C.L. Lawson. Software for C1 surface interpolation. In J.R. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[14] first defines a random sample of O(n ) vertices from the triangulation (n being the current number of vertices) and then determines which one of these is closest to p. Finally, an oriented walk is performed, starting with one triangle t adjacent to the chosen vertex. The oriented walk [24] [12] method selects one edge e of t, which separates the centroid of t and p in two distinct semi planes. Then, e is used to switch t to the other triangle adjacent to e. This simple process continues until t contains p. However it is only guaranteed to work for DTs [22] 8] and we have included ....

Lawson, C. L. (1977): Software for C1 Surface Interpolation. In J. R. Rice (ed), Mathematical Software III, Academic Press, New York, 161--194

....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: ....

....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. Software for (J surface interpolation. In J R Rice, editor, Mathematical S'ofiware III, pages 161 194. Academic Press, New York, 1977.

....a 1734 N. K. LEUNG AND R. J. RENKA data dependent triangulation, if it exists, for which the piecewise linear interpolant is convex. Equivalently, we seek the upper hull of the data points in R 3 [14] Lawson developed an incremental algorithm for constructing a Delaunay triangulation [8], which was implemented as ACM algorithm 624 [15] and Scott discovered that a small modification to that algorithm produces a convexity preserving triangulation (or returns a flag specifying nonconvex data) 18] Additional theory characterizing the convex piecewise linear interpolant is provided ....

....we compare a convexity preserving interpolant F c of a sample data set with an interpolant F s produced by SRFPACK [16] with the global gradient estimation method and no tension) which is not designed to preserve convexity. The sample data set consists of a set of 26 nodes borrowed from Lawson [8] (Table 1) with data values taken from the following convex test function: F (x, y) x 4 y 4 . 1748 N. K. LEUNG AND R. J. RENKA Fig. 4. Convex triangulation of 26 data points in the primal space. Fig. 5. Gradient feasibility diagram in the dual space. CONVEXITY PRESERVING INTERPOLATION ....

C. L. Lawson, Software for C 1 surface interpolation, in Mathematical Software III, J. R. Rice, ed., Academic Press, New York, 1977, pp. 161--194.

....visualization. This method consists of first dissecting the definition space into a suitable set of triangles with the given data points being the corners of the triangles. Then, each of the triangles is interpolated independently. Several criteria for an optimal triangulation are known [15]. One optimal triangulation, which is based on a Dirichlet tessellation of the data set, is called Delaunay triangulation . It can be computed with a divide and conquer algorithm of algorithmical complexity O(n logn) where n is the number of data points. If the pixels lie on a regular grid, ....

C.L. Lawson. Software for C 1 surface interpolation. In J.R. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, 1977.

.... 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. Software for C 1 surface interpolation. In Math. Software III, J. R. Rice, ed., Academic Press, 1977, 161--194.

.... best triangulation is of a great importance. In this connection, we can pose two main problems: 1) How do we find a triangulation (2) How do we measure the quality of various triangulations To find a triangulation is apparently of slight significance due to the Lawson s exchange procedure [Law72, Law77]. This procedure allows first to construct any reasonable triangulation (at least such which does not violate the supposed topology of the data) and then to optimise this initial triangulation using some criteria. Remark 2.4.2. It seems, Lawson s procedure has not yet been formally extended to ....

C.L. Lawson. Software for C 1 surface interpolation. In J.R. Rice, editor, Mathematical software III, 1977.

....) may be good in practice. Inserting a point in S reduces to locating the new point at all levels, computing its level i and inserting the new vertex at all levels j; 0 j i (which is sensitive to the degree of the new vertex once the location is done) The insertion using the standard algorithm [Law77]. 3 Worst case randomized analysis The analysis will rely on the randomization in the construction of the random subsets S i and the points of S are assumed to be inserted in a random order. In this section, no assumption applies to the data distribution, which can be in the worst case. As usual ....

C. L. Lawson. Software for C 1 surface interpolation. In J. R. Rice, editor, Math. Software III, pages 161--194. Academic Press, New York, NY, 1977.

....minimum of the six internal angles is not increased. It is this criterion which guarantees the well shaped ness of the Delaunay triangulation. The third property characterises the Delaunay triangulation as a dual of the well known Voronoi diagram [5] Since these three properties are equivalent [10], our subsequent discussion will use only the CIRCLE criterion. Two further terms are needed for the discussion of boundary concavities. Let DT be any supertriangle based Delaunay triangle algorithm, and suppose that a triangulation T (V; E) produced by DT has boundary points P 1 ; P 2 ; P ....

C. L. Lawson. Software for C1 surface interpolation. In J. R. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, 1987. 10

....line segment with endpoints at the bisector of (i; j) and the circumcenter of the unique triangle incident upon (i; j) 2. 2 Minimal Quality Requirements for the Control Volume Technique Although it is well known that the Delaunay Triangulation is optimal with respect to many interesting criteria[Law77] Raj94] Mus97] not all Delaunay meshes are appropriate for the purpose of solving PDEs with this discretization technique[For91] Indeed, the quality of a triangulation has been a major research topic in mesh generation, resulting in many definitions. This diversity stems in part from the ....

....triangle t = a; b; c) with boundary edge e = a; b) opposite the obtuse angle do 3 Let d be the orthogonal projection of c onto e into the triangulation. 4 Remove t from the triangulation 5 Create two new triangles (a; c; d) and (b; c; d) 6 Restore Delaunay Triangulation via Flip algorithm[Law77] 7 end while Project Flip has the following properties: ffl It terminates and produces triangles so that the circumcenters of each triangle are contained in the boundary. ffl The number of points inserted can be bounded by O(n) ffl While it is useful for constrained or conforming Delaunay ....

C. L. Lawson. Software for C 1 surface interpolation. In J. R. Rice, editor, Math. Software III, pages 161--194, New York, NY, 1977. Academic Press.

....as triangulations. Here the domain is decomposed into simplices (triangles in two and tetrahedra in three dimensions) so that the intersection of two simplices is either empty or a face of both. Applications of triangulations can be found in finite element analysis [Cave74] surface interpolation [Laws77], shape reconstruction [Bois88] and other research 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 ....

C. L. Lawson. Software for C 1 surface interpolation. In Math. Software III, J. R. Rice, ed., Academic Press, 1977, 161--194.

.... 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. Software for C 1 surface interpolation. In Math. Software III, J. R. Rice, ed., Academic Press, 1977, 161--194.

....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 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 ....

C. L. Lawson. Software for C 1 surface interpolation. Math. Software III, J. R. Rice, ed., Academic Press, NY, 1977, 161--194.

.... where k x and k y are constant and positive, a transformation from x; y to, say, x 0 ; y 0 can be performed so that diffusion tensor (17) becomes tensor (12) in the x 0 ; y 0 plane and a Delaunay triangulation can be constructed in the transformed plane by performing edge swapping (see [19]) Equation (9) can then be solved in the original coordinate system by mapping the Delaunay triangulation back into the x; y plane. Note that for tensor (17) a regular triangulation (which is a Delaunay triangulation) in the transformed plane will also be a regular triangulation in the original ....

C. L. Lawson. Software for C 1 Surface Interpolation. In J. R. Rice, editor, Mathematical Software III, pages 161--193, New York, 1977. Academic Press.

....triangulation . It is almost optimal for error bounds, yet can be constructed in O(N log N) time. Indeed, 50] shows that no other triangulation can reduce the error bounds by more than a factor of two, while many fast methods for constructing the Delaunay triangulation have been proposed [9, 19, 21, 23, 27, 28, 30, 33, 43, 44]. In this section, we describe the Delaunay triangulation and a fast method for its construction, following [44] 10 4.2 De nitions and data structures The Delaunay triangulation can be (and historically has been) de ned in many ways. Currently one popular de nition is in terms of the Voronoi ....

C. Lawson. Software for C 1 surface interpolation. In J. Rice, editor, Mathematical Software III. Academic Press, New York, 1977.

....given below. After computing the Constrained Delaunay triangulation of the modified PSLG, circumcenters of triangles whose radii are larger than h are inserted, one at a time. The Constrained Delaunay triangulation can be restored after each such Steiner point insertion using Lawson s algorithm [Law77] The process continues until no triangles with circumradii exceeding h exist, which Chew demonstrates always occurs eventually. 2.2 Treating the small angles If the input PSLG contains angles less than 30 degrees, finding a value for h so that the PSLG is decomposed into edges of length in the ....

C. L. Lawson. Software for C 1 surface interpolation. In J. R. Rice, editor, Math. Software III, pages 161--194, New York, NY, 1977. Academic Press.

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C.L. Lawson. Software for C1 surface interpolation. In J.R. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, 1977.

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C.L. Lawson. Software for C1 surface interpolation. In J.R. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, 1977.

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Charles L. Lawson. Software for C 1 Surface Interpolation. In John R. Rice, editior, Mathematical Software III, Academic Press, NY(Proc. of symp., Madison, WI, Mar.), pages 161--194, 1977.

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C. Lawson, Software for c1 surface interpolation. In J. Rice, ed., Mathematical Software III, pp. 161--194, Academic Press, New York, 1977.

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C.L. Lawson. Software for C surface interpolation. In J. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, 1977.

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C.L. Lawson. Software for C1 surface interpolation. In J.R. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, 1977.

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Lawson CL. Software for C surface interpolation. In Mathematical Software III, vol. 3, Rice JR (ed.). Academic Press: New York, NY, 1977.

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C. L. Lawson. Software for C surface interpolation. In J. R. Rice, editor, Math. Software III, pages 161194. Academic Press, 1977.

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C. Lawson. Software for C1 surface interpolation. In J. R. Rice, editor, Mathematical Software III, pages 161--194. Academic Press, New York, 1977.

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C.L. Lawson. Software for C surface interpolation. In J. Rice, editor, Mathematical Software III, pages 161-194. Academic Press, 1977.

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