271 research outputs found
Counting Realizations of Laman Graphs on the Sphere
We present an algorithm that computes the number of realizations of a Laman graph on a sphere for a general choice of the angles between the vertices. The algorithm is based on the interpretation of such a realization as a point in the moduli space of stable curves of genus zero with marked points, and on the explicit description, due to Keel, of the Chow ring of this space
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
Generic rigidity of reflection frameworks
We give a combinatorial characterization of generic minimally rigid
reflection frameworks. The main new idea is to study a pair of direction
networks on the same graph such that one admits faithful realizations and the
other has only collapsed realizations. In terms of infinitesimal rigidity,
realizations of the former produce a framework and the latter certifies that
this framework is infinitesimally rigid.Comment: 14 pages, 2 figure
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
Slider-pinning Rigidity: a Maxwell-Laman-type Theorem
We define and study slider-pinning rigidity, giving a complete combinatorial
characterization. This is done via direction-slider networks, which are a
generalization of Whiteley's direction networks.Comment: Accepted, to appear in Discrete and Computational Geometr
On the Number of Embeddings of Minimally Rigid Graphs
Rigid frameworks in some Euclidian space are embedded graphs having a unique
local realization (up to Euclidian motions) for the given edge lengths,
although globally they may have several. We study the number of distinct planar
embeddings of minimally rigid graphs with vertices. We show that, modulo
planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order
of , and .
For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety over the complex numbers . In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with hyperplanes yields at most
zero-dimensional components, and one finds this degree to be
. The lower bounds are related to inductive
constructions of minimally rigid graphs via Henneberg sequences.
The same approach works in higher dimensions. In particular we show that it
leads to an upper bound of for the number of spatial embeddings
with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to
rigid motions
Non-crossing frameworks with non-crossing reciprocals
We study non-crossing frameworks in the plane for which the classical
reciprocal on the dual graph is also non-crossing. We give a complete
description of the self-stresses on non-crossing frameworks whose reciprocals
are non-crossing, in terms of: the types of faces (only pseudo-triangles and
pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a
geometric condition on the stress vectors at some of the vertices.
As in other recent papers where the interplay of non-crossingness and
rigidity of straight-line plane graphs is studied, pseudo-triangulations show
up as objects of special interest. For example, it is known that all planar
Laman circuits can be embedded as a pseudo-triangulation with one non-pointed
vertex. We show that if such an embedding is sufficiently generic, then the
reciprocal is non-crossing and again a pseudo-triangulation embedding of a
planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation
embedding of a planar Laman circuit, the reciprocal is still non-crossing and a
pseudo-triangulation, but its underlying graph may not be a Laman circuit.
Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal
arise as the reciprocals of such, possibly singular, stresses on
pseudo-triangulation embeddings of Laman circuits.
All self-stresses on a planar graph correspond to liftings to piece-wise
linear surfaces in 3-space. We prove characteristic geometric properties of the
lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure
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