Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

3D Conforming Delaunay Triangulation

  • Siu-Wing ChengEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_716-1

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...


Delaunay refinement Protecting ball Radius-edge ratio Weighted Delaunay triangulation 
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Recommended Reading

  1. 1.
    Cheng S-W, Dey TK (2003) Quality meshing with weighted Delaunay refinement. SIAM J Comput 33(1):69–93CrossRefMathSciNetGoogle Scholar
  2. 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)Google Scholar
  3. 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)Google Scholar
  4. 4.
    Cheng S-W, Dey TK, Shewchuk JR (2012) Delaunay mesh generation. CRC, Boca RatonGoogle Scholar
  5. 5.
    Cohen-Steiner D, de Verdière ÉC, Yvinec M (2004) Conforming Delaunay triangulation in 3D. Comput Geom Theory Appl 28(2–3):217–233CrossRefzbMATHGoogle Scholar
  6. 6.
    Murphy M, Mount DM, Gable CW (2001) A point-placement strategy for conforming Delaunay tetrahedralization. Int J Comput Geom Appl 11(6):669–682CrossRefzbMATHMathSciNetGoogle Scholar
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    Pav SE, Walkington NJ (2004) Robust three-dimensional Delaunay refinement. In: Proceedings of the international meshing roundtable, Williamsburg, pp 145–156Google Scholar
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    Ruppert J (1995) A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. Algorithms 18(3):548–585CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Shewchuk JR (1998) Tetrahedral mesh generation by Delaunay refinement. In: Proceedings of the annual symposium on computational geometry, Minneapolis, pp 86–95Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering, Hong Kong University of Science and TechnologyHong KongChina