Poncelet pairs of a circle and parabolas from a confocal family and Painlev\'e VI equations

TL;DR

This paper investigates special pairs of a circle and a parabola from a confocal family that admit inscribed polygons with specific properties, revealing links to Painlevé VI equations through algebraic solutions.

Contribution

It establishes geometric conditions for Poncelet pairs involving confocal conics and connects these to explicit algebraic solutions of Painlevé VI equations.

Findings

Circle contains focus F iff all parabolas form 3-Poncelet pairs

Center of circle coincides with focus F iff all parabolas form 4-Poncelet pairs

Confocal family is not n-isoperiodic for n ≠ 3,4

Abstract

We study pairs of conics $(\mathcal{D},\mathcal{P})$, called \textit{$n$-Poncelet pairs}, such that an $n$-gon, called an \textit{$n$-Poncelet polygon}, can be inscribed into $\mathcal{D}$ and circumscribed about $\mathcal{P}$. Here $\mathcal{D}$ is a circle and $\mathcal{P}$ is a parabola from a confocal pencil $\mathcal{F}$ with the focus $F$. We prove that the circle contains $F$ if and only if every parabola $\mathcal{P}\in\mathcal{F}$ forms a $3$-Poncelet pair with the circle. We prove that the center of $\mathcal{D}$ coincides with $F$ if and only if every parabola $\mathcal{P}\in \mathcal{F}$ forms a $4$-Poncelet pair with the circle. We refer to such property, observed for $n=3$ and $n=4$, as \textit{$n$-isoperiodicity}. We prove that $\mathcal{F}$ is not $n$-isoperiodic with any circle $\mathcal{D}$ for $n$ different from $3$ and $4$. Using isoperiodicity, we construct explicit algebraic solutions to Painlev\'e VI equations.