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Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil

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Abstract

Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range and Blaschke products. We examine the behaviour of such polygons when the inscribed conic varies through a confocal pencil and discover cases when each conic from the confocal family is inscribed in an n-polygon, which is inscribed in the circle, with the same n. Complete geometric characterization of such cases for \(n\in \{4,6\}\) is given and proved that this cannot happen for other values of n. We establish a relationship of such families of Poncelet quadrangles and hexagons to solutions of a Painlevé VI equation.

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Notes

  1. There is a misprint in that example. Here we corrected the term \(I_3^2\) to \(I_3^3\)

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Acknowledgements

We are grateful to Corinna Ulcigrai and the referees for valuable remarks and comments, which helped us improve the presentation. The research was supported by the Australian Research Council, Discovery Project 190101838 Billiards within quadrics and beyond, the Science Fund of Serbia grant Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics, MEGIC, Grant No. 7744592 and the Simons Foundation grant no. 854861.

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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, USA

    Vladimir Dragović

  2. School of Mathematics and Statistics, The University of Sydney, Camperdown, Australia

    Milena Radnović

  3. Mathematical Institute SANU, Belgrade, Serbia

    Vladimir Dragović & Milena Radnović

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  1. Vladimir Dragović
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  2. Milena Radnović
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Correspondence to Vladimir Dragović.

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Dragović, V., Radnović, M. Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil. Geom Dedicata 218, 81 (2024). https://doi.org/10.1007/s10711-024-00929-9

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