On the Number of Higher Order Delaunay Triangulations

  • Dieter Mitsche
  • Maria Saumell
  • Rodrigo I. Silveira
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)

Abstract

Higher order Delaunay triangulations are a generalization of the Delaunay triangulation which provides a class of well-shaped triangulations, over which extra criteria can be optimized. A triangulation is order-k Delaunay if the circumcircle of each triangle of the triangulation contains at most k points. In this paper we study lower and upper bounds on the number of higher order Delaunay triangulations, as well as their expected number for randomly distributed points. We show that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delaunay triangulations, the maximum number is 2n − 3. Next we show that uniformly distributed points have an expected number of at least \(2^{\rho_1 n(1+o(1))}\) first order Delaunay triangulations, where ρ1 is an analytically defined constant (ρ1 ≈ 0.525785), and for k > 1, the expected number of order-k Delaunay triangulations (which are not order-i for any i < k) is at least \(2^{\rho_k n(1+o(1))}\), where ρk can be calculated numerically.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dieter Mitsche
    • 1
  • Maria Saumell
    • 2
  • Rodrigo I. Silveira
    • 2
  1. 1.Centre de Recerca MatemàticaUniversitat Autònoma de BarcelonaSpain
  2. 2.Dept. Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaSpain

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