An Upper Bound on the Average Size of Silhouettes

It is a widely observed phenomenon in computer graphics that the size of the silhouette of a polyhedron is much smaller than the size of the whole polyhedron. This paper provides, for the first time, theoretical evidence supporting this for a large class of objects, namely for polyhedra or, more generally, tessellated surfaces that approximate surfaces in some reasonable way. The approximated surfaces are two-manifolds that may be nonconvex and nondifferentiable and may have boundaries. The tessellated surfaces should, roughly speaking, have no short edges, have fat faces, and the distance between the mesh and the surface it approximates should never be too large. We prove that such tessellated surfaces of complexity n have silhouettes of expected size where the average is taken over all points of view. The viewpoints can be chosen at random at infinity or at random in a bounded region.