In the pursuit of realistic terrain models, Gudmundsson, Hammar, and van Kreveld introduced higher-order Delaunay triangulations. A usual Delaunay triangulation is a 0order Delaunay triangulation, thus unique for a non-degenerate point set, while order-k Delaunay triangulations can be non-unique when k ≥ 1. In this work, we propose an algorithm to list all order-k Delaunay triangulations of a g...

A polytope D, whose vertices belong to a lattice of rank d, is Delaunay if it can be circumscribed by an ellipsoid E with interior free of lattice points, and so that the vertices of D are the only lattice points on the quadratic surface E. If in addition E is uniquely determined by D, we call D a perfect Delaunay polytope. Thus, in the perfect case, the lattice points on E, which are the verti...

A lattice Delaunay polytope D is called perfect if it has the property that there is a unique circumscribing ellipsoid with interior free of lattice points, and with the surface containing only those lattice points that are the vertices of D. An inhomogeneous quadratic form is called perfect if it is determined by such a circumscribing ”empty ellipsoid” uniquely up to a scale factorComplete pro...

The Delaunay Tree is a hierarchical data structure that has been introduced in [7] and analyzed in [6, 4]. For a given set of sites S in the plane and an order of insertion for these sites, the Delaunay Tree stores all the successive Delaunay triangulations. As proved before, the Delaunay Tree allows the insertion of a site in logarithmic expected time and linear expected space, when the insert...

While several existing Delaunay refinement algorithms allow acute 3D piecewise linear complexes as input, algorithms producing conforming Delaunay tetrahedralizations (as opposed to constrained or weighted Delaunay tetrahedralizations) often involve cumbersome constructions and are rarely implemented. We describe a practical construction for both “collar” and “intestine”-based approaches to thi...

Aiming at Delaunay triangulation with islets constrains in terrain simulation. A general Delaunay triangulation algorithm for constrained data set with islets is proposed. The algorithm firstly constructs Constrained Delaunay Triangulation with constraint polygons which are inner boundary of islets, then according to topological relations within edge, surface, arc segment, applies bidirectional...

A b s t r a c t . Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of a...

We extend the notion of higher-order Delaunay triangulations to constrained higherorder Delaunay triangulations and provide various results. We can determine the order k of a given triangulation in O(min(nk log n log k, n log n)) time. We show that the completion of a set of useful order-k Delaunay edges may have order 2k − 2, which is worst-case optimal. We give an algorithm for the lowest-ord...

We show that, given a set of randomly distributed wireless nodes with density n, when the transmission range rn of wireless nodes satis£es πr n ≥ 3 log n+c(n) n , the localized Delaunay triangulation (LDel) [1] is the same as the Delaunay triangulation with high probability, where c(n) → ∞ as n goes in£nity. Our experiments show that the delivery rates of existing localized routing protocols ar...

Given a set S of points in the plane, a triangulation of S is a partition of the convex hull of S into triangles whose vertices are the points of S. A triangulation of S is said to be Delaunay if no point of S lies inside the triangles’ circumcircles. In this thesis, we study a generalization of these notions to a set S of disjoint segments in the plane. At first, we define a new family of diag...