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A comprehensive analysis of periodic trajectories of billiards withinellipses in the Euclidean plane is presented. The novelty of the approach isbased on a relationship recently established by the authors between periodicbilliard trajectories and extremal polynomials on the systems of $d$ intervalson the real line and ellipsoidal billiards in $d$-dimensional space. Even inthe planar case, systematically studied in the present paper it leads to newresults in characterizing $n$ periodic trajectories vs. so-called $n$ ellipticperiodic trajectories, which are $n$-periodic in elliptical coordinates. Thecharacterizations are done both in terms of the underlying elliptic curve anddivisors on it and in terms of polynomial functional equations, like Pell'sequation. This new approach also sheds light on some classical results. Inparticular we connect search for caustics which generate periodic trajectorieswith three classical classes of extremal polynomials on two intervals,introduced by Zolotarev and Akhiezer. The main classifying tool are windingnumbers, for which we provide several interpretations, including one in termsof numbers of points of alternance of extremal polynomials. The latter impliesimportant inequality between the winding numbers, which as a consequence,provides another proof of monotonicity of rotation numbers. A complete catalogof billiard trajectories with small periods is provided for $n=3, 4, 5, 6$along with an effective search for caustics. As a byproduct, an intriguingconnection between Cayle type conditions and discriminantly separablepolynomials has been observed for all those small periods.

Caustics of Poncelet polygons and classical extremal polynomials