Computing lines of curvature for implicit surfaces

Lines of curvature are important intrinsic characteristics of a curved surface used in a wide variety of geometric analysis and processing. Although their differential attributes have been examined in detail, their global geometric distribution and topological pattern are very difficult to compute over the whole surface because of umbilical points and unstable numerical computation. No studies have yet been carried out on this problem, especially for an implicit surface. In this paper, we present a scheme for computing and visualizing the lines of curvature defined on an implicit surface. A key structure is introduced, conveying significant structure information about lines of curvature to facilitate their investigation, rather than computing their whole net. Our current framework is confined to a collection of manageable structures, consisting of algorithms to locate some seed umbilical points, to compute the lines of curvature through them, and finally to assemble this structure. The numerical implementations are provided in detail and a novel evaluation function measuring the violation of umbilical points in an implicit surface, i.e. indicating how much a point is to be umbilical, is also presented. This paper is the continuation of [Che, W.J., Paul, J.-C., Zhang, X.P., 2007. Lines of curvature and umbilical points for implicit surfaces. Computer Aided Geometric Design 24 (7), 395–409].