On a discretization of confocal quadrics. II. A geometric approach to general parametrizations

TL;DR

This paper introduces a geometric discretization of confocal quadrics using factorizable orthogonal coordinate systems, leading to new discrete nets and explicit coordinate functions, with applications in elliptic functions and connections to IC nets.

Contribution

It presents a novel geometric discretization of confocal quadrics based on factorizable orthogonal systems, expanding the theory of discrete confocal coordinate systems.

Findings

Discrete confocal coordinate systems constructed via polarity.

Explicit formulas for discrete confocal quadrics coordinate functions.

Connections established with incircular nets and Euler-Poisson-Darboux system.

Abstract

We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system may be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. Various sequences correspond to various discrete parametrizations. The coordinate functions of discrete confocal quadrics are computed explicitly. The theory is illustrated with a variety of examples in two and three dimensions. These include confocal coordinate systems parametrized in terms of Jacobi elliptic functions. Connections with incircular (IC) nets and a generalized Euler-Poisson-Darboux system are established.