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This article examines projectively-invariant local geometric properties of smooth curves and surfaces. Oriented projective differential geometry is proposed as a general framework for establishing such invariants and characterizing the local projective shape of surfaces and their outlines. It is applied to two problems: (1) the projective generalization of Koenderink's famous characterization of convexities, concavities, and inflections of the apparent contours of solids bounded by smooth surfaces, and (2) the image-based construction of rim meshes, which provide a combinatorial description of the arrangement induced on the surface of an object by the contour generators associated with multiple cameras observing it.