The implicit structure of ridges of a smooth parametric surface

Given a smooth surface, a blue (red) ridge is a curve such that at each of its points, the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately --in which case a local orientation procedure of principal directions is needed, but no method developed so far certifies the topology of the curves reported. On the way to developing certified algorithms independent from local orientation procedures, we make the following fundamental contributions. For any smooth parametric surface, we exhibit the implicit equation P = 0 of the singular curve P encoding all ridges and umbilics of the surface (blue and red), and show how to recover the colors from factors of P. Exploiting second order derivatives of the principa...

• Type: Journal
• Year: 2006
• Where: CAGD
• Authors: Frédéric Cazals, Jean-Charles Faugère, Marc Pouget, Fabrice Rouillier
• Keywords: CAGD 2006, Local Orientation Procedures, Principal Curvatures, Smooth Surface