3,796 research outputs found
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
Universally Rigid Framework Attachments
A framework is a graph and a map from its vertices to R^d. A framework is
called universally rigid if there is no other framework with the same graph and
edge lengths in R^d' for any d'. A framework attachment is a framework
constructed by joining two frameworks on a subset of vertices. We consider an
attachment of two universally rigid frameworks that are in general position in
R^d. We show that the number of vertices in the overlap between the two
frameworks must be sufficiently large in order for the attachment to remain
universally rigid. Furthermore, it is shown that universal rigidity of such
frameworks is preserved even after removing certain edges. Given positive
semidefinite stress matrices for each of the two initial frameworks, we
analytically derive the PSD stress matrices for the combined and edge-reduced
frameworks. One of the benefits of the results is that they provide a general
method for generating new universally rigid frameworks.Comment: 16 pages, 4 figure
On affine rigidity
We define the notion of affine rigidity of a hypergraph and prove a variety
of fundamental results for this notion. First, we show that affine rigidity can
be determined by the rank of a specific matrix which implies that affine
rigidity is a generic property of the hypergraph.Then we prove that if a graph
is is -vertex-connected, then it must be "generically neighborhood
affinely rigid" in -dimensional space. This implies that if a graph is
-vertex-connected then any generic framework of its squared graph must
be universally rigid.
Our results, and affine rigidity more generally, have natural applications in
point registration and localization, as well as connections to manifold
learning.Comment: Updated abstrac
Toward the Universal Rigidity of General Frameworks
Let (G,P) be a bar framework of n vertices in general position in R^d, d <=
n-1, where G is a (d+1)-lateration graph. In this paper, we present a
constructive proof that (G,P) admits a positive semi-definite stress matrix
with rank n-d-1. We also prove a similar result for a sensor network where the
graph consists of m(>= d+1) anchors.Comment: v2, a revised version of an earlier submission (v1
Natural realizations of sparsity matroids
A hypergraph G with n vertices and m hyperedges with d endpoints each is
(k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le
kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a
linearly representable matroidal family.
Motivated by problems in rigidity theory, we give a new linear representation
theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the
representing matrix captures the vertex-edge incidence structure of the
underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars
Mathematica Contemporane
Graph connectivity and universal rigidity of bar frameworks
Let be a graph on nodes. In this note, we prove that if is
-vertex connected, , then there exists a
configuration in general position in such that the bar framework
is universally rigid. The proof is constructive and is based on a
theorem by Lovasz et al concerning orthogonal representations and connectivity
of graphs [12,13].Comment: updated 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
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