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