A Direct Method for Determining the Lower Convex Hull of a Finite Point Set in 3D

  • Thanh An Phan
  • Thanh Giang Dinh
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 358)


Determining the convex hull, its lower convex hull, and Voronoi diagram of a point set is a basic operation for many applications of pattern recognition, image processing, and data mining. To date, the lower convex hull of a finite point set is determined from the entire convex hull of the set. There arises a question “How can we determine the lower convex hull of a finite point set without relying on the entire convex hull?” In this paper, we show that the lower convex hull is wrapped by lower facets starting from an extreme edge of the lower convex hull. Then a direct method for determining the lower convex hull of a finite point set in 3D without the entire convex hull is presented. The actual running times on the set of random points (in the uniform distribution) show that our corresponding algorithm runs significantly faster than the incremental convex hull algorithm and some versions of the gift-wrapping algorithm.


Convex Hull Extreme Edge Gift-wrapping Algorithm Lower Convex Hull Pattern Recognition Voronoi Diagram 


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  1. 1.
    Akl, S.G., Toussaint, G.: Efficient convex hull algorithms for pattern recognition applications. In: 4th Int’l Joint Conf. on Pattern Recognition, Kyoto, Japan, pp. 483–487 (1978)Google Scholar
  2. 2.
    An, P.T., Trang, L.H.: An efficient convex hull algorithm for finite point sets in 3D based on the Method of Orienting Curves. Optimization 62(7), 975–988 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Day, A.M.: An implementation of an algorithm to find the convex hull of a set of three-dimensional points. ACM Transactions on Graphics 9(1), 105–132 (1990)CrossRefzbMATHGoogle Scholar
  4. 4.
    Luo, D.: Pattern Recognition and Image Processing. Woodhead Publishing (1998)Google Scholar
  5. 5.
    McMullen, P., Shephard, G.C.: Convex Polytopes and the Upper Bound Conjecture. Cambridge University Press, Cambridge (1971)zbMATHGoogle Scholar
  6. 6.
    Meethongjan, K., Dzulkifli, M., Rehman, A., Saba, T.: Face recognition based on fusion of Voronoi diagram automatic facial and wavelet moment invariants. International Journal of Video & Image Processing and Network Security 10(4), 1–8 (2010)Google Scholar
  7. 7.
    Na, H.S., Lee, C.N., Cheong, O.: Voronoi diagrams on the sphere. Computational Geometry 23(2), 183–194 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Okabe, A., Boots, B., Sugihara, K.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley & Sons Ltd (2000)Google Scholar
  9. 9.
    O’Rourke, J.: Computational Geometry in C, 2nd edn. Cambridge University Press (1998)Google Scholar
  10. 10.
    Preparata, F.P., Shamos, M.I.: Computational Geometry - An Introduction, 2nd edn. Second Edition. Springer, New York (1988)zbMATHGoogle Scholar
  11. 11.
    Sugihara, K.: Robust gift wrapping for the three-dimensional convex hull. Journal of Computer and System Sciences 49, 391–407 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Thanh An Phan
    • 1
    • 2
  • Thanh Giang Dinh
    • 2
    • 3
  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.CEMAT, Instituto Superior TécnicoUniversity of LisbonLisbonPortugal
  3. 3.Department of MathematicsVinh UniversityVinh CityVietnam

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