Poncelet–Darboux, Kippenhahn, and Szegő: projective geometry, matrices and orthogonal polynomials

We study algebraic curves that are envelopes of families of polygons supported on the unit circle $\mathbb{T}$. 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 Szegő polynomials. These 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 is a joint work with Markus Hunziker, Taylor Poe, and Brian Simanek, all at Baylor University.