TY - GEN

T1 - Lines through segments in 3D space

AU - Fogel, Efi

AU - Hemmer, Michael

AU - Porat, Asaf

AU - Halperin, Dan

PY - 2012

Y1 - 2012

N2 - Given a set of n line segments in three-dimensional space, finding all the lines that simultaneously intersect at least four line segments in is a fundamental problem that arises in a variety of domains. We refer to this problem as the lines-through-segments problem, or LTS for short. We present an efficient output-sensitive algorithm and its implementation to solve the LTS problem. The implementation is exact and properly handles all degenerate cases. To the best of our knowledge, this is the first implementation for the LTS problem that is (i) output sensitive and (ii) handles all degenerate cases. The algorithm runs in O((n 3 + I)logn) time, where I is the output size, and requires O(nlogn + J) working space, where J is the maximum number of output elements that intersect two fixed line segments; I and J are bounded by O(n 4) and O(n 2), respectively. We use Cgal arrangements and in particular its support for two-dimensional arrangements in the plane and on the sphere in our implementation. The efficiency of our implementation stems in part from careful crafting of the algebraic tools needed in the computation. We also report on the performance of our algorithm and its implementation compared to others. The source code of the LTS program as well as the input examples for the experiments can be obtained from http://acg.cs.tau.ac.il/ projects/lts.

AB - Given a set of n line segments in three-dimensional space, finding all the lines that simultaneously intersect at least four line segments in is a fundamental problem that arises in a variety of domains. We refer to this problem as the lines-through-segments problem, or LTS for short. We present an efficient output-sensitive algorithm and its implementation to solve the LTS problem. The implementation is exact and properly handles all degenerate cases. To the best of our knowledge, this is the first implementation for the LTS problem that is (i) output sensitive and (ii) handles all degenerate cases. The algorithm runs in O((n 3 + I)logn) time, where I is the output size, and requires O(nlogn + J) working space, where J is the maximum number of output elements that intersect two fixed line segments; I and J are bounded by O(n 4) and O(n 2), respectively. We use Cgal arrangements and in particular its support for two-dimensional arrangements in the plane and on the sphere in our implementation. The efficiency of our implementation stems in part from careful crafting of the algebraic tools needed in the computation. We also report on the performance of our algorithm and its implementation compared to others. The source code of the LTS program as well as the input examples for the experiments can be obtained from http://acg.cs.tau.ac.il/ projects/lts.

UR - http://www.scopus.com/inward/record.url?scp=84866635491&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-33090-2_40

DO - 10.1007/978-3-642-33090-2_40

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AN - SCOPUS:84866635491

SN - 9783642330896

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 455

EP - 466

BT - Algorithms, ESA 2012 - 20th Annual European Symposium, Proceedings

T2 - 20th Annual European Symposium on Algorithms, ESA 2012

Y2 - 10 September 2012 through 12 September 2012

ER -