Advertisement

Discrete & Computational Geometry

, Volume 14, Issue 4, pp 411–428 | Cite as

Linear-size nonobtuse triangulation of polygons

  • M. Bern
  • S. Michell
  • J. Ruppert
Article

Abstract

We give an algorithm for triangulatingn-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than π/2. The number of triangles in the triangulation is onlyO(n), improving a previous bound ofO(n2), and the running time isO(n log2n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.

Keywords

Steiner Point Simple Polygon Polygonal Region Straight Side Disk Packing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    I. Babuŝka and A. Aziz. On the angle condition in the finite element method.SIAM J. Numer. Anal. 13 (1976), 215–227.Google Scholar
  2. 2.
    B. S. Baker, E. Grosse, and C. S. Rafferty, Nonobtuse triangulation of polygons,Discrete Comput. Geom. 3 (1988), 147–168.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    R. E. Bank.PLTMG User's Guide. Philadelphia, PA, 1990.Google Scholar
  4. 4.
    J. L. Bentley and J. B. Saxe. Decomposable searching problems: 1. Static-to-dynamic transformation.J. Algorithms 1 (1980), 301–358.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Bern, L. P. Chew, D. Eppstein, and J. Ruppert. Dihedral bounds for mesh generation in high dimensions.Proc. 6th ACM-SIAM Symp. on Discrete Algorithms, 1995, pp. 189–196.Google Scholar
  6. 6.
    M. Bern, D. Dobkin, and D. Eppstein. Triangulating polygons without large angles.Proc. 8th Annual ACM Symp. on Computational Geometry, 1992, pp. 221–231,Internat. J. Comput. Geom. Appl.,5 (1995), 171–192.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Bern and D. Eppstein. Polynomial-size nonobtuse triangulation of polygons,Internat. J. Comput. Geom. Appl. 2 (1992), 241–255.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Bern and D. Eppstein, Mesh generation and optimal triangulation. Tech. Report CSL-92-1, Xerox PARC, Palo Alto, CA. Also inComputing in Euclidean Geometry, World Scientific, Singapore, 1992.CrossRefGoogle Scholar
  9. 9.
    M. Bern, D. Eppstein, and J. R. Gilbert. Provably good mesh generation.Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp. 231–241,J. Comput. System Sci.,48 (1994), 384–409.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    H. S. M. Coxeter,Introduction to Geometry. Wiley, New York, 1961.zbMATHGoogle Scholar
  11. 11.
    H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangulations.Proc. 8th Annual ACM Symp. on Computational Geometry, 1992, pp. 43–52.Google Scholar
  12. 12.
    H. Edelsbrunner and T. S. Tan. An upper bound for conforming Delaunay triangulations.Discrete Comput. Geom. 10 (1993), 197–213.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Eppstein, Faster circle packing with application to nonobtuse triangulation. Tech. Report 94-33. Department of Information and Computer Science, University of California, Irvine, CA, 1994. To appear inInternat. J. Comput. Geom. Appl. Google Scholar
  14. 14.
    S. Fortune, A sweepline algorithm for Voronoi diagrams,Algorithmica 2 (1987), 153–174.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    M. T. Goodrich, C. ÓDúlaing, and C. Yap, Computing the Voronoi diagram of a set of line segments in parallel.Algorithmica 9 (1993), 128–141.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    MATLAB Reference Guide, The MathWorks, Inc., Natick, MA, 1992.Google Scholar
  17. 17.
    E. Melissaratos and D. Souvaine, Coping with inconsistencies: a new approach to produce quality triangulations of polygonal domains with holes.Proc. 8th Annual ACM Symp. on Computational Geometry, 1992, pp. 202–211.Google Scholar
  18. 18.
    S. A. Mitchell. Refining a triangulation of a planar straight-line graph to eliminate large anglesProc. 34th Symp. on Foundations of Computer Science, 1993, pp. 583–591.Google Scholar
  19. 19.
    S. A. Mitchell. Finding a covering triangulation whose maximum angle is provably small. (Proc. 17th Annual Computer Science Conference).Austral. Comput. Sci. Comm. 16 (1994), 55–64.MathSciNetGoogle Scholar
  20. 20.
    J.-D. Müller. Proven angular bounds and stretched triangulations with the frontal Delaunay method.Proc. 11th AIAA Comp. Fluid Dynamics, Orlando, FL, 1993.Google Scholar
  21. 21.
    S. Müller, K. Kells, and W. Fichtner, Automatic rectangle-based adaptive mesh generation without obtuse angles,IEEE Trans. Computer-Aided Design 10 (1992), 855–863.CrossRefGoogle Scholar
  22. 22.
    F. Preparata and M. Shamos,Computational Geometry—an Introduction. Springer-Verlag, New York, 1985.Google Scholar
  23. 23.
    J. F. Randolph.Calculus and Analytic Geometry. Wadsworth, Belmont, CA, 1961, pp. 373–374.Google Scholar
  24. 24.
    J. Ruppert. A new and simple algorithm for quality two-dimensional mesh generation.Proc. 4th ACM-SIAM Symp. on Discrete Algorithms, 1993, pp. 83–92.Google Scholar
  25. 25.
    K. Shimada and D. C. Gossard. Computational methods for physically based FE mesh generation.Proc. IFIP TC5/WG5.3 8th Internat. Conf. on PROLAMAT, Tokyo, 1992.Google Scholar
  26. 26.
    D. D. Sleator and R. E. Tarjan. A data structure for dynamic trees,J. Comput. System Sci. 24 (1983), 362–381.MathSciNetCrossRefGoogle Scholar
  27. 27.
    W. D. Smith. Accurate circle configurations and numerical conformal mapping in polynomial time. Tech. Report 91-091-3-0058-6, NEC Research Center, Princeton, NJ, 1991.Google Scholar
  28. 28.
    G. Strang and G. J. Fix.An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.zbMATHGoogle Scholar
  29. 29.
    T.-S. Tan. An optimal bound for conforming quality triangulations.Proc. 10th ACM Symp. on Computational Geometry, 1994, pp. 240–249.Google Scholar
  30. 30.
    S.-H. Teng. Points, spheres, and separators: a unified geometric approach to graph partitioning. Ph.D. Thesis, CMU-CS-91-184, Carnegie Mellon University, Pittsburgh, PA, 1991.Google Scholar
  31. 31.
    S. A. Vavasis. Stable finite elements for problems with wild coefficients. Tech. Report TR93-1364, Department of Computer Science, Cornell University, Ithaca, NY, 1993.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • M. Bern
    • 1
  • S. Michell
    • 2
  • J. Ruppert
    • 3
  1. 1.Xerox Palo Alto Research CenterPalo AltoUSA
  2. 2.Applied and Numerical Mathematics DepartmentSandia National LaboratoriesAlbuquerqueUSA
  3. 3.IBM Almaden Research CenterSan JoseUSA

Personalised recommendations