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 trian...

Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space R triangulates the convex hull of P , provided that P satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay’s genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stron...

In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay triangulation, where the empty circumcircle property is defined by using the Euclidean metric in R...

Providing adequate software support for unstructured grid generation codes is challenging even for sequential implementations. In parallel implementations, program complexity increases by an order of magnitude due to dynamic, data{dependent, irregular computation and communication requirements. Even more challenging is the implementation of parallel methods that maintain the same quality guaran...

For n + 1 disjoint flats of dimension k in H, we produce a Delaunay cell which is a generalization of the Delaunay simplex associated to n + 1 points in H. Combinatorially, these Delaunay cells resemble truncated n-dimensional simplices. For certain classes of arrangements of flats in H, we prove that these Delaunay cells can be glued together to form a Delaunay complex, with the result that al...

In this paper, we give the definition for the voronoi diagram and its dual graph – Delaunay triangulation for 3D cuboids. We prove properties of the 3D Delaunay triangulation, and provide algorithms to construct and update the Delaunay triangulation. The Delaunay triangulation data structure is used to perform proximity searches for both static and kinetic cases. We describe experimental result...

In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay triangulation, where the empty circumcircle property is defined by using the Euclidean metric in R...