Discrete Koenigs nets and discrete isothermic surfaces

TL;DR

This paper introduces a geometric framework for discretizing Koenigs nets and isothermic surfaces using dual quadrilaterals, revealing new properties and invariants under projective transformations.

Contribution

It provides a novel geometric characterization of discrete Koenigs and isothermic nets based on dual quadrilaterals and co-planarity, advancing the understanding of their properties.

Findings

Discrete Koenigs nets characterized by co-planarity of diagonal intersection points

New invariance property under projective transformations

Discrete isothermic nets defined as circular Koenigs nets

Abstract

We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.