Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems

This 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.