Abstract
Chazelle [SIAM J Comput 21(4):671–696, 1992] gave a linear-time algorithm to compute the intersection of two convex polyhedra in three dimensions. We present a simpler algorithm to do the same.
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Notes
Throughout the paper, “below” means “below or is incident to” unless preceded by “strictly”.
References
Amato, N.M., Goodrich, M.T., Ramos, E.A.: A randomized algorithm for triangulating a simple polygon in linear time. Discrete Comput. Geom. 26(2), 245–265 (2001)
Chan, T.M.: Deterministic algorithms for 2-D convex programming and 3-D online linear programming. J. Algorithms 27(1), 147–166 (1998)
Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6, 485–524 (1991)
Chazelle, B.: An optimal algorithm for intersecting three-dimensional convex polyhedra. SIAM J. Comput. 21(4), 671–696 (1992)
Clarkson, K.L., Shor, P.W.: Application of random sampling in computational geometry. II. Discrete Comput. Geom. 4, 387–421 (1989)
Dobkin, D.P., Kirkpatrick, D.G.: A linear algorithm for determining the separation of convex polyhedra. J. Algorithms 6(3), 381–392 (1985)
Dobkin, D.P., Kirkpatrick, D.G.: Determining the separation of preprocessed polyhedra—A unified approach. In: Proceedings of the 17th International Colloquium on Automata, Languages and Programming, pp. 400–413 (1990)
Dyer, M., Megiddo, N., Welzl, E.: Linear programming. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, Chapter 45, 2nd edn. CRC Press, New York (2004)
Kirkpatrick, D.G.: Efficient computation of continuous skeletons. In: Proceedings of the 20th Annual Symposium on Foundations of Computer Science, pp. 18–27 (1979)
Kirkpatrick, D.G.: Optimal search in planar subdivisions. SIAM J. Comput. 12(1), 28–35 (1983)
Martin, A.K.: A simple primal algorithm for intersecting 3-polyhedra in linear time. Master’s thesis, Department of Computer Science, University of British Columbia. https://circle.ubc.ca/handle/2429/30114 or http://www.cs.ubc.ca/cgi-bin/tr/1991/TR-91-16 (1991)
Mulmuley, K.: Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs (1993)
Preparata, F.P., Hong, S.J.: Convex hulls of finite sets of points in two and three dimensions. Commun. ACM 20(2), 87–93 (1977)
Shamos M.I., Hoey D.: Closest-point problems. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, pp. 151–162 (1975)
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The author thanks Stefan Langerman for discussion on these problems.
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Editor in Charge: János Pach
A preliminary version of this work appeared in the Proceedings of the 31st International Symposium on Computational Geometry, 2015. Part of this work was done during the author’s visit to the Hong Kong University of Science and Technology.
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Chan, T.M. A Simpler Linear-Time Algorithm for Intersecting Two Convex Polyhedra in Three Dimensions. Discrete Comput Geom 56, 860–865 (2016). https://doi.org/10.1007/s00454-016-9785-3
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DOI: https://doi.org/10.1007/s00454-016-9785-3