Witness Computation for Solving Geometric Constraint Systems

In geometric constraint solving, the constraints are represented with an equation system F (U, X) = 0, where X denotes the unknowns and U denotes a set of parameters. The target solution for X is noted X-T. A witness is a couple (U-W, X-W) such that F (U-W, X-W) = 0. The witness is not the target solution, but they share the same combinatorial features, even when the witness and the target lie on two distinct connected components of the solution set of F (U, X) = 0. Thus a witness enables the qualitative study of the system: the detection of over-and under-constrained systems, the decomposition into irreducible subsystems, the computation of subsystems boundaries. This paper investigates the witness computation in various configurations. The witness computation will be studied under several numerical methods: Newton iterations from random seeds either in R and C, the Broyden-Fletcher-Goldfarb-Shanno method, the Nelder-Mead simplex method. The robustness and performances of these methods will be analyzed and compared. The paper also presents the numerical probabilistic method from which the witness method was originated, and shows how the witness can be used for detecting dependent parameters within systems of geometric constraints.