B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design, 8:123--142, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an edge or a face. Subdivision of a domain with a tetrahedral mesh is called tetrahedrization. It should be pointed out that a domain with a vertex set may be tetrahedrized in many differentways. Delanuay tetrahedrization has some nice properties, which make it suitable for many applications [12]. A Delanuay d simplicial complex Sigma is one in whichtheinterior of the hypersphere circumscribing any d simplex does not contain avertex of Sigma. 2.2 Volumetric Datasets Roughly speaking, a volumetric dataset is a discrete representation of a scalar field defined over some spatial . ....

....the base mesh must be convex. Cignoni et al. [2] have also proposed a refinement method, which selects the data point that introduces the maximum error with respect to the reference mesh. They used Delaunay tetrahedrization and local operations to modify the mesh when a new point is inserted [12]. Later, they extended their work to include non convex meshes obtained by the deformation of convex domains (curvilinear datasets) 3] The data structure used in the implementation of this study is very similar to the one used in [3] Optimization in Contraction Operations In this chapter, ....

B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design, 8:123--142, 1991.

....map (see below) to transform the triangulation problem in R to the problem of constructing the convex hull in R . This idea goes back to [7] details on the con structions of convex hulls in d dimensions can be found in [21] Another approach is based on local transformations or flips [28,46,52]. A variant of this method will be discussed in section 4.1 of this thesis. Lifting map. Identify 3 with the xlx2x3 space in 4, that is, the subspace x4 = 0. The lifting map is a geometric transform that projects points p = h, r2, r3) in 3 along the x4 axis onto the paraboloid of revolution U: ....

....it is indeed possible to get stuck with faces that are not Delaunay but are still not transformable. An example of such a non transformable face is a triangle whose two incident tetrahedra form a non convex union. This was first observed by Joe [45] however, he also proves the following lemma [46]. Lemma 4.1.1. If a single point p is added to the Delaunay triangulation of a point set S in R 3, then there exists a sequence of flips that results in the Delaunay triangulation of S U p . This result is the basis for the incremental flip algorithm in three dimensions. It was extended to ....

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B Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design, 8(2):123 142, 1991.

....the other diagonal of the quadrilateral, yielding a di erent triangulation of the same point set. Other recent connectivity results have been established for triangulations, including triangulations of polygons [13, 14] topological triangulations [5, 20] and triangulations in higher dimensions [15]. Other classes of objects, such as non crossing spanning trees and Euclidean matchings have also been studied [9, 10] One class of great importance to computational geometry, but for which no satisfactory enumeration method is yet known, is that of simple polygons. Given a set of n points S, one ....

B. Joe, `Construction of three-dimensional Delaunay triangulations using local transformations', Computer Aided Geom. Design 8:123-142, 1991. 19

....with n internal nodes. Under this bijection, flipping an edge in a triangulation corresponds precisely to a rotation in the corresponding binary tree [16, 11] Finally, we note that the flip operation also appears applied to triangulations in higher dimensions or to topological triangulations [1, 2, 6, 13, 14, 17]. In a previous paper [12] the authors studied several questions about flips in triangulations, mainly the question of how many flips are needed to transform a triangulation of a plane point set (or of a simple polygon) into another triangulation. Among other results, it was shown that two ....

B. Joe, Construction of three-dimensional Delaunay triangulations using local transformations, Computer Aided Geom. Design 8 (1991), 123--142.

....P n i=1 W i (X; Y; Z)z i P n i=1 W i (X; Y; Z) 45) where W i (X; Y; Z) X Gamma X i ) 2 (Y Gamma Y i ) 2 (Z Gamma Z i ) 2 ] Gamma 1 2 . The piecewise linear and piecewise cubic transformations in 3 D are obtained by partitioning the space of the images into tetrahedra [12] and mapping corresponding tetrahedral regions into each other using the trilinear transformation function shown in equations (30) 32) or the piecewise cubic functions given by Alfeld [1] The local weighted mean in 3 D is the same as in 2 D except that R = X Gamma X i ) 2 (Y Gamma ....

B. Joe, Construction of three-dimensional Delaunay triangulation using local transformations, Computer Aided Geometric Design, vol. 8, no. 2, pp. 123-142, 1991.

....types of single step flipping, namely T23, T32, T22, and T44, are illustrated in Figure 1. We follow Dr. Joe s notation in terms of the types of flipping as described in [7] Flipping has been proven to be a successful technique in creation of Delaunay triangulation based on point insertion method [5, 6]. Combinations of a sequence of basic flips have been used in improving triangulations towards optimal triangulations [7] Let S be a set of constraints consisting of vertices, edges, and faces (for simplicity, we assume that faces Silvaco International, 4701 Patrick Henry Drive, Santa Clara, ....

B. Joe (1991), Construction of three-dimensional Delaunay triangulations using local transformations, Comput. Aided Geom. Design, 8, pp. 123-142.

....ill shaped elements induced from poorly distributed node locations. Tetrahedral mesh is widely used in commercial applications such as finite element packages and computer aided geometric design systems. Although many algorithms are available for tetrahedral meshing, such as the Delaunay method [6,7,15] and the advancing front method [4,10] most of them are applicable only to isotropic graded meshes. In our method, an anisotropy controls the shape of the mesh over the domain. Typically, the element shapes are controlled by a prescribed anisotropy function ) z y x M M = which defines the ....

Barry Joe, "Construction of three-dimensional Delaunay triangulations using local transformations," Computer Aided Geometric Design 8, pp.123-142, 1991

....set of A p of approachable points w.r.t. the DT, and Theorem 6.2 remains true. The proof is slightly more challenging in this case but we omit the details from this version of the paper. The three dimensional Delaunay triangulation is maintained by simply doing some face edge or edge face ips [12]. As a result, what we need to do in the 3D case in order to update our nearest neighbors structure is the same as in the two dimensional unconstrained case and the kinetic maintenance algorithm works as is, the only di erence being that ips replace edges with facets (or vice versa) as opposed ....

B. Joe. Construction of three-dimensional delaunay triangulations using local transformations. Comput. Aided Geom. Design, 8:123-142, 1991.

....each element s circumsphere contains no other nodes in the mesh. This basic property leads to the development of nodal insertion algorithms where a new node is added to an existing Delaunay type mesh such that the new mesh is also of Delaunay type. We use an edge swapping nodal insertion algorithm [17, 18] which is then used within a general automatic mesh generation procedure. This general procedure involves many aspects such as boundary mesh generation and integrity, internal node generation, and refinement control. Along with the 3D automatic mesh generator, methods are developed to model the ....

....node is placed into the mesh, the elements around this node will rearrange themselves so as to meet this Delaunay criteria. In the mesh generation process, nodes will be inserted into the mesh one by one until the entire mesh satisfies a given quality criteria. We use an edge swapping algorithm [17, 18, 27] for the process of nodal insertion. The edgeswapping algorithm makes use of alternate element configurations to allow the old mesh to accommodate a new node so as to meet the Delaunay criteria. When a new node is inserted into an existing mesh, an advancing front of rearranging element ....

B. Joe, "Construction of three-dimensional Delaunay triangulations using local transformations ", Computer Aided Geometric Design, 8 (1991) 123--142.

....of intersection of the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance [Chazelle and Matousek 1992] Clarkson et al. 1993] Edelsbrunner and Shah 1992] Guibas et al. 1992] [Joe 1991] [Mulmuley 1994] Points are processed one at a time in a random order. In this article, we propose and analyze a strategy for processing points in a more efficient order. The result is a faster algorithm for distributions with interior points. An incremental algorithm for the convex hull ....

....the points. They locate a visible facet by a depth first search of the previous convex hulls. Consider the sequence of facets tested for a point. Quickhull may test the same sequence during successive partitions of the point into outside sets. Edelsbrunner and Shah [Edelsbrunner and Shah 1992] Joe [Joe 1991], and Boissonnat and Devillers Teillaud [Boissonnat and Teillaud 1993] use a similar method for Delaunay triangulations. They express their algorithm in terms of triangulations and the in sphere test. By the correspondence between Delaunay triangulation and convex hull, each triangle is a facet of ....

Joe, B. 1991. Construction of three-dimensional Delaunay triangulations using local transformations. Computer-Aided Geometric Design 8, 123--142.

....much fewer particles and the new methods and strategies developed for the class of simulations we are targeting here. Mesh Generation A new version of the automatic mesh generator described in [4,7] is used here. The algorithm is based on Delaunay methods [9,10] and uses edge swapping techniques [11,12] for the nodal insertion process. This new version was necessary because for this class of simulations we are faced with the need for repetitive generation (due to remeshing) of very large meshes with the number of elements of the order 1 million. The previous, pilot version suffered from both ....

B. Joe, "Construction of three-dimensional Delaunay triangulations using local transformations ", Computer Aided Geometric Design, 8 (1991) 123--142.

....O(n 2 ) facets. Hence, using the separating surfaces method on the implicit 3D Voronoi diagram yields the following result: The implicit Voronoi diagram V (S) of a set S of n points in 3D space can be constructed by computing the 3D Delaunay triangulation with the incremental algorithm by Joe [39], whose time complexity and storage is O(n 2 ) see also [48] Since the most demanding operation of the algorithm in terms of degree is the 3D insphere test, from Lemma 6 we have that the degree of the algorithm that computes V (S) is 5. As in the planar case, the topological structure of V ....

B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Comput. Aided Geom. Design, 8(2):123--142, May 1991.

....rational arithmetic. However, it may be too computationally expensive for large models. Both Bowyer s and Watson s techniques are presented using k dimensional space. Therefore they are suitable for both 2D and 3D applications. Lawson s method has been extended to 3D by Shenton et al. [66] and Joe [38]. The Delaunay triangulation is defined in the convex hull of the given node points. Therefore, if the object contains holes or concavities, some triangles may lie outside the object boundary. They have to be removed to give a valid triangulation. The Delaunay triangulation of a set of nodes ....

B. Joe, "Construction of three-dimensional Delaunay triangulations using local transformations," Computer Aided Geometric Design, vol 8, pp123-142, 1991.

....by polygonal faces and possibly with interior constraining faces and edges. Algorithms for mesh generation have been a research topic over the last two decades. Basically, three main families of algorithms have been described in the literature: octree methods [2, 1] Delaunay based methods [3, 4, 5, 6]; and advancing front methods [7, 8, 9, 10] A good survey of these methods can be found in Owen [11] Unlike the 2D case, 3D Delaunay triangulations do not have the property of maximizing the minimum angle. Nonetheless, they are very attractive from a robustness point of view due to the ....

....too close to an existing vertex or group of vertices. On the other hand, comparisons with floating point numbers generally need tolerances. The best compromise is obtained by using the predicate P 2 . An alternative approach for the cavity algorithm is the 3D flip algorithm created by Barry Joe [4, 5], which generalizes to 3D the well known 2D flip algorithm of Lawson [20] The flip algorithm is very elegant and has the advantage of keeping a triangulation at all times. However, it is more complex to implement. The 3D flip algorithm is based on the fact that there are only two ways for ....

Barry Joe. Construction of three-dimensional Delaunay triangulations using local transformation. Computer Aided Geometric Design, 8:123-- 142, 1991.

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B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design, 8:123--142, 1991.

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B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Comput. Aided Geom. Design, 8(2):123-142, May 1991. 17

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B. Joe, Construction of three-dimensional Delaunay triangulations using local transformations, Computer Aided Geom. Design 8 (1991), 123--142.

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Joe B. Construction of three-dimensional delaunay triangulations using local transformations. Computer Aided Geometric Design 1991; 8:123--142.

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B. Joe, Construction of three-dimensional Delaunay triangulations using local transformations, Computer Aided Geom. Design 8 (1991), 123-142.

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B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design 8:123--142, 1991.

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B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design, 10:123--142, 1989.

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B. Joe, "Construction of three-dimensional Delaunay triangulation using local transformations," Computer Aided Geometric Design, 8, 123-142, North-Holland, 1991.

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Joe, B.,"Construction of Three-Dimensional Delaunay Triangulations From Local Transformations", CAGD, Vol. 8, 1991, pp. 123-142.

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B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design, 8:123--142, 1991.

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B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations. Computer Aided Geometric Design, 8:123--142, 1991.

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