Polygon placement under translation and rotation

Avnaim, Francis; Boissonnat, Jean‐Daniel

doi:10.1051/ita/1989230100051

Cited by 31 publications

(31 citation statements)

References 13 publications

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“…Computing the event structure took approximately 65 minutes, and computed 8, 427, 881 events on the scaling domain [0. 2,2]. We see a similar asymptotic behavior in speed-up over recomputing the line intersections from scratch, as we do in 2D.…”

Section: Results From Uniform Scaling In 2dsupporting

confidence: 61%

“…These methods generate slices of C-space obstacles (C-obst) at fixed rotational resolution using the Minkowski sums and ignore the temporal and spatial coherence between slices. To avoid this problem, the methods proposed by Avnaim et al [2] and Brost [9] focus on the idea of contact surfaces, which have a close relationship to convolution (defined in Section 3). For nonconvex polygons, part of the contact region may belong to the interior of the C-obst.…”

Section: Related Workmentioning

confidence: 99%

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Dynamic Minkowski sums under scaling

Behar¹,

Lien²

2013

Computer-Aided Design

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In many common real-world and virtual environments, there are a significant number of repeated objects, primarily varying in size. Similarly, in many complex machines, there are a significant number of parts which also vary in size rather than shape. This repetition saves in both design and production costs. Recent research in robotics has also shown that exploiting workspace repetition can significantly increase efficiency. In this paper, we address the need to support computation reuse in fundamental operations. To this end, we propose an algorithm to reuse the computation of the Minkowski sum when an object is transformed by uniform scaling. The Minkowski sum is a fundamental operation in many areas of robotics, such as motion planning, computer vision, and mathematical morphology, and has been studied extensively over the last four decades. We present two methods for dynamically updating Minkowski sums under scaling, the first of which updates the sum under uniform scaling of arbitrary polygons and polyhedra, and the second of which updates the sum under non-uniform scaling of convex polyhedra. Ours are the first methods that study the M-sum under this type of transformation. Our results show speed gains between one and two orders of magnitude over recomputing the Minkowski sum from scratch for each repeated object, and we discuss applications for motion planning, CAD, rapid prototyping, and shape decomposition.

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Section: Results From Uniform Scaling In 2dsupporting

confidence: 61%

Section: Related Workmentioning

confidence: 99%

Dynamic Minkowski sums under scaling

Behar¹,

Lien²

2013

Computer-Aided Design

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“…For convex Q, the running time bound is O(mn 2 ), and for non-convex P and Q, O(m 3 n 3 (m + n) log(m + n)). Avnaim et al [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.…”

Section: Related Workmentioning

confidence: 99%

“…Repeatedly, for all triples h, i, j , geometric restriction sets U ij to be a subset, 4 U ij ∩ (U ih ⊕ U hj ). This corresponds to the rule that "a valid placement of P j relative to P i must also be a valid placement of P j relative to P h plus a valid placement of P h relative to P i ".…”

Section: Restrictionmentioning

confidence: 99%

Rotational polygon containment and minimum enclosure

Milenkovic

1998

Proceedings of the Fourteenth Annual Symposium on Computational Geometry - SCG '98

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An algorithm and a robust floating point implementation is given for rotational polygon containment: given polygons P 1 , P 2 , P 3 , . . . , P k and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C ∈ C of minimum area for which containment has a solution. The minimum enclosure is approximate: it bounds the minimum area between (1 − ε)A and A. Experiments indicate that finding the minimum enclosure is practical for k = 2, 3 but not larger unless optimality is sacrificed or angles ranges are limited (although these solutions can still be useful). Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles. In particular, it introduces nearest pair rounding, which allows all these calculations to be carried out in rounded floating point arithmetic.

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“…The polygon containment problem, that is, deciding whether an n-gon P can fit into an m-gon Q under translations and/or rotations, has been studied by various researchers in computational geometry (Chazelle 1983;Baker et al 1986;Fortune 1985; Avnaim and Boissonnat 1988). In the case where Q is convex, the best known algorithm runs in time O(m 2 n) when both translations and rotations are allowed (Chazelle 1983).…”

Section: Related Workmentioning

confidence: 99%

Geometric Sensing of Known Planar Shapes

Jia

Erdmann

1996

The International Journal of Robotics Research

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Industrial assembly involves sensing the pose (orientation and position) of a part. Efficient and reliable sensing strategies can be developed for an assembly task if the shape of the part is known in advance. In this article we investigate two problems of determining the pose of a polygonal part of known shape for the cases of a continuum and a finite number of possible poses respectively.The first problem, named sensing by inscription, involves determining the pose of a convex n-gon from a set of m supporting cones. An algorithm with running time O(nm) that almost always reduces to O(n + m log n) is presented to solve for all possible poses of the polygon. We prove that the number of possible poses cannot exceed 6n, given m ≥ 2 supporting cones with distinct vertices. Simulation experiments demonstrate that two supporting cones are sufficient to determine the real pose of the n-gon in most cases. Our results imply that sensing in practice can be carried out by obtaining viewing angles of a planar part at multiple exterior sites in the plane.On many occasions a parts feeder will have reduced the number of possible poses of a part to a small finite set. Our second problem, named sensing by point sampling, is concerned with a more general version: finding the minimum number of sensing points required to distinguish between n polygonal shapes with a total of m edges. In practice this can be implemented by embedding a series of point light detectors in a feeder tray or by using a set of mechanical probes that touch the feeder at a finite number of predetermined points. We show that this problem is equivalent to an NP-complete set-theoretic problem introduced as Discriminating Set, and present an O(n 2 m 2 ) approximation algorithm to solve it with a ratio of 2 ln n. Furthermore, we prove that one can use an algorithm for Discriminating Set with ratio c log n to construct an algorithm for Set Covering with ratio c log n + O(log log n). Thus the ratio 2 ln n is asymptotically optimal unless NP ⊂ DTIME(n poly log n ), a consequence of known results on approximating Set Covering. The complexity of subproblems of Discriminating Set is also analyzed, based on their relationship to a generalization of Independent Set called 3-Independent Set. Finally, simulation results suggest that sensing by point sampling is mostly applicable when poses are densely distributed in the plane.

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Polygon placement under translation and rotation

Cited by 31 publications

References 13 publications

Dynamic Minkowski sums under scaling

Dynamic Minkowski sums under scaling

Rotational polygon containment and minimum enclosure

Geometric Sensing of Known Planar Shapes

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