Isothermic nets with spherical parameter lines from discrete holomorphic maps

TL;DR

This paper introduces a novel method to construct discrete isothermic nets with spherical parameter lines using discrete holomorphic maps, generalizing classical smooth surface techniques and enabling efficient creation of complex discrete surfaces.

Contribution

It presents a new approach linking discrete holomorphic maps to isothermic nets, extending classical methods to discrete geometry and surface construction.

Findings

All discrete isothermic nets with spherical lines can be generated from discrete holomorphic maps.

The method allows construction of discrete tori from periodic holomorphic maps.

Provides an efficient discretization of classical surface curvature line techniques.

Abstract

We prove that all discrete isothermic nets with a family of planar or spherical lines of curvature can be obtained from special discrete holomorphic maps via lifted-folding. This novel approach is a generalization and discretization of a classical method to create planar curvature lines on smooth surfaces. In particular, this technique provides an efficient way to construct discrete isothermic topological tori composed of fundamental pieces from discrete periodic holomorphic maps.