Planar realizations of nonlinear Davenport-Schinzel sequences by segments
Discrete & Computational Geometry
Computing the antipenumbra of an area light source
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
I3D '92 Proceedings of the 1992 symposium on Interactive 3D graphics
A fast shadow algorithm for area light sources using backprojection
SIGGRAPH '94 Proceedings of the 21st annual conference on Computer graphics and interactive techniques
Fast computation of shadow boundaries using spatial coherence and backprojections
SIGGRAPH '94 Proceedings of the 21st annual conference on Computer graphics and interactive techniques
The visibility skeleton: a powerful and efficient multi-purpose global visibility tool
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
The common exterior of convex polygons in the plane
Computational Geometry: Theory and Applications
Fast and accurate hierarchical radiosity using global visibility
ACM Transactions on Graphics (TOG)
ACM Transactions on Graphics (TOG)
IEEE Computer Graphics and Applications
On the Number of Views of Polyhedral Scenes
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
Transversals to Line Segments in Three-Dimensional Space
Discrete & Computational Geometry
On incremental rendering of silhouette maps of a polyhedral scene
Computational Geometry: Theory and Applications
Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra
SIAM Journal on Computing
On the complexity of sets of free lines and line segments among balls in three dimensions
Proceedings of the twenty-sixth annual symposium on Computational geometry
Lines through segments in 3d space
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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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. 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 on a fixed plane by a segment or convex polygonal light source in the presence of convex polygonal or polyhedral obstacles in R^3. In particular, we show that a single segment light source may cast on a plane, in the presence of two disjoint triangles, four connected components of umbra and that two fat convex and disjoint obstacles of total complexity n can give rise to as many as @W(n) connected components of umbra. In a scene consisting of a segment light source and k disjoint convex polyhedra of total complexity n, we prove an @W(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; if the obstacles may intersect, we only prove an upper bound of O(n^2k^2). We also prove that, in the presence of k convex polyhedra of total complexity n, some of which are light sources, the umbra cast on a plane may have in the worst case @W(n^2k^3+nk^5) connected components and has complexity O(n^3k^3) (the polyhedra are supposed pairwise disjoint for lower bounds and possibly intersecting for the upper bounds). 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 boundary between full light and penumbra which is bounded by line segments and whose worst-case complexity is, as we show, in @W(nk+k^4) and O(nk@a(k)+k^4); moreover, if there are only O(1) light sources of total complexity m, then the complexity is in @W(n@a(k)+km+k^2) and O(n@a(k)+km@a(k)+k^2).