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...
The sad thing about spherical models of H3 is that the infinitely dense horizon obscures the large-scale centre! I like that you cut the {4,4,4} in half, presumably for this reason. I wonder about half-space models.
ReplyDeleteYep, that was one of the main reasons. The other nice thing about cutting in half is saving on printing costs. We should do half-ball models of these as well.
ReplyDeleteWe have played some with half-space models too, but haven't printed anything yet. Culling out the large parts in a nice way will be a challenge. I'll post some of the experimental pictures in a bit.