On Image Contours of Projective Shapes

International audienceThis paper revisits classical properties of the outlines of solid shapes bounded by smooth surfaces, and shows that they can be established in a purely projective setting, without appealing to Euclidean measurements such as normals or curvatures. In particular, we give new synthetic proofs of Koenderink's famous theorem on convexities and concavities of the image contour, and of the fact that the rim turns in the same direction as the viewpoint in the tangent plane at a convex point, and in the opposite direction at a hyperbolic point. This suggests that projective geometry should not be viewed merely as an analytical device for linearizing calculations (its main role in structure from motion), but as the proper framework for studying the relation between solid shape and its perspective projections. Unlike previous work in this area, the proposed approach does not require an oriented setting, nor does it rely on any choice of coordinate system or analytical considerations

The local projective shape of smooth surfaces and their outlines

This paper examines projectively invariant local properties of smooth curves and surfaces. Oriented projective differential geometry is proposed as a theoretical framework for establishing such invariants and describing the local shape of surfaces and their outlines. This framework is applied to two problems: a projective proof of Koenderink’s famous characterization of convexities, concavities, and inflections of apparent contours; and the determination of the relative orientation of rim tangents at frontier points.

The Local Projective Shape of Smooth Surfaces and their Outlines

This paper examines projectively invariant local properties of smooth curves and surfaces. Oriented projective differential geometry is proposed as a theoretical framework for establishing such invariants and describing the local shape of surfaces and their outlines. This framework is applied to two problems: a projective proof of Koenderink's famous characterization of convexities, concavities, and inflections of apparent contours; and the determination of the relative orientation of rim tangents at frontier points