Lines of curvature and umbilical points for implicit surfaces

This paper develops a method to analyze and compute the lines of curvature and their differential geometry defined on implicit surfaces. With our technique, we can explicitly obtain the analytic formulae of the associated geometric attributes of an implicit surface, e.g. torsion of a line of curvature and Gaussian curvature. Additionally, it can be used to directly derive the closed formulae of principal directions and corresponding principal curvature of an implicit surface. We also present a novel criterion for non-umbilical points and umbilical points on an implicit surface.