C L Lawson. Generation of a triangular grid with applications to contour plotting. Memo 299, Jet Propulsion Laboratory, Pasadena, California, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....general position assumption, will be given in section 4.3. We assume general position of the point sets throughout this section. 4. 1 Constructing Delaunay Triangulations The algorithm discussed in this section, is based on local transformations or flips (as defined below) and goes back to Lawson [51,52]. He introduced this method (also known as Lawsoh s flip method) for constructing two dimensional Delaunay triangulations in 1972. Given a finite point set S E 2, the method initially constructs an arbitrary triangulation T of S. This triangulation is then altered step by step through a sequence ....

C L Lawson. Generation of a triangular grid with applications to contour plotting. Memo 299, Jet Propulsion Laboratory, Pasadena, California, 1972.

.... 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, Laws77], divide and conquer [ShHo75, GuSt85] geometric transformation [Brow79] plane sweep [For87] and randomized incrementation [GuKS90] Recently, Edelsbrunner, Tan, and Waupotitsch devised a polynomial time algorithm that minimizes the maximum angle [EdTW92] This algorithm constructs a ....

....triangulation A of S. repeat T : A; for all pairs q; s 2 S do B : Edge insertion(A; qs) if B OE A then A : B; exit the for loop endif endfor until T = A. The edge insertion paradigm can be viewed as a generalization of the edge flipping paradigm that computes a Delaunay triangulation [Laws72, Laws77]. An edge flip inserts the diagonal of a convex quadrilateral formed by two neighboring triangles; the process halts when no edge flip improves the current triangulation. The simpler edge flipping paradigm, however, fails to compute global optima for maximum angle, height, eccentricity, and slope, ....

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C. L. Lawson. Generation of a triangular grid with applications to contour plotting. Jet Propul. Lab. Techn. Memo. 299, 1972.

.... 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 as edge flipping [Laws72, Laws77], divide andconquer [ShHo75, GuSt85] geometric transformation [Brow79] plane sweep [For87] and randomized incrementation [GuKS90] Recently, polynomial time algorithms have also been found for the minmax angle and the minmax edge length criteria [EdTW92, EdTa91] The method of [EdTW92] is most ....

....and the minmax angle criteria all tend to avoid thin and elongated triangles in the resulting optimal triangulations, they do not necessarily define the same optima. Indeed, four point examples can be constructed to show that the three criteria are pairwise different. The edge flipping strategy [Laws72, Laws77] applied to the maxmin height criterion does not always succeed in computing an optimal triangulation. For consider a regular pentagon abcde and the circle through the five points. Perturb a slightly to a point outside the circle and c and d slightly to points inside the circle so that h(c; db) ....

C. L. Lawson. Generation of a triangular grid with applications to contour plotting. Jet Propul. Lab. Techn. Memo. 299, 1972.

....to compute optimal triangulations; for instance, the min max angle criterion can actually be computed in O(n 2 log n) time and linear storage (Section 4.1) 2.3 Edge Flip Edge flip is a local optimization method that operates on two triangles whose union forms a convex polygon. It was used in [Laws72] to remove small angles: the edge bd shared by triangles abd and cbd is replaced or flipped when the smallest angle in these triangles is smaller than that of acb and acd. In effect, an edge flip replaces two existing triangles by two new ones. This operation was incorporated into a plane sweep ....

C. L. Lawson. Generation of a triangular grid with applications to contour plotting. Jet Propul. Lab. Techn. Memo. 299, Pasadena, CA, 1972.

....allows the addition of a node to an existing triangulation. The new node is connected to three or four existing nodes: three nodes if the new node lies inside a triangle, four nodes if the new node lies on an edge. The triangulation is made Delaunay by the use of a diagonal swapping algorithm [12]. For a sequence of multigrid grids, our formulation of the intergrid transfer operators demands that a node of a coarse grid appears in the finer grids. Two strategies can be used to satisfy this condition. A sequence of grids can be constructed by refining a coarse grid (adding nodes) or by ....

Lawson, C. L. (1972). Generation of a triangular grid with application to contour plotting. California Institute of Technology, Jet Propulsion Laboratory, Technical Memorandum, 299.

....lower boundary of conv(S U ) along with their faces, parallel to the x d 1 axis, into IR d , then we obtain the Delaunay triangulation of S. 3 The Randomized Incremental Flip Algorithm The idea of constructing Delaunay triangulations by local transformations or flips goes goes back to Lawson [24, 25]. He introduced this method (also known as the Lawson s flip algorithm) for constructing two dimensional Delaunay triangulations in 1972. Given a finite point set S 2 IR 2 , the method initially constructs an arbitrary triangulation T of S. This triangulation is then altered step by step ....

C. L. Lawson. Generation of a triangular grid with applications to contour plotting. Memo 9, Jet Propulsion Laboratory, Pasadena, California, 1972.

....of the shape 2 one that is invariant under scalings and translations a raster display, then we want to avoid triangles less than one pixel wide as these can cause undesirable artifacts [7] Many other alternative definitions of optimality have been proposed. See [3, 13] for surveys. Lawson [14] introduced the idea of using flips to maximize the minimum angle to improve a triangulation. By elementary geometry we can prove that this is the same flip rule as DT . Because DT is systematic the very simple flip algorithm (repeatedly apply the flip rule) will converge to the Delaunay ....

C. L. Lawson. Generation of a triangular grid with application to contour plotting. Technical Memorandum 299, California Institute of Technology Jet Propulsion Laboratory, 1972.

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C.L. Lawson, Generation of a triangular grid with application to contour plotting, California institute of technology, JPL, 299, 1972.

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