Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

Delaunay Triangulation and Randomized Constructions

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_711-1

Years and Authors of Summarized Original Work

  • 1989; Clarkson, Shor

  • 1993; Seidel

  • 2002; Devillers

  • 2003;Amenta, Choi, Rote

Problem Definition

The Delaunay triangulation and the Voronoi diagram are two classic geometric structures in the field of computational geometry. Their success can perhaps be attributed to two main reasons: Firstly, there exist practical, efficient algorithms to construct them; and secondly, they have an enormous number of useful applications ranging from meshing and 3D-reconstruction to interpolation.

Given a set S of n sites in some space \(\mathbb{E}\)


Delaunay triangulation Voronoi diagram Randomization Convex hull 
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Recommended Reading

  1. 1.
    Amenta N, Choi S, Rote G (2003) Incremental constructions con BRIO. In: Proceedings of the 19th annual symposium on computational geometry, San Diego, pp 211–219. DOI10.1145/777792.777824, http://page.inf.fu-berlin.de/~rote/Papers/pdf/Incremental+constructions+con+BRIO.pdf
  2. 2.
    Amenta N, Attali D, Devillers O (2012) A tight bound for the Delaunay triangulation of points on a polyhedron. Discret Comput Geom 48:19–38. DOI10.1007/s00454-012-9415-7, http://hal.inria.fr/hal-00784900
  3. 3.
    Attali D, Boissonnat JD, Lieutier A (2003) Complexity of the Delaunay triangulation of points on surfaces: the smooth case. In: Proceedings of the 19th annual symposium on computational geometry, San Diego, pp 201–210. DOI10.1145/777792.777823, http://dl.acm.org/citation.cfm?id=777823
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    Aurenhammer F, Klein R (2000) Voronoi diagrams. In: Sack JR, Urrutia J (eds) Handbook of computational geometry. Elsevier/North-Holland, Amsterdam, pp 201–290. ftp://ftp.cis.upenn.edu/pub/cis610/public_html/ak-vd-00.ps
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    Boissonnat JD, Teillaud M (1986) A hierarchical representation of objects: the Delaunay tree. In: Proceedings of the 2nd annual symposium computational geometry, Yorktown Heights, pp 260–268. http://dl.acm.org/citation.cfm?id=10543
  6. 6.
    Clarkson KL, Shor PW (1989) Applications of random sampling in computational geometry, II. Discret Comput Geom 4:387–421. DOI10.1007/BF02187740, http://www.springerlink.com/content/b9n24vr730825p71/
  7. 7.
    Devillers O (2002) The Delaunay hierarchy. Int J Found Comput Sci 13:163–180. DOI10.1142/S0129054102001035, http://hal.inria.fr/inria-00166711
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    Devillers O (2012) Delaunay triangulations, theory vs practice. In: Abstracts 28th European workshop on computational geometry, Assisi, pp 1–4. http://hal.inria.fr/hal-00850561, invited talk
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    Dwyer R (1993) The expected number of k-faces of a Voronoi diagram. Int J Comput Math 26(5):13–21. DOI10.1016/0898-1221(93)90068-7, http://www.sciencedirect.com/science/article/pii/0898122193900687
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    Fortune SJ (1987) A sweepline algorithm for Voronoi diagrams. Algorithmica 2:153–174. DOI10.1007/BF01840357, http://www.springerlink.com/content/n88186tl165168rw/
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    Frederick CO, Wong YC, Edge FW (1970) Two-dimensional automatic mesh generation for structural analysis. Internat J Numer Methods Eng 2:133–144. DOI10.1002/nme.1620020112/abstractGoogle Scholar
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    Golin MJ, Na HS (2002) The probabilistic complexity of the Voronoi diagram of points on a polyhedron. In: Proceedings of the 18th annual symposium on computational geometry. Barcelona http://www.cse.ust.hk/~golin/pubs/SCG_02.pdf
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    Lee DT (1978) Proximity and reachability in the plane. PhD thesis, Coordinated Science Lab., University of Illinois, Urbana. http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA069764
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    Okabe A, Boots B, Sugihara K (1992) Spatial tessellations: concepts and applications of Voronoi diagrams. Wiley, ChichesterGoogle Scholar
  15. 15.
    Seidel R (1993) Backwards analysis of randomized geometric algorithms. In: Pach J (ed) New trends in discrete and computational geometry, algorithms and combinatorics, vol 10. Springer, pp 37–68. http://ftp.icsi.berkeley.edu/ftp/pub/techreports/1992/tr-92-014.ps.gz

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Inria Nancy - Grand-EstVillers-lès-NancyFrance