Authors: | Dragović, Vladimir Radnović, Milena |
Affiliations: | Mechanics Mathematical Institute of the Serbian Academy of Sciences and Arts |
Title: | Is every triangle a trajectory of an elliptical billiard? | Issue Date: | 2024 | ISSN: | 2331-8422 | URL: | http://arxiv.org/abs/2405.08922v2 | 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. |
Description: | 23 pages, 33 figures |
Keywords: | Mathematics - Dynamical Systems; Mathematics - Dynamical Systems; Mathematical Physics; Mathematics - Complex Variables; Mathematics - Metric Geometry; Mathematics - Mathematical Physics; Nonlinear Sciences - Exactly Solvable and Integrable Systems; 51N20, 30C15, 37J35, 70H06 | Publisher: | arXiv |
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