Computing discrete shape operators on general meshes

Abstract

Discrete curvature and shape operators, which capture complete information about directional curvatures at a point, are essential in a variety of applications: simulation of deformable two-dimensional objects, variational modeling and geometric data processing. In many of these applications, objects are represented by meshes. Currently, a spectrum of approaches for formulating curvature operators for meshes exists, ranging from highly accurate but computationally expensive methods used in engineering applications to efficient but less accurate techniques popular in simulation for computer graphics.We propose a simple and efficient formulation for the shape operator for variational problems on general meshes, using degrees of freedom associated with normals. On the one hand, it is similar in its simplicity to some of the discrete curvature operators commonly used in graphics; on the other hand, it passes a number of important convergence tests and produces consistent results for different types of meshes and mesh refinement. IntroductionDiscrete curvature is a key ingredient in a variety of applications: simulation of deformable two-dimensional objects, variational modeling and geometric data processing. In these applications it is often necessary to approximate a solution of a continuous problem involving curvature-based energy or forces. Such energy may either capture the physics of the problem (bending energy for thin deformable objects) or our intuition about desirable behavior (variational approaches to surface modeling or curvature flow smoothing of complex geometry).Curvature and related surface Laplacian discretizations range from highly accurate but complex and computationally expensive high-order finite elements used in engineering to efficient and simple approximations for meshes used in graphics and interactive geometric modeling. Unfortunately, the latter type of approximations, while essential for interactive applications, lacks the predictability and convergence of more mathematically complex and computationally expensive formulations. The lack of convergence leads to mesh-dependent behavior, visual artifacts and fundamental difficulties with adaptive refinement and remeshing of variationally defined shapes.In this paper we present a simple and efficient discrete curvature operator exhibiting good convergence properties for general meshes. This operator is closely related to widely used discrete geometric operators on one hand, and classical nonconforming finite elements on the other. The discretization we propose uses mesh normals as additional degrees of freedom. One of the essential observations is that the normal field can be defined by scalars, assigned to edges in a natural and geometrically invariant way, similar to the definition of discrete

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