Current iterative numerical methods, such as continuation or Newton-Raphson, work only on systems for which the corresponding matrix is a square one. The geometric constraint systems need thus either to have no degrees of freedom, or to be a system the software can anchor, i. e. a rigid system. In this article, we propose a new iterative numerical approach which can handle both rigid and under-rigid geometric constraint systems. It is based on the translation of the system under the form of a particle-spring system where particles correspond to the geometric entities and springs to the constraints. We show that consistently over-constrained systems are also solved. We show that our approach is promising by giving results of a prototype implementation. We propose tracks for enhancements of the approach which could tackle its drawbacks (mainly stability).

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.

This 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.

For 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 boundary. 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.

The data model of an application, the nature and format of data stored across executions, is typically a very rigid part of its early specification, even when prototyping, and changing it after code that relies on it was written can prove quite expensive and error-prone. Code and data in a running Lisp image can be dynamically modified. A MOP-based persistence library can bring this dynamicity to the data model. This enables to extend the easy prototyping way of development to the storage of data and helps avoiding interruptions of service. This article presents the conditions to do this portably and transparently.

Most 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.