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Stress matrices and global rigidity of frameworks on surfaces

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Stress matrices and global rigidity of frameworks on surfaces. / Jackson, Bill; Nixon, Anthony.

In: Discrete and Computational Geometry, Vol. 54, No. 3, 10.2015, p. 586-609.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

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Jackson, B & Nixon, A 2015, 'Stress matrices and global rigidity of frameworks on surfaces', Discrete and Computational Geometry, vol. 54, no. 3, pp. 586-609. https://doi.org/10.1007/s00454-015-9724-8

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Vancouver

Jackson B, Nixon A. Stress matrices and global rigidity of frameworks on surfaces. Discrete and Computational Geometry. 2015 Oct;54(3):586-609. https://doi.org/10.1007/s00454-015-9724-8

Author

Jackson, Bill ; Nixon, Anthony. / Stress matrices and global rigidity of frameworks on surfaces. In: Discrete and Computational Geometry. 2015 ; Vol. 54, No. 3. pp. 586-609.

Bibtex

@article{1f810384d39848da97c2781554a0cc33,
title = "Stress matrices and global rigidity of frameworks on surfaces",
abstract = "In 2005, Bob Connelly showed that a generic framework in R d is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in R 3 . For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum-rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.",
keywords = "Rigidity, Global rigidity, Stress matrix, Framework on a surface",
author = "Bill Jackson and Anthony Nixon",
note = "Evidence of acceptance is on publisher pdf The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-015-9724-8",
year = "2015",
month = oct,
doi = "10.1007/s00454-015-9724-8",
language = "English",
volume = "54",
pages = "586--609",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "3",

}

RIS

TY - JOUR

T1 - Stress matrices and global rigidity of frameworks on surfaces

AU - Jackson, Bill

AU - Nixon, Anthony

N1 - Evidence of acceptance is on publisher pdf The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-015-9724-8

PY - 2015/10

Y1 - 2015/10

N2 - In 2005, Bob Connelly showed that a generic framework in R d is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in R 3 . For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum-rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.

AB - In 2005, Bob Connelly showed that a generic framework in R d is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in R 3 . For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum-rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.

KW - Rigidity

KW - Global rigidity

KW - Stress matrix

KW - Framework on a surface

U2 - 10.1007/s00454-015-9724-8

DO - 10.1007/s00454-015-9724-8

M3 - Journal article

VL - 54

SP - 586

EP - 609

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -