P.K. Agarwal, B. Aronov and M. Sharir, Motion planning for a convex polygon in a polygonal environment, in preparation.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[3,4] improve this to O(m 3 n 3 log(m n) Most recent work deals with finding the largest copy of a convex P that can be placed, which is equivalent to finding the minimum (scaled) enclosure. For convex Q the best running time is O(mn 2 log n) 1] and for non convex Q, O(m 2 n 2 ) [2]. Grinde and Cavalier use an extensive case analysis and linear programming to place a single convex P [15] or convex P 1 and P 2 [16] The running time of the first algorithm appears to be O(m 2 n 3 ) and it is not clear what the running time of the second is, but for one of its cases they ....

P.K. Agarwal, B. Aronov, M. Sharir, Motion planning for a convex polygon in a polygonal environment, in preparation, 1992.

....[4, 3] improve this to O(m 3 n 3 log(m n) Most recent work deals with finding the largest copy of a convex P that can be placed, which is equivalent to finding the minimum (scaled) enclosure. For convex Q the best running time is O(mn 2 log n) 1] and for non convex Q, O(m 2 n 2 ) [2]. Grinde and Cavalier use an extensive case analysis and linear programming to place a single convex P [11] or convex P1 and P2 [12] The running time of the first algorithm appears to be O(m 2 n 3 ) and it is not clear what the running time of the second is, but for one of its cases they ....

P.K. Agarwal, B. Aronov, and M. Sharir. Motion planning for a convex polygon in a polygonal environment. In preparation., 1992.

....[4, 3] improve this to O(m 3 n 3 log(m n) Most recent work deals with finding the largest copy of a convex P that can be placed, which is equivalent to finding the minimum (scaled) enclosure. For convex Q the best running time is O(mn 2 log n) 1] and for non convex Q, O(m 2 n 2 ) [2]. Grinde and Cavalier use an extensive case analysis and linear programming to place a single convex P [15] or convex P1 and P2 [16] The running time of the first algorithm appears to be O(m 2 n 3 ) and it is not clear what the running time of the second is, but for one of its cases they ....

P.K. Agarwal, B. Aronov, and M. Sharir. Motion planning for a convex polygon in a polygonal environment. In preparation., 1992.

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P.K. Agarwal, B. Aronov and M. Sharir, Motion planning for a convex polygon in a polygonal environment, in preparation.

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P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, Discrete Comput. Geom. 22 (1999) 201--221.

....the largest similar copy of P (under translation, rotation, and scaling) that can be placed inside Q. Using generalized Delaunay triangulation induced by P within Q, Chew and Kedem [67] obtained an O(m log n) time algorithm. Faster algorithms have been developed using parametric searching [4, 255]. The decision problem in this case can be defined as follows: Given a convex polygon B with m edges (a scaled copy of P ) and a planar polygonal environment Q with n edges, can B be placed inside Q (allowing translation and rotation) Each placement of B can be represented as a point in , using ....

....FP 6= they first compute a superset of the vertices of FP , in O(mn 6 (mn) log mn) time, and then spend O(m log n) time for each of these vertices to determine whether the corresponding placement of B is free, using a standard triangle range searching data structure. Recently, Agarwal et al. [4] gave an O(mn 6 (mn) log mn) expected time randomized algorithm to compute FP . Plugging these algorithms into the parametric searching machinery, one can obtain an O(m n 6 (mn) log mn log log mn) time deterministic algorithm, or an O(mn 6 (mn) log mn) expected time randomized algorithm, ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, Tech. Report CS-

....Our goal is to find the largest similar copy of P inside Q (allowing translation, rotation, and scaling of P ) see Figure 1. A restricted version of this problem, in which we just determine whether P can be placed inside Q without scaling, was solved by Chazelle [4] in O(mn time. See also [1, 6, 12] for other approaches to the more general problem, in which Q is an arbitrary polygonal region. We remark that the complexity of the algorithms for the general case is considerably higher, about O(m ) in [1] O(m ) in [12] and O(m in [6] Problems concerning the placement of one ....

....Q without scaling, was solved by Chazelle [4] in O(mn time. See also [1, 6, 12] for other approaches to the more general problem, in which Q is an arbitrary polygonal region. We remark that the complexity of the algorithms for the general case is considerably higher, about O(m ) in [1], O(m ) in [12] and O(m in [6] Problems concerning the placement of one polygon inside another are important in robotics and manufacturing. This restricted problem is also applicable to an approach to object recognition recently proposed by Basri and Jacobs [3] based on matching ....

P.K. Agarwal, B. Aronov and M. Sharir, Motion planning for a convex polygon in a polygonal environment, in preparation.

....translation, rotation, and scaling) that can be placed inside Q. Using generalized Delaunay triangulation induced by P within Q, Chew and Kedem [42] obtained an O(m 4 n 2 2 (n) log n) time algorithm. Faster algorithms can be developed using randomization and search parametric searching [3, 170]. The decision problem in this case can be de ned as follows: Given a convex polygon B with m edges (a scaled copy of P ) and a planar polygonal environment Q with n edges, can B be placed inside Q (allowing translation and rotation) Each placement of B can be represented as a point in R 3 , ....

....of an arrangement of O(mn) contact surfaces in R 3 . Leven and Sharir [126] have shown that the complexity of FP is O(mn 6 (mn) where s (n) is the maximum length of a Davenport Schinzel sequence of order s composed of n symbols [169] it is almost linear in n for any xed s) Agarwal et al. [3] gave an O(mn 6 (mn) log mn) expected time randomized algorithm to compute FP . Plugging these algorithms into the parametric searching machinery, one can obtain an O(m 2 n 6 (mn) log 3 mn log log mn) time deterministic algorithm, or an O(mn 6 (mn) log 4 mn) expected time randomized ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, Tech. Report CS-1997-17, Department of Computer Science, Duke University, 1997.

....similar copy of P (under translation, rotation, and scaling) that can be placed inside Q. Using generalized Delauney triangulation induced by P within Q, Chew and Kedem [58] obtained an O(m 4 n 2 2 ff(n) log n) time algorithm. Faster algorithms have been developed using parametric searching [3, 217]. The decision problem in this case is: Given a convex polygon B with m edges (a scaled copy of P ) and a planar polygonal environment Q with n edges, can B be placed inside Q (allowing translation and rotation) Each placement of B can be represented as a point in R 3 , using two coordinates ....

....FP 6= they first compute a superset of the vertices of FP , in O(mn 6 (mn) log mn) time, and then spend O(m log n) time for each of these vertices to determine whether the corresponding placement of B is free, using a standard triangle range searching data structure. Recently, Agarwal et al. [3] gave an O(mn 6 (mn) log mn) expected time randomized algorithm to compute FP . Plugging these algorithms into the parametric searching machinery, one can obtain an O(m 2 n 6 (mn) log 3 mn log log mn) time deterministic algorithm, or an O(mn 6 (mn) log 4 mn) expected time randomized ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, manuscript, 1996.

....translation, rotation, and scaling) that can be placed inside Q. Using generalized Delaunay triangulation induced by P within Q, Chew and Kedem [45] obtained an O(m 4 n 2 2 ff(n) log n) time algorithm. Faster algorithms can be developed using randomization and search parametric searching [3, 180]. The decision problem in this case can be defined as follows: Given a convex polygon B with m edges (a scaled copy of P ) and a planar polygonal environment Q with n edges, can B be placed inside Q (allowing translation and rotation) Each placement of B can be represented as a point in R 3 , ....

....of an arrangement of O(mn) contact surfaces in R 3 . Leven and Sharir [132] have shown that the complexity of FP is O(mn 6 (mn) where s (n) is the maximum length of a Davenport Schinzel sequence of order s composed of n symbols [179] it is almost linear in n for any fixed s) Agarwal et al. [3] gave an O(mn 6 (mn) log mn) expected time randomized algorithm to compute FP . Plugging these algorithms into the parametric searching machinery, one can obtain an O(m 2 n 6 (mn) log 3 mn log log mn) time deterministic algorithm, or an O(mn 6 (mn) log 4 mn) expected time randomized ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, Tech. Report CS-1997-17, Department of Computer Science, Duke University, 1997.

....al. 1] Basically, for each expanded triangle K i , their algorithm will compute the vertices, edges, and faces of U that lie on K i , by a straightforward incremental construction that inserts all the other K j s in a random order. Omitting all the details, which can be found in [1] see also [2]) we conclude the following. Theorem 5.3 Let S be a set of n triangles in R 3 with pairwise disjoint interiors, and let B be a ball. The boundary of the union of the Minkowski sums fs Phi B j s 2 Sg, can be computed in randomized expected O(n 2 ) time, for any 0. As mentioned in the ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, Discrete Comput. Geom. (in press).

....free configuration space, as a collection of cells in a corresponding 3 dimensional arrangement of surfaces. The running time of the whole algorithm is only slightly larger than the time needed to solve the fixed size placement problem. The best running time is O(mn 6 (mn) log 3 mn log 2 n) [11]; see also [232, 318] If Q is a convex n gon, the largest similar copy of P that can be placed inside Q can be computed in O(mn 2 log n) time [5] Diameter in 3D. Given a set S of n points in R 3 , determine the maximum distance between a pair of points in S. The problem is reduced to ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, Tech. Report CS-1997-17, Department of Computer Science, Duke University, 1997.

....similar copy of P (under translation, rotation, and scaling) that can be placed inside Q. Using generalized Delaunay triangulation induced by P within Q, Chew and Kedem [68] obtained an O(m 4 n 2 2 ff(n) log n) time algorithm. Faster algorithms have been developed using parametric searching [4, 257]. The decision problem Geometric Optimization April 30, Placement and Intersection 37 in this case can be defined as follows: Given a convex polygon B with m edges (a scaled copy of P ) and a planar polygonal environment Q with n edges, can B be placed inside Q (allowing translation and ....

....FP 6= they first compute a superset of the vertices of FP , in O(mn 6 (mn) log mn) time, and then spend O(m log n) time for each of these vertices to determine whether the corresponding placement of B is free, using a standard triangle range searching data structure. Recently, Agarwal et al. [4] gave an O(mn 6 (mn) log mn) expected time randomized algorithm to compute FP . Plugging these algorithms into the parametric searching machinery, one can obtain an O(m 2 n 6 (mn) log 3 mn log log mn) time deterministic algorithm, or an O(mn 6 (mn) log 4 mn) expected time randomized ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, Tech. Report CS-1997-17, Department of Computer Science, Duke University, 1997.

....Our goal is to find the largest similar copy of P inside Q (allowing translation, rotation, and scaling of P ) see Figure 1. A restricted version of this problem, in which we just determine whether P can be placed inside Q without scaling, was solved by Chazelle [4] in O(mn 2 ) time. See also [1, 6, 12] for other approaches to the more general problem, in which Q is an arbitrary polygonal region. We remark that the complexity of the algorithms for the general case is considerably higher, about O(m 2 n 2 ) in [1] O(m 3 n 2 ) in [12] and O(m 4 n 2 ) in [6] Problems concerning the ....

....without scaling, was solved by Chazelle [4] in O(mn 2 ) time. See also [1, 6, 12] for other approaches to the more general problem, in which Q is an arbitrary polygonal region. We remark that the complexity of the algorithms for the general case is considerably higher, about O(m 2 n 2 ) in [1], O(m 3 n 2 ) in [12] and O(m 4 n 2 ) in [6] Problems concerning the placement of one polygon inside another are important in robotics and manufacturing. This restricted problem is also applicable to an approach to object recognition recently proposed by Basri and Jacobs [3] based on ....

P.K. Agarwal, B. Aronov and M. Sharir, Motion planning for a convex polygon in a polygonal environment, in preparation.

....in R 3 is O(n 5=2 ) for any 0, where the constant of proportionality depends on . 4 Efficient Construction of the Union In this section we sketch a randomized algorithm for computing the boundary of U in expected time O(n 5=2 ) The algorithm is similar to the ones described in [1, 2]; see also [9, 17] Our algorithm can also preprocess U in time O(n 5=2 ) into a data structure of the same size, as in [2] so that a path between two query points in R 3 n U that does not intersect the interior of U can be computed in O(log n k) time, where k is the number of edges in ....

.... In this section we sketch a randomized algorithm for computing the boundary of U in expected time O(n 5=2 ) The algorithm is similar to the ones described in [1, 2] see also [9, 17] Our algorithm can also preprocess U in time O(n 5=2 ) into a data structure of the same size, as in [2], so that a path between two query points in R 3 n U that does not intersect the interior of U can be computed in O(log n k) time, where k is the number of edges in the path. Let K = fK 1 ; Kng be the set of expanded obstacles. Define C; Sigma as in the previous section. For each ....

P. K. Agarwal, B. Aronov, and M. Sharir, Motion planning for a convex polygon in a polygonal environment, to appear in Discrete Comput. Geom..

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