Can the Elliptic Billiard Still Surprise Us?
This work addresses a fundamental problem in dynamical systems and geometry for researchers, but it is incremental as it builds on known properties of the elliptic billiard.
The paper tackled the problem of discovering new properties in the well-studied elliptic billiard system, finding that 3-periodic trajectories conserve the inradius-to-circumradius ratio and generalizing invariants to arbitrary edge counts, with results including curves like ellipses, quartics, and a stationary point.
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!