Mean Value Estimation of Shape Operator on Triangular Meshes

Abstract

The principal curvatures, eigenvalues of the shape operator, are an important differential geometric features that characterize the object’s shape, as a matter of fact, it plays a central role in geometry processing and physical simulation. The shape operator is a local operator resulting from the matrix quotient of normal derivative with the metric tensor, and hence, its matrix representation is not symmetric in general. In this paper, the local differential property of the shape operator is exploited to propose a local mean value estimation of the shape operator on triangular meshes. In contrast to the stat-of-art approximation methods that produce a symmetric operator, the resulting estimation matrix is accurate and generally not symmet-ric. Various comparative examples are presented to demonstrate the accuracy of proposed estimation. The results show that the principle curvature arising from the estimated shape operator are accurate in comparison with the standard estimation in the literature.