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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