Elliptical Billiard tables
Summary:
Imagine playing billiards on an elliptical table. Can you hit the ball so that its trajectory is a triangle? Can every triangle appear as a billiards trajectory in some ellipse? As of 2024, the answer is Yes. The proof is a lovely tour of 18th century geometry, with a modern (1950’s) twist. I’ll prove this result, using a menagerie of facts about circles in and around ellipses. Digressions include: The Poncelet Porism, Pascal’s Hexagrammum mysticum, and Hartshorn’s beautiful representation of the moduli space of charge 2 instatons using poncelet ellipses
Presented at:
- UC Berkeley Many cheerful facts, spring 2025
🔗 Link to file
Some links:
An interactive site, exploring the set of traingular billiards for a fixed elliptical table:
A youtube video showing the phase portrait of the billiards map as the eccentricity changes. This is part of a playlist, showing phase portraits for other billiards tables. We can easily see the integrability of the elliptical billiards, and the failure of integrability of other billiards in the different videos. This is a great way to see KAM theory, and the breaking of integrability.s