Berg, M. De, Kreveld, M. Van, Overmars, M., & Schwartskopf, O. (1997). Computational geometry : Algorithms and applications. Springer Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....number of elements retrieved. Such a unique feature of XR tree makes it possible to most effectively skip both ancestors and descendants during a structural join if XR trees are built on two joining element sets. The idea of XR tree is motivated by an internal memory data structure: interval trees [4]. There are also works on interval management in external memory [1] and indexing time intervals [13] XR tree stands out among those proposed approaches in that it deals specifically with regions of XML elements while existing approaches manage arbitrary one dimensional intervals. By fully ....

M. D. Berg, M. V. Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, Germany, 1997.

....times the product of the sum of parallel edges, and the distance between them. satisfies q corresponds to the area of the intersection between CX(q) and DS. In Figure 4. 3, for example, the intersection region is hexagon AEFGCD, which can be obtained using a standard polygon intersection algorithm [BKOS97]. In particular, for 2 dimensional spaces the computation time is O(1) due to the fact that CX(q) and DS have constant complexities (i.e. they contain at most 6 and exactly 4 edges, respectively) After obtaining the intersection polygon, its area can be computed by decomposing the polygon into ....

....the extents of q at the current time 0 and q T , respectively. Further, we introduce the concept of the extended convex hull ECX(q,l) which enlarges CX(q) with length l on all directions. The extended convex hull is motivated by the Minkowski sum, a well studied concept in computational geometry [BKOS97], which is popular in query analysis [KF93, PSTW93, BBKK97, B00] A useful property of the Minkowski sum is that it facilitates the adaptation of the proposed (i.e. Euclidean) solutions to other metrics. Figure 5.1 shows the three types of ECX that correspond to the possible shapes of CX ....

Berg, M., Kreveld, M., Overmas, M., Schwarzkopf, O. Computational Geometry: Algorithms and Applications. Springer, ISBN: 3-540-61270-X, 1997.

....imagery and vector data. Each pair of corresponding control points from the two datasets indicates identical positions on each datasets. Transformations are calculated from the control point pairs. Other points in both datasets are aligned based on these transformations. The Delaunay Triangulation [5] and piecewise linear rubber sheeting [28] are utilized to find the appropriate transformations. The Delaunay Triangulation is discussed in Section 4.1, and rubber sheeting is explained in Section 4.2. Moreover, a novel technique to alleviate the spatial inconsistencies for those areas where we ....

M.d. Berg, M.v. Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications. Springer-Verlag, 1997.

....the prices of the individual loading configurations, and an ordering of all loading configurations that allow for sharing of computational effort and pruning the set of configurations for which the complete price has to be computed. The data structure is an adapted version of the kd tree [BKOS97] Each node of the tree corresponds to a set of loading configurations. The set associated with node p is denoted L p . In the root of the tree, we store the bounding box of all characteristic vectors. This involves storing two d dimensional vectors x (L) where k (L) min (k) x k ....

Berg, M. de, Kreveld, M. van, Overmars, M., and Schwarzkopf, O. Computational Geometry: Algorithms and Applications. Springer-Verlag, 1997.

....point in a solid angle defined by the two adjacent faces of viv i (Figure 6) It is worth remarking that, in order to improve the quality of the reconstructed mesh, we keep a Delaunay triangu lation whenever a new vertex is inserted in the mesh. The incremental insertion algorithm of Lawson [1] is applied. The sequence of operations refining inflating is repeated until no more subdivision and no more inflation is possible, as in the case of parallel faces. The parallel faces are the ones whose normal vector is almost orthogonal to the inflating direction (Figure 7) Growing front s ....

M. Berg and et.al. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2 edition,

.... the traversal around a vertex or around a facet is no longer uniquely defined [35] The winged edge data structure where the wings PCCW and NCCW are omitted has been called Doubly Connected Edge List (DCEL) by [23] though this name is now more commonly used for the halfedge data structure [6]. 3 3 In order to avoid confusion we will not use the name DCEL since it turned out to be ambiguous. In fact, the name is misleading when denoting halfedges and the possible variants of single linking. 6 opposite ccw cw facet vertex halfedge Fig. 4. FE structure. ccw cw facet ....

M. de Berg, M. van Krefeld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer, 1997.

....problems involving objects like points, lines, polygons, and polyhedra. Over the past twenty years, the eld has developed a rich collection of solutions to a huge variety of geometric problems including intersection problems, visibility problems, and proximity problems. See the textbooks [15, 23, 18, 21, 6, 1] and the handbook [10] for an overview. The standard approach taken in computational geometry is the development of provably good and ecient solutions to problems. However, implementing these algorithms is not easy. The most common problems are the dissimilarity between fast oating point ....

....cause runtime overhead. In the sequel we give examples of the use of the generic programming paradigm in cgal. 2 Generic Programming in Geometric Computing One of the hallmarks of geometry is the use of transformations. Indeed, geometric transformations link several geometric structures together [6, 1]. For example, duality relates the problem of computing the intersection of halfplanes containing the origin to that of computing the convex hull of their dual points. The Voronoi diagram of a set of points is also dually related to its Delaunay triangulation, and this triangulation can be ....

M. de Berg, M. van Kreveld, M. Overmars, and Otfried Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997.

....S, where jP j = m and jSj = n, we can construct a full (explicit) version of DP;q in O(n(n m) log (nm) time. Standard line sweep techniques suffice to compute the explicit diagram in the claimed time. We assume that the diagram is represented as a doubly connected edge list (abbreviated DCEL; see [BKOS]) or by an equivalent data structure, and we store depth information for each region of the diagram. It follows immediately that diagrams for all n points in S can be computed in O(n 2 (n m) log (nm) time. Theorem 9 Given a precomputed diagram DP;q , we can compute in O(m) time the smallest ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, Springer-Verlag, Germany, 1997.

....outside the triple T of the great grandparent C . Triangulation For balanced quadtrees in which adjacent cells di er in size by no more than a factor of 2 such as the distance tree shown in Fig. 2(a) cell vertices and centers can easily be triangulated into conforming meshes [3]. Each cell in such a tree has 0 to 4 smaller neighbors in d = 2 dimensions, so a triangulation can be built from the six possible cell con gurations shown in Fig. 2(c) Velocity extension Moving interfaces via the advection equation (2) requires a globally de ned velocity W which extends V ....

....each cell C whose edge length exceeds the minimum value of jGj on C. Since G is not a signed distance function, the resulting quadtree may be unbalanced and dicult to triangulate: neighboring cells vary in size by more than a factor of 2 (Fig. 2) We can balance the quadtree by brute force [3], or by modifying the cell splitting criterion: x an estimated gradient bound for krGk and split every cell C whose size exceeds the minimum of jGj= over C. If the bound krGk holds, then the resulting tree is balanced and easy to triangulate. Otherwise, we double and rebuild until satis ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational geometry: algorithms and applications. Springer-Verlag, Berlin, 1997.

.... and Preparata performs better than the modified algorithm by Chan, if the numberofintersection points is sublinear, although Chan s algorithm outperforms other line segmentintersection algorithms [2] In this paper we do not consider algorithms using a network oriented representation, e.g. [11, 14, 16, 23], since many GIS support only polygon oriented representations, e.g. ARC INFO [15] The remainder of this paper is structured as follows. In Section 2 we review the standard algorithm by Nievergelt and Preparata for overlaying twosetsof polygonal objects and showhow to modify Chan s algorithm ....

....transferred to disk. As long as no resulting region contains too many boundary segments, there is a fair chance that even for massive data sets both active segments and active regions fit into main memory. In contrast, if we aim at constructing a network oriented layer for the resulting partition [14, 23] we are forced to keep the complete data structure in main memory. Conceptually, the x queue can be thought of as an input stream that feeds the algorithm, and due to the size of the input data set this stream will be resident on secondary storage. To optimize access to this x queue its elements ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer, Berlin, 1997.

..... Algorithms that do not exploit occluder fusion are likely to rely on the presence of large occluders in the scene. A great amount of work has been done in visibility culling in both computer graphics and computational geometry. For those interested in the computational geometry literature, see [8, 9, 10]. For a survey of computer graphics work, see [20] We very briefly survey some of the recent work more directly related to our technique. Hierarchical occlusion maps [21] solve the visibility problem by using two hierarchies, an object space bounding volume hierarchy and another hierarchy of ....

....obscure the cells behind it. Its position with respect to the viewpoint. We transfer a cell s solidity to a neighboring cell based on how orthogonal the face that is shared between cells is to the view direction v (see Fig. 7) We also give preference to neighboring cells that are starshaped [8] with respect to the viewpoint and the shared face. That is, we attempt to force the cells in the front to have their interior, e.g. their center point, visible from the viewpoint along a ray that passes through the face shared by the two cells. The reason for this is to avoid projecting cells ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997.

..... Algorithms that do not exploit occluder fusion are likely to rely on the presence of large occluders in the scene. A great amount of work has been done in visibility culling in both computer graphics and computational geometry. For those interested in the computational geometry literature, see [8, 9, 10]. For a survey of computer graphics work, see [28] We very briefly survey some of the recent work more directly related to our technique. Hierarchical occlusion maps [29] solve the visibility problem by using two hierarchies, an object space bounding volume hierarchy and another hierarchy of ....

....incurred to the transfer. Its position with respect to the viewpoint. We transfer a cell s solidity to a neighboring cell based on how orthogonal the face that is shared between cells is to the view direction #v (see Fig. 6) We also give preference to neighboring cells that are star shaped [8] with respect to the viewpoint and the shared face. That is, we attempt to force the cells in the front to have their interior, e.g. their center point, visible from the viewpoint along a ray that passes through the face shared by the two cells. The reason for this is to avoid projecting cells ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997.

.... whether our chosen point pBV lies, or points pBV;i lie, outside the corresponding outer planes of the new volume (for outside ) inside all the corresponding inner planes (for inside ) or between the inner and outer planes (for intersection ) Similar ideas are used in robot motion planning [Berg97]. We chose to use the third listed approach. One interesting fact is that this (and the first listed) culling method against a view frustum works for arbitrarily shaped bounding volumes. The inside planes will always be six and parallel to the planes of the view frustum. But the outside planes ....

M. de berg, M. van Kreveld, M. Overmars, O. Schwarzkopf, "Computational Geometry - Algorithms and Applications", Springer-Verlag, Berlin, 1997. 26

....by sMBR # T # (c polyio vcand # cvertio) The CPU cost of the refinement step depends on the algorithm used for testing overlap. Detecting if two general polygons overlap can be done in O(n log n) using a plane sweep algorithm, where n is the total number of vertices in both polygons [5]. We assume that every candidate polygon has v cand vertices and use the following formula to estimate the CPU cost of the refinement step: sMBR # T # (vq vcand ) log (vq vcand ) # c polytest where v q is the number of vertices in the query polygon, and c polytest is a proportionality ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry -- Algorithms and Applications. Springer-Verlag, 1997.

....two closed meshes: T , which represents the boundary of a solid object, and the convex hull H T of T . Mesh H T contains T in its enclosed volume, and the vertices of H T are a subset of vertices of T . Mesh T is given in input, while mesh H T can be computed with any algorithm for 3D convex hull [6]. A tetrahedral mesh is a set of tetrahedra where any two distinct tetrahedra are either disjoint, or share exactly a face, or an edge, or a vertex. We consider a tetrahedral mesh filling the portion of space which separates T from H T . Figure 1 shows a two dimensional example. Mesh can be ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry Algorithms and Applications. Springer-Verlag, Berlin Heidelberg, 1997.

....are used to nd the next leftmost label; other data structures are only used to update the former ones eciently. The data structures are red black trees T , heaps H, and priority search trees P [18] These are also described in standard textbooks on algorithms [5] and computational geometry [6]. 4.1 Leftmost label query structures We use three data structures to nd the leftmost label position among the ones represented by H int , H right , and V int;right . They are: 1. For each segment in H right we store the x coordinate of its right endpoint. This corresponds to the right edge of ....

Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997.

....addition, we show that if diam(A) diam(B) where diam( denotes the diameter, then we have fatness(A B) fatness(A) diam(A) diam(A) diam(B) d 1 : Both bounds are tight in the worst case. Keywords: Computational geometry, Minkowski sums, fatness. 1 Introduction The Minkowski sum [4] of two sets A and B, which we denote by A B, is the vector sum of the two sets: A B : fa b : a 2 A and b 2 Bg; where a b denotes the vector sum of a and b. Minkowski sums play an important role in many areas. One example is motion planning [7] where the forbidden space of a translating ....

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997.

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Berg, M. De, Kreveld, M. Van, Overmars, M., & Schwartskopf, O. (1997). Computational geometry : Algorithms and applications. Springer Verlag.

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Berg, M., Kreveld, M., Overmars, M., Schwarzkopf, O. Computational Geometry: Algorithms and Applications. ISBN 3-54065620 -0. Springer, 1997

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M. Berg, M. Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2000.

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M. D. Berg. Computational Geometry: Algorithms and Applications. Springer Verlay, 2000.

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M. Berg, M. Kreveld, M.Overmars, and O.Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, 1997.

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M. Berg, M. Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications, chapter 7. Springer-Verlag, New York, NY, USA, 1996.

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M. Berg, M. Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry Algorithms and Applications. Berlin, Germany: Springer, 1997.

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de Berg, M., van Kreveld, M., Overmars, M., and Schwarzkopf, O. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997.

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