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Pairing symmetries for Euclidean and spherical frameworks

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Pairing symmetries for Euclidean and spherical frameworks. / Clinch, Katherine; Nixon, Anthony; Schulze, Bernd; Whiteley, Walter.

In: Discrete and Computational Geometry, Vol. 64, 01.09.2020, p. 483–518.

Research output: Contribution to journalJournal article

Harvard

Clinch, K, Nixon, A, Schulze, B & Whiteley, W 2020, 'Pairing symmetries for Euclidean and spherical frameworks', Discrete and Computational Geometry, vol. 64, pp. 483–518. https://doi.org/10.1007/s00454-020-00198-9

APA

Clinch, K., Nixon, A., Schulze, B., & Whiteley, W. (2020). Pairing symmetries for Euclidean and spherical frameworks. Discrete and Computational Geometry, 64, 483–518. https://doi.org/10.1007/s00454-020-00198-9

Vancouver

Clinch K, Nixon A, Schulze B, Whiteley W. Pairing symmetries for Euclidean and spherical frameworks. Discrete and Computational Geometry. 2020 Sep 1;64:483–518. https://doi.org/10.1007/s00454-020-00198-9

Author

Clinch, Katherine ; Nixon, Anthony ; Schulze, Bernd ; Whiteley, Walter. / Pairing symmetries for Euclidean and spherical frameworks. In: Discrete and Computational Geometry. 2020 ; Vol. 64. pp. 483–518.

Bibtex

@article{d0bea194b0cf4c2abe06477ecd0d099b,
title = "Pairing symmetries for Euclidean and spherical frameworks",
abstract = "We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in Rd. In particular, for a graph G=(V,E) and a framework (G, p), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset X⊂V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.",
author = "Katherine Clinch and Anthony Nixon and Bernd Schulze and Walter Whiteley",
year = "2020",
month = sep,
day = "1",
doi = "10.1007/s00454-020-00198-9",
language = "English",
volume = "64",
pages = "483–518",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",

}

RIS

TY - JOUR

T1 - Pairing symmetries for Euclidean and spherical frameworks

AU - Clinch, Katherine

AU - Nixon, Anthony

AU - Schulze, Bernd

AU - Whiteley, Walter

PY - 2020/9/1

Y1 - 2020/9/1

N2 - We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in Rd. In particular, for a graph G=(V,E) and a framework (G, p), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset X⊂V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.

AB - We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in Rd. In particular, for a graph G=(V,E) and a framework (G, p), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset X⊂V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.

U2 - 10.1007/s00454-020-00198-9

DO - 10.1007/s00454-020-00198-9

M3 - Journal article

VL - 64

SP - 483

EP - 518

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -