Cayley-type: Conditions for billiards within k quadrics in ℝ d

Abstract

The notions of reflection from outside, reflection from inside and signature of a billiard trajectory within a quadric are introduced. Cayley-type conditions for periodical trajectories for the billiard in the region bounded by k quadrics in ℝd and for the billiard ordered game within k ellipsoids in ℝd are derived.In a limit, the condition describing periodic trajectories of billiard systems on a quadric in ℝ d is obtained.