Elliptical billiard systems and the full Poncelet’s theorem in n dimensions

Abstract

In this work is presented a generalization of Poncelet’s theorem to n dimensions which is refered to as the full Poncelet’s theorem. The theorem states that if the reflections of a trajectory by a sequence of confocal quadrics lead to a closed skew polygon, then there exists an (n−1)‐parameter family of polygons having the same property. A physical realization and a projective geometrical proof of this theorem are given. If all the reflecting quadrics coincide, the above theorem reduces to the n‐dimensional Poncelet’s theorem presented by Chang and Friedberg. The geometrical proof is a finite construction based on a preliminary theorem which extends Hart’s lemma. The full Poncelet’s theorem may thus be extended to projective geometries over most fields, including discrete ones.