Stachel, H. (2022). On the Diagonals of Billiards. In ICGG 2022 - Proceedings of the 20th International Conference on Geometry and Graphics (pp. 19–33). https://doi.org/10.1007/978-3-031-13588-0_2
A billiard is the trajectory of a mass point in a domain with ideal physical reflections in the boundary e. If e is an ellipse, then the billiard’s sides are tangents of a confocal conic called caustic c. The variation of billiards in e with caustic c is called billiard motion. We recall and extend a classical result of Poncelet on the diagonals of billiards which envelope motion-invariant conics. Each billiard in e with caustic c is the flat pose of a Henrici framework. Its spatial poses define focal billiards in an ellipsoid with a fixed focal conic c. We prove that for even j the j-th diagonals are located on a motion-invariant one-sheeted hyperboloid.
