Coning, symmetry and spherical frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning, (b) the further transfer of these results to spherical space via associated rigidity matrices, and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix. Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric \M^{d} and the hyperbolic metric \H^{d}. This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among \E^{d}, cones in $E^{d+1}$, \SS^{d}, \M^{d}, and \H^{d}. We also consider the further extensions associated with the other Cayley-Klein geometries overlaid on the shared underlying projective geometry.Comment: 38 pages, 7 figure

The orbit rigidity matrix of a symmetric framework

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms - giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure

Rigidity of frameworks on expanding spheres

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterisations are established for the rigidity of generic frameworks for $d=1$ with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.Comment: 22 pages, 2 figures, updated reference

Lifting symmetric pictures to polyhedral scenes

Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d − 1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). In this paper we study the impact of symmetry on the lifting properties of pictures. We first use methods from group representation theory to show that the lifting matrix of a symmetric picture can be transformed into a block-diagonalized form. Using this result we then derive new symmetry-extended counting conditions for a picture with a non-trivial symmetry group in an arbitrary dimension to be minimally flat (i.e., ‘non-liftable’). These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the various symmetry operations of the picture. We then also transfer lifting results for symmetric pictures from Euclidean (d − 1)-space to Euclidean d-space via the technique of coning. Finally, we offer some conjectures regarding sufficient conditions for a picture realized generically for a symmetry group to be minimally flat

Geometry of Point-Hyperplane and Spherical Frameworks

In this thesis we show that the infinitesimal rigidity of point-hyperplane frameworks in Euclidean spaces is equivalent to the infinitesimal rigidity of bar-joint frameworks in spherical spaces with a set of joints (corresponding to the hyperplanes) located on a hyperplane. This is done by comparing the rigidity matrix of Euclidean point-hyperplane frameworks and the rigidity matrix of spherical frameworks. This result clearly shows how the first-order rigidity in projective spaces and Euclidean spaces are globally connected. This geometrically significant result is central to the thesis. This result leads to the equivalence of the first-order rigidity of point-hyperplane frameworks with that of bar-joint frameworks with a set of joints in a hyperplane in a Euclidean space (joint work). We also study the rigidity of point-hyperplane frameworks and characterize their rigidity in Euclidean spaces. We next highlight the relationship between point-line frameworks and slider mechanisms in the plane. Point-line frameworks are used to model various types of slider mechanisms. A combinatorial characterization of the rigidity of pinned-slider frameworks in the plane is derived directly as an immediate consequence of the analogous result for pinned bar-joint frameworks in the plane. Using fixed-normal point-line frameworks, we model a second type of slider system in which the slider directions do not change. Also, a third type of slider mechanism is introduced in which the sliders may only rotate around a fixed point but do not translate. This slider mechanism is defined using point-line frameworks with rotatory lines (no translational motion of the lines is allowed). A combinatorial characterization of the generic rigidity of these frameworks is coauthored in a joint work. Then we introduce point-hyperplane tensegrity frameworks in Euclidean spaces. We investigate the rigidity and the infinitesimal rigidity of these frameworks using tensegrity frameworks in spherical spaces. We characterize these different types of rigidity for point-hyperplane tensegrity frameworks and show how these types of rigidity are linked together. This leads to a characterization of the rigidity of a broader class of slider mechanisms in which sliders may move under variable distance constraints rather than fixed-distance constraints. Finally we investigate body-cad constraints in the plane. A combinatorial characterization of their generic infinitesimal rigidity is given. We show how angular constraints are related to non-angular constraints. This leads to a combinatorial result about the rigidity of a specific class of body-bar frameworks with point-point coincidence constraints in the space

The number of realisations of a rigid graph in Euclidean and spherical geometries

A graph is $d$-rigid if for any generic realisation of the graph in $\mathbb{R}^d$ (equivalently, the $d$-dimensional sphere $\mathbb{S}^d$), there are only finitely many non-congruent realisations in the same space with the same edge lengths. By extending this definition to complex realisations in a natural way, we define $c_d(G)$ to be the number of equivalent $d$-dimensional complex realisations of a $d$-rigid graph $G$ for a given generic realisation, and $c^*_d(G)$ to be the number of equivalent $d$-dimensional complex spherical realisations of $G$ for a given generic spherical realisation. Somewhat surprisingly, these two realisation numbers are not always equal. Recently developed algorithms for computing realisation numbers determined that the inequality $c_2(G) \leq c_2^*(G)$ holds for any minimally 2-rigid graph $G$ with 12 vertices or less. In this paper we confirm that, for any dimension $d$, the inequality $c_d(G) \leq c_d^*(G)$ holds for every $d$-rigid graph $G$. This result is obtained via new techniques involving coning, the graph operation that adds an extra vertex adjacent to all original vertices of the graph

Rigidity of Frameworks Supported on Surfaces

A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in \bR^2. A more general theory is developed for frameworks in \bR^3 whose vertices are constrained to move on a two-dimensional smooth submanifold \M. Furthermore, when \M is a union of concentric spheres, or a union of parallel planes or a union of concentric cylinders, necessary and sufficient combinatorial conditions are obtained for the minimal rigidity of generic frameworks.Comment: Final version, 28 pages, with new figure

Rigidity through a Projective Lens

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