International Journal of Computational Geometry & Applications

Polynomial equation systems arising from real applications often have associated combinatorial information, expressible as graphs and underlying matroids. To simplify the system and improve its numerical robustness before attempting to solve it with numeric-algebraic techniques, solvers can employ graph algorithms to extract substructures satisfying or optimizing various combinatorial properties. When there are underlying matroids, these algorithms can be greedy and efficient. In practice, correct and effective merging of the outputs of different graph algorithms to simultaneously satisfy their goals is a key challenge.

This paper merges and improves two highly effective but separate graph-based algorithms that preprocess systems for resolving the relative position and orientation of a collection of incident rigid bodies. Such collections naturally arise in many situations, for example in the recombination of decomposed large geometric constraint systems. Each algorithm selects a subset of incidences, one to optimize algebraic complexity of a parametrized system, the other to obtain a well-formed system that is robust against numerical errors. Both algorithms are greedy and can be proven correct by revealing underlying matroids. The challenge is that the output of the first algorithm is not guaranteed to be extensible to a well-formed system, while the output of the second may not have optimal algebraic complexity. Here we show how to reconcile the two algorithms by revealing well-behaved maps between the associated matroids.

We consider three related problems of robot movement in arbitrary dimensions: coverage, search, and navigation. For each problem, a spherical robot is asked to accomplish a motion-related task in an unknown environment whose geometry is learned by the robot during navigation. The robot is assumed to have tactile and global positioning sensors. We view these problems from the perspective of (non-linear) competitiveness as defined by Gabriely and Rimon. We first show that in 3 dimensions and higher, there is no upper bound on competitiveness: every online algorithm can do arbitrarily badly compared to the optimal. We then modify the problems by assuming a fixed clearance parameter. We are able to give optimally competitive algorithms under this assumption. We show that these modified problems have polynomial competitiveness in the optimal path length, of degree equal to the dimension.

A two-dimensional direction-length framework is a pair (G, p), where G = (V; D, L) is a graph whose edges are labeled as 'direction' or 'length' edges, and a map p from V to ℝ². The label of an edge uv represents a direction or length constraint between p(u) and p(v). The framework (G, p) is called globally rigid if every other framework (G, q) in which the direction or length between the endvertices of corresponding edges is the same, is 'congruent' to (G, p), i.e. it can be obtained from (G, p) by a translation and, possibly, a dilation by -1.

We show that labeled versions of the two Henneberg operations (0-extension and 1-extension) preserve global rigidity of generic direction-length frameworks. These results, together with appropriate inductive constructions, can be used to verify global rigidity of special families of generic direction-length frameworks.

For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t′ ∈ T such that d(x, t′) < d(y, t′). The set of non-dominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in Euclidean spaces, the Pareto envelopes and the convex hulls coincide. Chalmet et al. (1981) characterized the Pareto envelopes in the rectilinear plane (ℝ², d₁) and constructed them in O(n log n) time. In this note, we investigate the Pareto envelopes of point-sets in simple polygons P endowed with geodesic d₂- or d₁-metrics (i.e., Euclidean and Manhattan metrics). We show that Kuhn's characterization extends to Pareto envelopes in simple polygons with d₂-metric, while that of Chalmet et al. extends to simple rectilinear polygons with d₁-metric. These characterizations provide efficient algorithms for construction of these Pareto envelopes.

Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in ℝ^d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks.