Elliptical Billiards in the Minkowski Plane and Extremal Polynomials

A. K. Adabrah^a, V. Dragović^ba, M. Radnović<sup>cb

a</sup> Department of Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, 75080 Richardson TX, USA
^b Mathematical Institute SANU, Kneza Mihaila 36, 11001, Belgrade, p.p. 367, Serbia
^c School of Mathematics and Statistics, The University of Sydney, Carslaw F07, 2006 NSW, Australia

Abstract: We derive necessary and sufficient conditions for periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlying elliptic curve. Equivalent conditions are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. Similarities and differences with respect to the previously studied Euclidean case are indicated.

Keywords: Minkowski plane, elliptical billiards, elliptic curve, Akhiezer polynomials.

MSC: 14H70, 41A10, 70H06, 37J35, 26C05

Received: 05.06.2019
Accepted: 26.08.2019

Language: English

DOI: 10.20537/nd190401

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