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 transformations.
-- 2 fixed point cases (elliptic, hyperbolic, loxodromic) --
The surface above the plane is a Dupin cyclide, or an inversion of a cone, which touches the boundary at two fixed points. Except for elliptic (which fixes all points along a line in the space), these are the only fixed points. A given point in hyperbolic space will be confined to one cyclide (in a family of nested cyclides). Loxodromic transformations are a combination of elliptic and hyperbolic, and may be more or less twisted depending on the relative contribution of each.
-- 1 fixed point case (parabolic) --
The sphere is a horosphere that kisses the plane at the origin, which is also the center of the horosphere and the single fixed point of the transformation. A given point in hyperbolic space will be confined to one horosphere (in a family of nested horospheres). Parabolic transformations are conjugate to translations (where the fixed point is at infinity).
Pretty! But what shape is this two-pointed blob, and why?
ReplyDeleteDear John -
ReplyDeleteIn hyperbolic space the blob N = N(\gamma) is an R-neighbourhood of bi-infinite geodesic \gamma. Therefore N is invariant under all isometries of hyperbolic space that fix \gamma. The animation shows a (very pretty!) one-parameter family from that group of isometries.
In euclidean space, the blob N is (apparently!) one half of a "symmetric horn cyclide". It is obtained by spherically inverting a cone. See here:
https://en.wikipedia.org/wiki/Dupin_cyclide#/media/File:Zyklide-iv-keg1.svg
This is not really a surprise. For, think about what N looks like when \gamma is a vertical geodesic... and recall the classification of isometries of the upper half space model...
best,
saul
This article here:
ReplyDeletehttps://en.wikipedia.org/wiki/Dupin_cyclide
Says:
> In Maxime Bôcher's 1891 dissertation, Ueber die Reihenentwickelungen der Potentialtheorie, it was shown that the Laplace equation in three variables can be solved using separation of variables in 17 conformally distinct quadric and cyclidic coordinate geometries.
Which caught my eye due to numerology: There are also17 wallpaper groups. Probably nothing more than an excuse to look at Bôcher's work...
John Baez, It appears I've been spending too much time on Twitter, where I can get a hit of social-approval dopamine without effort of explanation :)
ReplyDeleteI added info to the post about the banana shape, aka Dupin cyclide. Also, thanks Saul Schleimer for the excellent answer while I was happily sleeping! The gif I'm going to post tonight is a loxodromic transformation when \gamma is a vertical geodesic, which will show a cone instead of a banana.
Roice Nelson - thanks! It will take me a while longer to understand this banana, but I'll keep trying:
ReplyDeleteThe surface above the plane is a Dupin cyclide, or an inversion of a cone, which touches the boundary at two fixed points.
What kind of inversion is this, exactly? Conformal inversion... about some point that's not the tip of the cone?
The wikipadia article on Dupin cyclides has a few illuminating illustrations in that regard, John Baez !
ReplyDeleteOkay.... I think one well-chosen sentence would describe the idea in a perfectly clear way. A sentence can be worth a thousand pictures.
ReplyDelete> Conformal inversion [...] about some point that's not the tip of the cone[.]
ReplyDeleteYes!https://lh3.googleusercontent.com/G8YaZYjP_xX1mZZp7z8WsntrSvK92gaOZNjXAqC533XjYXkZINR97CwdCnD82ME0nNN1Jh3JqC81t8vBjuvbn_D1RyOH3_iUgik=s0
Sorry I couldn't resist. When the sphere isn't in symmetrical position between the double cone halves it doesn't look as symmetrical but due to the conformal nature we should still get a doubly channeled surface.
ReplyDeleteJohn Baez, yes, conformal inversion in spheres and planes (that are geodesic surfaces in the model). A horn cyclide is the shape taken in the model by an equidistant surface from a core geodesic. This is true in the ball model as well, so edges in all the honeycomb images I've made are also horn cyclides (just less thickened and not "symmetric horn cyclides" in that case).
ReplyDeleteThis is similar to how isometries of Euclidean space can be built up from reflections in planes. In the Euclidean case, the equidistant surface is a cylinder. In the spherical case, the equidistant surface is again a Dupin cyclide (also generated by sphere inversions), but of a different type... a ring cyclide made from inversions of a torus.
So it's all very neat. Equidistant surfaces in the conformal models of all geometries are Dupin cyclides, which can be moved around isometrically by inversions in geodesic planes.
The following paper is pretty and a nice read to get more familiar with cyclides, even though it is about circles orthogonal to them.
Ortho-Circles of Dupin Cyclides
http://www.geometrie.tuwien.ac.at/odehnal/ocdc.pdf
Y'all are seriously fast commenters - I was behind :) Refurio Anachro, yep! Those must be the cyclides we get in the ball model.
ReplyDeleteOne other thing for now (I have to get back to work!). I didn't actually use inversions in the shader to get the banana. I applied a Mobius transformation to 3-component quaternions (a magical way to achieve isometries in the upper-half-space model). But Mobius transformations can be made from pairs of sphere inversions, so all the discussion here still applies.
ReplyDeleteThanks for the background info, Roice Nelson !
ReplyDeleteAand I had time to take a look at the paper you suggested, Roice Nelson . A nice one, thanks! Just make sure people don't look at fig.18 first thing in the morning, unless they are experienced topologists, who spent the night preparing for situations just like the one shown.
ReplyDeleteRoice Nelson what a coincidence! I am just playing with sections of hyperbolic tiling by those cyclides. Pretty wild images. That pattern is formed on the surface of your banana when it intersect tiling by Lambert cubes.
ReplyDeletehttps://lh3.googleusercontent.com/mmLit8ZOGrKf9LH0fEALYdNIP-n5OGfdEawGnl6MX9_4RSc8552wjCApHUlCGkw0dEO8yAYNkdn6SoKaUcparhYfBgRbMQkOmfE=s0
another such pattern from infinity of choices
ReplyDeletehttps://lh3.googleusercontent.com/uwe5syXW6TbHIYA-XWmRevws_iAImrwpn8K4vroe03RULDU5Rwz83tFwh-HdEqO-eoncSzzlSay8BKUUCvjnXAByfP4MRCsc6xE=s0
Beautiful as always Vladimir Bulatov! To map the banana to the plane, do you unwrap the hyperbolic cylinder? I haven’t yet thought about what the natural choice should be. I see circles in your pictures, which suggests your mapping is conformal.
ReplyDeleteThe mapping from cylinder to cone is simple exp(). It maps horizontal band into infinite sector, sector is wrapped around cone, cone is moebius transformed into banana. The patterns are periodic in vertical direction (the band) and may be periodic in horizontal direction if end points of banana axis are end points of some hyperbolic or loxodromic transform of the tiling. Those are not circles. But angles are preserved.
ReplyDelete