The Ballet of Triangle Centers on the Elliptic Billiard
This work addresses theoretical geometry problems for mathematicians, focusing on incremental discoveries in billiard dynamics.
The paper investigates the dynamic geometry of 3-periodic triangles in an Elliptic Billiard, revealing phenomena such as non-monotonic motion of triangle centers and complex loci like ellipses and quartics, with results including conserved perimeter and a stationary point.
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.