Properties of some Euclidean proximity graphs
Pattern Recognition Letters
Higher order Delaunay triangulations
Computational Geometry: Theory and Applications
A better upper bound on the number of triangulations of a planar point set
Journal of Combinatorial Theory Series A
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Random triangulations of planar point sets
Proceedings of the twenty-second annual symposium on Computational geometry
Generating realistic terrains with higher-order Delaunay triangulations
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
Optimization for first order Delaunay triangulations
Computational Geometry: Theory and Applications
On the number of higher order delaunay triangulations
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
On the number of higher order delaunay triangulations
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
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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 k1, the expected number of order-k Delaunay triangulations (which are not order-i for any ik) is at least $2^{\rho_k n(1+o(1))}$, where ρk can be calculated numerically.