Movement Problems for 2-Dimensional Linkages

Joseph, Deborah; Whitesides, Sue

doi:10.1137/0213038

Cited by 87 publications

(40 citation statements)

References 2 publications

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“…THEOREM 9.5.3 Complexity of reachability [HJW84] The reachability problem is PSPACE-hard for a planar linkage without obstacles and permitting the linkage to self-intersect.…”

Section: Hardness Resultsmentioning

confidence: 99%

Geometry and Topology of Polygonal Linkages

Connelly¹,

Demaine²

2004

Handbook of Discrete and Computational Geometry, Second Edition

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“…THEOREM 9.5.3 Complexity of reachability [HJW84] The reachability problem is PSPACE-hard for a planar linkage without obstacles and permitting the linkage to self-intersect.…”

Section: Hardness Resultsmentioning

confidence: 99%

Geometry and Topology of Polygonal Linkages

Connelly¹,

Demaine²

2004

Handbook of Discrete and Computational Geometry, Second Edition

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“…If one insists on a unidimensional folding, Hopcroft et al [8] observed that the minimum folded length can be at least 2 − ε for any ε > 0 with a suitable unit ruler. Moreover, determining the shortest interval, i.e., unidimensional case, into which a given ruler can be folded is NP-complete [8].…”

Section: Related Workmentioning

confidence: 99%

“…Moreover, determining the shortest interval, i.e., unidimensional case, into which a given ruler can be folded is NP-complete [8]. A 2-approximation algorithm [8] and a several polynomial-time approximation schemes [4] are available.…”

Section: Related Workmentioning

confidence: 99%

Nonconvex cases for carpenter's rulers

Chen

Dumitrescu

2015

Theoretical Computer Science

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We consider the carpenter's ruler folding problem in the plane, i.e., finding a minimum area shape with diameter 1 that accommodates foldings of any ruler whose longest link has length 1. An upper bound of π/3 − √ 3/4 = 0.614 . . . and a lower bound of 10 + 2 √ 5/8 = 0.475 . . . are known for convex cases. We generalize the problem to simple nonconvex cases: in this setting we improve the upper bound to 0.583 and establish the first lower bound of 0.073. A variation is to consider rulers with at most k links. The current best convex upper bounds are (about) 0.486 for k = 3, 4 and π/6 = 0.523 . . . for k = 5, 6. These bounds also apply to nonconvex cases. We derive a better nonconvex upper bound of 0.296 for k = 3, 4.

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“…Other authors provided more complete proofs of this fact during the period 1877-1902. The Peaucellier-Lipkin linkage is also used in computer science to prove theorems about workspace topology in robotics [36].…”

Section: What Is "Straight"? How Can We Draw a Straight Line?mentioning

confidence: 99%