International audienceThis paper describes new ways to tackle several important problems encountered in geometric constraint solving, in the context of CAD, and which are linked to the handling of under- and over-constrained systems. It presents a powerful decomposition algorithm of such systems. Our methods are based on the witness principle whose theoretical background is recalled in a first step. A method to generate a witness is then explained. We show that having a witness can be used to incrementally detect over-constrainedness and thus to compute a well-constrained boundary system. An algorithm is introduced to check if anchoring a given subset of the coordinates brings the number of solutions to a finite number. An algorithm to efficiently identify all maximal well-constrained parts of a geometric constraint system is described. This allows us to design a powerful algorithm of decomposition, called W-decomposition, which is able to identify all well-constrained subsystems: it manages to decompose systems which were not decomposable by classic combinatorial methods

International audienceFor more than a decade, the trend in geometric constraint systems solving has been to use a geometric decomposition/recombination approach. These methods are generally grounded on the invariance of systems under rigid motions. In order to decompose further, other invariance groups (e.g., scalings) have recently been considered. Geometric decomposition is grounded on the possibility to replace a solved subsystem with a smaller system called . This article shows the central property that justifies decomposition, without assuming specific types of constraints or invariance groups. The exact nature of the boundary system is given. This formalization brings out the elements of a general and modular implementation

International audienceThis paper deals with the resolution of geometric constraint systems encountered in CAD-CAM. The main results are that the witness method can be used to detect that a constraint system is over-constrained and that the computation of the maximal rigid subsystems of a system leads to a powerful decomposition method. In a first step, we recall the theoretical framework of the witness method in geometric constraint solving and extend this method to generate a witness. We show then that it can be used to incrementally detect over-constrainedness. We give an algorithm to efficiently identify all maximal rigid parts of a geometric constraint system. We introduce the algorithm of W-decomposition to identify all rigid subsystems: it manages to decompose systems which were not decomposable by classical combinatorial methods

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International audienceMost of the geometric constraints solvers consider systems of constraints well-constrained modulo the rigid motions group, and either halt on error when they encounter under-constrained sub-systems, or attempt to add parameterized constraints so as to get rid of the under-constriction. We studied transformations groups making well-constrained some problems that are usually considered as under-constrained. This leads to new algorithms which allow an homogeneous handling of systems of geometric constraints and thus a better adaptation to the needs of the user