Isothermic triangulated surfaces

TL;DR

This paper introduces a class of triangulated surfaces in Euclidean space called isothermic surfaces, which generalize classical differential geometry concepts to discrete meshes, and explores their properties and applications.

Contribution

It defines and characterizes isothermic triangulated surfaces, showing their Möbius invariance and connection to circle patterns and conformal equivalence, extending classical theory to discrete meshes.

Findings

Isothermic triangulated surfaces are Möbius invariant.

They can be characterized via circle patterns or conformal equivalence.

A discrete analog of minimal surfaces is developed using this framework.

Abstract

We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is M\"{o}bius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes. A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.