Ridges, Ravines and Singularities

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. We investigate relationships between these surface line features, singularities of the caustic generated by the surface normals, and the singularities of the distance function from the surface. We also propose a variational problem to model garment wrinkles and investigate relationships between singularities of a proper solution of the problem and singularities of the distance function