Nucleation-free 3D3D rigidity

When all non-edge distances of a graph realized in $\mathbb{R}^{d}$ as a {\em bar-and-joint framework} are generically {\em implied} by the bar (edge) lengths, the graph is said to be {\em rigid} in $\mathbb{R}^{d}$. For $d=3$, characterizing rigid graphs, determining implied non-edges and {\em dependent} edge sets remains an elusive, long-standing open problem. One obstacle is to determine when implied non-edges can exist without non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal with them. In this paper, we give general inductive construction schemes and proof techniques to generate {\em nucleation-free graphs} (i.e., graphs without any nucleation) with implied non-edges. As a consequence, we obtain (a) dependent graphs in $3D$ that have no nucleation; and (b) $3D$ nucleation-free {\em rigidity circuits}, i.e., minimally dependent edge sets in $d=3$. It additionally follows that true rigidity is strictly stronger than a tractable approximation to rigidity given by Sitharam and Zhou \cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial characterization. As an independently interesting byproduct, we obtain a new inductive construction for independent graphs in $3D$. Currently, very few such inductive constructions are known, in contrast to $2D$

Hyperbanana Graphs

A bar-and-joint framework is a finite set of points together with specified distances between selected pairs. In rigidity theory we seek to understand when the remaining pairwise distances are also fixed. If there exists a pair of points which move relative to one another while maintaining the given distance constraints, the framework is flexible; otherwise, it is rigid. Counting conditions due to Maxwell give a necessary combinatorial criterion for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R^2. However, the flexible "double banana" shows that Maxwell's conditions are not sufficient to guarantee rigidity in R^3. We present a generalization of the double banana to a family of hyperbananas. In dimensions 3 and higher, these are (infinitesimally) flexible, providing counterexamples to the natural generalization of Laman's theorem

Algorithms for detecting dependencies and rigid subsystems for CAD

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract based on v1

Rigidity for sticky disks

We study the combinatorial and rigidity properties of disc packings with generic radii. We show that a packing of n discs in the plane with generic radii cannot have more than 2n − 3 pairs of discs in contact. The allowed motions of a packing preserve the disjointness of the disc interiors and tangency between pairs already in contact (modelling a collection of sticky discs). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly 2n − 3 contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy–Alexandrov stress lemma. Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly et al. (Connelly et al. 2018 (http://arxiv.org/abs/1702.08442)) on the number of contacts in a jammed packing of discs with generic radii.PostprintPeer reviewe

The combinatorial geometry of stresses in frameworks

In this paper we formulate and prove necessary and sufficient geometric conditions for existence of generic tensegrities in the plane for arbitrary graphs. The conditions are written in terms of "meet-join" relations for the configuration spaces of fixed points and non-fixed lines through fixed points