Simultaneous Containment of Several Polygons: Analysis of the Contact Configurations

Devillers, Olivier

doi:10.1142/s0218195993000270

Cited by 5 publications

(7 citation statements)

References 8 publications

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“…Avnaim and Boissonnat [3], [2] use the Minkowski sum and convex decomposition to solve 1NN and 2NN, and Avnaim [2] gives a characterizationbased (and, hence, iteration-based) algorithm for 3NN. Devillers [9] gives faster algorithms for 2CN and 3CN. These running times are summarized in the following table:…”

Section: Related Workmentioning

confidence: 99%

“…O(m 3 n 3 log m) [9] O(m 14 n 6 log mn) [2] Avnaim and Boissonnat also give a solution to the 3NP problem, three nonconvex polygons in a parallelogram container, using time in O(m 60 log m) [3], [2]. Their 3NP algorithm is based on polygon unions, intersections, and Minkowski sum operations.…”

Section: Related Workmentioning

confidence: 99%

“…Their algorithm for 2CN runs in O(mn log n) time. Their algorithm for 3CN has running time in O(m 3 n log mn), an improvement of a factor of O(n 2 ) over previous algorithms, in particular [9]. The 3CN algorithm iterates over combinations of orientations for the lines which separate the convex polygons.…”

Section: Related Workmentioning

confidence: 99%

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Multiple Translational Containment Part I: An Approximate Algorithm

Daniels

Milenkovic

1997

Algorithmica

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We present an algorithm for finding a solution to the two-dimensional translational approximate multiple containment problem: find translations for k polygons which place them inside a polygonal container so that no point of any polygon is more than 2ε inside of the boundary of any other polygon. The polygons and container may be nonconvex. The value of ε is an input to the algorithm. In industrial applications the containment solution acts as a guide to a machine cutting out polygonal shapes from a sheet of material. If ε is chosen to be a fraction of the cutter's accuracy, then the solution to the approximate containment problem is sufficient for industrial purposes.Given a containment problem, we characterize its solution and create a collection of containment subproblems from this characterization. We solve each subproblem by first restricting certain two-dimensional configuration spaces until a steady state is reached, and then testing for a solution inside the configuration spaces. If necessary, we subdivide the configuration spaces to generate new subproblems. The running time of our algorithm is O((1/ε) k log(1/ε)k 6 s log s), where s is the largest number of vertices of any polygon generated by a restriction operation. In the worst case s can be exponential in the size of the input, but, in practice, it is usually not more than quadratic.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Multiple Translational Containment Part I: An Approximate Algorithm

Daniels

Milenkovic

1997

Algorithmica

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“…Since p 1 ∈ P 1 and p 2 ∈ P 2 are arbitrary, (5) and (6) imply that P 1 + t 1 and P 2 + t 2 lie on opposite sides of L + t 1 + v 2 .…”

Section: Cn Containmentmentioning

confidence: 99%

“…Avnaim and Boissonnat [2], [1] use the Minkowski sum and convex decomposition to solve 1NN and 2NN, and Avnaim gives an algorithm for 3NN. Devillers [5] gives faster algorithms for 2CN and 3CN. These running times are summarized in the following table:…”

Section: Introductionmentioning

confidence: 99%

Multiple Translational Containment Part II: Exact Algorithms

Milenkovic¹

1997

Algorithmica

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We present exact algorithms for finding a solution to the two-dimensional translational containment problem: find translations for k polygons which place them inside a polygonal container without overlapping. The term kCN denotes the version in which the polygons are convex and the container is nonconvex, and the term kNN denotes the version in which the polygons and the container are nonconvex. The notation (r, k)CN, (r, k)NN, and so forth refers to the problem of finding all subsets of size k out of r objects that can be placed in a container. The polygons have up to m vertices, and the container has n vertices, where n is usually much larger than m.We present exact algorithms for the following:is the time to solve a linear program with a variables and b constraints. All these results are improvements on previously known running times except for the last. The algorithm for kNN is slower asymptotically than the naive O((mn) 2k log n) algorithm, but is expected to be much faster in practice. The algorithm for 2CN is based on the use of separating line orientations as a means of characterizing the solution.The solution to 3CN also uses a separating line orientation characterization leading to a simple and robust "carrousel" algorithm. The kCN algorithm uses the idea of disassembling the layout to the left. Finally, the kNN algorithm uses the concept of subdivision trees and linear programming.

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Translational polygon containment and minimal enclosure using mathematical programming

Milenkovic

1999

International Transactions in Operational Research

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A new algorithm is given for the two-dimensional translational containment problem: ®nd translations for k polygons which place them inside a polygonal container without overlapping. Both the polygons and the container can be non-convex. The algorithm is based on mathematical programming principles. The containment algorithm is generalized to solve minimal enclosure problems: ®nd the minimal area enclosing square or minimal area enclosing rectangle for k translating polygons. The containment algorithm consists of new algorithms for restriction, evaluation, and subdivision of two-dimensional con®guration spaces. Restriction establishes lower bounds through relaxation and the solution of linear programs. Evaluation establishes upper bounds by ®nding potential solutions. Subdivision branches, when necessary, by introducing a cutting plane. The algorithm shows that either the upper bound of the objective (overlap) is exactly zero or the lower bound is greater than zero. Hence, it gives an exact solution to the containment problem. Experiments show that new containment algorithm clearly outperforms purely geometric containment algorithms. For data sets from the apparel industry, it can solve containment for up to ten nonconvex polygons in practice. An implementation of the algorithm given in this paper has been licensed by Gerber Garment Technologies, the largest provider of textile CAD/CAM software in the Intl.

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Simultaneous Containment of Several Polygons: Analysis of the Contact Configurations

Cited by 5 publications

References 8 publications

Multiple Translational Containment Part I: An Approximate Algorithm

Multiple Translational Containment Part I: An Approximate Algorithm

Multiple Translational Containment Part II: Exact Algorithms

Translational polygon containment and minimal enclosure using mathematical programming

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