Exactly Solvable and Integrable Systems

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New submissions for Thu, 16 May 24

[1]  arXiv:2405.08922 [pdf, ps, other]

Title: Is every triangle a trajectory of an elliptical billiard?

Authors: Vladimir Dragović, Milena Radnović

Comments: 18 pages, 16 figures

Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Complex Variables (math.CV); Metric Geometry (math.MG); Exactly Solvable and Integrable Systems (nlin.SI)

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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