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Maxwell-Laman counts for bar-joint frameworks in normed spaces

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Maxwell-Laman counts for bar-joint frameworks in normed spaces. / Kitson, Derek; Schulze, Bernd.
In: Linear Algebra and its Applications, Vol. 481, 15.09.2015, p. 313-329.

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

Harvard

Kitson, D & Schulze, B 2015, 'Maxwell-Laman counts for bar-joint frameworks in normed spaces', Linear Algebra and its Applications, vol. 481, pp. 313-329. https://doi.org/10.1016/j.laa.2015.05.007

APA

Kitson, D., & Schulze, B. (2015). Maxwell-Laman counts for bar-joint frameworks in normed spaces. Linear Algebra and its Applications, 481, 313-329. https://doi.org/10.1016/j.laa.2015.05.007

Vancouver

Kitson D, Schulze B. Maxwell-Laman counts for bar-joint frameworks in normed spaces. Linear Algebra and its Applications. 2015 Sept 15;481:313-329. Epub 2015 May 15. doi: 10.1016/j.laa.2015.05.007

Author

Kitson, Derek ; Schulze, Bernd. / Maxwell-Laman counts for bar-joint frameworks in normed spaces. In: Linear Algebra and its Applications. 2015 ; Vol. 481. pp. 313-329.

Bibtex

@article{08aaabb1a38b4af4bd400f5c0fa57ae8,
title = "Maxwell-Laman counts for bar-joint frameworks in normed spaces",
abstract = "The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces. Using this new matrix, we derive necessary Maxwell-Laman-type counting conditions for a well-positioned bar-joint framework in a real normed vector space to be infinitesimally rigid. Moreover, we derive symmetry-extended counting conditions for a bar-joint framework with a non-trivial symmetry group to be isostatic (i.e., minimally infinitesimally rigid). These conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework. Finally, we offer some observations and conjectures regarding combinatorial characterisations of 2-dimensional symmetric, isostatic bar-joint frameworks where the unit ball is a quadrilateral.",
keywords = "Rigidity matrix, Bar-joint framework, Infinitesimal rigidity, Minkowski geometry, Symmetric framework",
author = "Derek Kitson and Bernd Schulze",
year = "2015",
month = sep,
day = "15",
doi = "10.1016/j.laa.2015.05.007",
language = "English",
volume = "481",
pages = "313--329",
journal = "Linear Algebra and its Applications",
issn = "0024-3795",
publisher = "Elsevier Inc.",

}

RIS

TY - JOUR

T1 - Maxwell-Laman counts for bar-joint frameworks in normed spaces

AU - Kitson, Derek

AU - Schulze, Bernd

PY - 2015/9/15

Y1 - 2015/9/15

N2 - The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces. Using this new matrix, we derive necessary Maxwell-Laman-type counting conditions for a well-positioned bar-joint framework in a real normed vector space to be infinitesimally rigid. Moreover, we derive symmetry-extended counting conditions for a bar-joint framework with a non-trivial symmetry group to be isostatic (i.e., minimally infinitesimally rigid). These conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework. Finally, we offer some observations and conjectures regarding combinatorial characterisations of 2-dimensional symmetric, isostatic bar-joint frameworks where the unit ball is a quadrilateral.

AB - The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces. Using this new matrix, we derive necessary Maxwell-Laman-type counting conditions for a well-positioned bar-joint framework in a real normed vector space to be infinitesimally rigid. Moreover, we derive symmetry-extended counting conditions for a bar-joint framework with a non-trivial symmetry group to be isostatic (i.e., minimally infinitesimally rigid). These conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework. Finally, we offer some observations and conjectures regarding combinatorial characterisations of 2-dimensional symmetric, isostatic bar-joint frameworks where the unit ball is a quadrilateral.

KW - Rigidity matrix

KW - Bar-joint framework

KW - Infinitesimal rigidity

KW - Minkowski geometry

KW - Symmetric framework

U2 - 10.1016/j.laa.2015.05.007

DO - 10.1016/j.laa.2015.05.007

M3 - Journal article

VL - 481

SP - 313

EP - 329

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -