The Ballet of Triangle Centers on the Elliptic Billiard
Abstract
The dynamic geometry of the family of 3-periodics in the Elliptic Billiard is mystifying. Besides conserving perimeter and the ratio of inradius-to-circumradius, it has a stationary point. Furthermore, its triangle centers sweep out mesmerizing loci including ellipses, quartics, circles, and a slew of other more complex curves. Here we explore a bevy of new phenomena relating to (i) the shape of 3-periodics and (ii) the kinematics of certain Triangle Centers constrained to the Billiard boundary, specifically the non-monotonic motion some can display with respect to 3-periodics. Hypnotizing is the joint motion of two such non-monotonic Centers, whose many stops-and-gos are akin to a Ballet.
Cite
@article{arxiv.2002.00001,
title = {The Ballet of Triangle Centers on the Elliptic Billiard},
author = {Dan Reznik and Ronaldo Garcia and Jair Koiller},
journal= {arXiv preprint arXiv:2002.00001},
year = {2021}
}
Comments
22 pages, 16 figures, 3 tables, 15 videos. arXiv admin note: substantial text overlap with arXiv:2001.08041