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**1 - 3**of**3**### Angular rigidity in 3D: combinatorial characterizations and algorithms

"... Constraint-based CAD software, used by engineers to design sophisticated mechanical systems, relies on a wide range of geometrical constraints. In this paper we focus on one special case: angular constraints in 3D. We give a complete combinatorial characterization for generic minimal rigidity in two ..."

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Constraint-based CAD software, used by engineers to design sophisticated mechanical systems, relies on a wide range of geometrical constraints. In this paper we focus on one special case: angular constraints in 3D. We give a complete combinatorial characterization for generic minimal rigidity in two new models: lineplane-and-angle and body-and-angle structures. As an immediate consequence, we obtain efficient algorithms for analyzing angular rigidity. pairwise angular constraints between them. A body-andangle structure is composed of rigid bodies with lines and planes rigidly affixed to them; angular constraints are placed between identified lines or planes on a pair of bodies. See Figures 1a and 2a for examples. We restrict the angular constraints to lie in the range [0, π] and remark that this restriction does not limit our model, as an angle α larger than π may be associated to the “small ” angle 2π − α. For lack of space, we present here only the (0, π) case. 1

### Reconciling conflicting combinatorial preprocessors for geometric constraint systems

- 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 numericalgebraic techniques, solvers can employ gr ..."

Abstract
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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 numericalgebraic 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. The algorithms are essentially 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.