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

TL;DR

This paper investigates Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family, providing geometric characterizations for quadrilaterals and hexagons and linking them to Painlevé VI solutions.

Contribution

It offers a complete geometric characterization of Poncelet quadrilaterals and hexagons inscribed in a circle and circumscribed about confocal conics, and connects these configurations to Painlevé VI equations.

Findings

Complete characterization for n=4 and n=6 cases.

Such polygons do not exist for other n values.

Connection established between Poncelet polygons and Painlevé VI solutions.

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 Painlev\'e VI equation.