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

Abstract

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