On a discretization of confocal quadrics. I. An integrable systems approach

TL;DR

This paper introduces a novel discretization of confocal quadrics that preserves key geometric properties like separability and isothermicity, using an integrable approach based on the Euler-Poisson-Darboux equation.

Contribution

It presents the first integrable discretization of confocal quadrics maintaining separability and isothermic properties, with explicit gamma function formulas for coordinate functions.

Findings

Discrete nets preserve separability and isothermicity.

All two-dimensional subnets are Koenigs nets.

Introduces a discrete orthogonality property.

Abstract

Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a conformal parametrization along curvature lines, or, equivalently, supporting orthogonal Koenigs nets). Our construction is based on an integrable discretization of the Euler-Poisson-Darboux equation and leads to discrete nets with the separability property, with all two-dimensional subnets being Koenigs nets, and with an additional novel discrete analog of the orthogonality property. The coordinate functions of our discrete nets are given explicitly in terms of gamma functions.