Is every triangle a trajectory of an elliptical billiard?

Autor:	Dragović, Vladimir, Radnović, Milena
Rok vydání:	2024
Předmět:	Mathematics - Dynamical Systems Mathematical Physics Mathematics - Complex Variables Mathematics - Metric Geometry Nonlinear Sciences - Exactly Solvable and Integrable Systems 51N20 30C15 37J35 70H06
Druh dokumentu:	Working Paper
Popis:	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. Comment: 23 pages, 33 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2405.08922 Zobrazit plný text záznamu View this record from Arxiv