Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

Generic Remeshing of 3D Triangular Meshes with Metric-Dependent Discrete Voronoi Diagrams

Abstract: In this paper, we propose a generic framework for 3D surface remeshing. Based on a metric-driven Discrete Voronoi Diagram construction, our output is an optimized 3D triangular mesh with a user-defined vertex budget. Our approach can deal with a wide range of applications, from high-quality mesh generation to shape approximation. By using appropriate metric constraints, the method generates isotropic or anisotropic elements. Based on point sampling, our algorithm combines the robustness and theoretical strength of Delaunay criteria with the efficiency of an entirely discrete geometry processing. Besides the general described framework, we show the experimental results using isotropic, quadric-enhanced isotropic, and anisotropic metrics, which prove the efficiency of our method on large meshes at a low computational cost.

Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model.

Abstract: We study multicellular tumor spheroids by introducing a new three-dimensional agent-based Voronoi-Delaunay hybrid model. In this model, the cell shape varies from spherical in thin solution to convex polyhedral in dense tissues. The next neighbors of the cells are provided by a weighted Delaunay triangulation with on average linear computational complexity. The cellular interactions include direct elastic forces and cell-cell as well as cell-matrix adhesion. The spatiotemporal distribution of two nutrients---oxygen and glucose---is described by reaction-diffusion equations. Viable cells consume the nutrients, which are converted into biomass by increasing the cell size during the ${\mathrm{G}}_{1}$ phase. We test hypotheses on the functional dependence of the uptake rates and use computer simulations to find suitable mechanisms for the induction of necrosis. This is done by comparing the outcome with experimental growth curves, where the best fit leads to an unexpected ratio of oxygen and glucose uptake rates. The model relies on physical quantities and can easily be generalized towards tissues involving different cell types. In addition, it provides many features that can be directly compared with the experiment.

A faster divide-and-conquer algorithm for constructing delaunay triangulations

Abstract: An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its ź(n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn≤216, the range of the experiments. It is conjectured that the average number of edges it creates--a good measure of its efficiency--is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theLp metric for 1

Constrained Delaunay triangulations

Abstract: Given a set of n vertices in the plane together with a set of noncrossing edges, the constrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimal O(n log n) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (nonDelaunay) triangulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that should make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles in the plane and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.

Optimality of the Delaunay triangulation in źd

Abstract: In this paper we present new optimality results for the Delaunay triangulation of a set of points in ?d. These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ?d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.