Can the Elliptic Billiard Still Surprise Us?

Abstract

Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery of curves: ellipses, quartics, sextics, circles, and even a stationary point. Secondly, one notices this family conserves an intriguing ratio: Inradius-to-Circumradius. In turn this implies three conservation corollaries: (i) the sum of bounce angle cosines, (ii) the product of excentral cosines, and (iii) the ratio of excentral-to-orbit areas. Monge's Orthoptic Circle's close relation to 4-periodic Billiard trajectories is well-known. Its geometry provided clues with which to generalize 3-periodic invariants to trajectories of an arbitrary number of edges. This was quite unexpected. Indeed, the Elliptic Billiard did surprise us!

19 pages, 16 figures

References in corpus (1)

• Synthetic foundations of cevian geometry, III: The generalized orthocenter

Cited by in corpus (7)

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• Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?
• Mather $\beta$-function for ellipses and rigidity
• New Invariants of Poncelet-Jacobi Bicentric Polygons
• The Talented Mr. Inversive Triangle in the Elliptic Billiard
• Poncelet Plectra: Harmonious Curves in Cosine Space
• Non-Birkhoff periodic orbits in symmetric billiards