On the Size of the 3D Visibility Skeleton: Experimental Results
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Line transversals of convex polyhedra in R3
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
On the degree of standard geometric predicates for line transversals in 3D
Computational Geometry: Theory and Applications
On the complexity of umbra and penumbra
Computational Geometry: Theory and Applications
Linear data structures for fast ray-shooting amidst convex polyhedra
ESA'07 Proceedings of the 15th annual European conference on Algorithms
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
On the computation of 3D visibility skeletons
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
Line Transversals of Convex Polyhedra in $\mathbb{R}^3$
SIAM Journal on Computing
Lines through segments in 3d space
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in $\mathbb{R}^3$ with a total of $n$ edges consists of $\Theta(n^2)$ connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily degenerate scenes. More generally, we show that a set of $k$ possibly intersecting convex polyhedra with a total of $n$ edges admits, in the worst case, $\Theta(n^2k^2)$ connected components of maximal free line segments tangent to at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present an $O(n^2 k^2 \log n)$ time and $O(nk^2)$ space algorithm that, given a scene of $k$ possibly intersecting convex polyhedra, computes all the minimal free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines.