  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.
