As observed in chapter 5, is the oriented double cover of M . The image of X r is the parallel hypersurface of X at radius r . The union of the singular sets of parallel surfaces of X is the focal hypersurface F . Therefore the focal surface of X is the bifurcation set of the family of radial distance functions D . Let M 2 be a smooth surface.
R is stable.
For X B , the singularities of the focal set F are cuspidal edges, swallowtails, elliptic umbilics, and hyperbolic umbilics. The umbilic singularities occur precisely at the foci of the umbilic points of the immersion . Porteous calls the cuspidal edges of the focal surface ribs . This description does not include those ridge points with associated principal curvature zero, which correspond to ribs at infinity. For example, the focal surface of the monkey saddle has an elliptic umbilic at infinity.