It is known that the deformations of the apparent contours of a surface under perspective projection and viewer motion enable the recovery of the geometry of the surface, for example by utilising the epipolar parametrization. These methods break down with apparent contours that are singular i.e. with cusps. In this paper, we study this situation in detail and show how, nevertheless, the surface geometry (including the Gauss curvature and mean curvature of the surface) can be recovered by following the cusps. Indeed the formulae are much simpler in this case, and require lower spatio-temporal derivatives than in the general case of nonsingular apparent contours. We give a simulated example, and also show that following cusps does not by itself provide us with information on ego-motion.>