Formal Study of Plane Delaunay Triangulation

  • Jean-François Dufourd
  • Yves Bertot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6172)


This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe triangulations, we rely on a combinatorial hypermap specification framework we have been developing for years. We embed hypermaps in the plane by attaching coordinates to elements in a consistent way. We then describe what are legal and illegal Delaunay edges and a flipping operation which we show preserves hypermap, triangulation, and embedding invariants. To prove the termination of the algorithm, we use a generic approach expressing that any non-cyclic relation is well-founded when working on a finite set.


Computational Geometry Delaunay Triangulation Naive Algorithm Adjacent Triangle Delaunay Edge 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauer, G., Nipkow, T.: The 5 Colour Theorem in Isabelle/Isar. In: Carreño, V.A., Muñoz, C.A., Tahar, S. (eds.) TPHOLs 2002. LNCS, vol. 2410, pp. 67–82. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  3. 3.
    Bertrand, Y., Dufourd, J.-F.: Algebraic specification of a 3D-modeler based on hypermaps. Graphical Models and Image Processing 56(1), 29–60 (1994)CrossRefGoogle Scholar
  4. 4.
    Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development - Coq’Art: The Calculus of Inductive Constructions, Text in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2004)Google Scholar
  5. 5.
    Bertot, Y., Magaud, N., Zimmermann, P.: A proof of GMP square root. Journal of Automated Reasoning 29, 225–252 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Besson, F.: Fast Reflexive Arithmetic Tactics: the linear case and beyond. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 48–62. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Boissonnat, J.-D., Devillers, O., Pion, S., Teillaud, M., Yvinec, M.: Tiangulations in CGAL. Comp. Geom. - Th and Appl. 22(1-3), 5–9 (2002); Spec. iss. SOCG’00 (2002) Google Scholar
  8. 8.
    Brun, C., Dufourd, J.-F., Magaud, N.: Designing and proving correct a convex hull algorithm with hypermaps in Coq. (submitted 2009)Google Scholar
  9. 9.
    The Coq Development Team: The Coq Proof Assistant Reference Manual - Version 8.2. INRIA, France (2009),
  10. 10.
    Cori, R.: Un Code pour les Graphes Planaires et ses Applications. Astérisque 27 (1970); Société Math. de FranceGoogle Scholar
  11. 11.
    Dehlinger, C., Dufourd, J.-F.: Formalizing the trading theorem in Coq. Theoretical Computer Science 323, 399–442 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Dufourd, J.-F., Puitg, F.: Functional specification and prototyping with combinatorial oriented maps. Comp. Geometry - Th. and Appl. 16(2), 129–156 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Dufourd, J.-F.: Design and formal proof of a new optimal image segmentation program with hypermaps. Pattern Recognition 40, 2974–2993 (2007)CrossRefzbMATHGoogle Scholar
  14. 14.
    Dufourd, J.-F.: Polyhedra genus theorem and Euler Formula: A hypermap-formalized intuitionistic proof. Theor. Comp. Sc. 403, 133–159 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Dufourd, J.-F.: An Intuitionistic Proof of a Discrete Form of the Jordan Theorem Formalized in Coq with Hypermaps. Journal of Automated Reasoning 43, 19–51 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Dufourd, J.-F.: Reasoning formally with Split, Merge and Flip in plane triangulations (submitted 2009),
  17. 17.
    Dufourd, J.-F., Bertot, Y.: Formal proof of Delaunay by edge flipping (2010),
  18. 18.
    Edelsbrunner, H.: Triangulations and meshes in combinatorial geometry. In: Acta Numerica, pp. 1–81. Cambridge Univ. Press, Cambridge (2000)Google Scholar
  19. 19.
    Flato, E., et al.: The Design and Implementation of Planar Maps in CGAL. WAE 1999 16 (2000); In: Vitter, J.S., Zaroliagis, C.D. (eds.) WAE 1999. LNCS, vol. 1668, pp. 154–168. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  20. 20.
    Fousse, L., et al.: MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2) (2007)Google Scholar
  21. 21.
    Gonthier, G., Mahboubi, A., Rideau, L., Tassi, E., Théry, L.: A modular formalisation of finite group theory. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 86–101. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Gonthier, G.: Formal proof - the four-Colour theorem. Not. Am. Math. Soc. 55, 1382–1393 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Guibas, L., Stolfi, J.: Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams. ACM TOG 4(2), 74–123 (1985)zbMATHGoogle Scholar
  24. 24.
    Kettener, L., Mehlhorn, K., Pion, S., Scirra, S., Yap, C.: Classroom examples of robustness problems in geometric computations. Computational Geometry - Theory and Applications 40, 61–78 (2008)MathSciNetGoogle Scholar
  25. 25.
    Knuth, D.E.: Axioms and Hulls. LNCS, vol. 606. Springer, Heidelberg (1992)zbMATHGoogle Scholar
  26. 26.
    Meikle, L.I., Fleuriot, J.: Mechanical Theorem Proving in Computational Geometry. In: Hong, H., Wang, D. (eds.) ADG 2004. LNCS (LNAI), vol. 3763, pp. 1–18. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  27. 27.
    Melquiond, G., Pion, S.: Formally Certified Floating-Point Filters For Homogeneous Geometric Predicates. Theoretical Informatics and Applications, EDP Science 41(1), 57–69 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Obua, S., Nipkow, T.: Flyspeck II: the basic linear programs. Annals of Mathematics and Artificial Intelligence 56(3-4), 245–272 (2009)CrossRefzbMATHGoogle Scholar
  29. 29.
    Pichardie, D., Bertot, Y.: Formalizing Convex Hulls Algorithms. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 346–361. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  30. 30.
    Priest, D.: Algorithms for Arbitrary Precision Floating Point Arithmetic. In: Tenth Symposium on Computer Arithmetic, pp. 132–143. IEEE, Los Alamitos (1991)Google Scholar
  31. 31.
    Puitg, F., Dufourd, J.-F.: Formal specifications and theorem proving breakthroughs in geometric modelling. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 401–427. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  32. 32.
    Tutte, W.E.: Graph Theory. Encyclopedia of Mathematics and its Applications. Addison Wesley, Reading (1984)Google Scholar
  33. 33.
    Yap, C.-K., Pion, S.: Special Issue on Robust Geometric Algorithms and their Implementations. Computational Geometry - Theory and Applications 33(1-2) (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jean-François Dufourd
    • 1
  • Yves Bertot
    • 2
  1. 1.Université de Strasbourg, LSIIT, UMR CNRS-UdS 7005IllkirchFrance
  2. 2.INRIA-Centre de Sophia Antipolis MéditerranéeSophia-Antipolis CedexFrance

Personalised recommendations