R. Sibson, "Locally equiangular triangulation," The Computer Journal, vol. 21, pp. 243--245, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....adaptive clustering method for terrain modeling. Arbitrary samples taken from largescale terrain models are recursively selected according to their relevance. Continuous approximations of the terrain model are constructed based on the individual sets of selected sites using Sibson s interpolant [10]. We have implemented this algorithm using a Delaunay triangulation, i.e. the dual of a Voronoi diagram, as underlying data structure. Constructing a Delaunay triangulation requires less implementation than constructing the corresponding Voronoi diagram, since a lot of special cases (these where ....

....time, which requires the use of some acceleration method. For applications in dimensional spaces aces 49 the Delaunay triangulation consists of simplices whose circumscribed dimensional hyperspheres contain no other point. The adaptive clustering algorithm uses Sibson s interpolant [10] constructing the functions J . Sibson s interpolant is based on blending function values . associated with the points . that define the Voronoi diagram. The blending weights for Sibson s interpolant at a point 45 7 are computed by inserting temporarily into the Voronoi diagram and by ....

R. Sibson, locally equiangular triangulation. The Computer Journal, Vol. 21, No. 2, 1992, pp. 65--70.

....the above steps for determining child nodes, then the nodes compute a spanning tree with root node R. 2.3 Building Delaunay Triangulations with Local Properties Delaunay triangulations can be defined in terms of a locally enforceable property. A triangulation is said to be locally equiangular [23] if, for every convex quadrilateral formed by triangles 4acb and 4abd that share a common edge ab, the minimum internal angle of triangles 4acb and 4abd is at least as large as the minimum internal angle of triangles 4acd and 4cbd. This is illustrated in Figure 4. In [23] it was shown that a ....

....to be locally equiangular [23] if, for every convex quadrilateral formed by triangles 4acb and 4abd that share a common edge ab, the minimum internal angle of triangles 4acb and 4abd is at least as large as the minimum internal angle of triangles 4acd and 4cbd. This is illustrated in Figure 4. In [23] it was shown that a locally equiangular triangulation is a Delaunay triangulation. In a graph that is a triangulation, each node N can enforce the locally equiangular property for all quadrilaterals formed by N and its neighbors. In Figure 5, node N can detect that the locally equiangular ....

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R. Sibson. Locally equiangular triangulations. The Computer Journal, 21(3):243--245, 1977.

....The property holds for triangles 4abc and 4abd if the minimum internal angle is at least as large as the minimum internal angle of triangles 4acd and 4cdb. A Delaunay triangulations can be defined in terms of a locally enforceable property. A triangulation is said to be locally equiangular [15] if, for every pair of triangles 4acb and 4abd that share a common edge (in this case, ab) the minimum internal angle of triangles 4acb and 4abd is at least as large as the minimum internal angle of triangles 4acd and 4cbd. This property is illustrated in Figure 5. In [15] it was shown ....

....equiangular [15] if, for every pair of triangles 4acb and 4abd that share a common edge (in this case, ab) the minimum internal angle of triangles 4acb and 4abd is at least as large as the minimum internal angle of triangles 4acd and 4cbd. This property is illustrated in Figure 5. In [15] it was shown that a locally equiangular triangulation is a Delaunay triangulation. If a node knows the coordinates of its neighbors neighbors it can determine (and enforce) that the local triangles satisfy the locally equiangular property. We have devised and implemented a network protocol ....

R. Sibson. Locally equiangular triangulations. The Computer Journal, 21(3):243--245, 1977.

....and maybe the most famous triangulation for a point set S is the Delaunay triangulation, DT (S) See [5] 25] 31] 43] for extensive treatments and surveys. DT (S) contains for each triple of points in S the corresponding triangle provided its circumcircle is empty of points in S. Sibson [63] proved that DT (S) can be constructed from any triangulation T of S by applying a sequence of good edge ips. These are exchanges of diagonals in one of T s convex quadrilaterals Q such that after the ip the two new triangles are locally Delaunay, i.e. have circumcircles empty of vertices ....

R.Sibson: `Locally equiangular triangulations', The Computer Journal 21 (1978) pp. 243-245

....triangulation has furnished a number of problems of longstanding interest in computational geometry. These problems have applications to cartography, spatial data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [20, 24], minimizing the maximum angle [6] minimizing the minimum angle [7] minimizing the maximum aspect ratio [3] and minimizing the maximum edge length [5] The most longstanding open problem in computational geometry is the complexity of another optimal triangulation problem, the minimum weight ....

R. Sibson. Locally equiangular triangulations. Computer J. 21 (1978) 243--245.

.... if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 10, 11] Polynomial time algorithms have been developed in determining those triangulations [2, 7, 8, 15]. In computational geometry another important research object is to compute the minimum weight triangulation. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the edges in the triangulation. Despite the intensive study made during the lase two decades, it remains ....

R. Sibson, "Locally equiangular triangulations", Comput. J, 21, pp.243-245, 1978.

....a triangulation of the points. Most positive results are related to the Delaunay triangulation [Del34] It has been shown that among all triangulations of a given finite point set, the Delaunay triangulation optimizes various criteria. The Delaunay triangulation maximizes the minimum angle [Sib78], minimizes the maximum circumscribing circle [D AS89] and minimizes the maximum smallest enclosing circle [D AS89, Raj91] Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, ....

R. Sibson. Locally equiangular triangulations. Comput. J. 21 (1978), 243--245.

.... Though there is yet no general, formal and practical criterion to measure the quality of a triangulation, applicable to various classes of data, the main idea governing different criteria, is that long thin triangles should be avoided and that the triangles should be as equiangular as possible [Sch87, Sib78]. The argumentation is two fold: 1) Aestetic aspects. The PLIS constructed from thin triangles is not, in general, visually pleasing. 2) Numerical justification. In spline approximation theory thin triangles are undesirable because approximation errors usually depend on the thinness of the ....

....for a triangulation as well. Moreover, the Gaussian and mean curvatures take descriptive geometrical, visual and easily evaluated forms. Making use of the theory of polyhedral metrics we can give a geometrical interpretation of some known triangulations. Delaunay triangulations (DT) [Sch87, Sib78]. We can say they preserve intrinsic geometry of surfaces (within a certain limit) Indeed, as the triangles in DT are the most thick with respect to all other triangulations, hence, the deviation from tangent planes is not big, and thus the lengths and the angles are not to be greatly changed. ....

R. Sibson. Locally equiangular triangulations. Comp. J., 21:243--245, 1978.

....areas. An important type of triangulation is the Delaunay triangulation [Dela34] It is dual to the socalled Voronoi diagram [Voro08] The popularity of the two dimensional Delaunay triangulation is partly due to the fact that it optimizes various quality measures, including the smallest angle [Sibs78], the largest circumscribed circle [D AS89] the largest minimum enclosing circle [D AS89, Raja91] and the integral of the gradient squares [Ripp90] Algorithms that construct the Delaunay triangulation of a given set of n points in the plane in time O(n log n) can be found in Guibas, Stolfi ....

R. Sibson. Locally equiangular triangulations. Comput. J. 21 (1978), 243--245.

....defines a triangulation of the points. Most positive results are related to the Delaunay triangulation defined for finite point sets [Del34] It has been shown that among all triangulations of a given point set, the Delaunay triangulation optimizes various criteria. These include the maxmin angle [Sib78], the minmax circumscribed circle [D AS89] the minmax smallest enclosing circle [D AS89, Raj91] and the minimum integral of the gradient squared [Rip90] Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms ....

R. Sibson. Locally equiangular triangulations. Comput. J. 21 (1978), 243--245.

....This operation was incorporated into a plane sweep scheme (Section 2.1) to incrementally compute a locally optimal triangulation T (S) Laws77] that is, one that has no edge flip to improve its quality. It was found that this locally optimality actually implies that T (S) is a completion of D(S) [Dela34, Sibs78]. Since any two completions of D(S) have the same value for their smallest angles, T (S) is actually a max min angle triangulation (see [Edel87, page 301 303] There are various possibilities to implement the above scheme. One variant is to first compute an arbitrary triangulation, T (S) and ....

.... geometric transformation [Brow79] and plane sweep [Fort87] Furthermore, it can be computed in time O(n log n) with high probability by randomized incrementation [GKS92] Among all triangulations of a given point set S, the Delaunay triangulation optimizes criteria such as the max min angle [Sibs78], the min max circumscribed circle [D AS89] the min max smallest enclosing circle [D AS89, Raja91] and the minimum integral of the gradient squared [Ripp90] As a graph structure, the Delaunay triangulation of S, D(S) is the straight line dual of the so called Voronoi diagram [Voro07, Voro08, ....

R. Sibson. Locally equiangular triangulations. Comput. J. 21 (1978), 243--245.

.... algorithm allows the elimination of the smallest remaining angles by taking advantage of the fact that the edge swapping step locally maximizes the minimum angles over pairs of adjacent triangles as stated in the following theorem, which is a simplified version of a theorem stated by Sibson [20]. Theorem 3 Let A, B, C, X be non co circular points defining a convex quadrilateral as shown in Figure 4. Then, over the convex quadrilateral ABCX with diagonal CB, the Delaunay edge swapping step is performed (selecting the diagonal AX instead of CB) if and only if the point X lies strictly in ....

R. Sibson, "Locally equiangular triangulations," The Computer Journal, vol. 21, pp. 243--245, 1978.

....[4, 40, 26, 27, 43, 44] Finding the optimal triangulation is a particular mesh generation problem in computational geometry. The most often used optimization criteria [4] include maximizing the minimum angle among all elements of the partition (solved by the wellknown Delaunay triangulation [48]) minimizing the maximum angle [14] minimizing a maximum mincontainment ellipse [13] and minimizing total length (an outstanding open problem in the field [17] Variants of these problems allow one to add extra vertices, called Steiner points, in order to further improve the quality of the ....

....mesh elements when generate some mesh vertices. The typical example of the first approach is the sphere packing method [3, 5, 27, 37, 50, 28] The advancing front methods [6, 30, 31] are the most widely used second approach. 15 If we have the set of points in the domain, Delaunay triangulation [48] of the point set minimizes the smallest angle for two dimension domain. Assume that P is a point set in IR d . The simplex defined by (d 1) affinely independent points from P is a Delaunay simplex if the circum sphere of the simplex contains no point from P in its interior. The union of all ....

R. Sibson. Locally equiangular triangulations. In Computer Journal, volume 21, pages 243--245, 1978.

....dihedral angles greater than 180 o . In selecting different cutting surfaces for an edge, care is taken to avoid producing short edges, narrow subregions and small dihedral angles. After the operation, the object should consist of a collection of convex polyhedra. Then, Delaunay triangulation [69] can be used to generate elements inside each polyhedron. This method has a disadvantage that the decomposition of an object into convex polyhedra is not guaranteed. Subdivision using the medial axis transform can also be classified as a topology decomposition technique. The medial axis transform ....

....Each node is visited and lines are joined to its neighbouring nodes to form a well shaped quadrilateral. If a well shaped quadrilateral cannot be formed, then a triangle will be formed instead. However, the best known method for generating elements from a set of nodes is the Delaunay triangulation [69]. For a set of points P i , i=1, n in R 2 space, define a set V= V i , i=1, n where V i = x # R 2 : # x P i # # x P j # # j = i Chapter 1 Introduction 5 Section 1.1.3 V i denotes the set of points that are nearer to P i then to any other Ps. V i is an open convex ....

R. Sibson, "Locally equiangular triangulations," The Computer journal, vol 21, pp243-245, 1978.

....triangulation has furnished a number of problems of longstanding interest in computational geometry. These problems have applications to cartography, spatial data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [19, 23], minimizing the maximum angle [6] minimizing the minimum angle [7] minimizing the maximum aspect ratio [3] and minimizing the maximum edge length [5] The most longstanding open problem in computational geometry is the complexity of another optimal triangulation problem, the minimum weight ....

R. Sibson. Locally equiangular triangulations. Computer J. 21 (1978) 243--245.

....to least sharp. As a consequence, the sharpest angle determined by three vertices of a convex polygon can be found in linear time. 1. Introduction A celebrated result in computational geometry is that the Delaunay triangulation of a planar point set maximizes the minimum angle in any triangle [7]. More specifically, if the points are in general position (by which we mean no four points are cocircular) then the sequence of triangle angles, sorted from sharpest to least sharp, is lexicographically maximized over all such sequences constructed from triangulations of the points. In this ....

R. Sibson, Locally equiangular triangulations, Computer J. 21:243--245, 1978.

....Box 1, Kensington NSW 2033, Australia. T.Lambert unsw.edu.au Many other alternative definitions of optimality have been proposed. See [4, 13] for surveys. Amongst all triangulations, The Delaunay triangulation optimizes many triangulation measures. These include ffl maximizing the minimum angle [20], ffl minimizing the maximum circumscribed circle [5] ffl minimizing the maximum smallest enclosing circle 1 [5, 17] ffl minimizing the integral of the gradient squared [18, 16] Little [14] and Schumaker [19] have proposed that the triangles in a good triangulation should have large ....

R. Sibson. Locally equiangular triangulations. The Computer Journal, 21(3):243--245, 1978.

.... if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 15] Polynomial time algorithms have been developed for determining those triangulations [5, 12, 13, 21]. In computational geometry computing the minimum weight triangulation is another important research topic. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the edges in the triangulation. Despite the intensive study made during the lase two decades, it remains ....

R. Sibson, "Locally equiangular triangulations", Computer Journal 21 (1978) 243-245.

....mesh. In particular, it is desirable that the angles of each element are not too small [2, 17] The Delaunay triangulation and Delaunay re nement algorithms generate high quality meshes satisfying this criterion. Speci cally, Delaunay triangulation maximizes the minimum angle among all elements [16]; Delaunay re nement allows one to add extra vertices, called Steiner points, in order to further improve the quality of the mesh [14] SIFFEA contains a two dimensional mesh generator, which can generate exact Delaunay triangulations, constrained Delaunay triangulations, and conforming Delaunay ....

Sibson, R. Locally equiangular triangulations. Computer Journal 21 (1978), 243245.

....equivalent way to define a Delaunay triangulation is to say that it contains all triangles for which the circumscribing circle does not contain any data point in its interior. The Delaunay triangulation has several nice properties. For instance, it maximizes the minimum angle of the triangles [Sib78] which reduces robustness problems and aliasing problems when rendering. Intuitively, if the function is sufficiently sampled, Delaunay triangulation gives in general a good approximation because it connects the points by 2 This assumes the data points are in general position, otherwise some ....

R. Sibson. Locally equiangular triangulations. Comput. J., 21:243--245, 1978.

....of S. If for each triangle in T the inscribing circle contains no other point of S (except the three points that form the triangle) then T is the Delaunay Triangulation DT(S) If the above mentioned general position assumption holds, then the DT exists and it is unique. Furthermore, Sibson [24] has proved in 1978 that the DT is the optimal triangulation according to the Min Max Angle criterion. The DT also guarantees the smoothest piecewise linear approximation for a given set of samples [22] Property (ii) of the VD implies a duality between the VD and the DT, i.e. every vertex of the ....

R. Sibson, "Locally Equiangular Triangulations", Computer Journal, V. 21, No. 3, pp. 243-245, 1978.

....and flips performed to construct a new LOT are those adjacent to the new site. School of Computer Science and Engineering, University of New South Wales, PO Box 1, Kensington NSW 2033, Australia. T. Lambert unsw.edu.au If a triangulation is locally Delaunay then it is globally Delaunay [18], so the flip rule that selects the Delaunay triangulation of the quadrilateral (DT ) is systematic and local. In this paper I prove that DT is the only flip rule invariant under translation and rotation of the sites which is systematic and local. Definition. Given a collection of shapes such ....

R. Sibson. Locally equiangular triangulations. The Computer Journal, 21(3):243--245, 1978.

....interest in computational geometry is optimal triangulation of a planar point set [6] This problem finds application in cartography, spatial data analysis, and finite element methods. Optimization criteria include maximizing the minimum angle (solved by the well known Delaunay triangulation [24, 27]) minimizing the maximum angle [13] minimizing a maximum min containment ellipse [11] and minimizing total length (an outstanding open problem in the field [16, 20] Variants of these problems allow one to add extra vertices, called Steiner points, in order to further improve the quality of ....

R. Sibson. Locally equiangular triangulations. Computer Journal, 21:243--245, 1978.

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R. Sibson, "Locally equiangular triangulation," The Computer Journal, vol. 21, pp. 243--245, 1978.

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R. Sibson. Locally equiangular triangulations. The Computer Journal, 21(3):243--245, 1977.

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R. Sibson, "Locally equiangular triangulations," The Computer Journal, vol. 21, no. 3, pp. 243--245, 1977.

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Sibson, R. "Locally equiangular triangulations", Comput. J., 21, 243--245, 1978.

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R. Sibson, "Locally equiangular triangulations", Computer Journal 21 (1978) 243-245.

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R. Sibson. Locally equiangular triangulations. Computer J., 21:243--245, 1978.

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R. Sibson. Locally equiangular triangulations. Computer Journal , 21 (1978), 243-- 245.

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R. Sibson. Locally equiangular triangulations. Computer J., 21 (1978), 243--245.

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