The Geometry of Billiards in Ellipses and their Poncelet Grids

Abstract

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They define the associated Poncelet grid. If a billiard is periodic then it closes for any choice of the initial vertex on the ellipse. This gives rise to a continuous variation of billiards which is called billiard's motion though it is neither a Euclidean nor a projective motion. The extension of this motion to the associated Poncelet grid leads to new insights and invariants.

Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids

“…. , P N run to and fro along one component of e. According to [13,Definition 3.13], there exist conjugate billiards also in this case.…”

“…with Q ′′ i and Q ′′ i−1 as points of intersection of the principal axis with the sides through P ′′ i . Instead of a verification of Theorem 1 based on formulas from [13], we embed below the two isometric planar billiards as limiting poses in a continuous set of isometric spatial billiards in an ellipsoid.…”

“…where u i as the unit vector of the directed side P i P i+1 and n 0|i the normalvector of E at P i (cf. [13,Figure 3]). Thus, we obtain for focal billiards by virtue of ( 20)…”

“…Below we summarize some properties of the two types of billiards. For further details see, e.g., [13].…”

“…Based on the standard parametrizations of e and c, let t i denote the parameter of P i and t ′ i that of Q i . Then for N-periodic elliptic billiards the sequence of parameters t 1 , t [13,Definition 3.10]). The turning number τ of any periodic elliptic billiard counts the surroundings of e within one period.…”