A characterization of generically rigid frameworks on surfaces of revolution

A foundational theorem of Laman provides a counting characterization of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterization was obtained for frameworks in three dimensions whose vertices are constrained to concentric spheres or to concentric cylinders. Noting that the plane and the sphere have 3 independent locally tangential infinitesimal motions while the cylinder has 2, we obtain here a Laman-type theorem for frameworks on algebraic surfaces with a 1-dimensional space of tangential motions. Such surfaces include the torus, helicoids, and surfaces of revolution. The relevant class of graphs are the (2,1)-tight graphs, in contrast to (2,3)-tightness for the plane/sphere and (2,2)-tightness for the cylinder. The proof uses a new characterization of simple (2,1)-tight graphs and an inductive construction requiring generic rigidity preservation for 5 graph moves, including the two Henneberg moves, an edge joining move, and various vertex surgery moves. Read More: http://epubs.siam.org/doi/abs/10.1137/13091319

Stress matrices and global rigidity of frameworks on surfaces

In 2005, Bob Connelly showed that a generic framework in \bR^d is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in \bR^3. For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.Comment: Significant changes due to an error in the proof of Theorem 5.1 in the previous version which we have only been able to resolve for 'generic' surface

Symmetry adapted Assur decompositions

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs, and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure

Non-crossing frameworks with non-crossing reciprocals

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that if such an embedding is sufficiently generic, then the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation embeddings of Laman circuits. All self-stresses on a planar graph correspond to liftings to piece-wise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure

Double-distance frameworks and mixed sparsity graphs

A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint frameworks in a variety of such contexts. The main results are combinatorial characterisations for (i) frameworks restricted to surfaces with both Euclidean and geodesic distance constraints, (ii) frameworks in the plane with Euclidean and non-Euclidean distance constraints, and (iii) direction-length frameworks in the non-Euclidean plane

A comparison of vertex-splitting and spider-splitting for the study of three-dimensional rigidity

1 online resource (42 p.) : illustrationsIncludes abstract.Includes bibliographical references (p. 41-42).In two dimensions, generic rigidity is a combinatorial property of a framework, but extensions into three dimensions fail to completely characterize generic rigidity. It is therefore interesting to investigate two graph operations introduced by Walter Whiteley, vertex-splitting and spider-splitting, which are known to take a minimally rigid framework in three dimensions to a new minimally rigid framework with an additional vertex. We present algorithms for generating all possible graphs obtained by vertex-splitting, spider-splitting, and combinations of vertex-splitting and spider-splitting. For graphs with up to and including 8 vertices, the set of graphs obtained by spider-splitting is a subset of the set obtained by vertex-splitting. Additionally, the set produced by combinations of vertex-splitting and spider-splitting is equal to the set obtained by vertex-splitting. This suggests that as a method for generating rigid graphs, spider-splitting is inferior to vertex-splitting at all steps of iteration

The rigidity of infinite graphs

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces (\bR^d,\|\cdot \|_q), for $d\geq 2$ and $1 <q < \infty$. Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for  finite graphs in  (\bR^2,\|\cdot \|_2).  Also Tay's multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in (\bR^d,\|\cdot\|_2) is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms  it is shown that a generically rigid countable simple graph is the direct limit $G= \varinjlim G_k$ of an inclusion tower of finite graphs $G_1 \subseteq G_2 \subseteq  \dots$ for which the inclusions satisfy a relative rigidity property. For $d\geq 3$ a countable graph which is rigid for generic placements in  \bR^d may fail the stronger property of  sequential rigidity, while for $d=2$ the properties are equivalent