Invariants of Self-Intersected N-Periodics in the Elliptic Billiard
This work addresses a niche problem in mathematical billiard theory, providing incremental insights into invariants for specialized periodic orbits.
The paper tackled the geometry of self-intersected N-periodics in the elliptic billiard, finding new properties such as concyclic vertices with foci for 4-periodics, and identified two cases where a previously claimed invariant is variable.
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.