Skip to main content
Springer Nature Link
Log in

Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Constrained Delaunay tetrahedralizations (CDTs) are valuable for discretizing three-dimensional domains with constraints such as edges and polygons. But they are difficult to generate and maintain robustly when finite-precision coordinates yield vertices on a line that are not perfectly collinear and polygonal facets that are not perfectly flat. This work focuses on two key operations, polygon insertion and vertex insertion in CDTs. These operations suffice to incrementally construct and update a CDT from a Delaunay triangulation of the vertices. We experimentally compare two recent algorithms for inserting a polygon into a CDT: a bistellar flip algorithm of Shewchuk (Proc. 19th Annual Symposium on Computational Geometry, June 2003) and a cavity retriangulation algorithm of Si and Gärtner (Proc. Fourteenth International Meshing Roundtable, September 2005). We modify these algorithms to robustly succeed in practice for polygons whose vertices deviate from exact coplanarity. Vertex insertion in a CDT is much more complicated than in a Delaunay tetrahedralization. Adding a single vertex into a CDT may not yield a new CDT. Multiple vertices may need to be inserted together to ensure the existence of a CDT. We propose a new algorithm for vertex insertion. Given a new vertex to be inserted into a CDT, this algorithm adds one or more Steiner points incrementally. It guarantees a new CDT including that vertex.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Notes

  1. http://www.tetgen.org.

  2. http://www-roc.inria.fr/gamma/gamma/gamma.php.

References

  1. Bowyer A (1981) Computing Dirichlet tessellations. Comput J 24(2):62–166

    Article  MathSciNet  Google Scholar 

  2. Chazelle B (1984) Convex partition of a polyhedra: a lower bound and worst-case optimal algorithm. SIAM J Comput 13(3):488–507

    Article  MATH  MathSciNet  Google Scholar 

  3. Cheng SW, Dey TK, Edelsbrunner H, Facello MA, Teng SH (2000) Sliver exudation. J Assoc Comput Mach 47(5):883–904

    Article  MathSciNet  Google Scholar 

  4. Cheng SW, Dey TK, Shewchuk JR (2012) Delaunay mesh generation. CRC Press, Boca Raton

    Google Scholar 

  5. Chew LP (1990) Building Voronoi diagrams for convex polygons in linear expected time. Tech Rep PCS-TR90-147. Department of Mathematics and Computer Science, Dartmouth College

  6. Chew PL (1989) Guaranteed-quality triangular meshes. Tech Rep TR 89-983. Department of Computer Science, Cornell University

  7. Clarkson KL, Shor PW (1989) Applications of random sampling in computational geometry, II. Discret Computat Geom 4(1):387–421

    Article  MATH  MathSciNet  Google Scholar 

  8. Devillers O, Pion S, Teillaud M (2001) Walking in a triangulation. In: Proceedings of the 17th Annual Symposium on Computational Geometry. Medford, Massachusetts, pp 106–114

  9. Edelsbrunner H, Mücke EP (1990) Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans Graph 9(1):66–104

    Article  MATH  Google Scholar 

  10. Fortune S, Van Wyk CJ (1996) Static analysis yields efficient exact integer arithmetic for computational geometry. ACM Trans Graph 15(3):223–248

    Article  Google Scholar 

  11. Guigue P, Devillers O (2003) Fast and robust triangle–triangle overlap test using orientation predicates. J Graph Tool 8(1):25–32

    Google Scholar 

  12. Hermeline F (1980) Une Methode Automatique de Maillage en Dimension n. Ph.D. thesis, Université Pierre et Marie Curie, Paris

  13. Hermeline F (1982) Triangulation Automatique d’un Polyèdre en Dimension N. RAIRO Analyse Numérique 16(3):211–242

    MATH  MathSciNet  Google Scholar 

  14. Lee DT, Lin AK (1986) Generalized Delaunay triangulations for planar graphs. Discret Comput Geom 1:201–217

    Google Scholar 

  15. Miller GL, Talmor D, Teng SH, Walkington NJ, Wang H (1996) Control volume meshes using sphere packing: generation, refinement and coarsening. In: Proceedings of the 5th International Meshing Roundtable. Pittsburgh, Pennsylvania, pp 47–61

  16. Ruppert J (1995) A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J Algorithms 18(3):548–585

    Article  MATH  MathSciNet  Google Scholar 

  17. Schönhardt E (1928) Über die Zerlegung von Dreieckspolyedern in Tetraeder. Mathematische Annalen 98:309–312

    Article  MathSciNet  Google Scholar 

  18. Schroeder WJ, Shephard MS (1988) Geometry-based fully automatical mesh generation and the Delaunay triangulation. Int J Numer Methods Eng 26(11):2503–2515

    Article  MATH  MathSciNet  Google Scholar 

  19. Seidel R (1982) Voronoi diagrams in higher dimensions (1982). Diplomarbeit, Institut für Informationsverarbeitung, Technische Universität Graz

  20. Shewchuk JR (1997) Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discret Comput Geom 18(3):305–363

    Article  MATH  MathSciNet  Google Scholar 

  21. Shewchuk JR (1998) A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations. In: Proceedings of the 14th Annual Symposium on Computational Geometry, pp 76–85

  22. Shewchuk JR (1998) Tetrahedral mesh generation by Delaunay refinement. In: Proceedings of the 14th Annual Symposium on Computational Geometry. Minneapolis, Minnesota, pp 86–95

  23. Shewchuk JR (2000) Mesh generation for domains with small angles. In: Proceedings of the 16th Annual Symposium on Computational Geometry. Hong Kong, pp 1–10

  24. Shewchuk JR (2000) Sweep algorithms for constructing higher-dimensional constrained Delaunay triangulations. In: Proceedings of the 16th Annual Symposium on Computational Geometry. Hong Kong, pp 350–359

  25. Shewchuk JR (2002) Constrained Delaunay tetrahedralizations and provably good boundary recovery. In: Proceedings of the 11th International Meshing Roundtable. Ithaca, New York, pp 193–204

  26. Shewchuk JR (2003) Updating and constructing constrained Delaunay and constrained regular triangulations by flips. In: Proceedings of the 19th Annual Symposium on Computational Geometry, pp 181–190

  27. Shewchuk JR (2008) General-dimensional constrained Delaunay triangulations and constrained regular triangulations I: combinatorial properties. Discret Comput Geom 39(1–3):580–637

    Article  MATH  MathSciNet  Google Scholar 

  28. Si H (2008) Adaptive tetrahedral mesh generation by constrained Delaunay refinement. Int J Numer Methods Eng 75(7):856–880

    Article  MATH  MathSciNet  Google Scholar 

  29. Si H, Gärtner K (2005) Meshing piecewise linear complexes by constrained Delaunay tetrahedralizations. In: Hanks BW (ed) Proceedings of the 14th International Meshing Roundtable, pp 147–163

  30. Si H, Gärtner K (2011) 3D boundary recovery by constrained Delaunay tetrahedralization. Int J Numer Methods Eng 85(11):1341–1364

    Article  MATH  Google Scholar 

  31. Watson DF (1981) Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Comput J 24(2):167–172

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstraße 39, 10117, Berlin, Germany

    Hang Si

  2. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94720-1776, USA

    Jonathan Richard Shewchuk

Authors
  1. Hang Si
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jonathan Richard Shewchuk
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Hang Si.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Si, H., Shewchuk, J.R. Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates. Engineering with Computers 30, 253–269 (2014). https://doi.org/10.1007/s00366-013-0331-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-013-0331-0

Keywords