Discrete Isothermic Nets Based on Checkerboard Patterns

TL;DR

This paper explores the discrete differential geometry of checkerboard patterns in quadrilateral nets, establishing a framework for defining and analyzing principal, Koenigs, and isothermic nets with applications to discrete minimal surfaces.

Contribution

It introduces a unified approach to discrete isothermic nets based on checkerboard patterns, linking them with Moebius geometry and transformations.

Findings

Discrete Koenigs nets are dualizable with equal Laplace invariants.

Isothermic nets are invariant under dualization and Moebius transformations.

Framework enables construction of discrete minimal surfaces.

Abstract

This paper studies the discrete differential geometry of the checkerboard pattern inscribed in a quadrilateral net by connecting edge midpoints. It turns out to be a versatile tool which allows us to consistently define principal nets, Koenigs nets and eventually isothermic nets as a combination of both. Principal nets are based on the notions of orthogonality and conjugacy and can be identified with sphere congruences that are entities of Moebius geometry. Discrete Koenigs nets are defined via the existence of the so called conic of Koenigs. We find several interesting properties of Koenigs nets, including their being dualizable and having equal Laplace invariants. Koenigs nets that are also principal are defined as isothermic nets. We prove that the class of isothermic nets is invariant under both dualization and Moebius transformations. Among other things, this allows a natural construction of discrete minimal surfaces and their Goursat transformations.