H. Edelsbrunner and N. R. Shah. Incremental topological ipping works for regular triangulations. In Proc. 8th ACM Symp. Comp. Geometry, pages 43-52, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....d dimensions does not work here. 2 Preliminaries 2.1 Delaunay Triangulation Delaunay triangulations are widely used to generate d dimensional simplicial meshes, especially the triangular and tetrahedral meshes. There are abundant well studied algorithms to construct the Delaunay triangulations [8,9]. We will extensively use the following properties about Delaunay triangulations in our algorithm. After inserting a new vertex p, every new d dimensional simplex created in the Delaunay triangulation of the new vertex set has p as one of its vertices. The new triangulation can be updated by ....

H. Edelsbrunner, N. R. Shah, Incremental topological ipping works for regular triangulations, Algorithmica 15.

....is an essential, although often dicult, step in the numerical solution of partial di erential equations on complex problem domains. Much work has been done on the development and implementation of algorithms to improve the quality of a mesh through topological changes, such as edge or face ipping [7, 13, 14], alone or in combination with geometric changes, such as vertex smoothing [2, 5, 18] These approaches usually demand that the initial mesh be valid; however, recent work has begun to consider the problem of recovering a valid mesh from a topologically correct mesh that contains inverted, or ....

H. Edelsbrunner and N. Shah. Incremental topological ipping works for regular triangulations. In Proceedings of the 8th ACM Symposium on Computational Geometry, pages 43-52, 1992.

....to the Voronoi diagram of S [10] and the regular and Delaunay [4] tetrahedralizations for S coincide. In this paper we discuss REGTET, a Fortran 77 program for computing regular tetrahedralizations (or Delaunay tetrahedralizations in the absence of weights) with incremental topological ipping [6] and lexicographical manipulations [3] A copy of REGTET that includes instructions for its execution can be obtained from http: math.nist.gov JBernal. 2 Topological Flipping Let T be a tetrahedralization for S, let t be a tetrahedron in T , and let p a point in S that is not a vertex of t. ....

....0 the tetrahedra whose vertex sets are fq j 1 ; q j 2 ; q j 3 ; pg, and fq j 2 ; q j 3 ; q j 4 ; pg, respectively. It then follows that T 1 consists of t and t 0 , and T 2 exactly of t 1 . 3 Lexicographical Manipulations Program REGTET which is based on an algorithm by Edelsbrunner and Shah [6] computes a regular tetrahedralization for the set S by adding the points in S one at a time into a regular tetrahedralization for the set of previously added points. This implies that before any points in S are added a regular tetrahedralization must be rst constructed by REGTET with vertices ....

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Edelsbrunner, H., Shah, N. R.: Incremental topological ipping works for regular triangulations. Algorithmica 15(3) (1996) 223-241

.... combinatorial properties Delaunay triangulation local improvement le: optriang date: November 15, 1999 2 The Delaunay triangulation is a special instance of regular triangulations, which are obtained by projecting the lower part of a convex hull in 3 space; see e.g. Edelsbrunner and Shah [26]. Some more optimality properties of DT (S) can be derived from this fact; see Musin [55] DT (S) is the geometric dual of the famous Voronoi diagram of a point set S and can be computed in O(n log n) time and O(n) space by various di erent approaches; see e.g. 5] 25] 31] Su and Drysdale ....

H.Edelsbrunner, N.R.Shah: `Incremental topological ipping works for regular triangulations', Algorithmica 15 (1996) pp. 223-241 7 date: November 15, le: optriang

....the replacement of their common bounding edge by 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 ....

H. Edelsbrunner and N. R. Shah, `Incremental topological ipping works for regular triangulations,' Algorithmica 15:223-241, 1996.

....ipping the adjacent edge of the two pairs of non Delaunay triangles, as in the right most part of Figure 1. Many solutions have been proposed to compute DT [1] most of them fall into three broad classes: on line (or incremental insertion) incremental construction, and divide conquer On line [12] methods start with a tetrahedron which contains the pointset, then they insert the points in P one at a time: the tetrahedron which contains the point that is currently being added is partitioned into sub tetrahedra by inserting it as a new vertex. The empty circumsphere criterion is tested ....

H. Edelsbrunner and N. R. Shah. Incremental topological ipping works for regular triangulations. In Proceedings of the 8th Annual ACM Symposium on Computational Geometry, pages 43-52, June 1992.

....a sphere of a vertex outside the star vanishes. We say the weight of u at that time is critical. To prevent the weighted square distance to become negative, we locally change the triangulation by a ip, which replaces two by three tetrahedra that occupy the same space, or vice versa. We refer to [8] for details on maintaining a weighted Delaunay triangulation by ipping. We may compute the critical weights of u in increasing order by breadth rst search using a priority queue. At any moment during the process, the prestar of u consists of all tetrahedra in the initial weighted Delaunay ....

H. Edelsbrunner and N. R. Shah. Incremental topological ipping works for regular triangulations. Algorithmica 15 (1996), 223-241.

.... class has been studied extensively in the geometry literature where its meshes are known as regular triangulations [3] and also as coherent triangulations [12] The fast algorithms for Delaunay triangulations extend with minor modi cation to the larger class of weighted Delaunay triangulations [9]. Previous work. The generation of meshes with wellshaped triangles in R 2 is reasonably well understood. Bern, Eppstein and Gilbert prove that quad tree decompositions can be used to generate meshes free of badly shaped triangles that adapt to the local density of input speci cations [2] ....

....The Delaunay triangulation of S, denoted as Del S, is the 3 complex consisting of all Delaunay tetrahedra and their triangles, edges, and vertices. Delaunay triangulations are popular meshes for several reasons. If S is in general position then Del S is unique and can be eciently constructed [4, 9]. The changes caused by deleting or inserting a point are typically local. Del S contains all edges of a minimum spanning tree, and for each p 2 S it contains the edge to the closest point. Delaunay triangulations are optimal with respect to smallest containing spheres of tetrahedra, see [17] ....

[Article contains additional citation context not shown here]

H. Edelsbrunner and N. R. Shah. Incremental topological ipping works for regular triangulations. Algorithmica 15 (1996), 223-241.

.... class has been studied extensively in the geometry literature where its meshes are known as regular triangulations [3] and also as coherent triangulations [12] The fast algorithms for Delaunay triangulations extend with minor modi cation to the larger class of weighted Delaunay triangulations [9]. Previous work. The generation of meshes with wellshaped triangles in R 2 is reasonably well understood. Bern, Eppstein and Gilbert prove that quad tree decompositions can be used to generate meshes free of badly shaped triangles that adapt to the local density of in put speci cations [2] ....

....The Delaunay triangulation of S, denoted as Del S, is the 3 complex consisting of all Delaunay tetrahedra and their triangles, edges, and vertices. Delaunay triangulations are popular meshes for several reasons. If S is in general position then Del S is unique and can be e ciently constructed [4, 9]. The changes caused by deleting or inserting a point are typically local. Del S contains all edges of a minimum spanning tree, and for each p 2 S it contains the edge to the closest point. Delaunay triangulations are optimal with respect to smallest containing spheres of tetrahedra, see [17] ....

[Article contains additional citation context not shown here]

H. Edelsbrunner and N. R. Shah. Incremental topological ipping works for regular triangulations. Algorithmica 15 (1996), 223-241.

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H. Edelsbrunner and N. R. Shah. Incremental topological ipping works for regular triangulations. In Proc. 8th ACM Symp. Comp. Geometry, pages 43-52, 1992.

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H. Edelsbrunner and N. Shah. Incremental topological ipping works for regular triangulations. In Proceedings of the 8th ACM Symposium on Computational Geometry, pages 43-52, 1992.

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