A Conformal Energy for Simplicial Surfaces

Summary

A new functional for simplicial surfaces is suggested. It is invariant with respect to Möbius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. As an application a bending energy for discrete thin-shells is derived.

1. Introduction

In the variational description of surfaces, several functionals are of primary importance:

• • The area A = ∫ dA, where dA is the area element, is preserved by isometries.

• • The total Gaussian curvature g = ∫ K dA, where K is the Gaussian curvature, is a topological invariant.

• • The total mean curvature M = ∫ H dA, where H is the mean curvature, depends on the external geometry of the surface.

• • The Willmore energy W = ∫ H² dA is invariant with respect to Möbius transformations.

Geometric discretizations of the first three functionals for simplicial surfaces are well known. For the area functional the discretization is obvious. For the local Gaussian curvature the discrete analog at a vertex v is defined as the angle defect.