409 research outputs found
A Constructive Characterisation of Circuits in the Simple (2,2)-sparsity Matroid
We provide a constructive characterisation of circuits in the simple
(2,2)-sparsity matroid. A circuit is a simple graph G=(V,E) with |E|=2|V|-1 and
the number of edges induced by any is at most 2|X|-2.
Insisting on simplicity results in the Henneberg operation being enough only
when the graph is sufficiently connected. Thus we introduce 3 different join
operations to complete the characterisation. Extensions are discussed to when
the sparsity matroid is connected and this is applied to the theory of
frameworks on surfaces to provide a conjectured characterisation of when
frameworks on an infinite circular cylinder are generically globally rigid.Comment: 22 pages, 6 figures. Changes to presentatio
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
One brick at a time: a survey of inductive constructions in rigidity theory
We present a survey of results concerning the use of inductive constructions
to study the rigidity of frameworks. By inductive constructions we mean simple
graph moves which can be shown to preserve the rigidity of the corresponding
framework. We describe a number of cases in which characterisations of rigidity
were proved by inductive constructions. That is, by identifying recursive
operations that preserved rigidity and proving that these operations were
sufficient to generate all such frameworks. We also outline the use of
inductive constructions in some recent areas of particularly active interest,
namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar
frameworks. We summarize the key outstanding open problems related to
inductions.Comment: 24 pages, 12 figures, final versio
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
A characterisation of generically rigid frameworks on surfaces of revolution
A foundational theorem of Laman provides a counting characterisation of the
finite simple graphs whose generic bar-joint frameworks in two dimensions are
infinitesimally rigid. Recently a Laman-type characterisation 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-Henneberg 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 characterisation 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.Comment: 23 pages, 5 figures. Minor revisions - most importantly, the new
version has a different titl
Assur decompositions of direction-length frameworks
A bar-joint framework is a realisation of a graph consisting of stiff bars linked by universal joints. The framework is rigid if the only bar-length preserving continuous motions of the joints arise from isometries. A rigid framework is isostatic if deleting any single edge results in a flexible framework. Generically, rigidity depends only on the graph and we say an Assur graph is a pinned isostatic graph with no proper pinned isostatic subgraphs. Any pinned isostatic graph can be decomposed into Assur components which may be of use for mechanical engineers in decomposing mechanisms for simpler analysis and synthesis. A direction-length framework is a generalisation of bar-joint framework where some distance constraints are replaced by direction constraints. We initiate a theory of Assur graphs and Assur decompositions for direction-length frameworks using graph orientations and spanning trees and then analyse choices of pinning set
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
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