1,793 research outputs found
Operations preserving the global rigidity of graphs and frameworks in the plane
AbstractA straight-line realization of (or a bar-and-joint framework on) graph G in Rd is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in Rd if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in R2. If G is triangle-reducible, which is a subfamily of globally rigid graphs that includes Cauchy graphs as well as Grünbaum graphs, the constructed realization will also be infinitesimally rigid.Our algorithm relies on the inductive construction of globally rigid graphs which uses edge additions and one of the Henneberg operations. We also show that vertex splitting, which is another well-known operation in combinatorial rigidity, preserves global rigidity in R2
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
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
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
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
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
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