Computational geometry: an introduction
Computational geometry: an introduction
Skewed projections with an application to line stabbing in R3
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Arrangements of lines in 3-space: a data structure with applications
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Determining sector visibility of a polygon
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Stabbing and ray shooting in 3 dimensional space
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Finding stabbing lines in 3-dimensional space
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
Finding a line transversal of axial objects in three dimensions
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Maximum stabbing line in 2D plane
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
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Algorithms are developed for determining if a set of polyhedral objects in R3 can be intersected by a common transversal (stabbing) line. It can be determined in &Ogr;(n) time if a set of n lines in space has a line transversal, and such a transversal can be found in the same time bound. For a set of n line segments, the complexity of finding such a transversal becomes &Ogr;(nlogn). Finally, for a set of polyhedra with a total of n vertices, we give a &Ogr;(n5) algorithm for determining the existence of, and computing, a line transversal. Helly-type theorems for lines and segments are also given. In particular, it is shown that if every six of a set of lines in space are intersected by a common transversal, then the entire set has a common transversal.