We prove that the expected size of the 3D Delaunay triangulation of n points evenly distributed on a cylinder is Θ(n log n). This shows that the n √ n behavior of the cylinder-example of Erickson [9] is pathological. Key-words: Delaunay triangulation, random distribution, random sample, surface reconstruction Ce travail préliminaire a été joint avec un travail parallèle de Jeff Erickson et sera...

This article deals with dependencies of a piecewise linear vector field and the triangulation of the domain. It shows that the topology of the field may depend on the triangulation and gives a suitable choice to obtain a simple topology by changes of the triangulation. The main point is the appearance of pairs of critical points with positive and negative Poincaré index. Many of these occurence...

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...

A (geometric) triangulation in the plane is a maximal connected plane graph with straight edges. It is thus a plane graph whose bounded faces are triangles. For a xed set of vertices, there are, in general, exponentially many w ays to form a triangulation. Various criteria related to the geometry of triangles are used to deene what one could mean by a triangulation that is optimal over all poss...

Given a planar point set, we consider three classes of optimal triangulations: (1) the minimum weight triangulation with angular constraints (constraints on the minimum angle and the maximum angle in a triangulation), (2) the angular balanced triangulation which minimizes the sum of the ratios of the maximum angle to the minimum angle for each triangle, and (3) the area balanced triangulation w...

In this paper we consider the tolerance of a geometric or combinatorial structure associated to a set of points as a measure of how much the set of points can be perturbed while leaving the (topological or combinatorial) structure essentially unchanged. We concentrate on studying the Delaunay triangulation and show that its tolerance can be computed in O(n) time if the triangulation is given as...

In this paper we construct and study a new 15-vertex triangulationX of the complex projective plane CP. The automorphism group of X is isomorphic to S4 ×S3. We prove that the triangulation X is the minimal by the number of vertices triangulation of CP admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for simplices of X and show that the automorphism...

We introduce the Dyck path triangulation of the cartesian product of two simplices ∆n−1×∆n−1. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of ∆rn−1 × ∆n−1 using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of t...

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...

When reconstructing a surface from irregularly spaced data, sampled from a closed surface in 3D, we need to decide how to identify a good triangulation. As a measure of quality we consider various differential geometrical properties, such as integral Gaussian curvature, integral mean curvature and area. We furthermore study a non-functional approach, which is based on a mapping procedure. A loc...