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Lines through Segments in 3D Space

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Algorithms – ESA 2012 (ESA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7501))

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Abstract

Given a set \(\mathcal S\) of n line segments in three-dimensional space, finding all the lines that simultaneously intersect at least four line segments in \(\mathcal S\) 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 .

This work has been supported in part by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL—Computational Geometry Learning), by the Israel Science Foundation (grant no. 1102/11), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.

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Authors and Affiliations

  1. The Blavatnik School of Computer Science, Tel Aviv University, Israel

    Efi Fogel, Michael Hemmer, Asaf Porat & Dan Halperin

Authors
  1. Efi Fogel
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  2. Michael Hemmer
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  3. Asaf Porat
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  4. Dan Halperin
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Leah Epstein

  2. Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56100, Pisa, Italy

    Paolo Ferragina

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Fogel, E., Hemmer, M., Porat, A., Halperin, D. (2012). Lines through Segments in 3D Space. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_40

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  • DOI: https://doi.org/10.1007/978-3-642-33090-2_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33089-6

  • Online ISBN: 978-3-642-33090-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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