Cusps of caustics by reflection in ellipses

TL;DR

This paper investigates the cusps of caustics formed by reflected rays inside ellipses, proving a conjecture for circles and describing cusp locations for general ellipses.

Contribution

It proves the four-cusp conjecture for caustics in circular billiards and provides explicit cusp locations for general ellipses, advancing understanding of caustic geometry.

Findings

Proved the four-cusp conjecture for circular billiards.

Explicitly described four cusp locations in general ellipses.

Identified at least four cusps in caustics for elliptical billiards.

Abstract

This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point $O$ inside an elliptic billiard table, one considers the family of rays emanating from $O$ and the caustic $\Gamma_n$ of the reflected family after $n$ reflections off the ellipse, for each positive integer $n$. It is known that $\Gamma_n$ has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of $\Gamma_n$, though we do not prove that these are the only cusps.