B´ ezier Curve and Surface Fitting of 3D Point Clouds Through Genetic Algorithms, Functional Networks and Least-Squares Approximation AkemiG´alvez 1 , Andr´ es Iglesias 1 , Angel Cobo 1 , Jaime Puig-Pey 1 , and Jes´ us Espinola 2 1 Department of Applied Mathematics and Computational Sciences, University of Cantabria, Avda. de los Castros, s/n, E-39005, Santander, Spain akemi.galvez@postgrado.unican.es, {iglesias,acobo,puigpeyj}@unican.es 2 Faculty of Sciences, National University Santiago Ant´ unez de Mayolo, Per´ u espinolj@gmail.com Abstract. This work concerns the problem of curve and surface ﬁtting. In particular, we focus on the case of 3D point clouds ﬁtted with B´ ezier curves and surfaces. Because these curves and surfaces are parametric, we are confronted with the problem of obtaining an appropriate parameter- ization of the data points. On the other hand, the addition of functional constraints introduces new elements that classical ﬁtting methods do not account for. To tackle these issues, two Artiﬁcial Intelligence (AI) tech- niques are considered in this paper: (1) for the curve/surface parameter- ization, the use of genetic algorithms is proposed; (2) for the functional constraints problem, the functional networks scheme is applied. Both approaches are combined with the least-squares approximation method in order to yield suitable methods for B´ ezier curve and surface ﬁtting. To illustrate the performance of those methods, some examples of their application on 3D point clouds are given. 1 Introduction Fitting curves and surfaces to measurement data plays an important role in real problems such as manufacturing of car bodies, ship hulls, airplane fuselage, and other free-form objects. One typical geometric problem in Reverse Engineering is the process of converting dense data points captured from the surface of an object into a boundary representation CAD model [17,19]. Most of the usual models for ﬁtting in Computer Aided Geometric Design (CAGD) are free-form parametric curves and surfaces, such as B´ ezier, Bspline and NURBS. Curve/surface ﬁtting methods are mainly based on the least-squares approx- imation scheme, a classical optimization technique that (given a series of mea- sured data) attempts to ﬁnd a function which closely approximates the data (a “best ﬁt”). Suppose that we are given a set of n p data points {(x i ,y i )} i=1,...,np , and we want to ﬁnd a function f such that f (x i ) ≈ y i , ∀i =1,...,n p . The typ- ical approach is to assume that f has a particular functional structure which depends on some parameters that need to be calculated. The procedure is to O. Gervasi and M. Gavrilova (Eds.): ICCSA 2007, LNCS 4706, Part II, pp. 680–693, 2007. c  Springer-Verlag Berlin Heidelberg 2007