Teillaud, M., Towards Dynamic Randomized Algorithms in Computational Geometry, Lecture Notes in Computer Science 758, Springer, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Before we embark on discussing the relevance of the above notion for constructing bivariate splines, a few remarks on Definition 2 will be in order. The concept of a Delaunay configuration seems to be new. Although it is implicit in several papers on higher order Voronoi diagrams, see e.g. [17], it does not seem to have been studied as a separate entity. It is immediately clear from the well known properties of 12 13 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 E 527 35 36 Figure 10: Delaunay configurations and circumscribed circles. X B X I Figure 11: ....

Teillaud, M., Towards Dynamic Randomized Algorithms in Computational Geometry, Lecture Notes in Computer Science 758, Springer, 1993.

....three. Before we embark on discussing the relevance of the above notion for constructing multivariate splines, a few remarks on Definition 5 are in order. The concept of a Delaunay configuration seems to be new. Although it is implicit in several papers on higher order Voronoi diagrams, see e.g. [138], it 378 M. Neamtu ########### ########### ########### ########### ########### ########### ########### ########### ########### ########### ########### ########### ########### ########### X B X I Fig. 21. Boundary and interior knots of a Delaunay configuration of degree three. does ....

Teillaud, M., Towards Dynamic Randomized Algorithms in Computational Geometry, Lecture Notes in Computer Science, 758, Springer, 1993.

....11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 E 526 35 36 Figure 10. Delaunay configurations and circumscribed circles. Delaunay configuration seems to be new. Although it is implicit in several papers on higher order Voronoi diagrams, see e.g. [17], it does not seem to have been studied as a separate entity. It is immediately clear from the well known properties of Delaunay triangulations that these are Delaunay configurations of degree n = 0, i.e. # 0 = #. It should be noted that Delaunay configurations for n # 1 do not form a partition ....

Teillaud, M., Towards Dynamic Randomized Algorithms in Computational Geometry, Lecture Notes in Computer Science 758, Springer, 1993.

....Fig. 2) Fig. 2: Spatial adjacency within a line Voronoi diagram. Therefore, the only map state changes of the dynamic Voronoi data structure produced by events are the changes in the Delaunay triangulation. These events are ruled by the Delaunay triangulation empty circumcircle criterion (see [Teill93a] and Fig. 3) Fig. 3: The Delaunay triangulation empty circumcircle criterion When a point comes inside a circumcircle or goes out from a circumcircle (we call it a topological event ) the boundary between the two triangles inscribed in the circumcircle switches (see Fig. 2) Within the ....

Teillaud, M: Towards dynamic randomized algorithms in Computational Geometry, Lecture Notes in Computer Science no. 758, 156 pp., 1993.

....n 2 Delta algorithm for the geometric conversion of the ordinary Voronoi diagram into the Laguerre diagram. We exhibit a property of the Voronoi diagram in the Laguerre geometry analogous to the empty circumcircle property of the dual of the ordinary Voronoi diagram (the Delaunay triangulation [10]) This works extends the applicability of an empty circle criterion without assuming that the circles do not intersect. This is particularly useful when there is a need for the topology induced by the Laguerre diagram i.e. the spatial adjacency relationships among sites. Moreover, we give a ....

M. Teillaud, Towards dynamic randomized algorithms in computational geometry, Springer-Verlag, New-York, 1993.

....to a node, natural trees are not trees in the strict sense of graph theory. However, the term tree is retained both because these data structures are used analogously to trees in searching and sorting and because the term is already embedded in the literature for this type of structure, e.g. Teillaud (1993) refers to his structure as a Delaunay Tree . We prefer the appellation natural because the principles of natural tree construction are more closely associated with the overlapping behavior of natural neighbour circumcircles than with the disjoint Delaunay triangles. root 1 2 3 3 3 3 3 3 2 2 ....

....and its validity is corroborated by recently published work which partially overlaps our work; those approaches to the construction of the Delaunay triangulation through the use of a tree structure differs in small but significant details. In particular, Knuth (1991, p. 73 and following) and Teillaud (1993, p. 37 and following) do not use the notion of a neighbourhood group as their algorithms establish a hierarchy, and they are primarily concerned with the construction of the Delaunay tesselation. The essence of this contribution is the application of natural trees to the problems of ....

Teillaud, M., 1993, Towards Dynamic Randomized Algorithms in Computational Geometry, Lecture Notes in Computer Science 758, (Berlin: Springer-Verlag).

....graph and the way it can be used to compute the trapezoidal map of a set S of line segments in the plane, then we show how the structure must be modified to allow us to perform splits and unions of trapezoidal maps. 3. 1 The Influence Graph for trapezoidal maps The Influence graph [BDS 92, Tei93] is based on the idea of maintaining the history of the construction by an incremental algorithm [BT86] It allows semi dynamic constructions, with only insertions, and dynamic constructions, with insertions and deletions, in particular cases [DMT92] and in a general setting, it can be augmented ....

.... and dynamic constructions, with insertions and deletions, in particular cases [DMT92] and in a general setting, it can be augmented to obtain such dynamic constructions [DY93] However, for the trapezoidal map, it has been shown that a standard Influence graph was sufficient (see [DTY92, Tei93] or better [BY95] for a revised algorithm) to allow deletions. RR n2486 6 M. Teillaud (d) d) b) b) b) c) c) a) a) a) a) a) a) a) a) Complete trapezoids (b) Left incomplete trapezoids (c) Right incomplete trapezoids (d) Doubly incomplete trapezoids Figure 3: ....

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M. Teillaud. Towards dynamic randomized algorithms in computational geometry, volume 758 of Lecture Notes in Computer Science. SpringerVerlag, 1993.

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