Linear interpolation of shape operators for umbilical points through local parametrization

Abstract

Curvature analysis is frequently employed in shape interrogation. Umbilical points are of particular interest in curvature analysis because of their identical normal curvatures in all tangential directions. Locating the umbilical points is the basis of geometric analysis. On the one hand, as singularities, umbilical points severely hinder the analysis (e.g., in nets of curvature lines). On the other hand, they provide qualitative information about the intrinsic shape of a surface and are therefore desirable quantities in some applications. In this study, we develop a straightforward and effective method to detect generic umbilical points on triangular meshes. This method is applicable to any type of admissible parametrization. We propose two local parametrization schemes–orthogonal projection and conformal transformation–to be used with the proposed method. Furthermore, we systematically analyze our method and prove its convergence behavior. The algorithm used in our approach is flexible and straightforward to implement for triangular meshes of arbitrary topology.