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

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

Sparse Hypergraphs and Pebble Game Algorithms

A hypergraph G=(V,E) is (k,ℓ)-sparse if no subset V′⊂V spans more than k|V′|−ℓ hyperedges. We characterize (k,ℓ)-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behavior in terms of the sparsity parameters k and ℓ. Our constructions extend the pebble games of Lee and Streinu [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs

Periodic Body-And-Bar Frameworks

Periodic body-and-bar frameworks are abstractions of crystalline structures made of rigid bodies connected by fixed-length bars and subject to the action of a lattice of translations. We give a Maxwell–Laman characterization for minimally rigid periodic body-and-bar frameworks in terms of their quotient graphs. As a consequence we obtain efficient polynomial time algorithms for their recognition based on matroid partition and pebble games

Rigidity of frameworks on expanding spheres

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterisations are established for the rigidity of generic frameworks for $d=1$ with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.Comment: 22 pages, 2 figures, updated reference

A Constructive Characterisation of Circuits in the Simple (2,2)-sparsity Matroid

We provide a constructive characterisation of circuits in the simple (2,2)-sparsity matroid. A circuit is a simple graph G=(V,E) with |E|=2|V|-1 and the number of edges induced by any $X \subsetneq V$ is at most 2|X|-2. Insisting on simplicity results in the Henneberg operation being enough only when the graph is sufficiently connected. Thus we introduce 3 different join operations to complete the characterisation. Extensions are discussed to when the sparsity matroid is connected and this is applied to the theory of frameworks on surfaces to provide a conjectured characterisation of when frameworks on an infinite circular cylinder are generically globally rigid.Comment: 22 pages, 6 figures. Changes to presentatio

Generic rigidity with forced symmetry and sparse colored graphs

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.Comment: 21 pages, 2 figure