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[GIF] Caught (Voronoi cells of stereographically projected pattern)

POSTED BY: Clayton Shonkwiler
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J. M.
J. M.
Posted 8 years ago

Yes, the implementation in my reply is intended for en masse application of the stereographic transformation. If I were implementing Stereo[], I would maintain both implementations, and use an argument check so that the applicable method is used.

That is: Stereo[pt_?VectorQ] := (* stuff *) would be the one-point case, corresponding to your implementation, and Stereo[pt_?MatrixQ] := (* stuff *) would be the many-point case, corresponding to my implementation.
POSTED BY: J. M.
J. M.
J. M.
Posted 8 years ago

Here are some quick tips for your future artwork:

  1. The operation RotationMatrix[?, {1, 0, 0}].# & /@ pts can be recast into a more efficient form with a little linear algebra. Briefly, if you are mapping a premultiplication by a matrix over a list of points, this is equivalent to postmultiplying the list of points by the transpose of the matrix. That is, A.# & /@ list is just the same as list.Transpose[A].

In this case, it is very fortuitous that the transpose of RotationMatrix[] takes on a very convenient form. To sum up, the compact version of this operation is pts.RotationMatrix[-?, {1, 0, 0}].

  1. You can redefine your function Stereo[] so that it acts on a list of points en masse, instead of you having to map it over each point:

    Stereo[pts_?MatrixQ] := Drop[pts, None, -1]/(1 - pts[[All, -1]])

With these considerations, we can redo the heart of your code as

Stereo[Flatten[pts, 1].RotationMatrix[-?, {1, 0, 0}]]

with Stereo[] defined for a list of points.

Anyway, nice artwork, as always! :)
POSTED BY: J. M.

Ah, that's a good point! Even after all these years, I think I still have a bit of a mental block about representations of vectors in Mathematica, but of course you're quite right that a list of vectors can be interpreted as a matrix whose rows are the given vectors.

I don't like your redefinition of Stereo[] as much: with your definition I can't apply it to a single point, only to a list of points. Of course, in this particular example I am applying to to a list of points, but this is a function that I reuse constantly, including in circumstances when I want to apply it to single points.
POSTED BY: Clayton Shonkwiler

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