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 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.
Highlights
A billiard is the trajectory of a mass point within a domain with ideal physical reflections in the boundary
One basis for the investigations was the theory of confocal conics
These papers focus on invariants of periodic billiards when the vertices vary on the ellipse while the caustic remains fixed
Summary
A billiard is the trajectory of a mass point within a domain with ideal physical reflections in the boundary. Already for two centuries, billiards in ellipses have attracted the attention of mathematicians, beginning with J.-V. Tabachnikov published a book on various aspects of billiards, including their role as completely integrable systems [29]. In several publications and in the book [13], V. In higher dimensions, from the viewpoint of dynamical systems. Computer animations of billiards in ellipses, which were carried out by Reznik [24], stimulated a new vivid interest on this well studied topic, where algebraic and analytic methods are meeting
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