Discrete Isothermic Nets Based on Checkerboard Patterns

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 Mobius 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. Isothermic nets can be defined as Koenigs nets that are also principal nets. We prove that the class of isothermic nets is invariant under both dualization and Mobius transformations. Among other things, this allows a natural construction of discrete minimal surfaces and their Goursat transformations.

Automated summary

This paper investigates the discrete differential geometry of a checkerboard pattern formed by connecting edge midpoints in a quadrilateral net. This approach allows for the consistent definition of principal nets, Koenigs nets, and isothermic nets. Various properties of Koenigs nets are discovered, including their dualizability and equal Laplace invariants. Isothermic nets, which are both Koenigs nets and principal nets, are proven to be invariant under dualization and Mobius transformations, enabling the construction of discrete minimal surfaces and their Goursat transformations.