Almost nothing can be deduced about a general three-dimensional surface given only its occluding contours in an image, yet contour information is easily and effectively used by us to infer the shape of a surface. Therefore, implicit in the perceptual analysis of occluding contour must lie various assumptions about the viewed surfaces. The assumptions that seem most natural are (a) that the distinction between convex and concave segments reflects real properties of the viewed surface; and (b) that contiguous portions of contour arise from contiguous parts of the viewed surface - i.e. there are no invisible obscuring edges. It is proved that, for smooth surfaces, these assumptions are essentially equivalent to assuming that the viewed surface is a generalized cone. Methods are defined for finding the axis of such a cone, and for segmenting a surface constructed of several cones into its components, whose axes can then be found separately. These methods provide one link between an uninterpreted figure extracted from an image, and the 3-D representation theory of Marr & Nishihara (1977).