Poncelet pairs of a circle and parabolas from a confocal family and Painlevé VI equations

TL;DR

The paper analyzes circular–parabolic Poncelet configurations drawn from a confocal pencil of parabolas with a fixed focus $F$. Using Cayley’s condition and a detailed algebraic machinery, it proves that the confocal family is $3$-isoperiodic with a circle precisely when the circle contains $F$, and $4$-isoperiodic precisely when the circle’s center is $F$, while excluding $n$-isoperiodicity for all other $n$. It then leverages these isoperiodic structures to produce explicit algebraic solutions to Painlevé VI equations via an algebro-geometric framework based on elliptic curves and the Okamoto transformation, recovering known results for $n=3$ (Hitchin) and providing new algebraic realizations for $n=4$. The work highlights a deep link between Poncelet-type problems, isoperiodicity in confocal pencils, and special function theory, with implications for integrable systems and algebraic solutions of Painlevé VI. Overall, the authors establish a precise dichotomy for isoperiodicity in this geometric setting and translate it into explicit Painlevé VI solutions, enriching both classical projective geometry and modern integrable systems theory.

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 Painlevé VI equations.

Figures (9)

• Figure 1: An illustration of Poncelet's Theorem. $(\mathcal{D}_n,\mathcal{P})$ is a $n$-Poncelet pair. Two $n$-Poncelet polygons are inscribed in $\mathcal{D}_n$ and circumscribed about $\mathcal{P}$.
• Figure 2: Example \ref{['ex.2.1']}: the algebraic curves $Q^3(1/2,x,y)=0$ and $Q^4(1/2,x,y)=0$.
• Figure 3: Both $\mathcal{D}$ and $\mathcal{D}'$ contain the focus $F$ of $\mathcal{P}$. Two 3-Poncelet pairs $(\mathcal{D},\mathcal{P})$ and $(\mathcal{D}',\mathcal{P})$ with real and imaginary 3-Poncelet triangles, respectively.
• Figure 4: An illustration of $3$-isoperiodicity of $\mathcal{F}$ with $\mathcal{D}(E)$.
• Figure 5: A geometric proof of $3$-isoperiodicity.

Theorems & Definitions (69)

• Definition 1.1
• Definition 1.2
• Theorem 2.1: Cayley 1853, Cayley1853aCayley1853b
• Remark 2.1
• Lemma 2.1
• proof
• Corollary 2.1
• Lemma 2.2
• Proposition 2.1
• proof