On Cayley conditions for billiards inside ellipsoids

All the segments (or their continuations) of a billiard trajectory inside an ellipsoid of $\Rset^n$ are tangent to n-1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated to periodic billiard trajectories verify certain algebraic conditions. Cayley found them in the planar case. Dragović and Radnović generalized them to any dimension. We rewrite the original matrix formulation of these generalized Cayley conditions as a simpler polynomial one. We find several remarkable algebraic relations between caustic parameters and ellipsoidal parameters that give rise to nonsingular periodic trajectories.

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