On applications of SVD and sevd for nurbs identification

NURBS stands for Non-Uniform Rational B-Splines, the popular mathematical tool used for computer-aided geometric design (CAGD) of free-form curves and surfaces. The geometrical shape of NURBS curves and surfaces is mainly controlled by their control points and the corresponding weights. This chapter concerns the identification of NURBS curves and surfaces from discrete points measured on coordinate measuring machines (CMMs), laser scanners or other digitizing equipments. The complete process is divided into two steps. The weights of the control points are first identified with singular value decomposition (SVD) or symmetric eigen value decomposition (SEVD) techniques through a linear homogeneous system. The control points are then solved with linear least squares fitting techniques. Both interpolation (exact) and fitting (approximate) solutions are studied. The emphasis of the present chapter is on SVD and SEVD applications during the first step for the identification of the weights.