J. Matousek. On vertical ray shooting in arrangements. Comput. Geom. Theory Appl., 2(5):279--285, March 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....above some query point. Vertical point hyperplane distances in the dual space are the same as the corresponding vertical hyperplane point distances in the primal space. Thus, we can solve this problem by vertical ray shooting in the dual space. We will use the following result of Matousek [28]. Lemma 9.6 (Matousek [28] We can preprocess a set of n points in IR , in time and space O(n n) so that the k nearest neighbors to a query hyperplane can be found in time O(k log n) We make use of a technique developed by Chazelle et al. 9] for answering simplex range queries. Given ....

....Vertical point hyperplane distances in the dual space are the same as the corresponding vertical hyperplane point distances in the primal space. Thus, we can solve this problem by vertical ray shooting in the dual space. We will use the following result of Matousek [28] Lemma 9. 6 (Matousek [28]) We can preprocess a set of n points in IR , in time and space O(n n) so that the k nearest neighbors to a query hyperplane can be found in time O(k log n) We make use of a technique developed by Chazelle et al. 9] for answering simplex range queries. Given a data structure used to ....

J. Matousek. On vertical ray-shooting in arrangements. Comput. Geom. Theory. Appl., 2:279--285, 1993.

.... n log n [101] c oriented polytopes n log n [96] s convex polytopes s 2 n 2 log 2 n [12] d = 3 Halfplanes m n=m 1=3 [7] Terrain m n= p m [7, 70] Triangles m n=m 1=4 [8] Spheres n 3 log 2 n [4] Hyperplanes m n=m 1=d [7] d 3 Hyperplanes n d log d Gamma n log n [71, 182] Convex polytope m n=m 1=bd=2c [7, 185] Convex polytope n bd=2c log bd=2c Gamma n log n [185] Table 9. Asymptotic upper bounds for ray shooting queries. Steiner points) with O(log n) crossing number can be constructed in O(n log n) time. See [5, 195, 102, 177, 250] and the references ....

J. Matousek, On vertical ray shooting in arrangements, Comput. Geom. Theory Appl., 2 (1993), 279--285.

.... n log n [104] c oriented polytopes n log n [100] s convex polytopes s 2 n 2 log 2 n [12] d = 3 Halfplanes m n=m 1=3 [7] Terrain m n= p m [7, 73] Triangles m n=m 1=4 [8] Spheres n 3 log 2 n [4] Hyperplanes m n=m 1=d [7] d 3 Hyperplanes n d log d Gamma n log n [75, 202] Convex polytope m n=m 1=bd=2c [7, 205] Convex polytope n bd=2c log bd=2c Gamma n log n [205] Table 9. Asymptotic upper bounds for ray shooting queries, with polylogarithmic factors omitted. in the query time. Goodrich and Tamassia [143] have developed a dynamic ray shooting data ....

J. Matousek, On vertical ray shooting in arrangements, Comput. Geom. Theory Appl., 2 (1993), 279--285.

....also supported by the National Science Foundation under grant DMS 9627683 and by the U. S. Army Research Office under grant DAAH04 961 0013. and halfspace emptiness queries. Emptiness query data structures are used to solve several geometric problems, including point location [12] ray shooting [2, 12, 24, 27], nearest and farthest neighbor queries [2] linear programming queries [23, 7] depth ordering [5] collision detection [11] and output sensitive convex hull construction [23, 8] Most previous range searching lower bounds are presented in the so called semigroup arithmetic model, originally ....

....these problems. Is there a reduction from hyperplane queries to halfspace queries that only increases the dimension by a constant factor (preferably two) Finally, can our techniques be applied to other closely related problems, such as linear programming queries [23, 7] and ray shooting queries [2, 12, 24, 27] Acknowledgments I thank Pankaj Agarwal for suggesting this problem for study. ....

J. Matousek. On vertical ray shooting in arrangements. Comput. Geom. Theory Appl. 2(5):279--285, Mar. 1993.

....Perhaps the simplest type of query is an emptiness query (also called an existential query [6, 43] which asks whether the query range contains any points in the set. Emptiness query data structures have been used to solve several geometric problems, including point location [19] ray shooting [2, 19, 39, 42], nearest and farthest neighbor queries [2] linear programming queries [38, 11] depth ordering [9] collision detection [18] and output sensitive convex hull construction [38, 12] This paper presents the first nontrivial lower bounds on the complexity of data structures that support emptiness ....

.... from hyperplane queries to halfspace queries that only increases the dimension by a constant factor (preferably two) Finally, can our techniques be applied to other closely related problems, such as nearest neighbor queries [2] linear programming queries [38, 11] and ray shooting queries [2, 19, 39, 42] Acknowledgments I thank Pankaj Agarwal for suggesting studying the complexity of online emptiness problems. ....

J. Matousek. On vertical ray shooting in arrangements. Comput. Geom. Theory Appl. 2(5):279--285, Mar. 1993.

....above some query point. Vertical point hyperplane distances in the dual space are the same as the corresponding vertical hyperplane point distances in the primal space. Thus, we can solve this problem by vertical ray shooting in the dual space. We will use the following result of Matousek [28]. Lemma 9.6 (Matousek [28] We can preprocess a set of n points in IR d , in time and space O(n d = log d Gamma1 n) so that the k nearest neighbors to a query hyperplane can be found in time O(k log n) 2 We make use of a technique developed by Chazelle et al. 9] for answering simplex ....

....Vertical point hyperplane distances in the dual space are the same as the corresponding vertical hyperplane point distances in the primal space. Thus, we can solve this problem by vertical ray shooting in the dual space. We will use the following result of Matousek [28] Lemma 9. 6 (Matousek [28]) We can preprocess a set of n points in IR d , in time and space O(n d = log d Gamma1 n) so that the k nearest neighbors to a query hyperplane can be found in time O(k log n) 2 We make use of a technique developed by Chazelle et al. 9] for answering simplex range queries. Given a data ....

J. Matousek. On vertical ray-shooting in arrangements. Comput. Geom. Theory. Appl., 2:279--285, 1993.

....and the query is an annulus, using transforms and in Section 2. 4, we get the following detection problem: Given n points, is there a point in between two parallel query planes in IR 3 By duality, this becomes Does a vertical line segment in IR 3 intersect any of a set of n planes In [19] it is shown that a set of n hyperplanes in IR d can be stored in O(n d = log d Gamma1 n) space so that a vertical ray shooting query can be answered in O(log n) time. We can use this to solve our detection problem in M(n) O(n 3 = log 2 n) space and O(log n) time. We conclude: Theorem ....

Matousek, J. (1993). On vertical ray-shooting in arrangements. Computational Geometry: Theory and Applications, 2, 279--285.

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J. Matousek. On vertical ray shooting in arrangements. Comput. Geom. Theory Appl., 2(5):279--285, March 1993.

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