Rapport (Rapport De Recherche) Année : 2007

On the Complexity of Umbra and Penumbra

Résumé

Computing 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. 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 polygonal or polytopal obstacles. In particular, we show that a 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 complexity n can give rise to Ω(n) connected components. In a scene consisting of a segment light source and k disjoint polytopes of total complexity n, we prove an Ω(nk2+k4) lower bound on the maximum number of connected components of umbra and a O(nk3) upper bound on its complexity. We also prove that, in the presence of k disjoint polytopes of total complexity n, some of which are light sources, the umbra cast on a plane may have Ω(n2k3+nk5) connected components and has complexity O(n3k3). These bounds, the first ones in terms of both k and n, prove that the umbra is much more intricate than the full light boundary whose worst-case complexity is, as we show, in Ω(nk+k4) and O(nkα(k2)+k4). We also improve these last bounds when the number of light sources is bounded.
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Dates et versions

inria-00186262 , version 1 (08-11-2007)
inria-00186262 , version 2 (08-11-2007)

Identifiants

  • HAL Id : inria-00186262 , version 2

Citer

Julien Demouth, Olivier Devillers, Hazel Everett, Marc Glisse, Sylvain Lazard, et al.. On the Complexity of Umbra and Penumbra. [Research Report] RR-6347, INRIA. 2007, pp.28. ⟨inria-00186262v2⟩
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