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Symmetric Versions of Laman's Theorem

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Symmetric Versions of Laman's Theorem. / Schulze, Bernd.
In: Discrete and Computational Geometry, Vol. 44, No. 4, 12.2010, p. 946-972.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Schulze, B 2010, 'Symmetric Versions of Laman's Theorem', Discrete and Computational Geometry, vol. 44, no. 4, pp. 946-972. https://doi.org/10.1007/s00454-009-9231-x

APA

Schulze, B. (2010). Symmetric Versions of Laman's Theorem. Discrete and Computational Geometry, 44(4), 946-972. https://doi.org/10.1007/s00454-009-9231-x

Vancouver

Schulze B. Symmetric Versions of Laman's Theorem. Discrete and Computational Geometry. 2010 Dec;44(4):946-972. doi: 10.1007/s00454-009-9231-x

Author

Schulze, Bernd. / Symmetric Versions of Laman's Theorem. In: Discrete and Computational Geometry. 2010 ; Vol. 44, No. 4. pp. 946-972.

Bibtex

@article{ea82015c64ea42328f899d1dabeb9c42,
title = "Symmetric Versions of Laman's Theorem",
abstract = "Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are {"}fixed{"} by various symmetry operations of the framework. For the group C-3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46: 762-773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with C-3 symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg's theorem and Crapo's theorem for C-3 which provide alternate characterizations of {"}generically{"} isostatic graphs with C-3 symmetry. As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups C-2 and C-s, in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups C-2 and C-s which are generated by a half-turn and a reflection in the plane, respectively.",
keywords = "Generic rigidity , Infinitesimal rigidity , Bar-and-joint frameworks , Symmetric frameworks , Laman graphs , Henneberg construction, 3Tree2 partition",
author = "Bernd Schulze",
year = "2010",
month = dec,
doi = "10.1007/s00454-009-9231-x",
language = "English",
volume = "44",
pages = "946--972",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "4",

}

RIS

TY - JOUR

T1 - Symmetric Versions of Laman's Theorem

AU - Schulze, Bernd

PY - 2010/12

Y1 - 2010/12

N2 - Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are "fixed" by various symmetry operations of the framework. For the group C-3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46: 762-773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with C-3 symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg's theorem and Crapo's theorem for C-3 which provide alternate characterizations of "generically" isostatic graphs with C-3 symmetry. As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups C-2 and C-s, in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups C-2 and C-s which are generated by a half-turn and a reflection in the plane, respectively.

AB - Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are "fixed" by various symmetry operations of the framework. For the group C-3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46: 762-773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with C-3 symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg's theorem and Crapo's theorem for C-3 which provide alternate characterizations of "generically" isostatic graphs with C-3 symmetry. As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups C-2 and C-s, in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups C-2 and C-s which are generated by a half-turn and a reflection in the plane, respectively.

KW - Generic rigidity

KW - Infinitesimal rigidity

KW - Bar-and-joint frameworks

KW - Symmetric frameworks

KW - Laman graphs

KW - Henneberg construction

KW - 3Tree2 partition

U2 - 10.1007/s00454-009-9231-x

DO - 10.1007/s00454-009-9231-x

M3 - Journal article

VL - 44

SP - 946

EP - 972

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -