5,201 research outputs found
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
Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces
A result due in its various parts to Hendrickson, Connelly, and Jackson and
Jord\'an, provides a purely combinatorial characterisation of global rigidity
for generic bar-joint frameworks in . The analogous conditions
are known to be insufficient to characterise generic global rigidity in higher
dimensions. Recently Laman-type characterisations of rigidity have been
obtained for generic frameworks in when the vertices are
constrained to lie on various surfaces, such as the cylinder and the cone. In
this paper we obtain analogues of Hendrickson's necessary conditions for the
global rigidity of generic frameworks on the cylinder, cone and ellipsoid.Comment: 13 page
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
Characterizing the universal rigidity of generic frameworks
A framework is a graph and a map from its vertices to E^d (for some d). A
framework is universally rigid if any framework in any dimension with the same
graph and edge lengths is a Euclidean image of it. We show that a generic
universally rigid framework has a positive semi-definite stress matrix of
maximal rank. Connelly showed that the existence of such a positive
semi-definite stress matrix is sufficient for universal rigidity, so this
provides a characterization of universal rigidity for generic frameworks. We
also extend our argument to give a new result on the genericity of strict
complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio
Iterative Universal Rigidity
A bar framework determined by a finite graph and configuration in
space is universally rigid if it is rigid in any . We provide a characterization of universally rigidity for any
graph and any configuration in terms of a sequence of affine
subsets of the space of configurations. This corresponds to a facial reduction
process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure
- …