Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?

Abstract

We analyze the family of 3-periodic (triangular) trajectories in an Elliptic Billiard. Specifically, the loci of their Triangle Centers such as the Incenter, Barycenter, etc. Many points have ellipses as loci, but some are also quartics, self-intersecting curves of higher degree, and even a stationary point. Elegant proofs have surfaced for locus ellipticity of a few classic centers, however these are based on laborious case-bycase analysis. Here we present two rigorous methods to detect when any given Center produces an elliptic locus: a first one which is a hybrid of numeric and computer algebra techniques (good for fast detection only), and a second one based on the Theory of Resultants, which computes the implicit two-variable polynomial whose zero set contains the locus.

Parabola-Inscribed Poncelet Polygons Derived from the Bicentric Family

“…We can roughly divide it into three groups: (i) the study of point loci over certain triangle families [18,19,27], (ii) proving that loci of certain Poncelet triangle families are of a given curve type [9,13,21,23], and (iii) proving properties and invariants over N ≥ 3 Poncelet families [2,4,6,22]. Also related is the Steiner-Soddy Poncelet family which are the polar image of the so-called Brocard porism with respect to the circumcircle [10].…”

Invariant Center Power and Elliptic Loci of Poncelet Triangles

“…Furthermore, over said family, the locus of both X 3 and X 5 are concentric, axisaligned ellipses with semi-axes are given by [10]:…”

“…In [10] it was shown that over confocal 3-periodics, 29 triangle centers (out of the first 100 in [13]) trace out ellipses. Explicit expressions are given for the semi-axes of each locus.…”

Loci of Poncelet Triangles in the General Closure Case

“…Regarding the confocal family, loci of both incenter and excenters 1 are ellipses with reciprocal aspect ratios [7,19]. Loci of other notable centers such as the circumcenter, orthocenter, etc., are also elliptic [7,6,10]. Certain loci are non-conic (e.g., the symmedian point, the Fermat point, etc.)…”