Is every triangle a trajectory of an elliptical billiard?

TL;DR

This paper demonstrates that every triangle, parallelogram, and certain polygonal lines can be realized as billiard trajectories within a unique ellipse, revealing deep connections between classical geometric theorems and billiard dynamics.

Contribution

It establishes the existence and uniqueness of boundary ellipses for various polygons using Marden's Theorem, linking billiard trajectories to classical geometric results.

Findings

Every triangle has a unique inscribed ellipse with the triangle as a billiard trajectory.

Every parallelogram is a billiard trajectory within a unique ellipse.

Darboux butterflies are also billiard trajectories with explicitly calculated foci.

Abstract

Using Marden's Theorem from geometric theory of polynomials, we show that for every triangle there is a unique ellipse such that the triangle is a billiard trajectory within that ellipse. Since $3$-periodic trajectories of billiards within ellipses are examples of the Poncelet polygons, our considerations provide a new insight into the relationship between Marden's Theorem and the Poncelet Porism, two gems of exceptional classical beauty. We also show that every parallelogram is a billiard trajectory within a unique ellipse. We prove a similar result for the self-intersecting polygonal lines consisting of two pairs of congruent sides, named "Darboux butterflies". In each of three considered cases, we effectively calculate the foci of the boundary ellipses.