Work

Is every triangle a trajectory of an elliptical billiard?

Year: 2025

Type: article

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.

Source: Nonlinearity

DOI: 10.1088/1361-6544/adb121

Authors: Vladimir Dragović, Milena Radnović

Institutions: The University of Texas at Dallas, Mathematical Institute of the Serbian Academy of Sciences and Arts, The University of Sydney

Language: English

Topic: Mathematics and Applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,325

Funders: Simons Foundation, Australian Research Council