Poncelet-Darboux, Kippenhahn, and Szeg\H{o}: interactions between projective geometry, matrices and orthogonal polynomials

TL;DR

This paper explores the relationship between algebraic curves, projective geometry, matrices, and orthogonal polynomials on the unit circle, providing new insights, clarifications, and a formal foundation for classical geometric theorems and their modern interpretations.

Contribution

It offers a rigorous framework connecting Poncelet curves, orthogonal polynomials, and matrix theory, clarifies misconceptions, and introduces new results and perspectives on classical geometric properties.

Findings

Characterization of minimal class Poncelet-type curves.

Curves inscribed in polygon families can have cusps and intersect the circle.

Counterexamples to existing assumptions about convexity and tangency.

Abstract

We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szeg\H{o} polynomials. Our results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions. This paper contains new results, some old results presented from a different perspective or with a different proof, and a formal foundation for our analysis. We give a rigorous definition of the Poncelet property, of curves tangent to a family of polygons, and of polygons associated with Poncelet curves. As a result, we are able to clarify some misconceptions that appear in the literature and present counterexamples to some existing assertions along with necessary modifications to their hypotheses to validate them. For instance, we show that curves inscribed in some families of polygons supported on T are not necessarily convex, can have cusps, and can even intersect the unit circle. Two ideas play a unifying role in this work. The first is the utility of OPUC and the second is the advantage of working with tangent coordinates. This latter idea has been previously exploited in the works of B. Mirman, whose contribution we have tried to put in perspective.