Home > Research > Publications & Outputs > Coning, symmetry, and spherical frameworks

Research

Associated organisational units

Links

Text available via DOI:

Keywords

View graph of relations

Coning, symmetry, and spherical frameworks

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

Standard

Coning, symmetry, and spherical frameworks. / Schulze, Bernd; Whiteley, Walter.
In: Discrete and Computational Geometry, Vol. 48, No. 3, 01.10.2012, p. 622-657.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Schulze, B & Whiteley, W 2012, 'Coning, symmetry, and spherical frameworks', Discrete and Computational Geometry, vol. 48, no. 3, pp. 622-657. https://doi.org/10.1007/s00454-012-9427-3

APA

Schulze, B., & Whiteley, W. (2012). Coning, symmetry, and spherical frameworks. Discrete and Computational Geometry, 48(3), 622-657. https://doi.org/10.1007/s00454-012-9427-3

Vancouver

Schulze B, Whiteley W. Coning, symmetry, and spherical frameworks. Discrete and Computational Geometry. 2012 Oct 1;48(3):622-657. doi: 10.1007/s00454-012-9427-3

Author

Schulze, Bernd ; Whiteley, Walter. / Coning, symmetry, and spherical frameworks. In: Discrete and Computational Geometry. 2012 ; Vol. 48, No. 3. pp. 622-657.

Bibtex

@article{6a341e4a1ef64d1a90b5f396c23e3b55,
title = "Coning, symmetry, and spherical frameworks",
abstract = "In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53–70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561–598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric Md and the hyperbolic metric ℍ d .This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among Ed , cones in Ed+1 , Sd , Md , and ℍ d . We also consider the further extensions associated with the other Cayley–Klein geometries overlaid on the shared underlying projective geometry.",
keywords = "Bar-and-joint frameworks , Infinitesimal rigidity , Finite motions, Symmetry, Coning to spherical frameworks, Cayley-Klein geometries",
author = "Bernd Schulze and Walter Whiteley",
year = "2012",
month = oct,
day = "1",
doi = "10.1007/s00454-012-9427-3",
language = "English",
volume = "48",
pages = "622--657",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "3",

}

RIS

TY - JOUR

T1 - Coning, symmetry, and spherical frameworks

AU - Schulze, Bernd

AU - Whiteley, Walter

PY - 2012/10/1

Y1 - 2012/10/1

N2 - In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53–70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561–598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric Md and the hyperbolic metric ℍ d .This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among Ed , cones in Ed+1 , Sd , Md , and ℍ d . We also consider the further extensions associated with the other Cayley–Klein geometries overlaid on the shared underlying projective geometry.

AB - In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53–70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561–598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric Md and the hyperbolic metric ℍ d .This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among Ed , cones in Ed+1 , Sd , Md , and ℍ d . We also consider the further extensions associated with the other Cayley–Klein geometries overlaid on the shared underlying projective geometry.

KW - Bar-and-joint frameworks

KW - Infinitesimal rigidity

KW - Finite motions

KW - Symmetry

KW - Coning to spherical frameworks

KW - Cayley-Klein geometries

U2 - 10.1007/s00454-012-9427-3

DO - 10.1007/s00454-012-9427-3

M3 - Journal article

VL - 48

SP - 622

EP - 657

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -