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The orbit rigidity matrix of a symmetric framework

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The orbit rigidity matrix of a symmetric framework. / Schulze, Bernd; Whiteley, Walter.
In: Discrete and Computational Geometry, Vol. 46, No. 3, 10.2011, p. 561-598.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Schulze, B & Whiteley, W 2011, 'The orbit rigidity matrix of a symmetric framework', Discrete and Computational Geometry, vol. 46, no. 3, pp. 561-598. https://doi.org/10.1007/s00454-010-9317-5

APA

Schulze, B., & Whiteley, W. (2011). The orbit rigidity matrix of a symmetric framework. Discrete and Computational Geometry, 46(3), 561-598. https://doi.org/10.1007/s00454-010-9317-5

Vancouver

Schulze B, Whiteley W. The orbit rigidity matrix of a symmetric framework. Discrete and Computational Geometry. 2011 Oct;46(3):561-598. doi: 10.1007/s00454-010-9317-5

Author

Schulze, Bernd ; Whiteley, Walter. / The orbit rigidity matrix of a symmetric framework. In: Discrete and Computational Geometry. 2011 ; Vol. 46, No. 3. pp. 561-598.

Bibtex

@article{06b04ce5b8474078a301c7ecfdf4f397,
title = "The orbit rigidity matrix of a symmetric framework",
abstract = "A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions.With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms—giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.",
keywords = "Bar and joint frameworks, Symmetry, Orbits, Infinitesimal motion, Finite motion, Mechanism",
author = "Bernd Schulze and Walter Whiteley",
year = "2011",
month = oct,
doi = "10.1007/s00454-010-9317-5",
language = "English",
volume = "46",
pages = "561--598",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "3",

}

RIS

TY - JOUR

T1 - The orbit rigidity matrix of a symmetric framework

AU - Schulze, Bernd

AU - Whiteley, Walter

PY - 2011/10

Y1 - 2011/10

N2 - A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions.With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms—giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.

AB - A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions.With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms—giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.

KW - Bar and joint frameworks

KW - Symmetry

KW - Orbits

KW - Infinitesimal motion

KW - Finite motion

KW - Mechanism

U2 - 10.1007/s00454-010-9317-5

DO - 10.1007/s00454-010-9317-5

M3 - Journal article

VL - 46

SP - 561

EP - 598

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -