Isometries of hyperbolic 3-space
Isometries of hyperbolic 3-space 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 Update based on comments I should have said a little about the banana shape. It is explained some in the shader comments, which I'll copy here. Also, see the comment thread below! This shader shows basic isometries (length preserving transformations) of hyperbolic 3-space in the upper half space model. The z=0 plane is the boundary plane-at-infinity. There are four classes of transformations: parabolic, elliptic, hyperbolic, and loxodromic. These may fix 1 or 2 ideal points on the boundary plane. In general, any Mobius transformation applied to the boundary plane will extend to an isometry of hyperbolic 3-space, but all can be built by composition of the basic transformati...
This is lovely.
ReplyDeleteThanks! Some software I wrote can paste a picture on various hyperbolic tilings, so if you ever get the urge to have a pic of you and/or Des on one, let me know :D
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