On a Discretization of Confocal Quadrics. A Geometric Approach to General Parametrizations

Alexander I. Bobenko, Wolfgang K. Schief, Yuri B. Suris, Jan Techter

Research output: Contribution to journalArticlepeer-review

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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 nets and a generalized Euler-Poisson-Darboux system are established.

Original languageEnglish
Pages (from-to)10180-10230
Number of pages51
JournalInternational Mathematics Research Notices
Volume2020
Issue number24
DOIs
StatePublished - 1 Dec 2020
Externally publishedYes

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