H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangualtions. In Proc. 8th Annual ACM Symposium on Computational Geometry, pages 43--52, June 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is invalid (see Figure 11) 5 Implementation Details and Examples In this section we describe the implementation details of our algorithm. First, a complex corresponding to the molecule is constructed. Second, we sweep a hyperplane across the complex and record the topology changes, or flips [25], that occur, and perform those flips that are between given initial and final probe radii. For each topology, we compute the trimming curves for all of the intersections of atoms. These trimming curves, together with the atom center locations, then allow us to reconstruct the molecular surface ....

....with the atom center locations, then allow us to reconstruct the molecular surface patches in 3 space. The first stage in the implementation is a preprocessing step. We construct the complex C, and then let the hyperplane H sweep along the r axis from r = Gamma1 to r = 1 and record the flips [25] that occur dynamically. Second, once an array of flips sorted by r is obtained, we perform the flips that are between the initial probe radius r 1 and the destination probe radius r 2 . Figure 12 illustrates an example of how the smoothed molecular surface 18 Fig. 12. Several snapshots in two ....

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

....essential, although often difficult, step in the numerical solution of partial differential 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 flipping [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 flipping works for regular triangulations. In Proceedings of the 8th ACM Symposium on Computational Geometry, pages 43--52, 1992.

....circle, and the integral of the gradient squares[20, 45] Moreover, the Delaunay triangulation contains the nearest neighbors graph and the minimal spanning tree[39] On the other hand, the Delaunay criterion of a triangulation can be satisfied with out much overhead. Edelsbrunner and Shah[17] have shown that it takes at most O(n log n n [d 21 ) number of edge flippings to transform an arbitrary triangulation to a Delaunay triangulation. Construction of Delaunay triangulations (and Voronoi diagrams) for a point set is well studied in computational geometry, and the algorith mic ....

....of a bad shaped triangle is added as a Steiner point if it does not lie in the diametral circle of any segments in the current Delaunay triangulation. Otherwise the midpoint of such segment is added. A directed acyclic graph (DAG) is used to store the history of the Delaunay triangulation[17, 32, 45], such that the running time for point insertion to the De launay triangulation is proportional to the logarithm of the size of current Delaunay triangulation on average. And the expected running time for the algorithm is O( n q k)log(n q k) where n is the size of the PSLG and k is the total ....

H. Edelsbrunner and N. R. Shah, "Incremental topological flipping works for regular triangu- lations," Proc. 8th ACM $ymp. Cornput. Geom., 43-52, 1992.

....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: ....

....operation that replaces the triangulation that corresponds to the upper boundary by the one that corresponds to the lower boundary. Observe that a flip is defined with a direction, so that it tends toward the Delaunay triangulation of the set S. This idea can be extended to arbitrary dimensions [28]. Pk Pk Pi Figure 4.2: The two triangulations of set S = Pi,Pj,Pk,P,P . The triangulation to the left consists of two tetrahedra: ti,j,k, and ti,j,k, while the triangulation to the right consists three tetrahedra: t, i,j, t, j,k, and t, k,i. 42 Local ....

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H Edelsbrunner and N R Shah. Incremental topological flipping works for regular triangulations. In Proceedings of the Eighth Annual 'ympasium an Computational Geometry, pages 43 52, 1992.

....has size within a constant factor of the size of any almost good meshes for the same domain. Section 6 concludes the paper with discussions. 2 Preliminaries 2. 1 Delaunay Triangulation Due to their numerous desirable properties, and the abundance of the well studied algorithms to construct them [1,6], Delaunay triangulations are widely used in generating tetrahedral meshes. The following properties about Delaunay triangulations are extensively used in this paper. After inserting a new vertex p, all new tetrahedra created in the Delaunay triangulation of the new vertex set are incidentonp. And ....

Edelsbrunner, H., and Shah, N. R. Incremental topological flipping works for regular triangulations. Algorithmica15(1996).

....T 2 (gap triangulation) that fills the gaps between the tiles. To determine T 2 , we apply a Delaunay Triangulation algorithm to the boundary points of the pseudo tiles 1 . The algorithm we use employ a variant of the randomized incremental flip algorithm developed by Edelsbrunner and Shah [2]. Based on the characteristics of the data, we use two different variants of the Delaunay Triangulation: 1. For functional data (i.e. each point of the input data sets lies on the graph of a bivariate function f(x,y) we project the points in the x y plane and apply the 2D Delaunay ....

Edelsbrunner, H., Shah, N., "Incremental topological flipping works for regular triangulations," In the Proceedings of the 8th Annual Symposium on Computational Geometry, 1992, 43-52.

....the final shape of the solid. Rather than sweep a set of connected points, the connected set will serve as the skeleton of the part. with the radius function at each vertex determining the distance of the solid s boundary from the skeleton. Using weighted points is an idea originating from Edelsbrunner s (1992) use of them in visualizing and analysing molecular models (Edelsbrunner, Facello Liang 1998, Facello 1996, Bajaj, Pascucci, Holt Netravali 1998) The functions that this conceptual modeler must perform are shown in Figure 1.2. The function structure (FS) in Figure 1.2 is for a piece of ....

....shown in Figure 2.6. When a flip occurs, some of the tetrahedra neighboring the flipped tetrahedra may become nonregular. These, too, must be flipped, leading to possibly more nonregular tetrahedra. However, it has been shown that only a finite number of flips are required for any vertex insertion (Edelsbrunner Shah 1992). One important note is that the triangulation used by the modeler does not match the exact definition of a regular triangulation. A regular triangulation does 24 not contain any edges connecting any point whose sphere is contained entirely in the union of spheres of other input points. These ....

Edelsbrunner, H. & Shah, N. R. (1992), Incremental topological flipping works for regular triangulations, in `Eighth Annual Computational Geometry Symposium ', ACM, Berlin, pp. 43--52.

....Case Input We now define a notion of the expected case for a dynamic geometry problem. The definition we use has been popularized in a sequence of papers by Mulmuley [30, 31, 32] and is a generalization of the commonly occurring randomized incremental algorithms from computational geometry [14, 16, 25, 27, 29, 39]. We discuss the latter concept first, to motivate our definition of the expected case. An incremental algorithm maintains the solution to some problem as parts of the problem are added one at a time; for instance we might compute the minimum spanning tree of a point set by adding points one by ....

....may or may not behave well in the worst case, but typically has good behavior when the order in which the points are added is chosen uniformly among all possible permutations. Randomized incremental algorithms have been used to construct a number of geometric configurations including convex hulls [14, 16, 39], Voronoi diagrams and Delaunay triangulations [25, 27] linear programming optima [39] and intersection graphs of line segments [29] An important feature of these algorithms is that they do not depend on any special properties of the input point set, such as even distribution on the unit ....

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

....Case Input We now define a notion of the expected case for a dynamic geometry problem. The definition we use has been popularized in a sequence of papers by Mulmuley [25, 26, 27] and is a generalization of the commonly occurring randomized incremental algorithms from computational geometry [12, 14, 20, 22, 24, 33]. We discuss the latter concept first, to motivate our definition of the expected case. A randomized incremental algorithm is just an incremental algorithm (in which points are added one at a time while the solution is maintained for the set of points seen so far) which may or may not behave well ....

....seen so far) which may or may not behave well in the worst case, but which has good behavior when the order in which the points are added is chosen uniformly among all possible permutations. This sort of algorithm has been to construct a number of geometric configurations including convex hulls [12, 14, 33], Voronoi diagrams and Delaunay triangulations [20, 22] linear programming optima [33] and intersection graphs of line segments [24] An important feature of these algorithms is that they do not depend on any special properties of the input point set, such as even distribution on the unit ....

[Article contains additional citation context not shown here]

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

....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 ....

H. Edelsbrunner and N.R. Shah, Incremental Topological Flipping Works for Regular Triangulations, Algorithmica 15 (1996), 223-241.

....change per insertion is O(1) The total change per deletion can be analysed by a similar argument that examines the graph before the deletion, and computes the probability of each edge being removed in the deletion. # We also need the following special case of more general convex hull bounds [12]. Lemma 2.2. The expected number of convex hull vertices that change per update is O(1) Proof. We bound the change per insertion; deletions follow a symmetric argument. In each insertion, the only vertex that can be added is the inserted point, so we need only worry about removed vertices. ....

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

....as special cases some small k values such as k = 4 and k = 5. 4.2 Higher dimensions In higher dimensions, flipping, edge completion and shelling algorithms generalize but things became more difficult. The flipping must be done in a higher dimension, which makes it more intricate to implement [ES96]. Edge completion transforms into facet completion, and must deal with the triangulation of non simply connected polyhedra. Shelling is the easiest method to generalize. Furthermore, the increase in the average value of k with the dimension reinforces its advantage over alternative candidates in ....

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

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H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangualtions. In Proc. 8th Annual ACM Symposium on Computational Geometry, pages 43--52, June 1992.

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

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H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangulations. Algorithmica, 15:223--241, 1996. URL: http: // graphics. stanford. edu/ courses/ cs468-02-winter/

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H. Edelsbrunner and N.R. Shah, Incremental Topological Flipping Works for Regular Triangulations, Algorithmica 15 (1996), 223-241.

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EDELSBRUNNER H., SHAH N. R.: Incremental topological flipping works for regular triangulations. In Proceedings of 8th Annual ACM Symposium on Computational Geometry (1992), pp. 43--52.

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H. Edelsbrunner and N.R. Shah, Incremental Topological Flipping Works for Regular Triangulations, Algorithmica 15 (1996), 223-241.

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

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

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Edelsbrunner H, Shah NR. Incremental topological flipping works for regular triangulations. Algorithmica 1996;15:223--241.

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

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H. Edelsbruner and N. R. Shah, "Incremental topological flipping works for regular triangulations", 8th ACM Symp. Comp. Geom. 1992, 43-52.

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H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangulations. In 8th Annual Comp. Geometry, pages 6-43, 1992.

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

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