Recognising a curved surface from its outline in a single view is a major open problem in computer vision. This paper shows techniques for recognising a significant class of surfaces from a single perspective view. The approach uses geometrical facts about bitangencies, creases, and inflections to compute descriptions of the surface’s shape from its image outline. These descriptions are unaffected by the viewpoint or the camera parameters. We show, using images of real scenes, that these representations identify surfaces from their outline alone. This leads to fast and effective recognition of curved surfaces. The techniques we describe work for surfaces that have a rotational symmetry, or are projectively equivalent to a surface with a rotational symmetry, and can be extended to an even larger class of surfaces. All the results in this paper are for the case of full perspective. The results additionally yield techniques for identifying the line in the image plane corresponding to the axis of a rotationally symmetric surface, and for telling whether a surface is rotationally symmetric or not from its outline alone.