Computing constrained minimum-width annuli of point sets

  • Mark de Berg
  • Prosenjit Bose
  • David Bremner
  • Suneeta Ramaswami
  • Gordon Wilfong
Session 11A: Invited Lecture
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1272)


We study the problem of determining whether a manufactured disc of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and that contains all the probe points. Our algorithms run in O(n log n) time.


Voronoi Diagram Minimum Width Outer Circle Voronoi Region Voronoi Edge 
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  1. 1.
    P.K. Agarwal B. Aronov, and M. Sharir. Computing envelopes in four dimensions with applications. In Proceedings of the 10th Annual ACM Symposium on Computational Geometry, pages 348–358, 1994.Google Scholar
  2. 2.
    P.K. Agarwal and M. Sharir. Efficient randomized algorithms for some geometric optimization problems. In Proceedings of the 11th Annual ACM Symposium on Computational Geometry, pages 326–335, 1995.Google Scholar
  3. 3.
    P.K. Agarwal, M. Sharir, and S. Toledo. Applications of parametric searching in geometric optimization. Journal of Algorithms, 17:292–318, 1994.Google Scholar
  4. 4.
    F. Aurenhammer. Improved algorithms for discs and balls using power diagrams. Journal of Algorithms, 9:151–161, 1988.Google Scholar
  5. 5.
    Mark de Berg, Prosenjit Bose, David Bremner, Suneeta Ramaswami, and Gordon Wilfong. Computing constrained minimum-width annuli of point sets. Technical Report SOCS-96-7, McGill University, 1996.Google Scholar
  6. 6.
    P. Bose and L. Devroye. Intersections with random geometric objects. Technical report, McGill University, 1996.Google Scholar
  7. 7.
    K. Q. Brown. Geometric transforms for fast geometric algorithms. PhD thesis, Carnegie-Mellon University, Pittsburgh, PA, 1980.Google Scholar
  8. 8.
    C.A. Duncan, M.T. Goodrich, and E.A. Ramos. Efficient approximation and optimization algorithms for computational metrology. In Proceedings of 8th ACMSIAM Symp. Discrete Algorithms, pages 121–130, 1997.Google Scholar
  9. 9.
    H. Ebara, N. Fukuyama, H. Nakano, and Y. Nakanishi. Roundness algorithms using the Voronoi diagrams. In Proc. of the First CCCG, page 41, 1989.Google Scholar
  10. 10.
    S. J. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153–174, 1987.Google Scholar
  11. 11.
    L. W. Foster. GEO-METRICS II: The application of geometric tolerancing techniques. Addison-Wesley Publishing Co., 1982.Google Scholar
  12. 12.
    L. J. Guibas and R. Seidel. Computing convolutions by reciprocal search. Discrete Comput. Geom., 2:175–193, 1987.Google Scholar
  13. 13.
    D. G. Kirkpatrick. Optimal search in planar subdivisions. SIAM J. Computing, 12:28–35, 1983.Google Scholar
  14. 14.
    N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM, 30:852–865, 1983.Google Scholar
  15. 15.
    J. O'Rourke. Computational Geometry in C. Cambridge University Press, 1994.Google Scholar
  16. 16.
    F. Preparata and M. I. Shamos. Computational Geometry. Springer-Verlag, 1985.Google Scholar
  17. 17.
    T. J. Rivlin. Approximation by circles. Computing, 21:93–104, 1979.Google Scholar
  18. 18.
    U. Roy and X. Zhang. Establishment of a pair of concentric circles with the minimum radial separation for assessing roundness errors. Computer Aided Design, 24(3):161–168, 1992.Google Scholar
  19. 19.
    M.I. Shamos and D. Hoey. Closest-point problems. In Proc. Sixteenth Annual IEEE FOCS pages 151–162, October 1975.Google Scholar
  20. 20.
    M. Smid and R. Janardan. On the width and roundness of a set of points in the plane. In Proc. Seventh CCCG, pages 193–198, Quebec City, Quebec, August 1995.Google Scholar
  21. 21.
    C.K. Yap.An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete Comput. Geom., 2:365–393, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Mark de Berg
    • 1
  • Prosenjit Bose
    • 2
  • David Bremner
    • 3
  • Suneeta Ramaswami
    • 3
  • Gordon Wilfong
    • 4
  1. 1.Department of Computer ScienceUtrecht UniversityUtrechtThe Netherlands
  2. 2.Départment de Mathematique et InformatiqueUniveristé du Québec á Trois-RivièresTrois-RivièresCanada
  3. 3.School of Computer ScienceMontreal, QuébecCanada
  4. 4.Bell LaboratoriesLucent TechnologiesMurray HillUSA

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