Curvatures of Smooth and Discrete Surfaces

TL;DR

This paper explores definitions of Gauss and mean curvature for polyhedral surfaces, emphasizing discretizations that preserve fundamental integral relations such as the Gauss-Bonnet theorem and curvature force balances.

Contribution

It introduces discretization methods for curvatures on polyhedral surfaces that maintain key integral geometric properties.

Findings

Discretizations preserve Gauss-Bonnet theorem.

Methods maintain mean-curvature force balance.

Framework applicable to smooth and discrete surfaces.

Abstract

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.