On dimensional rigidity of bar-and-joint frameworks

Alfakih, Abdo Y.

doi:10.1016/j.dam.2006.11.011

Cited by 39 publications

(45 citation statements)

References 12 publications

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“…Theorem 1 (Alfakih [2]). A bar framework (G, p) is universally rigid if and only if it is dimensionally rigid and has no affine motion.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Universal rigidity of bar frameworks via the geometry of spectrahedra

Alfakih

2016

J Glob Optim

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A bar framework (G, p) in dimension r is a graph G whose nodes are points p 1 , . . . , p n in R r and whose edges are line segments between pairs of these points. Two frameworks (G, p) and (G, q) are equivalent if each edge of (G, p) has the same (Euclidean) length as the corresponding edge of (G, q). A pair of nonadjacent vertices i and j of (G, p) is universally linked if ||p i − p j || = ||q i − q j || in every framework (G, q) that is equivalent to (G, p). Framework (G, p) is universally rigid iff every pair of non-adjacent vertices of (G, p) is universally linked.

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“…Theorem 1 (Alfakih [2]). A bar framework (G, p) is universally rigid if and only if it is dimensionally rigid and has no affine motion.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Universal rigidity of bar frameworks via the geometry of spectrahedra

Alfakih

2016

J Glob Optim

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“…Another connection is to the notion of dimensional rigidity, which was introduced by Alfakih [1]. A framework (G, p) in E d is dimensionally rigid if there are no equivalent frameworks with a higher dimensional span.…”

Section: Remarksmentioning

confidence: 99%

Affine Rigidity and Conics at Infinity

Connelly

Gortler

Theran

2017

International Mathematics Research Notices

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We prove that if a framework of a graph is neighborhood affine rigid in d-dimensions (or has the stronger property of having an equilibrium stress matrix of rank n − d − 1) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.

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“…Hence, the terms "framework G(p)" and "framework G(X)" can be used interchangeably. For more details see [3].…”

Section: A Characterization Of Equivalent Frameworkmentioning

confidence: 99%

“…Lemma 3.1 (Alfakih [3]) Let G(p) be a given framework with n vertices in R r for some r ≤ n − 2; and let Z be the Gale matrix of G(p). Further, let U and W be the matrices whose columns form orthonormal bases of the nullspace and the rangespace ofX, whereX is the projected Gram matrix associated withp.…”

Section: A Characterization Of Equivalent Frameworkmentioning

confidence: 99%

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On bar frameworks, stress matrices and semidefinite programming

Alfakih

2010

Math. Program.

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A bar framework G(p) in r-dimensional Euclidean space is a graph G = (V, E) on the vertices 1, 2, . . . , n, where each vertex i is located at point p i in R r . Given a framework G(p) in R r , a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||qThis problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework G(q) is restricted to be in the same r-dimensional space or not. The stress matrix S of a bar framework G(p) plays a key role in these and other related problems.In this paper, we show that semidefinite programming (SDP) can be effectively used to address the universal rigidity problem. In particular, we use the notion of non-degeneracy of SDP to obtain a sufficient condition for universal rigidity, and to re-derive the known sufficient condition for generic universal rigidity. We present new results concerning positive semidefinite stress matrices and we use a semidefinite version of Farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix S. *

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On dimensional rigidity of bar-and-joint frameworks

Cited by 39 publications

References 12 publications

Universal rigidity of bar frameworks via the geometry of spectrahedra

Universal rigidity of bar frameworks via the geometry of spectrahedra

Affine Rigidity and Conics at Infinity

On bar frameworks, stress matrices and semidefinite programming

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