Periodic body-and-bar frameworks

Borcea, Ciprian S.; Streinu, Ileana; Tanigawa, Shin‐ichi

doi:10.1145/2261250.2261303

Cited by 9 publications

(9 citation statements)

References 36 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, when compared with the canonical linear representation of the union of frame matroids (cf. §3), linear matroids of gain graphs proposed in the context of rigidity [37,31,22,5,4,5] much rely on algebraic structures of the underlying groups. The primary motivation of this paper is to propose a new class of matroids of gain graphs, which forms the foundation in the study of symmetry-forced rigidity, as does the union of graphic matroids in classical rigidity problem.…”

Section: Applications To Rigidity Theorymentioning

confidence: 99%

“…We now briefly take a look at how the linear matroid given in the last subsection §7.1 arises in the context of rigidity of body-bar frameworks. The following modeling is based on [5]. A body-bar framework is a structure consisting of rigid bodies connected by bars, and it can be represented by a triple (H, B, q), where…”

Section: Symmetry-forced Rigidity Of Body-bar Frameworkmentioning

confidence: 99%

See 1 more Smart Citation

Matroids of gain graphs in applied discrete geometry

Tanigawa

2015

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Applications To Rigidity Theorymentioning

confidence: 99%

Section: Symmetry-forced Rigidity Of Body-bar Frameworkmentioning

confidence: 99%

Matroids of gain graphs in applied discrete geometry

Tanigawa

2015

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The basic theory of periodic frameworks from the point of view of rigidity and flexibility can be found in [3,4,6]. For wider or complementary aspects of periodic framework theory we suggest [50,11] and references therein.…”

Section: Ultrarigidity Of Periodic Frameworkmentioning

confidence: 99%

“…Sums over facepaths or face-cycles are assumed to be written according to the orientation rule given through tetrads. With this convention, we rewrite the relations (5) and (6) obtained in Section 2 as: (11) Applying (10) we obtain the following two conditions on the stress s:…”

Section: Stress Induced By a Periodic Liftingmentioning

confidence: 99%

Liftings and stresses for planar periodic frameworks

Borcea

Streinu

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

Self Cite

View full text Add to dashboard Cite

We formulate and prove a periodic analog of Maxwell's theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudotriangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.

show abstract

“…Due to their applications in fields such as crystallography, materials science, and engineering, the rigidity and flexibility of periodic structures has seen an increased interest in recent years (see e.g. [3,4,13,16,19,21]). In particular, combinatorial characterisations of generic rigid and globally rigid periodic bar-joint frameworks under fixed lattice representations in R 2 were obtained in [21] and [13], respectively.…”

Section: Introductionmentioning

confidence: 99%

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Kaszanitzky

Király

Schulze

2021

Discrete Comput Geom

View full text Add to dashboard Cite

Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

show abstract

Periodic body-and-bar frameworks

Cited by 9 publications

References 36 publications

Matroids of gain graphs in applied discrete geometry

Matroids of gain graphs in applied discrete geometry

Liftings and stresses for planar periodic frameworks

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Contact Info

Product

Resources

About