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: This paper shows that other classical triangulation algorithms, such as the greedy triangulating, and more notably, the minimum weight triangulations, also approximate the complete graph in this sense, and designs an algorithm for constructing extremely sparse planar graphs that approximate thecomplete graph.
Abstract: Chew and Dobkin et. al. have shown that the Delaunay triangulation and its variants are sparse approximations of the complete graph, in that the shortest distance between two sites within the triangulation is bounded by a constant multiple of their Euclidean separation. In this paper, we show that other classical triangulation algorithms, such as the greedy triangulation, and more notably, the minimum weight triangulation, also approximate the complete graph in this sense. We also design an algorithm for constructing extremely sparse (nontriangular) planar graphs that approximate the complete graph.
117 citations
TL;DR: The aim of the method is to show the possibility of generating adaptive surface meshes suitable for the finite element method, directly from an approximated boundary representation of an object created with CAD software, based on the Delaunay method.
Abstract: The aim of the method proposed here is to show the possibility of generating adaptive surface meshes suitable for the finite element method, directly from an approximated boundary representation of an object created with CAD software. First, we describe the boundary representation, which is composed of a simple triangulation of the surface of the object. Then we will show how to obtain a conforming size-adapted mesh. The size adaptation is made considering geometrical approximation and with respect to an isotropic size map provided by an error estimator. The mesh can be used “as is” for a finite element computation (with shell elements), or can be used as a surface mesh to initiate a volume meshing algorithm (Delaunay or advancing front). The principle used to generate the mesh is based on the Delaunay method, which is associated with refinement algorithms, and smoothing. Finally, we will show that not using the parametric representation of the geometrical model allows us to override some of the limitations of conventional meshing software that is based on an exact representation of the geometry.
116 citations
TL;DR: The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes, a characterization of the combinatorial properties ofCDTs and weighted CDTs, the proof of several optimality properties of CDT when they are used for piecewiselinear interpolation, and a simple and useful condition that guarantees that a domain has a CDT.
Abstract: Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions.
The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes (geometric domains that incorporate nonconvex faces with “internal” boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.
116 citations
TL;DR: It is shown that any set of points in the plane has a Hamiltonian triangulation, and it is proved that certain nondegenerate point sets do not admit asequential triangulations.
Abstract: High-performance rendering engines are often pipelined; their speed is bounded by the rate at which triangulation data can be sent into the machine. An ordering such that consecutive triangles share a face, which reduces the data rate, exists if and only if the dual graph of the triangulation contains a Hamiltonian path. We (1) show thatany set ofn points in the plane has a Hamiltonian triangulation; (2) prove that certain nondegenerate point sets do not admit asequential triangulation; (3) test whether a polygonP has a Hamiltonian triangulation in time linear in the size of its visibility graph; and (4) show how to add Steiner points to a triangulation to create Hamiltonian triangulations that avoid narrow angles.
116 citations