B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551--581, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Hence, the time needed for the second phase of the algorithm is bounded by O(nx log 2 n) E O( 5] l;p)log 2 n) where we sum over all polygons 7 with which we query, and lo is the complexity of the answer to the query. The time bound stated in the lemma now follows from the fact that ] o [5 ] = kvi k and y]p ip k. 3.2 Union Range Queries Let R be a set of n axis parallel rectangles in the plane. In this subsection we devise a data structure for so called union range queries on R. Such queries ask for the part of the union of R inside an axis parallel query polygon. More ....

....storage, such that the interections of a query line segment with the envelope can be found in O( 1)log(n l) time. A segment can be inserted into the structure in O( o( time. Proof: It is well known that the upper envelope of. line segments has complexity O( n) 12] Chazelle and Guibas [5] gave a structure for segment intersection queries in a simple polygon that uses linear storage and has O( i 1) log( query time. To insert a segment we first determine the new upper envelope in O( a( time. by querying with the segment. Then we completely rebuild the structure, which can ....

B. Chazelle and L.J. Guibas, Visibility and intersection problems in plane geometry, Discr. J Comp. Geometry 4 (1989), pp. 551-581.

....of the more widely studied problems in computational geometry. In the plane this has led to many efficient solutions, both for general scenes (where the objects are arbitrary line segments [1, 10, 17, 21] or curved segments [3] and for special cases (such as ray shooting inside a simple polygon [5, 8]) In 3 dimensional space, however, the ray shooting problem is still far from resolved. When the origin of the query ray is fixed and the objects are the faces of a polyhedral terrain, then an efficient solution exists [12] For arbitrary query rays, we know of only three results in the ....

B. Chazelle and L.J. Guibas, Visibility and Intersection Problems in Plane Geometry, Discr. 4 Comp. Geometry 4 (1989), pp. 551-581.

....can be identified by partitioning space into regions, and the problem can be solved in each region separately. Recursively continuing the partition leads to a hierarchical decomposition of a geometric space or object. Many such decompositions have been introduced to solve a variety of problems [5, 13]. Most data structures for geometric search problems are in fact hierarchical decompositions. Guibas and Hershberger [11] introduced a hierarchical decomposition of a simple polygon to efficiently answer shortest path queries within the polygon. Their structure is a hierarchy of regions, the root ....

B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Cornput. Geom., 4:551-581, 1989.

....circular arcs and straight line segments (also called a splinegon) of linear complexity. See Figure 2 for an example. Shooting inside a splinegon can be done relatively easy in time O(log 2 n) using O(n log n) storage by adapting the shooting algorithm for simple polygons by Chazelle and Guibas [2]. Using the special nature of the splinegon involved, the storage can be reduced to linear. Details axe left to the reader. If we hit a circle arc, the corresponding disk is the required answer. On the other hand, ff we hit l (in point p) then I does not intersect any disk of D above l. Figure ....

Chazelle, B., and L.J. Guibas, Visibility and intersection problems in plane geometry, Proc. 1 *t ann. stmp. on Comp. Geometrll (1985), pp. 135-146.

....polygons P and Q requires Omega Gammaqui n) time, even if points in P Gamma Q and Q Gamma P are given. Second, Overmars and van Leeuwen s data structure has been adapted for other purposes that do not have a vertical bias including implicit storage of arrangements [4, 5] ray shooting [1], etc. so that an affirmative answer simplifies and speeds up these applications by a constant factor. Third, it is natural to look for common tangents in situations where no separating line exists. In the next section, we show that tangents for disjoint convex polygons can be computed in ....

Bernard Chazelle and Leonidas J. Guibas. Visibility and intersection problems in plane geometry. Discrete & Computational Geometry, 4:551--581, 1989.

....have been studied in the literature. The ray shooting problem (where, given a polygon and a ray emanating from a given point in a given direction, the object is to find the first intersection, if any, of the ray with the polygon boundary) has been studied by Chazelle [7] Chazelle and Guibas [9], and Guibas et al. [17] The guarding problem (where, given a polygon, the object is to station guards at the vertices in order to cover the whole polygon) and its many variations has been studied by Chv atal [12] Aggarwal [1] O Rourke [26] and Shermer [30] The visibility polygon problem ....

B.Chazelle and L. Guibas. Visibility and Intersection Problems in Plane Geometry. Discrete and Computational Geometry, 4, pp. 551-581, 1989.

....di erent visibility problems have been studied in the literature. The ray shooting problem (given a polygon and a ray emanating from a given point in a given direction, nd the rst intersection, if any, of the ray with the polygon boundary) has been studied by Chazelle [10] Chazelle and Guibas [12], Guibas et al. [17] and Hershberger and Suri [20] The guarding problem (given a polygon, nd a placement of guards at the vertices to cover the whole polygon) and its many variations has been studied by Chv atal [15] Aggarwal [1] O Rourke [27] and Shermer [31] A survey paper by Shermer [32] ....

....able to circumvent this problem by preprocessing each visibility region for ray shooting. Given a simple n vertex polygon, with O(n) preprocessing time and space, we can determine the rst intersection point between a query ray and the boundary of the polygon in O(log n) time (Chazelle and Guibas [12], Guibas et al. [17] and Hershberger and Suri [20] Therefore, since the total complexity of all the visibility regions is O(m (m n) when the set of xed points has size m and the polygon has n vertices, with O(m (m n) time and space, we can preprocess all the visibility regions for ....

B. Chazelle and L. Guibas. Visibility and intersection problems in plane geometry. Discrete and Computational Geometry, 4, pp. 551-581, 1989.

....can be identified by partitioning space into regions, and the problem can be solved in each region separately. Recursively continuing the partition leads to a hierarchical decomposition of a geometric space or object. Many such decompositions have been introduced to solve a variety of problems [5, 13]. Most data structures for geometric search problems are in fact hierarchical decompositions. Guibas and Hershberger [11] introduced a hierarchical decomposition of a simple polygon to efficiently answer shortest path queries within the polygon. Their structure is a hierarchy of regions, the root ....

B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551--581, 1989.

....of the region the viewpoint is currently in. The description is usually based on the boundary of the region, but not necessary on the tightest boundary. V Figure 1: A region with the constant visibility. The discontinuity lines and the visibility were intensively studied not only in 2D [21, 1, 6, 16], but mostly in 3D computer graphics [11] Main areas where we can meet them are computer vision [7, 8] and global illumination [10, 3, 20, 4] Similar structure as we use was utilised in 3D in order to maintain the view from a viewpoint moving along a straight line [13] We review nowadays ....

B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. In Joseph O'Rourke, editor, Proceedings of the Symposium on Computational Geometry, pages 135--146, Baltimore, MD, June 1985. ACM Press.

....spatial subdivision techniques for rendering acceleration. We broadly refer to these methods as visibility precomputations, since by performing work offline they reduce the effort involved in solving the hidden surface problem. Much attention has focused on computing exact visibility (e.g. [5, 12, 16, 19, 22]) that is, computing an exact description of the visible elements of the scene data for every qualitatively distinct region of viewpoints. Such complete descriptions may be combinatorially complex and difficult to implement [16, 18] even for highly restricted viewpoint regions (e.g. viewpoints ....

B. Chazelle and L.J. Guibas. Visibility and intersection problems in plane geometry. In Proc. *
ACM Symposium on Computational Geometry, pages 135--146, 1985.

....a polygon (Chazelle [5] This research is partially supported by DGES grant PB96 0005 C02 01. y Corresponding author 1 Finally, the problem of computing the first intersection of an arbitrary ray in O(log n) time has been solved with O(n log n) preprocessing time by Chazelle and Guibas [6] and with optimal O(n) preprocessing time by Guibas et al. 12] There are also a number of variations on this subject. Among them, circular visibility was introduced by Agarwal and Sharir [2] Besides being a natural extension of linear visibility, circular visibility can model some physical ....

Bernard Chazelle and Leonidas J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551--581, 1989.

....where each face of either map M i is the locus of points dual to lines emanating from e and hitting first (the interior of) some fixed edge of P i . It is easy to show that each M i is a convex subdivision having O(n) faces, edges and vertices, and that it can be computed in O(n log n) time; see [14]. Now let us fix a length d 0, and consider the subproblem of determining whether a line segment of length d can be placed inside P so that it also intersects e. It is easy to show that if such a placement exists then there also exists a placement in which the segment either passes through two ....

B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry, Discrete Comput. Geom. 4 (1989), 551--589.

....of the polygon P being the only opaque object, a point p is said to be weakly visible from the trajectory if and only if p is visible from some point (depending on p) on the trajectory. Many algorithms have been developed for computing the weak visibility of a polygon from a line segment [4, 6 9, 14]. Our solutions will make substantial use of several structures of the weak visibility of P from the trajectory. The rest of the paper consists of 3 sections. Section 2 reviews some notations and preliminary results needed by our algorithms. Sections 3 describes in detail our algorithm for the ....

B. Chazelle and L. Guibas. Visibility and intersection problems in plane geometry. Discrete and Computational Geometry, 4:551-581, 1989.

....disjoint polyhedra. Since many ray shooting queries may be applied to a given set of obstacles, it is desirable to preprocess the obstacles so that query processing is as efficient as possible. The problem of ray shooting in a simple polygon with n vertices has been studied by Chazelle and Guibas [13], Chazelle et al. 12] Goodrich and Tamassia [18] and Hershberger and Suri [19] The main result for this special case of the problem is that ray shooting queries can be answered in O(log n) time with an O(n) size data structure. The problem becomes significantly more difficult for multiple A ....

B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. In Proc. 1st Annu. ACM Sympos. Comput. Geom., pages 135--146, 1985.

....a set S of objects, we can partition the parametric space R d Theta S d Gamma1 into cells so that all points within each cell correspond to rays that hit the same object first; this partition is called the visibility map of S. Using this approach and some other techniques, Chazelle and Guibas [73] showed that a ray shooting query in a simple polygon can be answered in O(log n) time using O(n) space. Simpler data structures were subsequently proposed by Chazelle et al. 66] and Hershberger and Suri [151] Following a similar approach, Pocchiola and Vegter [218] showed that a ray shooting ....

B. Chazelle and L. J. Guibas, Visibility and intersection problems in plane geometry, Discrete Comput. Geom., 4 (1989), 551--581. 48 Pankaj Agarwal and Jeff Erickson

....a vertex v of P 1 before hitting an edge a of P 1 . Let v(fl) denote the vertex of P 1 that the rays corresponding to points on the edge fl intersect before crossing the boundary of P 1 . By considering leftward directed rays, define a similar map M 2 for P 2 . By a result of Chazelle and Guibas [8], each M i is a convex planar subdivision having O(n) faces, edges, and vertices. Let Gamma 1 ; Gamma 2 denote the set of edges in M 1 and in M 2 , respectively. The intersection point of an edge fl 1 2 Gamma 1 and an edge fl 2 2 Gamma 2 is the dual of the line passing through v(fl 1 ) and ....

B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry, Discrete Comput. Geom. 4 (1989), 551--589.

....USA. A preliminary version of this paper appeared in Proc. 4th ACM SIAM Symp. on Discrete Algorithms, 1993, pp. 260 270. 1 Introduction 2 The ray shooting problem has received much attention in the last few years because of its applications in computer graphics and other geometric problems [1, 4, 5, 6, 9, 10, 14, 17, 21]. Most of the work to date studies the planar case, where Gamma is a collection of line segments in R 2 . Chazelle and Guibas proposed an optimal algorithm for the special case where Gamma is the boundary of a simple polygon [17] Their algorithm answers a ray shooting query in O(log n) time ....

....graphics and other geometric problems [1, 4, 5, 6, 9, 10, 14, 17, 21] Most of the work to date studies the planar case, where Gamma is a collection of line segments in R 2 . Chazelle and Guibas proposed an optimal algorithm for the special case where Gamma is the boundary of a simple polygon [17]. Their algorithm answers a ray shooting query in O(log n) time using O(n) space; simpler algorithms, with the same asymptotic performance bounds, were recently developed in [14, 22] If Gamma is a collection of arbitrary segments in the plane, the best known algorithm answers a ray shooting ....

B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry, Discrete Comput. Geom. 4 (1989), 551--589.

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B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551--581, 1989.

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B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551-581, 1989.

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B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551581, 1989.

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B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551--581, 1989.

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B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551--581, 1989. 10

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B. Chazelle and L. J. Guibas, "Visibility and Intersection Problems in Plane Geometry ", Discrete and Computational Geometry 4(6) (1989) 551--581.

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B. Chazelle and L. J. Guibas. Visibility and intersection problems in plane geometry. Discrete Comput. Geom., 4:551-581, 1989.

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B. Chazelle and L. J. Guibas. Visibility and Intersection Problems in Plane Geometry. Discrete & Computational Geometry, 4(6):551--581, 1989.

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