Here is the first of a set of animated gifs showing basic isometries of hyperbolic 3-space. This one displays a loxodromic transformation. I'll post one gif a day for a bit, but the shader rendering them all is ready for your tinkering! https://www.shadertoy.com/view/MstcWr pic.twitter.com/0JsK08BqjB
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Cool idea! I've seen a version of that with the white circle in the middle filled in. I'd need to track that version down to make it work, but it probably wouldn't be terribly difficult.
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oh ha, the link you provided has the image I was thinking of :)
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