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arXiv · 2405.08922

Is every triangle a trajectory of an elliptical billiard?

Vladimir Dragović·Milena Radnović
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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.

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Vladimir Dragović, Milena Radnović. 2024-05-14. Is every triangle a trajectory of an elliptical billiard?. https://arxiv.org/abs/2405.08922

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