Discrete schemes for Gaussian curvature and their convergence

TL;DR

This paper surveys discrete schemes for Gaussian curvature, introduces a new convergent scheme for certain vertices, and compares their asymptotic errors, advancing the understanding of discrete curvature approximation.

Contribution

It presents a new discrete scheme for Gaussian curvature that converges at regular vertices with valence ≥ 5 and analyzes convergence limitations at valence 4.

Findings

The new scheme converges at regular vertices with valence ≥ 5.

It is impossible for a scheme to converge at valence 4 vertices.

Asymptotic errors of various schemes are compared.

Abstract

In this paper, several discrete schemes for Gaussian curvature are surveyed. The convergence property of a modified discrete scheme for the Gaussian curvature is considered. Furthermore, a new discrete scheme for Gaussian curvature is resented. We prove that the new scheme converges at the regular vertex with valence not less than 5. By constructing a counterexample, we also show that it is impossible for building a discrete scheme for Gaussian curvature which converges over the regular vertex with valence 4. Finally, asymptotic errors of several discrete scheme for Gaussian curvature are compared.