These images were all generated with a shader that you can interact with in realtime . I leveraged the python script " shadertoy-render " to capture video, and ffmpeg to turn the video into animated gifs.

Some questions I received when posting these on social media and my answers are at the bottom. I'll gladly extend, so send comments or questions to roice3@gmail.com.

We show each transformation class in two conjugate positions. The column on the right can be considered the canonical position.

You should imagine these transformations as acting on the entire upper half space, but we draw their effect on just two surfaces:

There are four classes of transformations: elliptic, hyperbolic, loxodromic, and parabolic. These are also the classes of Möbius transformations of the complex plane. Any Möbius transformation applied to the boundary plane in the model will extend to an isometry of hyperbolic 3-space. The first three classes fix two ideal points on the boundary plane. Parabolic transformations fix just one ideal point.

Here are a set of animated gifs demonstrating basic isometries (length preserving transformations) of hyperbolic 3-space, in the upper half space model .

Questions and Answers

How far out are those points at the extremities?

The two fixed points (and every point on the boundary plane in fact) are infinitely far away. They are called “ideal” points and not part of hyperbolic space.

Could you put another bean beneath?

There is a family of concentric beans (I like to call them bananas). If you added a second one in the pic, it would either be hidden inside this one or block it. You could add bananas for any other 2 ideal fixed points, but they'd be associated with a different, conjugate transformation.

Is there a transformation which turns the cone into a plane, while turning the plane it touces into a cone?

There is not. The plane is the “plane at infinity”, and all hyperbolic isometries will keep its points ideal. Likewise, the finite points on the cone will always remain finite.

Are you saying that your 3-dimensional projection makes it look like a plane and a cone?

Yes, the model necessarily warps space, analogous to how we can't represent the earth on a flat piece of paper without warping. In the space itself, that cone is actually a cylinder and the plane is the visual sphere, infinitely far away.

Lots of questions like "What is that banana-shaped blob?"

If you've made it this far, it will be easier to point you to these comments! Also, see the paper "Ortho-circles of Dupin Cyclides".

What's a good reference for hyperbolic geometry?

My intro to hyperbolic geometry came from the wonderful book Visual Complex Analysis. That introduces one to the 2-dimensional case but I always find it easiest to think in lower dimensions and then use dimensional analogy anyway.