Caustics of Poncelet Polygons and Classical Extremal Polynomials

dc.contributor.authorDragović, Vladimir
dc.contributor.authorRadnović, Milena
dc.contributor.utdAuthorDragović, Vladimir
dc.descriptionDue to copyright restrictions and/or publisher's policy full text access from Treasures at UT Dallas is limited to current UTD affiliates (use the provided Link to Article).
dc.description.abstractA comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line and ellipsoidal billiards in d-dimensional space. Even in the planar case systematically studied in the present paper, it leads to new results in characterizing n periodic trajectories vs. so-called n elliptic periodic trajectories, which are n-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell's equation. This new approach also sheds light on some classical results. In particular, we connect the search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which, as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for n = 3, 4, 5, 6 along with an effective search for caustics. As a byproduct, an intriguing connection between Cayley-type conditions and discriminantly separable polynomials has been observed for all those small periods.
dc.description.departmentSchool of Natural Sciences and Mathematics
dc.description.sponsorshipSerbian Ministry of Education, Science, and Technological Development, Project 174020; Australian Research Council, Project DP190101838
dc.identifier.bibliographicCitationDragović, Vladimir, and Milena Radnović. 2019. "Caustics of Poncelet Polygons and Classical Extremal Polynomials." Regular & Chaotic Dynamics 24(1): 1-35, doi: 10.1134/S1560354719010015
dc.publisherPleiades Publishing Inc
dc.rights©2019 Pleiades Publishing, Ltd.
dc.source.journalRegular & Chaotic Dynamics
dc.subjectPoncelet's theorem
dc.subjectBilliards, Elliptical
dc.subjectCayley conditions
dc.subjectPolynomials, Extremal
dc.subjectCurves, Elliptic
dc.subjectTrajectories (Mechanics)
dc.subjectCaustics (Optics)
dc.subjectPell's equation
dc.subjectChebyshev polynomials
dc.subjectZolotarev polynomials
dc.subjectAkhiezer polynomials
dc.titleCaustics of Poncelet Polygons and Classical Extremal Polynomials


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