Efficient view point selection for silhouettes of convex polyhedra

AbstractSilhouettes of polyhedra are an important primitive in application areas such as machine vision and computer graphics. In this paper, we study how to select view points of convex polyhedra such that the silhouette satisfies certain properties. Specifically, we give algorithms to find all projections of a convex polyhedron such that a given set of edges, faces and/or vertices appear on the silhouette.We present an algorithm to solve this problem in O(k2) time for k edges. For orthogonal projections, we give an improved algorithm that is fully adaptive in the number l of connected components formed by the edges, and has a time complexity of O(klogk+kl). We then generalize this algorithm to edges and/or faces appearing on the silhouette

Between umbra and penumbra

International audienceComputing shadow boundaries is a difficult problem in the case of non-point light sources. A point is in the umbra if it does not see any part of any light source; it is in full light if it sees entirely all the light sources; otherwise, it is in the penumbra. While the common boundary of the penumbra and the full light is well understood, less is known about the boundary of the umbra. In this paper we prove various bounds on the complexity of the umbra and the penumbra cast by a segment or polygonal light source on a plane in the presence of polygon or polytope obstacles. In particular, we show that a single segment light source may cast on a plane, in the presence of two triangles, four connected components of umbra and that two fat convex obstacles of total complexity n can engender Omega(n) connected components of umbra. In a scene consisting of a segment light source and k disjoint polytopes of total complexity n, we prove an Omega(nk^2+k^4) lower bound on the maximum number of connected components of the umbra and a O(nk^3) upper bound on its complexity. We also prove that, in the presence of k disjoint polytopes of total complexity n, some of which being light sources, the umbra cast on a plane may have Omega(n^2k^3 + nk^5) connected components and has complexity O(n^3k^3). These are the first bounds on the size of the umbra in terms of both k and n. These results prove that the umbra, which is bounded by arcs of conics, is intrinsically much more intricate than the full light/penumbra boundary which is bounded by line segments and whose worst-case complexity is in Omega(n alpha(k) +km +k^2) and O(n alpha(k) +km alpha(k) +k^2), where m is the complexity of the polygonal light source