Invariants of Self-Intersected N-Periodics in the Elliptic Billiard

TL;DR

This paper investigates the geometry and invariants of self-intersected N-periodic trajectories in elliptic billiards, revealing new geometric facts and examining the invariance of known quantities, with explicit formulas for specific cases.

Contribution

It provides new geometric insights into self-intersected N-periodics and tests the invariance of known quantities, identifying cases where invariants vary.

Findings

Self-intersected 4-periodics have vertices concyclic with the foci.

Explicit expressions derived for low-N cases.

Identified cases where presumed invariants are actually variable.

Abstract

We study self-intersected N-periodics in the elliptic billiard, describing new facts about their geometry (e.g., self-intersected 4-periodics have vertices concyclic with the foci). We also check if some invariants listed in "Eighty New Invariants of N-Periodics in the Elliptic Billiard" (2020), arXiv:2004.12497, remain invariant in the self-intersected case. Toward that end, we derive explicit expressions for many low-N simple and self-intersected cases. We identify two special cases (one simple, one self-intersected) where a quantity prescribed to be invariant is actually variable.