On foci of ellipses inscribed in cyclic polygons
| Autor: | Hunziker, Markus, Martinez-Finkelshtein, Andrei, Poe, Taylor, Simanek, Brian |
|---|---|
| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 19 pages, 9 figures |
| Databáze: | arXiv |
| Externí odkaz: |
|