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 principal curvatures, we also derive a zero dimensional system coding the so-called turning points, from which elliptic and hyperbolic ridge sections of the two colors can be derived. Both contributions exploit properties of the Weingarten map of the surface in the specific parametric setting and require computer algebra. Algorithms exploiting the structure of P for algebraic surfaces are developed in a companion paper.