133 research outputs found
Computational Geometry Column 39
The resolution of a decades-old open problem is described: polygonal chains
cannot lock in the plane.Comment: 4 pages, 2 figures. To appear in SIGACT News and in Int. J. Comp.
Geom. App
A short proof of rigidity of convex polytopes
We present a much simplified proof of Dehn's theorem on the infinitesimal
rigidity of convex polytopes. Our approach is based on the ideas of Trushkina
and Schramm.Comment: to appear in Siberian Journal of Mathematics; 5 pages 2 figure
A necessary condition for generic rigidity of bar-and-joint frameworks in -space
A graph is -sparse if each subset with induces at most edges in . Maxwell showed in
1864 that a necessary condition for a generic bar-and-joint framework with at
least vertices to be rigid in is that should have a
-sparse subgraph with edges. This necessary
condition is also sufficient when but not when . Cheng and
Sitharam strengthened Maxwell's condition by showing that every maximal
-sparse subgraph of should have edges when
. We extend their result to all .Comment: There was an error in the proof of Theorem 3.3(b) in version 1 of
this paper. A weaker statement was proved in version 2 and then used to
derive the main result Theorem 4.1 when . The proof technique was
subsequently refined in collaboration with Hakan Guler to extend this result
to all in Theorem 3.3 of version
Generic Rigidity Matroids with Dilworth Truncations
We prove that the linear matroid that defines generic rigidity of
-dimensional body-rod-bar frameworks (i.e., structures consisting of
disjoint bodies and rods mutually linked by bars) can be obtained from the
union of graphic matroids by applying variants of Dilworth
truncation times, where denotes the number of rods. This leads to
an alternative proof of Tay's combinatorial characterizations of generic
rigidity of rod-bar frameworks and that of identified body-hinge frameworks
Mixed Volume Techniques for Embeddings of Laman Graphs
Determining the number of embeddings of Laman graph frameworks is an open
problem which corresponds to understanding the solutions of the resulting
systems of equations. In this paper we investigate the bounds which can be
obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is
to provide the methods to study the mixed volume of suitable systems of
polynomial equations obtained from the edge length constraints. While in most
cases the resulting bounds are weaker than the best known bounds on the number
of embeddings, for some classes of graphs the bounds are tight.Comment: Thorough revision of the first version. (13 pages, 4 figures
Henneberg constructions and covers of cone-Laman graphs
We give Henneberg-type constructions for three families of sparse colored
graphs arising in the rigidity theory of periodic and other forced symmetric
frameworks. The proof method, which works with Laman-sparse finite covers of
colored graphs highlights the connection between these sparse colored families
and the well-studied matroidal (k, l)-sparse families.Comment: 14 pages, 2 figure
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
Recognizing Planar Laman Graphs
Laman graphs are the minimally rigid graphs in the plane. We present two algorithms for recognizing planar Laman graphs. A simple algorithm with running time O(n^(3/2)) and a more complicated algorithm with running time O(n log^3 n) based on involved planar network flow algorithms. Both improve upon the previously fastest algorithm for general graphs by Gabow and Westermann [Algorithmica, 7(5-6):465 - 497, 1992] with running time O(n sqrt{n log n}).
To solve this problem we introduce two algorithms (with the running times stated above) that check whether for a directed planar graph G, disjoint sets S, T subseteq V(G), and a fixed k the following connectivity condition holds: for each vertex s in S there are k directed paths from s to T pairwise having only vertex s in common. This variant of connectivity seems interesting on its own
From graphs to tensegrity structures: Geometric and symbolic approaches
A form-finding problem for tensegrity structures is studied; given an
abstract graph, we show an algorithm to provide a necessary condition for it to
be the underlying graph of a tensegrity in (typically )
with vertices in general position. Furthermore, for a certain class of graphs
our algorithm allows to obtain necessary and sufficient conditions on the
relative position of the vertices in order to underlie a tensegrity, for what
we propose both a geometric and a symbolic approach.Comment: 17 pages, 8 figures; final versio
- …