Topic
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.
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TL;DR: In this article, a suitable procedure is presented to characterize the neighbourhood in disperse micro-structures by measuring the Delaunay neighbour distances and neighbour numbers, and a plot of variation coefficients is suggested to differentiate between various random distributions of particles.
Abstract: A suitable procedure is presented to characterize the neighbourhood in disperse micro-structures by measuring the Delaunay neighbour distances and neighbour numbers. Results are discussed from simulation experiments with three germ-grain models, regular arrangements and a seam-like model. A plot of variation coefficients is suggested to differentiate between various random distributions of particles.
30 citations
TL;DR: It is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time $O(n^2 )$.
Abstract: It is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time $O(n^2 )$. The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modifications the algorithm works for arbitrary normed metrics.
30 citations
01 Jan 2009
TL;DR: It is proved that, when the resolution is fine enough (this level is reached very fast in practice), the output mesh becomes homeomorphic to the input while preserving all input features.
Abstract: Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes. These domains include polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all non-manifold spaces. The algorithm is guaranteed to capture the input topology at the expense of four tests, some of which are computationally intensive and hard to implement. The goal of this paper is to present the theory that justifies a refinement algorithm with a single disk test in place of four tests of the previous algorithm. The algorithm is supplied with a resolution parameter that controls the level of refinement. We prove that, when the resolution is fine enough (this level is reached very fast in practice), the output mesh becomes homeomorphic to the input while preserving all input features. Moreover, regardless of the refinement level, each k-manifold element in the input complex is meshed with a triangulated k-manifold. Boundary incidences among elements maintain the input structure. Implementation results reported in a companion paper corroborate our claims. Research supported by NSF, USA (CCF-0430735 and CCF-0635008) and RGC, Hong Kong, China (HKUST 6181/04E). Department of Computer Science and Engineering, HKUST, Hong Kong. Email: scheng@cse.ust.hk Department of Computer Science and Engineering, Ohio State University, Ohio, USA. Email: tamaldey@cse.ohio-state.edu Department of Computer Science and Engineering, Ohio State University, Ohio, USA. Email: levinej@cse.ohio-state.edu
30 citations
TL;DR: It is shown that the interpolant generated by the kth-order standard coordinates have C^k continuity on the Delaunay spheres, and a formula for the gradient of the standard coordinates is derived.
Abstract: There are two types of the discontinuity of the original Voronoi-based interpolants: one appears on the data sites and the other on the Delaunay spheres. Some techniques are known for reducing the first type of the discontinuity, but not for the second type. This is mainly because the second type of the discontinuity comes from the coordinate systems used for the interpolants. This paper proposes a sequence of new coordinate systems, called the kth-order standard coordinates, for all nonnegative integers k, and shows that the interpolant generated by the kth-order standard coordinates have C^k continuity on the Delaunay spheres. The previously known Voronoi-based interpolants coincide with the cases k=0 and k=1. Hence, the standard coordinate systems constructed in this paper can reduce the second type of the discontinuity as much as we want. In addition, this paper derives a formula for the gradient of the standard coordinates.
30 citations
TL;DR: One-dimensional and two-dimensional distributions of the characteristics introduced previously for the forms of Delaunay simplices - tetrahedricity and octahedrality - have been investigated in computer models of a crystal, a liquid and an amorphous solid as discussed by the authors.
Abstract: One-dimensional and two-dimensional distributions of the characteristics introduced previously for the forms of Delaunay simplices - tetrahedricity and octahedricity - have been investigated in computer models of a crystal, a liquid and an amorphous solid. It has been established that in the absence of thermal perturbations (in the proper structure of the liquid and in an amorphous substance) there exists a distinguishable class of simplices with five almost equal edges and the sixth being longer. This class of simplices named isopentacmons, in turn includes the types of good tetrahedra and good quartoctahedra (a quarter of octahedron). In disordered systems the fraction of tetrahedra relative to quartoctahedra exceeds substantially that in the FCC crystal.
30 citations