Pankaj K. Agarwal, Boris Aronov, and Micha Sharir. Line traversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pages 483492, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A be a family of convex polytopes in R 3 with a total of n f faces. The combinatorial complexity of the space of line transversals to A is O(n 3 f log n) A very recent result by Agarwal, Aronov and Sharir bounds the complexity of the space of line transversals to balls in R 3 . Theorem 17 [2]. Let A be a family of n balls in R 3 . The combinatorial complexity of the space of line transversals to A is O(n 3 ) for any 0. In higher dimensions, there are no published tight or nearly tight bounds on the combinatorial complexity of the space of transversals. Problem 4. What ....

Agarwal, P. K., Aronov, B., and Sharir, M. Line traversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACMSIAM Sympos. Discrete Algorithms (1997), pp. 483-492.

.... that finding such a line can be reduced to computing the convex hull of a set of n points in R , which, combined with parametric searching, leads to an O(n n) time algorithm for finding a smallest cylinder enclosing S; see e.g. 245] The bound has recently been improved by Agarwal et al. [3] to O(n ) by showing that the combinatorial complexity of the space of all lines that intersect all the balls of B is O(n ) and by designing a different algorithm, also based on parametric searching, whose decision procedure calculates this space of lines and determines whether it is ....

....all the balls of B is O(n ) and by designing a different algorithm, also based on parametric searching, whose decision procedure calculates this space of lines and determines whether it is nonempty. Faster algorithms have been developed for some special cases [122, 245] Agarwal et al. [3] also gave an O(n=ffi ) time algorithm to compute a cylinder of radius (1 ffi)r containing all the points of S, where r is the radius of the smallest cylinder enclosing S. Note that this problem is different from those considered in the two previous subsections. The problem analogous ....

P. K. Agarwal, B. Aronov, and M. Sharir, Line traversals of balls and smallest enclosing cylinders in three dimensions, Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, 1997, pp. 483-- 492.

....be computed in time O(n 3 1 19 ) for any 0 [2] The same paper also presents near linear algorithms that compute an approximation to the minimum width enclosing spherical shell in any dimension. There has also been some work on computing the smallest cylinder enclosing a point set in R 3 [1, 14]. Agarwal et al. 1] developed an O(n 3 ) time algorithm, for any 0, for computing the smallest enclosing cylinder. They also proposed a (1 ) approximation algorithm (i.e. an algorithm that produces an enclosing cylinder whose radius is at most (1 ) times the minimum radius) that runs ....

....3 1 19 ) for any 0 [2] The same paper also presents near linear algorithms that compute an approximation to the minimum width enclosing spherical shell in any dimension. There has also been some work on computing the smallest cylinder enclosing a point set in R 3 [1, 14] Agarwal et al. [1] developed an O(n 3 ) time algorithm, for any 0, for computing the smallest enclosing cylinder. They also proposed a (1 ) approximation algorithm (i.e. an algorithm that produces an enclosing cylinder whose radius is at most (1 ) times the minimum radius) that runs in O(n= 2 ) ....

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P. K. Agarwal, B. Aronov, and M. Sharir, Line traversals of balls and smallest enclosing cylinders in three dimensions, Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, 1997, 483-492.

....of n pairwise disjoint discs in the plane admitting bn=2c geometric permutations. We then extend this construction to obtain a family of n pairwise disjoint balls in IR d admitting Omega Gamma n d Gamma1 ) geometric permutations. The constructions are inspired by those of Agarwal et al. [1]. Consider a line through the origin O in IR 2 . Let ffl be a small quantity to be fixed later and let R 1 0. Place two discs of radius R 1 tangent to at O from the opposite sides of , and then move them apart slightly, perpendicular to , such that any line that passes through O and ....

P. K. Agarwal, B. Aronov, and M. Sharir. Line traversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pages 483--492, 1997.

....time O(n 3 Gamma 1 19 ffi ) for any ffi 0 [2] The same paper also presents near linear algorithms that compute an approximation to the minimum width enclosing spherical shell in any dimension. There has also been some work on computing the smallest cylinder enclosing a point set in R 3 [1, 13]. Agarwal et al. 1] developed an O(n 3 ffi ) time algorithm, for any ffi 0, for computing the smallest enclosing cylinder. They also proposed a (1 ) approximation algorithm (i.e. an algorithm that produces an enclosing cylinder whose radius is at most (1 ) times the minimum radius) that ....

.... ffi ) for any ffi 0 [2] The same paper also presents near linear algorithms that compute an approximation to the minimum width enclosing spherical shell in any dimension. There has also been some work on computing the smallest cylinder enclosing a point set in R 3 [1, 13] Agarwal et al. [1] developed an O(n 3 ffi ) time algorithm, for any ffi 0, for computing the smallest enclosing cylinder. They also proposed a (1 ) approximation algorithm (i.e. an algorithm that produces an enclosing cylinder whose radius is at most (1 ) times the minimum radius) that runs in O(n= 2 ) ....

[Article contains additional citation context not shown here]

P. K. Agarwal, B. Aronov, and M. Sharir, Line traversals of balls and smallest enclosing cylinders in three dimensions, Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, 1997, 483--492.

....real parameters. We can therefore define the set of lines tangent to a sphere S i and lying above (resp. below) S i as a surface patch fl i (resp. fl 0 i ) in R 4 . Define Gamma = ffl i j 1 i ng and Gamma 0 = ffl 0 i j 1 i ng. If the lines are parameterized carefully, Agarwal et al. [10] showed that S ( Gamma; Gamma 0 ) is the set of lines intersecting all the spheres of S and that the combinatorial complexity of S ( Gamma; Gamma 0 ) is O(n 3 ) for any 0. However, a construction of Pellegrini [289] implies that the combinatorial complexity of the overlay of the two ....

....54 R 3 , then the space of line transversals of S has n 3 2 O( p log n) complexity. The bound was slightly improved by Agarwal [4] to O(n 3 log n) He reduced the problem to bounding the complexity of a family of cells in an arrangement of O(n) hyperplanes in R 5 . Agarwal et al. [10] proved that the complexity of the space of line transversals for a set of n balls in R 3 is O(n 3 ) Their argument works even if S is a set of homothets of a convex region of constant description complexity in R 3 . 14.4 Geometric optimization In the past few years, many problems in ....

P. K. Agarwal, B. Aronov, and M. Sharir, Line traversals of balls and smallest enclosing cylinders in three dimensions, Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, 1997, pp. 483-- 492.

.... such a line can be reduced to computing the convex hull of a set of n points in R 9 , which, combined with parametric searching, leads to an O(n 4 log O(1) n) time algorithm for finding a smallest cylinder enclosing S; see e.g. 247] The bound has recently been improved by Agarwal et al. [3] to O(n 3 ) by showing that the combinatorial complexity of the space of all lines that intersect all the balls of B is O(n 3 ) and by designing a different algorithm, also based on parametric searching, whose decision procedure calculates this space of lines and determines whether it is ....

....all the balls of B is O(n 3 ) and by designing a different algorithm, also based on parametric searching, whose decision procedure calculates this space of lines and determines whether it is nonempty. Faster algorithms have been developed for some special cases [123, 247] Agarwal et al. [3] also gave an O(n=ffi 2 ) time algorithm to compute a cylinder of radius (1 ffi)r containing all the points of S, where r is the radius of the smallest cylinder enclosing S. Note that this problem is different from those considered in the two previous subsections. The problem analogous ....

P. K. Agarwal, B. Aronov, and M. Sharir, Line traversals of balls and smallest enclosing cylinGeometric Optimization April 30, References 43 ders in three dimensions, Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, 1997, pp. 483-- 492.

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Pankaj K. Agarwal, Boris Aronov, and Micha Sharir. Line traversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pages 483492, 1997.

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Pankaj K. Agarwal, Boris Aronov, and Micha Sharir. Line traversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pages 483--492, 1997. 22

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Pankaj K. Agarwal, Boris Aronov, and Micha Sharir. Line traversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pages 483492, 1997.

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Pankaj K. Agarwal, Boris Aronov, and Micha Sharir. Line traversals of balls and smallest enclosing cylinders in three dimensions. In Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pages 483492, 1997.

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