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 wellknown. 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!

Invariant Center Power and Elliptic Loci of Poncelet Triangles

“…Properties of 3-periodics in the confocal pair (elliptic billiard) were studied in [11,18]. A few results and their subsequent proofs include the following: the elliptic locus of the incenter [7,19], circumcenter [6,7], invariant sum of cosines [1,2], and invariant ratio of outer-to-orbit polygon areas [3].…”

“…The power of the center wrt the circumcircle (dashed red) and Euler's circle (dashed green) is invariant. live is that the sum of angle cosines is invariant [11,18]. The Mittenpunkt X 9 is stationary at O [18].…”

“…live is that the sum of angle cosines is invariant [11,18]. The Mittenpunkt X 9 is stationary at O [18]. The semi-axes of the inner ellipse are given by [7]:…”

The geometry of billiards in ellipses and their poncelet grids

“…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 (see, e.g., [2,3,11,[21][22][23] and many further 0123456789(). : V,-vol references in [24]).…”

Family Ties

“…Classic invariants include Joachmisthal's constant J (all trajectory segments are tangent to a confocal caustic) and perimeter L [26]. A few properties detected experimentally [21] and later proved can be divided into two groups: (i) loci of triangle centers (we use the X k notation in [17]), and (ii) invariants. In terms of loci, the following results have been proved: (i) the locus of the incenter [9,23], barycenter [25], circumcenter [7,9], orthocenter [11] and many others are ellipses; (ii) a special triangle center known as the Mittenpunkt X 9 is stationary [24].…”