An Improved Bound for the Rigidity of Linearly Constrained Frameworks

Jackson, Bill; Nixon, Anthony; Tanigawa, Shin‐ichi

doi:10.1137/20m134304x

Cited by 5 publications

(7 citation statements)

References 9 publications

(11 reference statements)

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“…However, a possibly more tractable and more widely applicable problem is to move to linearly constrained frameworks (see, for example, [6]). These frameworks, in the 3-dimensional case, model 'generic surfaces', but are easily adaptable to arbitrary dimension where some interesting results are known [6,12].…”

Section: Discussionmentioning

confidence: 99%

Rigidity of symmetric frameworks on the cylinder

Nixon¹,

Schulze²,

Wall³

2022

Preprint

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A bar-joint framework (G, p) is the combination of a finite simple graph G = (V, E) and a placement p : V → R d . The framework is rigid if the only edge-length preserving continuous deformations of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in R 3 restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.

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Section: Discussionmentioning

confidence: 99%

Rigidity of symmetric frameworks on the cylinder

Nixon¹,

Schulze²,

Wall³

2022

Preprint

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“…In the case where the linear constraints are generic, a 2-dimensional analogue of Laman's theorem (closely analogous to Theorem 4) was proved by Streinu and Theran [46] and this has been extended to all dimensions, under additional hypotheses on the dimension of the affine subspaces each vertex is restricted to, first in [81] and then in [79]. Moreover if one restricts to body-bar frameworks or to 2-dimensions but allows nongeneric linear constraints, as in Theorem 4, then combinatorial characterisations are also known [82].…”

Section: Linear Constraints As Slidersmentioning

confidence: 98%

“…While giving an emphasis here to sliders viewed as points at infinity, there are multiple other strands of mathematical and applied work that connect to sliders, and points constrained to follow lines or plane [43,46,79]. See below for stronger connections.…”

Section: Joints At Infinity and Slidersmentioning

confidence: 99%

Rigidity through a Projective Lens

2021

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In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

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“…It would be an interesting future project to extend our analysis to higher dimensions. While for bar-joint frameworks little is known when d ≥ 3, in the linearly constrained case characterisations are known when suitable assumptions are made on the affine subspaces defined by the linear constraints [5,10].…”

Section: Rigidity Theorymentioning

confidence: 99%

“…Streinu and Theran [24] proved a characterisation of generic rigidity for linearly constrained frameworks in R 2 . The articles [5,10] provide an analogous characterisation for generic rigidity of linearly constrained frameworks in R d as long as the dimensions of the affine subspaces at each vertex are sufficiently small (compared to d), and [8] characterises the stronger notion of global rigidity in this context.…”

Section: Introductionmentioning

confidence: 99%