Classification of symmetric periodic trajectories in ellipsoidal billiards

Abstract

We classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of Rn+1 without any symmetry of revolution. SPTs are defined as periodic trajectories passing through some symmetry set. We prove that there are exactly 22n(2n+1-1) classes of such trajectories. We have implemented an algorithm to find minimal SPTs of each of the 12 classes in the 2D case (R2) and each of the 112 classes in the 3D case (R3). They have periods 3, 4, or 6 in the 2D case and 4, 5, 6, 8, or 10 in the 3D case. We display a selection of 3D minimal SPTs. Some of them have properties that cannot take place in the 2D case.