The rigidity of periodic body–bar frameworks on the three-dimensional fixed torus

Ross, Elissa

doi:10.1098/rsta.2012.0112

Cited by 14 publications

(41 citation statements)

References 28 publications

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“…It was proved that the symmetry-forced generic rigidity (i.e., symmetry-forced rigidity on generic configurations subject to the symmetry) can be checked by computing the rank of linear matroids defined on the edge sets of the underlying quotient gain graphs, and thus can be analyzed as in a conventional manner. After this concept has been emerged, characterizing in terms of the underlying quotient gain graphs were proved by Ross [30,31] for periodic 2-dimensional barjoint frameworks and periodic 3-dimensional body-bar frameworks with fixed lattice metric and by Malestein and Theran [23,22] for crystallographic 2-dimensional bar-joint frameworks with flexible lattice metric.…”

Section: Applications To Rigidity Theorymentioning

confidence: 99%

Matroids of gain graphs in applied discrete geometry

Tanigawa

2015

Trans. Amer. Math. Soc.

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A Γ-gain graph is a graph whose oriented edges are labeled invertibly from a group Γ. Zaslavsky proposed two matroids of Γ-gain graphs, called frame matroids and lift matroids, and investigated linear representations of them. Each matroid has a canonical representation over a field F if Γ is isomorphic to a subgroup of F × in the case of frame matroids or Γ is isomorphic to an additive subgroup of F in the case of lift matroids. The canonical representation of the frame matroid of a complete graph is also known as a Dowling geometry, as it was first introduced by Dowling for finite groups Γ.In this paper, we extend these matroids in two ways. The first one is extending the rank function of each matroid, based on submodular functions over Γ. The resulting rank function generalizes that of the union of frame matroids or lift matroids. Another one is extending the canonical linear representation of the union of d copies of a frame matroid or a lift matroid, based on linear representations of Γ on a d-dimensional vector space. We show that linear matroids of the latter extension are indeed special cases of the first extensions, as in the relation between Dowling geometries and frame matroids. We also discuss an attempt to unify the extension of frame matroids and that of lift matroids.This work is motivated from recent research on the combinatorial rigidity of symmetric graphs. As special cases, we give several new results on this topic, including combinatorial characterizations of the symmetry-forced rigidity of generic body-bar frameworks with point group symmetries or crystallographic symmetries and the symmetric parallel redrawability of generic bar-joint frameworks with point group symmetries or crystallographic symmetries.

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Section: Applications To Rigidity Theorymentioning

confidence: 99%

Matroids of gain graphs in applied discrete geometry

Tanigawa

2015

Trans. Amer. Math. Soc.

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“…Periodic frameworks in the plane have been studied by a number of groups. Whether the lattice is fixed [20], partially variable or fully flexible [12,3] there is a natural rigidity matrix which can be made square by pinning. Moreover periodic frameworks have been understood combinatorially using gain graphs.…”

Section: Extensions To Matrices For Other Constraint Systemsmentioning

confidence: 99%

Symmetry Adapted Assur Decompositions

et al. 2014

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Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.

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“…A modern treatment can be found in works of Graver, Servatius and Servatius [8] and Whiteley [23], [24]. More recently, significant progress has been made in topics such as global rigidity ( [5], [7], [10]) and the rigidity of periodic frameworks ( [4], [15], [19], [20]) in addition to newly emerging themes such as symmetric frameworks [21] and frameworks supported on surfaces [17]. In this article we consider rigidity properties of both finite and infinite bar-joint frameworks (G, p) in R d with respect to polyhedral norms.…”

Section: Introductionmentioning

confidence: 99%

Finite and Infinitesimal Rigidity with Polyhedral Norms

Kitson

2015

Discrete Comput Geom

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We characterise finite and infinitesimal rigidity for bar-joint frameworks in R d with respect to polyhedral norms (i.e. norms with closed unit ball P, a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R d which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in R d in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R 2 .

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The rigidity of periodic body–bar frameworks on the three-dimensional fixed torus

Abstract: We present necessary and sufficient conditions for the generic rigidity of body–bar frameworks on the three-dimensional fixed torus. These frameworks correspond to infinite periodic body–bar frame-works in with a fixed periodic lattice.

Cited by 14 publications

References 28 publications

Matroids of gain graphs in applied discrete geometry

Matroids of gain graphs in applied discrete geometry

Symmetry Adapted Assur Decompositions

Finite and Infinitesimal Rigidity with Polyhedral Norms

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