Exploring self-intersected N-periodics in the elliptic billiard

Abstract

This is a continuation of our simulation-based investigation of 𝑁 -periodic trajectories in the elliptic billiard. With a special focus on self-intersected trajectories we (i) describe new properties of 𝑁 = 4 family, (ii) derive expressions for quantities recently shown to be conserved, and to support further experimentation, we (iii) derive explicit expressions for vertices and caustic semi-axes for several families. Finally, (iv) we include links to several animations of the phenomena.

Poncelet Plectra: Harmonious Curves in Cosine Space

“…[15]. The N = 5 case, shown in Figure 1, turns out to be algebraically opaque: the axes of the caustic in terms of the outer ellipse's are roots of a degree-6 polynomial [7,Proposition 45 and 46]. In practice they are computed numerically (leading to conjectures).…”

The Talented Mr. Inversive Triangle in the Elliptic Billiard

“…For all N , the focus-inversive perimeter, sum of cosines ‡ , and sum of focus-to-vertex distances are experimentally invariant [15]. ‡ Sum of focus-inversive cosines is variable for N=4 simple and a certain self-intersected N=6 [6]. Video.…”