Elliptic Billiards and Ellipses Associated to the 3-Periodic Orbits

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].…”

“…6, the excentral family comprises the excentral triangles to 3-periodics in the elliptic billiard. Abusing notation, here, we let a, b denote the axes of said elliptic billiard (i.e., the caustic to the excentral family), and a e , b e denote the axes of outer ellipse E, which in [7] was derived as:…”

“…In [7,19], it is shown that in the confocal pair the locus of the incenter X 1 is an ellipse. Experimentally, we can strengthen Proposition 9:…”

“…Our very first experimental result was that over 3-periodics in the confocal pair, the locus of the incenter X 1 was an ellipse [16]. This was subsequently proved [7,19]. Considering the space of choices of nested ellipse pairs is 5d (5 parameters for each minus 4d homothethy group, minus 1d Cayley condition), little did we know how rare a phenomenon that was (the space of confocal ellipses is a 1d): Fig.…”

The Iris billiard: Critical geometries for global chaos

“…These systems exhibit three behaviors: 1) Regular (i.e. periodic or quasi-periodic orbits, as found in circular 2,3 elliptic 4 or confocal elliptic 5 billiards). 2) Ergodic, with orbits that fill the entire phase space, as found in the Sinai 6 , Bunimovich stadium 7 and cardioid 8 billiards.…”