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On Foci of Ellipses Inscribed in Cyclic Polygons

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Part of the Operator Theory: Advances and Applications book series (OT,volume 285)

Abstract

Given a natural number n ≥ 3 and two points a and b in the unit disk \(\mathbb {D}\) in the complex plane, it is known that there exists a unique elliptical disk having a and b as foci that can also be realized as the intersection of a collection of convex cyclic n-gons whose vertices fill the whole unit circle \(\mathbb {T}\). What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of n. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix.

Keywords

  • Orthogonal polynomials
  • Poncelet Ellipses
  • Blaschke products

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Notes

  1. 1.

    We rotate τ clockwise because of the complex conjugation in the Szegő recursion.

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Acknowledgements

Author Andrei Martinez-Finkelshtein was partially supported by Simons Foundation Collaboration Grants for Mathematicians (grant 710499) and by the Spanish Government–European Regional Development Fund (grant MTM2017-89941-P), Junta de Andalucía (research group FQM-229 and Instituto Interuniversitario Carlos I de Física Teórica y Computacional), and by the University of Almería (Campus de Excelencia Internacional del Mar CEIMAR).

Author Brian Simanek graciously acknowledges support from Simons Foundation Collaboration Grant 707882.

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Hunziker, M., Martinez-Finkelshtein, A., Poe, T., Simanek, B. (2021). On Foci of Ellipses Inscribed in Cyclic Polygons. In: Gesztesy, F., Martinez-Finkelshtein, A. (eds) From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory . Operator Theory: Advances and Applications, vol 285. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-75425-9_12

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