Recently, V. Alexandrov proposed an intriguing sufficient condition for rigidity, which we will call transverse rigidity. We show that transverse rigidity is actually equivalent to the known sufficient condition for rigidity called prestress stabilit
y. Indeed this leads to a novel interpretation of the prestress condition.
This note gives a detailed proof of the following statement. Let $din mathbb{N}$ and $m,n ge d + 1$, with $m + n ge binom{d+2}{2} + 1$. Then the complete bipartite graph $K_{m,n}$ is generically globally rigid in dimension $d$.
Let $mathbf{p}$ be a configuration of $n$ points in $mathbb{R}^d$ for some $n$ and some $d ge 2$. Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a pat
h that has the same endpoints. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing $mathbf{p}$ given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when $mathbf{p}$ will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has been already been used for the simpler problem of unlabeled edge lengths.
We introduce a new family of algebraic varieties, $L_{d,n}$, which we call the unsquared measurement varieties. This family is parameterized by a number of points $n$ and a dimension $d$. These varieties arise naturally from problems in rigidity theo
ry and distance geometry. In those applications, it can be useful to understand the group of linear automorphisms of $L_{d,n}$. Notably, a result of Regge implies that $L_{2,4}$ has an unexpected linear automorphism. In this paper, we give a complete characterization of the linear automorphisms of $L_{d,n}$ for all $n$ and $d$. We show, that apart from $L_{2,4}$ the unsquared measurement varieties have no unexpected automorphisms. Moreover, for $L_{2,4}$ we characterize the full automorphism group.
Let $G$ be a $3$-connected graph with $n$ vertices and $m$ edges. Let $mathbf{p}$ be a randomly chosen mapping of these $n$ vertices to the integer range $[1..2^b]$ for $bge m^2$. Let $mathbf{l}$ be the vector of $m$ Euclidean lengths of $G$s edges u
nder $mathbf{p}$. In this paper, we show that, WHP over $mathbf{p}$, we can efficiently reconstruct both $G$ and $mathbf{p}$ from $mathbf{l}$. In contrast to this average case complexity, this reconstruction problem is NP-HARD in the worst case. In fact, even the labeled version of this problem (reconstructing $mathbf{p}$ given both $G$ and $mathbf{l}$) is NP-HARD. We also show that our results stand in the presence of small amounts of error in $mathbf{l}$, and in the real setting with approximate length measurements. Our method is based on older ideas that apply lattice reduction to solve certain SUBSET-SUM problems, WHP. We also rely on an algorithm of Seymour that can efficiently reconstruct a graph given an independence oracle for its matroid.
We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell-Calladine index theorem we derive
a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origamis vertices. This supports the recent result by Tachi which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero energy deformations in the bulk that may be used to reconfigure the origami sheet.
We extend the mathematical theory of rigidity of frameworks (graphs embedded in $d$-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously
it must remain inside a small ball, a property we call almost-rigidity; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost-rigidity. We briefly discuss how the theory may be applied to tensegrities.
Let $mathbf{p}$ be a configuration of $n$ points in $mathbb{R}^d$ for some $n$ and some $d ge 2$. Each pair of points has a Euclidean length in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair lengths corresponding t
o the edges of $G$. In this paper, we study the question of when a generic $mathbf{p}$ in $d$ dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair lengths together with knowledge of $d$ and $n$. In this setting the lengths are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point-pair gave rise to which length, nor is data about $G$ given. We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair lengths (together with $d$ and $n$) iff it is determined by the labeled edge lengths.
Recent progress in advanced additive manufacturing techniques has stimulated the growth of the field of mechanical metamaterials. One area particular interest in this subject is the creation of auxetic material properties through elastic instability.
This paper focuses on a novel methodology in the analysis of auxetic metamaterials through analogy with rigid link lattice systems. Our analytic methodology gives extremely good agreement with finite element simulations for both the onset of elastic instability and post-buckling behaviour including Poissons ratio. The insight into the relationships between mechanisms within lattices and their mechanical behaviour has the potential to guide the rational design of lattice based metamaterials.
We prove that if a framework of a graph is neighborhood affine rigid in $d$-dimensions (or has the stronger property of having an equilibrium stress matrix of rank $n-d-1$) then it has an affine flex (an affine, but non Euclidean, transform of space
that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.
