Advertisement

Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator

  • Jonathan Richard Shewchuk
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1148)

Abstract

Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

Keywords

Delaunay Triangulation Refinement Stage Refinement Algorithm Roundoff Error Exact Arithmetic 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Francis Avnaim, Jean-Daniel Boissonnat, Olivier Devillers, Franco P. Preparata, and Mariette Yvinec. Evaluating Signs of Determinants Using Single-Precision Arithmetic. To appear in Algorithmica, 1995.Google Scholar
  2. 2.
    Marshall Bern and David Eppstein. Mesh Generation and Optimal Triangulation. Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, volume 1, pages 23–90. World Scientific, Singapore, 1992.Google Scholar
  3. 3.
    L. Paul Chew. Guaranteed-Quality Mesh Generation for Curved Surfaces. Proceedings of the Ninth Annual Symposium on Computational Geometry, pages 274–280. Association for Computing Machinery, May 1993.Google Scholar
  4. 4.
    Kenneth L. Clarkson. Safe and Effective Determinant Evaluation. 33rd Annual Symposium on Foundations of Computer Science, pages 387–395. IEEE Computer Society Press, October 1992.Google Scholar
  5. 5.
    Rex A. Dwyer. A Faster Divide-and-Conquer Algorithm for Constructing Delaunay Triangulations. Algorithmica 2(2):137–151, 1987.CrossRefGoogle Scholar
  6. 6.
    Steven Fortune. A Sweepline Algorithm for Voronoï Diagrams. Algorithmica 2(2):153–174, 1987.CrossRefGoogle Scholar
  7. 7.
    -. Voronoï Diagrams and Delaunay Triangulations. Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, volume 1, pages 193–233. World Scientific, Singapore, 1992.Google Scholar
  8. 8.
    Steven Fortune and Christopher J. Van Wyk. Efficient Exact Arithmetic for Computational Geometry. Proceedings of the Ninth Annual Symposium on Computational Geometry, pages 163–172. Association for Computing Machinery, May 1993.Google Scholar
  9. 9.
    Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir. Randomized Incremental Construction of Delaunay and Voronoï Diagrams. Algorithmica 7(4):381–413, 1992.CrossRefGoogle Scholar
  10. 10.
    Leonidas J. Guibas and Jorge Stolfi. Primitives for the Manipulation of General Subdivisions and the Computation of Voronoï Diagrams. ACM Transactions on Graphics 4(2):74–123, April 1985.CrossRefGoogle Scholar
  11. 11.
    C. L. Lawson. Software for C 1 Surface Interpolation. Mathematical Software III (John R. Rice, editor), pages 161–194. Academic Press, New York, 1977.Google Scholar
  12. 12.
    D. T. Lee and B. J. Schachter. Two Algorithms for Constructing a Delaunay Triangulation. International Journal of Computer and Information Sciences 9(3):219–242, 1980.CrossRefGoogle Scholar
  13. 13.
    Scott A. Mitchell. Cardinality Bounds for Triangulations with Bounded Minimum Angle. Sixth Canadian Conference on Computational Geometry, 1994.Google Scholar
  14. 14.
    Ernst P. Mücke, Isaac Saias, and Binhai Zhu. Fast Randomized Point Location Without Preprocessing in Two-and Three-dimensional Delaunay Triangulations. Proceedings of the Twelfth Annual Symposium on Computational Geometry, pages 274–283. Association for Computing Machinery, May 1996.Google Scholar
  15. 15.
    Jim Ruppert. A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation. Journal of Algorithms 18(3):548–585, May 1995.Google Scholar
  16. 16.
    Jonathan Richard Shewchuk. Robust Adaptive Floating-Point Geometric Predicates. Proceedings of the Twelfth Annual Symposium on Computational Geometry, pages 141–150. Association for Computing Machinery, May 1996.Google Scholar
  17. 17.
    Daniel Dominic Sleator and Robert Endre Tarjan. Self-Adjusting Binary Search Trees. Journal of the Association for Computing Machinery 32(3):652–686, July 1985.Google Scholar
  18. 18.
    Peter Su and Robert L. Scot Drysdale. A Comparison of Sequential Delaunay Triangulation Algorithms. Proceedings of the Eleventh Annual Symposium on Computational Geometry, pages 61–70. Association for Computing Machinery, June 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Jonathan Richard Shewchuk
    • 1
  1. 1.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations