2 m + 2]]], 3],
        Partition[
         Flatten[Riffle[Partition[Range[m + 2, 2 m + 2], 2, 1],
           Reverse@Range[2, m + 1]]], 3]]],
      VertexColors -> Join[#, Reverse[#]] &[
       Table[Blend[cols, 1 - 5/4 i], {i, 0, 1, 1/m}]]],
     {x, -2 r/n, r, 2 r/n}]},
   PlotRange -> {{-\[Pi], \[Pi]}, {0, 2 \[Pi]}}, ImageSize -> 540, 
   Background -> cols[[1]]],
  {s, 0, 2 r/n}]
 ]
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