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

Vladimir Dragović, Milena Radnović

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 $n\in\{4,6\}$ 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.

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

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 -polygon, which is inscribed in the circle, with the same . Complete geometric characterization of such cases for is given and proved that this cannot happen for other values of . We establish a relationship of such families of Poncelet quadrangles and hexagons to solutions of a Painlevé VI equation.

Paper Structure

This paper contains 15 sections, 30 theorems, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Poncelet porism and Cayley's conditions
  3. Isoperiodic families of Poncelet polygons
  4. Quadrangles
  5. Hexagons
  6. Do isoperiodic families exist for $n\ne4,6$?
  7. Poncelet pairs of conics and Blaschke ellipses
  8. Properties of the Blaschke transformations
  9. Poncelet triangles and Blaschke ellipses
  10. Poncelet quadrangles and Blaschke ellipses
  11. View towards Painlevé VI equations
  12. Picard solutions and Okamoto transformation
  13. From isoperiodic families to explicit solutions to a Painlevé VI equation
  14. Quadrangles
  15. Hexagons

Key Result

Proposition 2.4

Circle $\Gamma$ and conic $\mathcal{C}(t)$ are an $n$-Poncelet pair if and only if for smooth and non-smooth cubics $\mathcal{E}_t$, i.e. for any $t\not\in\{a,b\}$, where $p_n(t)$ is defined in eq:pnt.

Figures (5)

  • Figure 1: Two heptagons inscribed in a circle and circumscribed about an ellipse.
  • Figure 2: A circle and any conic with the foci on the circle are a $4$-Poncelet pair.
  • Figure 3: Each conic from the confocal family is inscribed in quadrangles whose vertices belong to the circle. The foci are symmetric to each other with respect to the circle.
  • Figure 4: Each conic from the confocal family is inscribed in hexagons whose vertices belong to the circle. The foci are symmetric with respect to the circle.
  • Figure 5: A triangle inscribed in the circle

Theorems & Definitions (44)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 34 more