This paper presents some important issues and potential research tracks for Geometric Constraint Solving: the use of the simplicial Bernstein base to reduce the wrapping effect in interval methods, the computation of the dimension of the solution set with methods used to measure the dimension of fractals, the pitfalls of graph based decomposition methods, the alternative provided by linear algebra, the witness configuration method, the use of randomized provers to detect dependences between constraints, the study of incidence constraints, the search for intrinsic (coordinate-free) formulations and the need for formal specifications.

Les objets geometriques sont souvent modelises par des systemes d'equations et d'inequations particuliers. Toutefois, tout systeme d'equations et d'inequations definit un objet geometrique. Cette these s'interesse a la possibilite de specifier un objet geometrique par un systeme de contraintes qui sera resolu par un solveur. Elle explicite aussi les difficultes que rencontre un tel solveur et apporte des debuts de solutions.

Geometric constraints solving often relies on graphbased methods to decompose systems of geometric constraints. These methods have intrinsic and unavoidable limitations which are overcome by the witness method presented here.

Many solutions exist to rebuild a three-dimensional object represented by a set of points. The purpose of our work is to provide an automatic reconstruction from an unorganized cloud, describing an unknown shape, in the aim to compute its volume. The approach employed in this paper consists in filling the object's interior with isosurfaces of potential fields and to use their fusion property in order to find the full volume and the continuous shape of the sampled object. Thus, the first step of our reconstruction is to search a correct interior for the object described by the set of points. Then, comes the positioning of implicit primitives into the cloud, deep inside of it and close to the boundary. A controlled fusion of the isosurfaces guarantees that no holes are present, such that we obtain a complete shape filling.

Bernstein based interval analysis permits to trace algebraic curves and surfaces. In this paper, we propose to use the classical de Casteljau algorithm to improve the efficiency of the Bernstein based method. The proposed tracing method gives significant results with functions of high degree. These results are illustrated and compared with other interval analysis approaches.