Abstract
We present an algorithm that, given a set of n parallel line segments in the plane, finds a convex polygon whose boundary intersects each segment at least once, or determines that none exists. Our algorithm runs in O(n log n) steps and linear space, which is optimal. Our solution involves a reduction to a bipartite stabbing problem, using a “point-sweeping” or “chain-unwrapping” technique. We use geometric duality to solve bipartite stabbing.
Research supported by NSF grant CCR-8810568.
Research supported by an NSF Graduate Fellowship.
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References
M. Atallah and C. Bajaj, Efficient algorithms for common transversals, Information Processing Letters, 25 (2), 6 May 1987, 87–90.
M. Ben-Or, Lower bounds for algebraic computation trees, Proc. 15th ACM Symp, on Theory Comput., Boston, 1983, 80–86.
H. Edelsbrunner, H.A. Maurer, F.P. Preparata, A.L. Rosenberg, E. Welzl, and D. Wood, Stabbing line segments, BIT, 22, 1982, 274–281.
H. Edelsbrunner, Finding transversals for sets of simple geometric figures, Theoretical Comp. Sci., 35, 1985, 55–69.
M.T. Goodrich and J.S. Snoeyink, Stabbing parallel segments with a convex polygon, submitted to Computer Vision, Graphics and Image Proc.
B. Grünbaum, On common transversals, Arch. Math. 9, 1958, 465–469.
M. Katchalski, T. Lewis, and A. Liu, Geometric permutations and common transversals, Disc. & Computational Geom., 1, 1986, 371–377.
J. O'Rourke, Computational Geometry Column #3, Computer Graphics 21 (5), October 1987, 314–315.
T. Pavlidis, A Vectorizer and Feature Extractor for Document Recognition, Comput. Vision, Graphics, Image Process 35, 1986, 111–127.
F.P. Preparata, and M.I. Shamos, Computational Geometry, Springer Verlag, New York, 1985.
F.P. Preparata, An optimal real time algorithm for planar convex hulls, Comm. ACM, 22(7), July 1979, 402–405.
M.I. Shamos, and D. Hoey, Geometric intersection problems, Proc. 17th IEEE Symp. on Foundations of Computer Science, Houston, 1976, 208–215.
J. Stolfi, Oriented projective geometry, Proc. of the 3rd ACM Symp. on Computational Geometry, Waterloo, 1987, 76–85.
A. Tamir, Problem 4-2 (New York University, Dept. of Statistics and Operations Research), Problems Presented at the Fourth NYU Computational Geometry Day (3/13/87).
M.R. Ward, L. Rossol, and S.W. Holland, CONSIGHT: An Adaptive Robot with Vision, Robotics Today, Summer 1979, 26–32.
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© 1989 Springer-Verlag Berlin Heidelberg
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Goodrich, M.T., Snoeyink, J.S. (1989). Stabbing parallel segments with a convex polygon. In: Dehne, F., Sack, J.R., Santoro, N. (eds) Algorithms and Data Structures. WADS 1989. Lecture Notes in Computer Science, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51542-9_21
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DOI: https://doi.org/10.1007/3-540-51542-9_21
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