Approximation of the Willmore energy by a discrete geometry model

TL;DR

This paper proves that a specific discrete energy model for triangulated surfaces converges to the continuous Willmore energy, bridging discrete differential geometry and smooth surface analysis.

Contribution

It establishes the $ ext{Gamma}$-convergence of a discrete energy to the Willmore energy, providing a rigorous mathematical foundation for discrete models in geometry processing.

Findings

Discrete energy converges to Willmore energy

Variants of the discrete energy are discussed in computer graphics

Provides rigorous proof of convergence

Abstract

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of $\Gamma$-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.