Authors: | Dragović, Vladimir Radnović, Milena |
Affiliations: | Mechanics Mathematical Institute of the Serbian Academy of Sciences and Arts |
Title: | Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil | Journal: | Geometriae Dedicata | Volume: | 218 | Issue: | 3 | First page: | 81 | Issue Date: | 1-Jun-2024 | Rank: | ~M23 | ISSN: | 0046-5755 | DOI: | 10.1007/s10711-024-00929-9 | 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∈{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 Painlevé VI equation. |
Keywords: | Cayley-type conditions | Elliptic curves | Isoperiodic confocal families | Okamoto transformations | Painlevé VI equations | Poncelet polygons | Publisher: | Springer Link | Project: | Science Fund of Serbia grant Inte- grability and Extremal Problems in Mechanics, Geometry and Combinatorics, MEGIC, Grant No. 7744592 |
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