# Complex caustics of the elliptic billiard

## TL;DR

This paper extends the study of elliptic billiards into the complex domain, revealing more caustics and orbit properties than in the real case, with implications for understanding complex dynamical systems.

## Contribution

It introduces the concept of complex caustics in elliptic billiards and establishes new results on their quantity and properties in the complex setting.

## Key findings

- Existence of multiple complex caustics for given elliptic billiards.
- Triangular orbits are circumscribed about specific confocal ellipses in the complex domain.
- A bound on the number of caustics for orbits with fixed sides.

## Abstract

The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exist exactly two confocal ellipses such that the triangular orbits of E are circumscribed about one of them, and each tangent line to one of those ellipses is a side of a triangular orbit. We also give a bound on the number of caustics for orbits with a fixed number of sides.