Hyperbolic Voronoi Diagram

  • Zahra Nilforoushan
  • Ali Mohades
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3984)


Voronoi diagrams are among the most extensively studied objects in computational geometry with useful applications in different areas of science. To understand impacts of non-Euclidean geometry on computational geometry, this paper investigates the Voronoi diagram in hyperbolic space specially the one in the Poincaré hyperbolic disk, which is a 2-dimensional manifold with negative curvature. We first prove some lemma in Poincaré hyperbolic disk and then give an incremental algorithm to construct Voronoi diagram.


Voronoi Diagram Computational Geometry Negative Curvature Hyperbolic Geometry Incremental Algorithm 
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  1. 1.
    Alt, H., Schwarzkopf, O.: The Voronoi diagram of curved objects. In: Proc. 11th Annu. ACM Sympos. Comput. Geom., pp. 89–97 (1995)Google Scholar
  2. 2.
    Anderson, J.W.: Hyperbolic Geometry. Springer, New York (1999)zbMATHGoogle Scholar
  3. 3.
    Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.R., Urrutia, J. (eds.) Hand book of Computational Geometry, pp. 201–290. Elsevier Science publishers B. V., North Holand (2000)CrossRefGoogle Scholar
  4. 4.
    Boissonat, J.-D., Yvinec, M.: Non-Euclidean metrics, §18.5 in Algorithmic Geometry, pp. 449–454. Cambridge University Press, Cambridge (1998)Google Scholar
  5. 5.
    Chew, L.P., Drysdale, R.L.: Voronoi diagram based on convex distance functions. In: Proc. 1st Ann. Symp. Comp. Geom., pp. 235–244 (1985)Google Scholar
  6. 6.
    Drysdale, S.: Voronoi Diagrams: Applications from Archaology to Zoology. Regional Geometry Institute. Smith College (July 19, 1993)Google Scholar
  7. 7.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Heidelberg (1987)zbMATHGoogle Scholar
  8. 8.
    Françcois, A.: Voronoi diagrams of semi-algebraic sets. Ph.D Thesis. Department of Computer Science. The University of British Colombia (January 2004)Google Scholar
  9. 9.
    Goodman-Strauss, C.: Compass and Straightedge in the Poincaré Disk. Amer. Math. Monthly 108, 33–49 (2001)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Green, P.J., Sibson, R.: Computing Dirichlet Tesselation in the plane. The Computer Journal 21, 168–173 (1978)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Karavelas, M.: 2D Segment Voronoi Diagrams. CGAL User and Reference Manual. All parts. Chapter 43 (December 20, 2004)Google Scholar
  12. 12.
    Karavelas, M.I., Yvinec, M.: The voronoi diagram of planar convex objects. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 337–348. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Kim, D.-S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set: 2. Geometry. Computer Aided Geometric Design 18, 563–585 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Koltun, V., Sharir, M.: Polyhedral Voronoi diagrams of polyhedra in three dimensions. In: Proc. 18th Annu. ACM Sympos. Comput. Geom., pp. 227–236 (2002)Google Scholar
  15. 15.
    Koltun, V., Sharir, M.: Three dimensional Euclidean Voronoi diagrams of lines with a fixed number of orientations. In: Proc. 18th Annu. ACM Sympos. Comput. Geom., pp. 217–226 (2002)Google Scholar
  16. 16.
    Lee, D.T.: Two-dimensional Voronoi diagrams in the Lp metric. JASM 27(4), 604–618 (1980)zbMATHGoogle Scholar
  17. 17.
    Morgan, F.: Riemannian Geometry: A Beginner’s Guide. A K Peters. Ltd (1993)Google Scholar
  18. 18.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial tesselations: concepts and applications of Voronoi diagrams, 2nd edn. John Wiley & Sons Ltd., Chichester (2000)zbMATHGoogle Scholar
  19. 19.
    Onishi, K., Takayama, N.: Construction of Voronoi diagram on the Upper halfplane. IEICE TRANS. Fundamentals E00-X(2) (Febrauary 1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zahra Nilforoushan
    • 1
  • Ali Mohades
    • 1
  1. 1.Faculty of Math. and Computer Sc.AmirKabir University of Tech.TehranIran

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