On foci of ellipses inscribed in cyclic polygons

TL;DR

This paper explores how orthogonal polynomials on the unit circle can be used to derive formulas for ellipses inscribed in cyclic polygons, linking geometric properties with matrix eigenvalues.

Contribution

It introduces a novel approach using orthogonal polynomials and matrix eigenvalues to find explicit formulas for ellipses inscribed in cyclic polygons.

Findings

Derived formulas for elliptical disks using orthogonal polynomials

Connected geometric inscribed ellipses with matrix eigenvalues

Provided methods for specific values of n

Abstract

Given a natural number $n\geq3$ and two points $a$ and $b$ in the unit disk $\mathbb D$ in the complex plane, it is known that there exists a unique elliptical disk having $a$ and $b$ as foci that can also be realized as the intersection of a collection of convex cyclic $n$-gons whose vertices fill the whole unit circle $\mathbb T$. What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of $n$. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix.