A perfect inhomogeneous quadratic form is defined by that it can be reconstructed in a unique way from its arithmetic minimum and all integral vectors on which it is attained. Such a form defines an integral Delaunay polytope that has a remarkable property of having a unique circumscribed ellipsoid whose interior is free of integral points. A polytope with this uniqueness property is called a p...

This paper discusses a problem for determining whether a given plane graph is a Delaunay graph, i.e., whether it is topologically equivalent to a Delaunay triangulation. There exists a theorem which characterizes Delaunay graphs and yields a polynomial time algorithm for the problem only by solving a certain linear inequality system. The theorem was proved by Rivin based on arguments of hyperbo...

The edge-flip technique can be used for transforming any existing triangular mesh into one that satisfies the Delaunay condition. Although several implementations for generating Delaunay triangulations are known, to the best of our knowledge no full parallel GPU-based implementation just dedicated to transform any existent triangulation into a Delaunay triangulation has been reported yet. In th...

Georges Voronoi (1908-09) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with L-type domains. A form is perfect if it can be reconstructed from all representations of its arithmetic minimum. Two forms have the same L-type if Delaunay tilings of their lattices are affinely equivalent. Delaunay (1937-38) asked about pos...

In this paper we discuss the automatic construction of quality nonobtuse boundary Delaunay triangulations of polygons such as needed for control volume or nite element method applications. These are Delaunay triangulations whose smallest angles are bounded and, in addition, whose boundary triangles do not have obtuse angles opposite to any boundary or interface edge. The method we propose in th...

In the study of depth functions it is important to decide whether we want such a function to be sensitive to multimodality or not. In this paper we analyze the Delaunay depth function, which is sensitive to multimodality and compare this depth with others, as convex depth and location depth. We study the stratification that Delaunay depth induces in the point set (layers) and in the whole plane...

In many computational applications involving 3D geometric modeling, a space decomposition of the problem domain whose elements satisfy some quality constraints is often required. The Delaunay tetrahedrization is a popular choice due to its many important geometric properties and to the existence of efficient algorithms to compute and to maintain it. However, the Delaunay tetrahedrization has a ...

Four constructions of constant mean curvature (CMC) hypersurfaces in S are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all values of the mean curvature. Second, ...

Delaunay meshing is a popular technique for mesh generation. Usually, the mesh has to be refined so that certain fidelity and quality criteria are met. Delaunay refinement is achieved by dynamically inserting and removing points in/from a Delaunay triangulation. In this paper, we present a robust parallel algorithm for computing Delaunay triangulations in three dimensions. Our triangulator offe...

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hype...