Sphere vs spheroid - This polar view looks almost like the inside view of a sphere. When the marble is at 90° you get a cone as the second picture shows, and rings when it's straight ahead at 180°. The big picture has a marble almost at 180°.

Spherical billiards are much simpler because every ray "follows" a great-circle, and stays in a plane through the sphere's center. So to understand a sphere, it's sufficient to study the paths in a circle.

Let's think about what we'd see when we'd highlight our viewpoint, say, with a glow-worm camera. Every white blob then corresponds to a closed path. There's a closed path straight ahead, one gives a triangle, another a square, and so on. Not only do you get a closed path for every regular polygon, there's one for every star polygon!

Every fraction of 180° gives a closed path, but irrational angles, e.g. sqrt(2), don't give closed paths. So we should see a gap for every irrational number. Actually, we only see gaps for some of them, most are covered up by the fact that i don't know how to render a pointlike highlight.

For a clearer view, have a look at Thomae's function, which is exactly about this idea. And, you can see a distorted version of it, hidden in the original picture.

https://en.wikipedia.org/wiki/Thomae%27s_function

One might think that this polar view of a prolate spheroid should be understood by looking at ellipses instead of circles. But the poles are special points! At it, there are no rays that pass between the foci of the ellipse. And that means, none of these paths can belong to the oscillating class, they're all rotating ones. And those behave exactly like circular ones, only mildly distorted.

But when you move the camera away from the pole, even if only a tiny bit, things look much different! The breaking of rotational symmetry has a most dramatic effect: The pearls sitting on the peaks of Thomae's function get pulled apart into spirals! I made the marble in picture 3 transparent so we can see through the blobs.

I've got even more pictures at hand and stories to tell, stay tuned!

This is part 2 of a series of appendices to a post on Visual Insight:
http://blogs.ams.org/visualinsight/2015/04/15/sphere-in-mirrored-spheroid/

top, Appendix 1 is my announcement on g+ here: