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.

曲率分析经常用于形状询问。脐带点在曲率分析中特别重要，因为它们在所有切线方向上具有相同的法向曲率。脐点的定位是几何分析的基础。一方面，作为奇点，脐点严重阻碍了分析（例如，在曲率线网中）。另一方面，它们提供有关表面固有形状的定性信息，因此在某些应用中是理想的数量。在这项研究中，我们开发了一种简单有效的方法来检测三角形网格上的通用脐点。该方法适用于任何类型的允许参数化。我们提出了两种局部参数化方案——正交投影和共形变换——与所提出的方法一起使用。此外，我们系统地分析了我们的方法并证明了其收敛行为。我们的方法中使用的算法对于任意拓扑的三角形网格来说是灵活且简单的。