12 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
Combinatorics of Bricard's octahedra
We re-prove the classification of motions of an octahedron — obtained by Bricard at the beginning of the XX century — by means of combinatorial objects satisfying some elementary rules. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic
Zero-Sum Cycles in Flexible Non-triangular Polyhedra
Finding necessary conditions for the geometry of flexible polyhedra is a classical problem that has received attention also in recent times. For flexible polyhedra with triangular faces, we showed in a previous work the existence of cycles with a sign assignment for their edges, such that the signed sum of the edge lengths along the cycle is zero. In this work, we extend this result to flexible non-triangular polyhedra
FlexRiLoG -- A SageMath Package for Motions of Graphs
In this paper we present the SageMath package FlexRiLoG (short for flexible
and rigid labelings of graphs). Based on recent results the software generates
motions of graphs using special edge colorings. The package computes and
illustrates the colorings and the motions. We present the structure and usage
of the package
On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere
We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with 3+3 vertices where no two vertices coincide or are antipodal
Zero-sum cycles in flexible polyhedra
We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, that is, can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and (Formula presented.). We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one-dimensional analog, which is trivial to solve
Flexible circuits in the d-dimensional rigidity matroid
A bar-joint framework in is rigid if the only
edge-length preserving continuous motions of the vertices arise from isometries
of . It is known that, when is generic, its rigidity
depends only on the underlying graph , and is determined by the rank of the
edge set of in the generic -dimensional rigidity matroid
. Complete combinatorial descriptions of the rank function of
this matroid are known when , and imply that all circuits in
are generically rigid in when .
Determining the rank function of is a long standing open
problem when , and the existence of non-rigid circuits in
for is a major contributing factor to why this
problem is so difficult. We begin a study of non-rigid circuits by
characterising the non-rigid circuits in which have at most
vertices.Comment: 15 pages, 4 figure