One Brick at a Time: A Survey of Inductive Constructions in Rigidity Theory

Nixon, Anthony; Ross, Elissa

doi:10.1007/978-1-4939-0781-6_15

Cited by 15 publications

(17 citation statements)

References 58 publications

(91 reference statements)

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“…From this angle, the two planes are projected onto the lines separating the colored regions in the figure, and the line s 5 ¼ s 6 ¼ 0 along which these planes intersect becomes the line passing through the origin that is normal to the plane of the figure. Henneberg moves [27,36,37] are rigidity-preserving transformations of graphs that are useful in rigidity theory. We consider only the Henneberg-1 move, which in three dimensions is just the addition of a new vertex, and attach it to the original graph with three new edges.…”

Section: H1 Triangulations and Butterfliesmentioning

confidence: 99%

Branches of Triangulated Origami Near the Unfolded State

Chen

Santangelo

2018

Phys. Rev. X

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Origami structures are characterized by a network of folds and vertices joining unbendable plates. For applications to mechanical design and self-folding structures, it is essential to understand the interplay between the set of folds in the unfolded origami and the possible 3D folded configurations. When deforming a structure that has been folded, one can often linearize the geometric constraints, but the degeneracy of the unfolded state makes a linear approach impossible there. We derive a theory for the second-order infinitesimal rigidity of an initially unfolded triangulated origami structure and use it to study the set of nearly unfolded configurations of origami with four boundary vertices. We find that locally, this set consists of a number of distinct "branches" which intersect at the unfolded state, and that the number of these branches is exponential in the number of vertices. We find numerical and analytical evidence that suggests that the branches are characterized by choosing each internal vertex to either "pop up" or "pop down." The large number of pathways along which one can fold an initially unfolded origami structure strongly indicates that a generic structure is likely to become trapped in a "misfolded" state. Thus, new techniques for creating self-folding origami are likely necessary; controlling the popping state of the vertices may be one possibility.

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Section: H1 Triangulations and Butterfliesmentioning

confidence: 99%

Branches of Triangulated Origami Near the Unfolded State

Chen

Santangelo

2018

Phys. Rev. X

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“…Also let ′′ be formed from by deleting two non-adjacent edges , from and adding a new vertex and edges , , , for , , , ∈ . This operation is known as -replacement [15,18]. As outlined above, when a circuit is sufficiently connected we will show that either a 1-reduction or an inverse -replacement can be applied to form a new circuit.…”

Section: F I G U R E 4 Base Graphs On Nine Verticesmentioning

confidence: 88%

A constructive characterisation of circuits in the simple (2,1)‐sparse matroid

McCourt

Nixon

2018

Journal of Graph Theory

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A simple graph G=(V,E) is a (2, 1)‐circuit if |E|=2|V| and |E(H)|≤2|V(H)|−1 for every proper subgraph H of G. Motivated, in part, by ongoing work to understand unique realisations of graphs on surfaces, we derive a constructive characterisation of (2, 1)‐circuits. The characterisation uses the well‐known 1‐extension and X‐replacement operations as well as several summation moves to glue together (2, 1)‐circuits over small cutsets.

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“…One indication of the potential difficulty to overcome here is that the minimum vertex degree in the graph may be 4. The analogue of the Henneberg moves for degree 4 vertices are known as X and V-replacement [7,11,21,35]. V-replacement is known to not preserve (2, ℓ)-sparsity.…”

Section: Further Workmentioning

confidence: 99%

“…X-replacement has been used to some effect in [11] so it is plausible that it could be used for the cases in question here. However, while X-replacement (as an operation on frameworks) is easy to understand in the plane (the generic argument is based on the simple fact that, generically, two lines intersect (see also [11])), the question whether the corresponding operation in 3-dimensions preserves generic rigidity is still open [7,21] (two generic lines in 3D need not intersect!). This difficulty also arises for X-replacements on frameworks supported on surfaces since two lines will typically not intersect in a point on the surface.…”

Section: Further Workmentioning

confidence: 99%

Symmetry-forced rigidity of frameworks on surfaces

Nixon

Schulze

2015

Geom Dedicata

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A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to frameworks that are symmetric with respect to some point group but are otherwise generic, and secondly to frameworks in Euclidean 3-space that are constrained to lie on 2-dimensional algebraic varieties. We combine these two settings and consider the rigidity of symmetric frameworks realised on such surfaces. First we establish necessary conditions for a framework to be symmetry-forced rigid for any group and any surface by setting up a symmetry-adapted rigidity matrix for such frameworks and by extending the methods in Jordán et al. (2012) to this new context. This gives rise to several new symmetry-adapted rigidity matroids on group-labelled quotient graphs. In the cases when the surface is a sphere, a cylinder or a cone we then also provide combinatorial characterisations of generic symmetry-forced rigid frameworks for a number of symmetry groups, including rotation, reflection, inversion and dihedral symmetry. The proofs of these results are based on some new Henneberg-type inductive constructions on the group-labelled quotient graphs that correspond to the bases of the matroids in question. For the remaining symmetry groups in 3-space-as well as for other types of surfaces-we provide some observations and conjectures.

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One Brick at a Time: A Survey of Inductive Constructions in Rigidity Theory

Cited by 15 publications

References 58 publications

Branches of Triangulated Origami Near the Unfolded State

Branches of Triangulated Origami Near the Unfolded State

A constructive characterisation of circuits in the simple (2,1)‐sparse matroid

Symmetry-forced rigidity of frameworks on surfaces

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