A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing minkowski sums of regular polygons. In Proc. of the 3rd Canad. Conf. Comput. Geom., pages 74--77, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a valid separating translation. To obtain the valid placements for all polygons, we have to consider the constraints for all pairs of parts simultaneously. The problems of computing Minkowski sums and configuration spaces for pairs of polygons have been extensively studied in the literature [42, 33, 55]. Here, we used this concept mainly for illustration. In practice, the constraints for a pair of polygons can be derived directly without the need of explicitly computing the configuration space obstacle [70] 2.2.2 Embedding Pairwise Constraints A simultaneous placement of all polygons P # ....

....have been proposed (for example, cf. 42, 52, 57] In our context, Minkowski sums and convex decompositions have to be computed in order to obtain linear constraints for pairs of parts. In the planar case, both the Minkowski sum and the equally useful convolution of polygons have been studied in [33, 55]. For computation of convex decompositions in two dimensions, cf. 49] In the case of polyhedral parts, preprocessing becomes more intricate. Guibas [20] computes convolutions of convex polyhedra by reciprocal search. Kaul [32] considers Minkowski sums of regular polyhedra. In the context of ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing minkowski sums of regular polygons. In Proc. of the 3rd Canad. Conf. Comput. Geom., pages 74--77, 1991.

....n vertices, then P Phi Q is a portion of the arrangement of O(mn) segments, where each segment is the Minkowski sum of a vertex of P and an edge of Q, or vice versa. Therefore the size of P Phi Q is O(m 2 n 2 ) and it can be computed within that time; this bound is tight in the worst case [20] (see Figure 2) If both P and Q are convex, then P Phi Q is a convex polygon with at most m n vertices, and it can be computed in O(m n) time [26] If only P is convex, then a result of Kedem et al. 21] implies that P Phi Q has Theta(mn) vertices (see Figure 3) Such a Minkowski sum can be ....

A. Kaul, M. A. O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. In Proc. 3rd Canad. Conf. Comput. Geom., pages 74--77, 1991.

....then P Q is a portion of the arrangement (see Section 2) of mn segments, where each segment is the Minkowski sum of a vertex of P and an edge of Q, or vice versa. Therefore the size of P Q is O(m 2 n 2 ) and it can be computed within that time; this bound is tight in the worst case [9]; see Figure 1. The sum has lower worst case complexity when one of the polygons or both are convex; see for example Figure 2. We devised and implemented three algorithms for computing the Minkowski sum of two polygonal sets based on the CGAL software library [1] Our main goal was to produce a ....

A. Kaul, M. A. O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. In Proc. 3rd Canad. Conf. Comput. Geom., pages 74-77, Aug. 1991.

....set with n vertices, then P Q is a portion of the arrangement of O(mn) segments, where each segment is the Minkowski sum of a vertex of P and an edge of Q, or vice versa. Therefore the size of P Q is O(m 2 n 2 ) and it can be computed within that time; this bound is tight in the worst case [20] (see Figure 2) If both P and Q are convex, then P Q is a convex polygon with at most m n vertices, and it can be computed in O(m n) time [26] If only P is convex, then a result of Kedem et al. 21] implies that P Q has (mn) vertices (see Figure 3) Such a Minkowski sum can be computed in ....

A. Kaul, M. A. O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. In Proc. 3rd Canad. Conf. Comput. Geom., pages 74-77, 1991.

....to point sets, for example, A = fa j a 2 Ag and A t = fa t j a 2 Ag: The Minkowski sum [28, 18, 29, 30] of two point sets (of R 2 in the case of this paper) is de ned A B = fa b j a 2 A; b 2 Bg: Let jAj denote the number of edges of A. It is well known that jA Bj = jAj 2 jBj 2 ) [20]. Although the notation may have been di erent, the following lemma probably goes back to Minkowski. Lemma 2.1 The intersection (P i t i ) P j t j ) is non empty if and only if t j t i 2 P i P j . If we set U ij = P i P j ; 0 i j k; 5) the complement of the overlap region, ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the 3rd Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

.... b j a 2 A; b 2 Bg: For a point set A, let A denote the set complement of A and define GammaA = f Gammaa j a 2 Ag. For a vector t, define A t = fa t j a 2 Ag. Note that A t = A Phi ftg. Let jAj denote the number of vertices of A. It is well known that jA Phi Bj = Theta(jAj 2 jBj 2 ) [21]. 1.3.2 Configuration Spaces. This paper presents algorithms for translating k polygonal regions P 1 ; P 2 ; P k into a polygonal container C without overlap. If we denote P 0 = C to be the complement of the container region, then containment is 3 equivalent to the placement of k 1 ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the 3rd Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....an edge of the other polygon candidate edges. If there are n vertices in P and m vertices in Q, then there are O(mn) candidate edges . A natural idea for generating the Minkowski sum is to calculate the arrangement [Ede87] of the candidate edges in O(m 2 n 2 log nm) time. The algorithms in [KOS91] and [AST92] for calculating the Minkowski sum of two simple polygons followed this idea. Kaul et al. KOS91] introduced the concept of vertex edge supporting pairs which reduces the number of candidate edges. In the worst case, the Minkowski sum of two simple polygon can have O(m 2 n 2 ) ....

....are O(mn) candidate edges . A natural idea for generating the Minkowski sum is to calculate the arrangement [Ede87] of the candidate edges in O(m 2 n 2 log nm) time. The algorithms in [KOS91] and [AST92] for calculating the Minkowski sum of two simple polygons followed this idea. Kaul et al. KOS91] introduced the concept of vertex edge supporting pairs which reduces the number of candidate edges. In the worst case, the Minkowski sum of two simple polygon can have O(m 2 n 2 ) edges and the same number of holes. 3.4.3 Starshaped Polygons There is a class of polygons called starshaped ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the Third Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....side) the only possible path to achieve this target passes through the line segment emanating from the hole in the Minkowski sum [38] and an edge of Q, or vice versa. Therefore the size of P Q is O(m 2 n 2 ) and it can be computed within that time; this bound is tight in the worst case [55]; see Figure 5. The sum has lower worst case complexity when one of the polygons or both are convex. We devised and implemented three algorithms for computing the Minkowski sum of two polygonal sets [4] 38] Our main goal was to produce a robust and exact implementation. This goal was achieved ....

A. Kaul, M. A. O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. In Proc. 3rd Canad. Conf. Comput. Geom., pages 74-77, Aug. 1991.

....code in our program with the linear programming functionalities of packages cdd and lrs. We conclude with directions of further research. A preliminary account of this work has appeared as [Emi00] 2 Related work Most existing work on Minkowski sums limits itself to low dimensions [GS86, KOS91, Ski97] or to special cases like zonotopes [Epp96] Among the former, we note the result in [GS86] that settles the 3 dimensional case by showing that Minkowski addition has complexity bounded by the sum of input and output sizes. As for zonotopes, they are the hardest inputs on which Minkowski ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proc. Canadian Conf. on Comput. Geometry, pages 7477, 1991.

....to prune the search space. An asymptotic complexity analysis is found in section 5. Section 6 describes the implementation and illustrates its performance. The paper concludes with directions of further research. 2 Related work Most existing work on Minkowski sums limits itself to low dimensions [11, 12, 13], or to special cases like zonotopes [7] More importantly, computing explicitly the Minkowski sum is to be avoided; we shall opt for a direct method reminiscent to [1] Sections 5 and 6 discuss asymptotic and practical complexity. Still, we should underline that our algorithm has complexity ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proc. Canadian Conf. on Computational Geometry, pages 7477, 1991.

....considering Minkowski sums of the form P Phi R, where P is the set of positions of a reference point and R is the shape of the robot itself. P is a set of feasible positions if and only if P Phi R does not contain any part of an obstacle. In this context, Minkowski sums have been considered by [2, 6, 7, 8, 10, 11, 12, 13, 15, 16], sometimes under the name of configuration space obstacles . Computing the unbounded face of the Minkowski sum of two simple polygons was considered in [15] In many cases, it is of particular importance to protect the boundary of a Minkowski sum: if a mobile obstacle (e.g. a person) enters ....

A. Kaul, M. A.O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. Proc. 3rd Canad. Conf. Comp.Geom., pp. 74--77, 1991.

....and an edge of the other polygon candidate edges. If there are n vertices in P and m vertices in Q, then there are O(mn) candidate edges. A natural idea for generating the Minkowski Sum is to calculate the arrangement [8] of the candidate edges in O(m 2 n 2 log nm) time. The algorithms in [13] and [1] for calculating the Minkowski sum of two simple polygons followed this idea. Kaul et. al [13] introduced the concept of vertex edge supporting pairs which reduces the number of candidate edges. In the worst case, the Minkowski sum of two simple polygon can have O(m 2 n 2 ) edges and ....

....then there are O(mn) candidate edges. A natural idea for generating the Minkowski Sum is to calculate the arrangement [8] of the candidate edges in O(m 2 n 2 log nm) time. The algorithms in [13] and [1] for calculating the Minkowski sum of two simple polygons followed this idea. Kaul et. al [13] introduced the concept of vertex edge supporting pairs which reduces the number of candidate edges. In the worst case, the Minkowski sum of two simple polygon can have O(m 2 n 2 ) edges and the same number of holes. There is a class of polygons called starshaped polygons which are not as ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the Third Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....the cube, and take the union of all the resulting infinite number of spheres. The solid formed by that union would be a cube bigger than the original by the radius of the sphere, but with rounded corners. This shape is a Minkowski or vector sum of the cube and the sphere, written cube Phi sphere [2]. The Minkowski sum of any two solid shapes is a well defined set. The reference point for the shape that moves about doesn t have to be its centroid it can be anywhere, though obviously the result will, in general, be translated to a different place in each case. Minkowski sums are both useful ....

A. Kaul, M.A. O'Connor, & V. Srinivasan, "Computing Minkowski Sums of Regular Polygons", in Proceedings of the Third Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....point set A, let A denote the set complement of A and define GammaA = f Gammaa j a 2 Ag. For a vector t, define A t = fa t j a 2 Ag. Note that A t = A Phi ftg. Let jAj denote the number of vertices of A. Known complexities of jA Phi Bj are: Theta(jAj 2 jBj 2 ) for nonconvex A and B [13], Theta(jAjjBj) for convex A [11] Theta(jAj) for convex A, Theta(jAj jBj) for convex A and B [12] 2.1.2 Configuration Spaces. A containment algorithm translates k polygonal regions P1 ; P2 ; Pk into a polygonal container C without overlap. If we denote P0 = C to be the ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the 3rd Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....a gap between them. We are currently working on trim placement algorithms. These will utilize compaction to open up gaps where trim pieces almost fit. This is a computer oriented way of solving the problem that humans solve by experience. 2 The Minkowski Sum The Minkowski sum [3] 2] 8] 9] 1] [4] is an important part of the preprocessing necessary for fast panel placement and compaction. Given two planar point sets A and B, the Minkowski sum and difference are defined as follows: A B = fa b j a 2 A and b 2 Bg and A Gamma B = fa Gamma b j a 2 A and b 2 Bg: The Minkowski sum of two ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the Third Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....following is known about A Phi B. Type Size Time to Construct Both A and B are convex: O(m n) 7] O(m n) 7] Either A or B is convex: O(mn) 5] O(mn log mn) 5] Both A and B are star shaped polygons: O(mnff(mn) 12] O( mn) log mn) 8] 12] Both A and B are general polygons: O(m 2 n 2 ) 10][9] O(m 2 n 2 log mn) 9] 1] For a point set A, let A denote the set complement of A and define GammaA = f Gammaa j a 2 Ag. For a vector t, define A t = fa t j a 2 Ag and A Gamma t = fa Gamma t j a 2 Ag. Section 1.1 gave the various parts of the containment problem: C; P i , 1 i k; V i , ....

....A Phi B. Type Size Time to Construct Both A and B are convex: O(m n) 7] O(m n) 7] Either A or B is convex: O(mn) 5] O(mn log mn) 5] Both A and B are star shaped polygons: O(mnff(mn) 12] O( mn) log mn) 8] 12] Both A and B are general polygons: O(m 2 n 2 ) 10] 9] O(m 2 n 2 log mn)[9][1] For a point set A, let A denote the set complement of A and define GammaA = f Gammaa j a 2 Ag. For a vector t, define A t = fa t j a 2 Ag and A Gamma t = fa Gamma t j a 2 Ag. Section 1.1 gave the various parts of the containment problem: C; P i , 1 i k; V i , 1 i k; and U ij , 1 ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the Third Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....input polygons. The Minkowski sum, on the other hand, can be combinatorially much larger, since intersections of different convolution edges that are on the Minkowski sum boundary must be explicitly stored. A worst case example of a Minkowski sum with complexity Theta(m 2 n 2 ) is provided in [8]. P Q P Q FP(P; Q) Figure 2: A larger example of the convolution and the fiber product (the fiber product is scaled to fit within the page) 2.1 Cycles of the convolution The convolution consists of a number of closed polygonal trips that we call cycles. They may overlap in the plane, ....

A. Kaul, M. A. O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. In Proc. 3rd Canad. Conf. Comput. Geom., pages 74--77, 1991.

....let u and v denote the number of vertices in P and Q respectively. It is well known that the combinatorial complexity of the boundary of P Psi Q, namely the overall number of edges and vertices on the boundary of P Psi Q, is O(u 2 v 2 ) and this bound is tight in the worst case (see, e.g. [11]) However, for our purposes, we only need to know the outer boundary of P Psi Q. Since it is defined by at most O(uv) segments, its complexity is at most O(uvff(uv) 17] where ff(n) is the extremely slowly growing functional inverse of Ackermann s function. Figure 2(a) shows two polygons P and ....

A. Kaul, M. A. O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. In Proc. 3rd Canad. Conf. Comput. Geom., pages 74--77, 1991.

....b j a 2 A; b 2 Bg: For a point set A, let A denote the set complement of A and define GammaA = f Gammaa j a 2 Ag. For a vector t, define A t = fa t j a 2 Ag. Note that A t = A Phi ftg. Let jAj denote the number of vertices of A. It is well known that jA Phi Bj = Theta(jAj 2 jBj 2 ) [21]. 1.3.2 Configuration Spaces. This paper presents algorithms for translating k polygonal regions P 1 ; P 2 ; P k into a polygonal container C without overlap. If we denote P 0 = C to be the complement of the container region, then containment is equivalent to the placement of k 1 ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the 3rd Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

....which is a polygonal set [4] Let P and Q be two polygonal sets, not necessarily connected, with k and n vertices respectively. The boundary of P Phi Q comes from an arrangement of O(nk) line segments, which has complexity bounded by O(n 2 k 2 ) and this bound is tight in the worst case [10, 14]. In applications such as motion planning [5] and assembly planning [18] however, we only need to know the face complexity the number of segments that bound a single face of the complement of the Minkowski sum P Phi Q in the worst case. Figure 1 depicts the outer face of a sum P Phi Q. ....

A. Kaul, M. A. O'Connor, and V. Srinivasan. Computing Minkowski sums of regular polygons. In Proc. 3rd Canad. Conf. Comput. Geom., pages 74--77, 1991.

....of polygons P and Q is a subset of a candidate edge. If there are n vertices in P and m vertices in Q, then there are O(mn) candidate edges. A natural idea for generating the Minkowski sum is to calculate the arrangement [11] of the candidate edges in O(m 2 n 2 log nm) time. The algorithms in [16] and [1] for calculating the Minkowski sum of two simple polygons followed this idea. Kaul et al. introduced the concept of vertex edge supporting pairs which reduces the number of candidate edges. Despite this reduction, they show that in the worst case, the Minkowski sum of two simple polygons ....

Kaul, A., O'Connor, M.A., and Srinivasan, V. Computing Minkowski Sums of Regular Polygons. In Thomas Shermer, editor, Proceedings of the Third Canadian Conference on Computational Geometry, pages 74--77, Vancouver, British Columbia, 1991. Simon Frasier University.

....time of these algorithms therefore depends on the worst case running time of these operations. In practice, the worst case is usually a gross over estimate. In theory, for nonconvex polygons A and B with jAj and jBj vertices, jA[Bj, jA Bj 2 O(jAj Delta jBj) and jA PhiBj 2 Theta(jAj 2 jBj 2 ) [14, 4], but, in practice, often jA [ Bj, jA Bj jAj jBj and jA Phi Bj jAjjBj. On the one hand, new vertices are created by edge edge intersections, but, on the other hand, many vertices are discarded when the boundary of the union or intersection is computed. Har Peled et al. 13] have recently ....

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Proceedings of the 3rd Canadian Conference on Computational Geometry, Vancouver, British Columbia, 1991.

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Kaul, A., O'Connor, M. A., and Srinivasan, V., "Computing Minkowski Sums of Regular Polygons", Proc. 3'rd Canadian Conference on Computational Geometry, pp. 74-77, 1991.

No context found.

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Thomas Shermer, editor, Proceedings of the Third Canadian Conferenceon Computational Geometry, pages 74--77, Vancouver, British Columbia, 1991. Simon Frasier University.

No context found.

A. Kaul, M.A. O'Connor, and V. Srinivasan. Computing Minkowski Sums of Regular Polygons. In Thomas Shermer, editor, Proceedings of the Third Canadian Conference on Computational Geometry, pages 74--77, Vancouver, British Columbia, 1991. Simon Frasier University.

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