This paper continues our work on visual representation s of three-dimensional surfaces [Brady and Yuille 1984b]. The theoretical component of our work is a study of classes of surface curves as a source of constraint n the surface on which they lie, and as a basis for describing it. We analyze bounding contours, surface intersections, lines of curvature, and asymptotes. Our experimental work investigates whether the information suggested by our theoretical study can be computed reliably and efficiently. We demonstrate algorithms that compute lines of curvature of a (Gaussian smoothed) surface; determine planar patches and umbilic regions; extract axes of surfaces of revolution and tube surfaces. We report preliminary results on adapting the curvature primal sketch algorithms of Asada and Brady [1984] to detect and describe surface intersections.