An Upper Bound on the Average Size of Silhouettes

Marc Glisse (1) , Sylvain Lazard (1)

1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility

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 non-convex and non-differentiable 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 \(O(\sqrt{n})\) 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.

Mots clés: Silhouette; apparent boundary; rim; profile; contour generator

Domaines: Géométrie algorithmique [cs.CG]

Marc Glisse, Sylvain Lazard. An Upper Bound on the Average Size of Silhouettes. Discrete and Computational Geometry, 2008, 40 (2), pp.241-257. ⟨10.1007/s00454-008-9089-3⟩. ⟨inria-00336571⟩