Applied to the general formulation of pole curves, least-square fining basically corresponds to minimize a quadratic error function d(P, t) (a sum of squared Euclidean norms) which depends on the set of control points P and nodes t linked to data points. We propose an iterative algorithm alternating between optimizing d(P, t) over t and P. Node corrections are achieved through the projection of data points on the approximation curve. Control points are updated relatively to the gradient of the error function. The method is convergent and directly applicable to every type of pole curve. Experimental results are proposed with Bezier and B-spline curves to emphasize the efficiency of this method.

Solid model abstraction is an integral part of parallel and process design. It is required for simulation and optimisation of the design and consists of retrieving a simplified model from the solid one, with appropriate dimension reduction and details removal. Unfortunately, current CAD systems do not provide the means for easy simplification of forms. Ongoing research efforts on abstraction yield some possible approaches. These attempt to generate the abstract model using expert systems, medial axis transforms or medial surfaces. A feature-based approach is presented. Analysts usually think about objects in terms of sets of forms rather than in terms of geometrical and topological entities. So, given an object and its description in terms of form features, the approach constructs a simplified model of this object using morphological information of the form features. The abstraction process has two parts: simplification and idealization. The simplification part removes any non-pertinent features from the initial model, while the idealization part idealizes the resulting objects according to the goal of the analysis.