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3D Conforming Delaunay Triangulation

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Encyclopedia of Algorithms

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Years and Authors of Summarized Original Work

1998; Shewchuk

2004; Cohen-Steiner, de Verdière, Yvinec

2006; Cheng, Poon

2012; Cheng, Dey, Shewchuk

Problem Definition

A three-dimensional domain with piecewise linear boundary elements can be represented as a piecewise linear complex (PLC) of linear cells – vertices, edges, polygons, and polyhedra – that satisfy the following properties [4]. First, no vertex lies in the interior of an edge and every two edges are interior-disjoint. Second, the boundary of a polygon or polyhedra are union of cells in the PLC. Third, if two cells f and g intersect, the intersection is a union of cells in the PLC with dimensions lower than f or g. A triangulation of an input PLC is conformingif every edge and polygon appear as a union of segments and triangles in the triangulation. Additional Steiner vertices are often necessary. The 3D conforming Delaunay triangulation problem is to construct a triangulation of an input PLC that is both conforming and...

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Recommended Reading

  1. Cheng S-W, Dey TK (2003) Quality meshing with weighted Delaunay refinement. SIAM J Comput 33(1):69–93

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  2. Cheng S-W, Poon S-H (2006) Three-dimensional Delaunay mesh generation. Discret Comput Geom 36(3):419–456; conference version in Proceedings of the ACM-SIAM symposium on discrete algorithms (2003)

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  3. Cheng S-W, Dey TK, Ramos EA, Ray T (2005) Quality meshing for polyhedra with small angles. Int J Comput Geom Appl 15(4):421–461; conference version in Proceedings of the annual symposium on computational geometry (2004)

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  4. Cheng S-W, Dey TK, Shewchuk JR (2012) Delaunay mesh generation. CRC, Boca Raton

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  5. Cohen-Steiner D, de Verdière ÉC, Yvinec M (2004) Conforming Delaunay triangulation in 3D. Comput Geom Theory Appl 28(2–3):217–233

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  6. Murphy M, Mount DM, Gable CW (2001) A point-placement strategy for conforming Delaunay tetrahedralization. Int J Comput Geom Appl 11(6):669–682

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  7. Pav SE, Walkington NJ (2004) Robust three-dimensional Delaunay refinement. In: Proceedings of the international meshing roundtable, Williamsburg, pp 145–156

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Shewchuk JR (1998) Tetrahedral mesh generation by Delaunay refinement. In: Proceedings of the annual symposium on computational geometry, Minneapolis, pp 86–95

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong, China

    Siu-Wing Cheng

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  1. Siu-Wing Cheng
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Correspondence to Siu-Wing Cheng .

Editor information

Editors and Affiliations

  1. Dept. Electrical Engineering & Computer Science, Northwestern University, Evanston, Illinois, USA

    Ming-Yang Kao

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Cheng, SW. (2014). 3D Conforming Delaunay Triangulation. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-3-642-27848-8_716-1

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  • DOI: https://doi.org/10.1007/978-3-642-27848-8_716-1

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  • Publisher Name: Springer, Boston, MA

  • Online ISBN: 978-3-642-27848-8

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