Ridges and ravines on implicit surfaces

Surface creases provide us with important information about the shapes of objects and can be intuitively defined as curves on a surface along which the surface bends sharply. Our mathematical description of such surface creases is based on a study of extrema of the principal curvatures along their curvature lines. On a smooth generic surface we define ridges to be the local positive maxima of the maximal principal curvature along its associated curvature line and ravines to be the local negative minima of the minimal principal curvature along its associated curvature line. The ridges and ravines are important for shape analysis and possess remarkable mathematical properties. For example, they correspond to the end points of shape skeletons. In this paper we derive formulas to detect the ridges and ravines on a surface given in implicit form. We also propose an algorithm for obtaining piecewise linear approximation of ridges and ravines as intersection curves of two implicit surfaces.