155 research outputs found
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
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
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
Geometric auxetics
We formulate a mathematical theory of auxetic behavior based on one-parameter
deformations of periodic frameworks. Our approach is purely geometric, relies
on the evolution of the periodicity lattice and works in any dimension. We
demonstrate its usefulness by predicting or recognizing, without experiment,
computer simulations or numerical approximations, the auxetic capabilities of
several well-known structures available in the literature. We propose new
principles of auxetic design and rely on the stronger notion of expansive
behavior to provide an infinite supply of planar auxetic mechanisms and several
new three-dimensional structures
Deformations of crystal frameworks
We apply our deformation theory of periodic bar-and-joint frameworks to
tetrahedral crystal structures. The deformation space is investigated in detail
for frameworks modelled on quartz, cristobalite and tridymite
Stretchability of Star-Like Pseudo-Visibility Graphs
We present advances on the open problem of characterizing vertex-edge visibility graphs (ve-graphs), reduced by results of O\u27Rourke and Streinu to a stretchability question for pseudo-polygons. We introduce star-like pseudo-polygons as a special subclass containing all the known instances of non-stretchable pseudo-polygons. We give a complete combinatorial characterization and a linear-time decision procedure for star-like pseudo-polygon stretchability and star-like ve-graph recognition. To the best of our knowledge, this is the first problem in computational geometry for which a combinatorial characterization was found by first isolating the oriented matroid substructure and then separately solving the stretchability question. It is also the first class (as opposed to isolated examples) of oriented matroids for which an efficient stretchability decision procedure based on combinatorial criteria is given. The difficulty of the general stretchability problem implied by Mnev\u27s Universality Theorem makes this a result of independent interest in the theory of oriented matroids
Pseudo-Triangulations, Rigidity and Motion Planning
This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. Β© 2005 Springer Science+Business Media, Inc
Large scale rigidity-based flexibility analysis of biomolecules
KINematics And RIgidity (KINARI) is an on-going project for in silico flexibility analysis of proteins. The new version of the software, Kinari-2, extends the function- ality of our free web server KinariWeb, incorporates advanced web technologies, emphasizes the reproducibility of its experiments, and makes substantially improved tools available to the user. It is designed specifically for large scale experiments, in particular, for (a) very large molecules, including bioassemblies with high degree of symmetry such as viruses and crystals, (b) large collections of related biomolecules, such as those obtained through simulated dilutions, mutations, or conformational changes from various types of dynamics simulations, and (c) is intended to work as seemlessly as possible on the large, idiosyncratic, publicly available repository of biomolecules, the Protein Data Bank. We describe the system design, along with the main data processing, computational, mathematical, and validation challenges under- lying this phase of the KINARI project
Extremal Configurations of Hinge Structures
We study body-and-hinge and panel-and-hinge chains in R^d, with two marked
points: one on the first body, the other on the last. For a general chain, the
squared distance between the marked points gives a Morse-Bott function on a
torus configuration space. Maximal configurations, when the distance between
the two marked points reaches a global maximum, have particularly simple
geometrical characterizations. The three-dimensional case is relevant for
applications to robotics and molecular structures
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