Topological changes in the apparent contour of convex sets

We analyse the topological changes occuring in the apparent contour of a collection of disjoint convex sets from a moving viewpoint, and relate them to the classical visual event surfaces, swept by tritangent rays and limiting bitangent rays. Specifically, we show that all topological changes occur as the viewpoint crosses these surfaces or the plane containing some flat object, and that for a subset of these surfaces, such crossings do trigger a topological change in the apparent contour. We also show that these so-called visual event surfaces are likely to be two-dimensional, in the sense that their closure has empty interior. Last, we sketch how this characterization of visual event surfaces led to improved algorithms for the computation of the boundaries of soft shadows induced by area-light sources. Convexity and disjointedness allow elementary proofs and getting rid of the classical distinction between the smooth and polyhedral realms: our results hold for scenes mixing sets that are flat, smooth, piecewise smooth, have infinitely many singularities, are in arbitrarily degenerate position...