On the Number of Embeddings of Minimally Rigid Graphs

Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.88^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety $CM^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the complex numbers $C$. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with $2n-4$ hyperplanes yields at most $deg(CM^{2,n})$ zero-dimensional components, and one finds this degree to be $D^{2,n}={1/2}{{2n-4}\choose {n-2}}$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular we show that it leads to an upper bound of $2 D^{3,n}= {\frac{2^{n-3}}{n-2}}{{n-6}\choose{n-3}}$ for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions

Deformations of crystal frameworks

We apply our deformation theory of periodic bar-and-joint frameworks to tetrahedral crystal structures. The deformation space is investigated in detail for frameworks modelled on quartz, cristobalite and tridymite

Liftings and stresses for planar periodic frameworks

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 pseudo-triangulations, 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.Comment: An extended abstract of this paper has appeared in Proc. 30th annual Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201

Expansive periodic mechanisms

A one-parameter deformation of a periodic bar-and-joint framework is expansive when all distances between joints increase or stay the same. In dimension two, expansive behavior can be fully explained through our theory of periodic pseudo-triangulations. However, higher dimensions present new challenges. In this paper we study a number of periodic frameworks with expansive capabilities in dimension $d\geq 3$ and register both similarities and contrasts with the two-dimensional case

Geometric auxetics

We formulate a mathematical theory of auxetic behavior based on one-parameter deformations of periodic frameworks. Our approach is purely geometric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behavior to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures

Extremal Configurations of Hinge Structures

We study body-and-hinge and panel-and-hinge chains in R^d, with two marked points: one on the first body, the other on the last. For a general chain, the squared distance between the marked points gives a Morse-Bott function on a torus configuration space. Maximal configurations, when the distance between the two marked points reaches a global maximum, have particularly simple geometrical characterizations. The three-dimensional case is relevant for applications to robotics and molecular structures

Line transversals to disjoint balls

We prove that the set of directions of lines intersecting three disjoint balls in $R^3$ in a given order is a strictly convex subset of $S^2$. We then generalize this result to $n$ disjoint balls in $R^d$. As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.Comment: 21 pages, includes figure

On tangents to quadric surfaces

We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric of rank at least three, and called, for intuitive reasons: a basket. Lines in any ruling of the latter will be common tangents. These considerations are then restricted to spheres in Euclidean three-space, and result in a complete answer to the question over the reals: ``When do four spheres allow infinitely many common tangents?''.Comment: 50 page

How Far Can You Reach?

The problem of computing the maximum reach configurations of a 3D revolute-jointed manipulator is a long-standing open problem in robotics. In this paper we present an optimal algorithmic solution for orthogonal polygonal chains. This appears as a special case of a larger family, fully characterized here by a technical condition. Until now, in spite of the practical importance of the problem, only numerical optimization heuristics were available, with no guarantee of obtaining the global maximum. In fact, the problem was not even known to be computationally solvable, and in practice, the numerical heuristics were applicable only to small problem sizes. We present elementary and efficient (mostly linear) algorithms for four fundamental problems: (1) finding the maximum reach value, (2) finding a maximum reach configuration (or enumerating all of them), (3) folding a given chain to a given maximum position, and (4) folding a chain in a way that changes the endpoint distance function monotonically. The algorithms rely on our recent theoretical results characterizing combinatorially the maximum of panel-and-hinge chains. They allow us to reduce the first problem to finding a shortest path between two vertices in an associated simple triangulated polygon, and the last problem to a simple version of the planar carpenter\u27s rule problem. Copyright © by SIAM