Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil (Q6540610)

The main object of this paper are Poncelet polygons inscribed in a given circle and circumscribed about conics from a confocal family, which is a setting that emerges from the study of the numerical range and Blaschke products. In this paper, the authors 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 \(n\). They found that in some cases all such Poncelet polygons have the same number of sides, even as the inscribed conic changes in the confocal pencil. They establish a relationship of such families of Poncelet quadrangles and hexagons to solutions of a Painlevé VI equation. This new phenomenon substantially improves and modifies the understanding and intuition related to injectivity and monotonicity of induced rotational numbers. They give elegant classical geometric characterisations of those families for \(n=4\) and \(n=6\) and show that no other natural \(n\) gives rise to such families. They also discern both analogy and disparity of the obtained results with the case when the circumscribed conic is not a circle but an ellipse which also belongs to the confocal family. \N\NThis paper is organized as follows. Section 1 is an introduction to the subject and statement of the result. In Section 2, the authors review the Poncelet porism and analytic conditions for closure in their setting. In particular, they analyse those conditions for polygons with three, four and five sides. Section 3 deals with isoperiodic families of Poncelet polygons. At the beginning of this section, the authors introduce isoperiodic families. Each such family consists of a circle and a confocal family of conics, such that each polygon inscribed in the circle and circumscribed about any of the confocal conics is closed with \(n\) sides. Section 4 is devoted to the following question: do isoperiodic families exist for \(n=4,6\)? Section 5 provides a further study of Blaschke ellipses. In particular, the authors characterize the Blaschke ellipse as the only conic from a confocal pencil, with both focal points in the interior of the unit disk, which is \(3\)-Poncelet inscribed in the unit circle. This result also answers negatively the following question: Is it possible to have a triangle inscribed in a circle and circumscribed about an ellipse intersecting the circle whose foci are within the circle ? Similarly, they characterize the Blaschke ellipse as the only conic from a confocal pencil, with both focal points in the interior of the unit disk, which is \(4\)-Poncelet inscribed in the unit circle. In Section 6, the obtained isoperiodic confocal families are used for the construction of explicit solutions to a Painlevé VI equation.