241 research outputs found
Finite and infinitesimal rigidity with polyhedral norms
We characterise finite and infinitesimal rigidity for bar-joint frameworks in
R^d with respect to polyhedral norms (i.e. norms with closed unit ball P a
convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown
to be equivalent for finite frameworks in R^d which are well-positioned with
respect to P. An edge-labelling determined by the facets of the unit ball and
placement of the framework is used to characterise infinitesimal rigidity in
R^d in terms of monochrome spanning trees. An analogue of Laman's theorem is
obtained for all polyhedral norms on R^2.Comment: 26 page
Maxwell-Laman counts for bar-joint frameworks in normed spaces
The rigidity matrix is a fundamental tool for studying the infinitesimal
rigidity properties of Euclidean bar-joint frameworks. In this paper we
generalize this tool and introduce a rigidity matrix for bar-joint frameworks
in arbitrary finite dimensional real normed vector spaces. Using this new
matrix, we derive necessary Maxwell-Laman-type counting conditions for a
well-positioned bar-joint framework in a real normed vector space to be
infinitesimally rigid. Moreover, we derive symmetry-extended counting
conditions for a bar-joint framework with a non-trivial symmetry group to be
isostatic (i.e., minimally infinitesimally rigid). These conditions imply very
simply stated restrictions on the number of those structural components that
are fixed by the various symmetry operations of the framework. Finally, we
offer some observations and conjectures regarding combinatorial
characterisations of 2-dimensional symmetric, isostatic bar-joint frameworks
where the unit ball is a quadrilateral.Comment: 17 page
Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant
under the action of a Fuchsian group of isometries (i.e. a group of isometries
leaving globally invariant a totally geodesic surface, on which it acts
cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric
to a hyperbolic metric with conical singularities of positive singular
curvature on a compact surface of genus greater than one. We prove that these
metrics are actually realised by exactly one convex Fuchsian polyhedron (up to
global isometries). This extends a famous theorem of A.D. Alexandrov.Comment: Some little corrections from the preceding version. To appear in Les
Annales de l'Institut Fourie
Infinitesimal rigidity in normed planes
We prove that a graph has an infinitesimally rigid placement in a
non-Euclidean normed plane if and only if it contains a -tight spanning
subgraph. The method uses an inductive construction based on generalised
Henneberg moves and the geometric properties of the normed plane. As a key
step, rigid placements are constructed for the complete graph by
considering smoothness and strict convexity properties of the unit ball.Comment: 26 page
The rigidity of infinite graphs
A rigidity theory is developed for the Euclidean and non-Euclidean placements
of countably infinite simple graphs in R^d with respect to the classical l^p
norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and
Henneberg combinatorial characterisations of generic infinitesimal rigidity for
finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation
of the rigidity of generic finite body-bar frameworks in d-dimensional
Euclidean space is generalised to the non-Euclidean l^p 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 of an inclusion tower of
finite graphs for which the inclusions satisfy a relative rigidity property.
For d>2 a countable graph which is rigid for generic placements in R^d may fail
the stronger property of sequential rigidity, while for d=2 the equivalence
with sequential rigidity is obtained from the generalised Laman
characterisations. Applications are given to the flexibility of non-Euclidean
convex polyhedra and to the infinitesimal and continuous rigidity of compact
infinitely-faceted simplicial polytopes.Comment: 51 page
Symmetric isostatic frameworks with or distance constraints
Combinatorial characterisations of minimal rigidity are obtained for
symmetric 2-dimensional bar-joint frameworks with either or
distance constraints. The characterisations are expressed in
terms of symmetric tree packings and the number of edges fixed by the symmetry
operations. The proof uses new Henneberg-type inductive construction schemes.Comment: 20 pages. Main theorem extended. Construction schemes refined. New
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