A Discrete Surface Theory

TL;DR

This paper introduces a novel discrete surface theory for 3-valent embedded graphs in 3D space, defining curvatures that mirror classical properties and exploring convergence to smooth surfaces through subdivision methods.

Contribution

It presents a new discrete surface framework with curvature definitions that emulate classical theory and investigates convergence to smooth surfaces via Goldberg-Coxeter subdivision.

Findings

Defined Gauss and mean curvatures for discrete surfaces

Established properties analogous to classical surface theory

Analyzed convergence of subdivided discrete surfaces

Abstract

In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean curvature of discrete surfaces are defined which satisfy properties corresponding to the classical surface theory. We also discuss the convergence of a family of subdivided discrete surfaces of a given 3-valent discrete surface by using the Goldberg-Coxeter construction. Although discrete surfaces in general have no corresponding smooth surfaces, we may find one as the limit.