Ridge extraction and its application to surface meshing

Abstract

Geometrical feature lines, such as $${{\mathcal{C}}}^1$$-discontinuity lines, lines of curvature, ridges, asymptotic lines, etc., provide useful information about the surface geometry. The inclusion of these lines into a surface mesh contributes to improving the quality of the geometrical approximation, independently from the element size. In particular, if the element size is locally large with respect to the minimal radius of curvature, then the geometrical approximation of the surface will generally be poor. In this case, the presence of ridges in the mesh improves the geometrical approximation. This paper proposes a new method to extract the extremal graph whose graph nodes are the umbilics and the extremal points, and edges are extremal lines. A function is also introduced to classify the extremal lines to be able to select the ridges from a parametric surface. The resulting graph of ridges is then simplified and integrated as a constraint in a classical surface mesh generation via an indirect approach. A numerical example will illustrate the pertinence of this method.