On a discretization of confocal quadrics

Alexander I. Bobenko, Yuri B. Suris, Jan Techter

Published 2015-11-05Version 1

Confocal quadrics lie at the basis of the system of elliptic coordinates. We suggest a discretization which respects two crucial properties of elliptic coordinates: separability and Koenigs parametrization of all two-dimensional coordinate surfaces (that is, parametrization along conjugate lines having equal Laplace invariants). Actually, one even has an isothermic parametrization, which means an additional orthogonality property. Our discretization is based on the discrete Euler-Darboux equations and leads to discrete nets with the separability property and with all two-dimensional subnets being Koenigs. The coordinate functions of our discrete nets are given explicitly in terms of gamma function.