The Convergence of Discrete Uniformizations for Genus Zero Surfaces

TL;DR

This paper extends the mathematical theory of discrete conformality to genus-zero surfaces, showing that discrete uniformizations approximate continuous ones through a reduction to planar cases using stereographic projections.

Contribution

It generalizes existing convergence results of discrete uniformizations from higher genus surfaces to genus-zero surfaces by employing stereographic projections.

Findings

Discrete uniformizations approximate continuous uniformizations for genus-zero surfaces.

The approach reduces the problem to planar cases via stereographic projection.

Theoretical proof of convergence for genus-zero surfaces.

Abstract

The notion of discrete conformality proposed by Luo and Bobenko-Pinkall-Springborn on triangle meshes has rich mathematical theories and wide applications. Gu et al. proved that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus $\geq 1$, given that the approximating triangle meshes are reasonably good. In this paper, we generalize this result to the remaining case of genus-zero surfaces, by reducing it to planar cases via stereographic projections.