HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation

On the Complexity of Umbra and Penumbra

Julien Demouth 1 Olivier Devillers 2 Hazel Everett 1 Marc Glisse 1 Sylvain Lazard 1 Raimund Seidel 3
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
2 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , INRIA Futurs
Abstract : 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 $\Omega(n)$ connected components. 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 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 are 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 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 $\Omega(nk +k^4)$ and $O(nk\alpha(k^2) +k^4)$. We also improve these last bounds when the number of light sources is bounded.
Complete list of metadata

Contributor : Rapport de Recherche Inria Connect in order to contact the contributor
Submitted on : Thursday, November 8, 2007 - 3:58:11 PM
Last modification on : Friday, January 21, 2022 - 3:10:57 AM
Long-term archiving on: : Tuesday, September 21, 2010 - 3:10:49 PM


Files produced by the author(s)


  • HAL Id : inria-00186262, version 2



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⟩



Record views


Files downloads