On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
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
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
Transversals to Line Segments in Three-Dimensional Space
Discrete & Computational Geometry
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Computing shadow boundaries is a difficult problem in the case of non-pointlight sources. A point is in the umbra if it does not see any part of anylight 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 penumbraand the full light is well understood, less is known about the boundary of theumbra. In this paper we prove various bounds on the complexity of the umbra andthe penumbra cast by a segment or polygonal light source on a plane in the presence ofpolygon or polytope obstacles. In particular, we show that a single segment light source may cast on a plane, in thepresence of two triangles, four connected components of umbra and that two fatconvex obstacles of total complexity n can engender Ω(n) connectedcomponents of umbra. In a scene consisting of a segment light source and kdisjoint polytopes of total complexity n, we prove an Ω(nk2+k4)lower bound on the maximum number of connected components of the umbra and a O(nk3) upper bound on its complexity. We also prove that, in the presence of kdisjoint polytopes of total complexity n, some of which being light sources,the umbra cast on a plane may have Ω(n2k3 +nk5) connected components and has complexity O(n3k3).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 whichis bounded by linesegments and whose worst-case complexity is in Ω(nα(k) +km +k2) and O(nα(k) + kmα(k) +k2), where m is the complexity of the polygonallight source.