Identity of simultaneous estimates of control points and of their estimates by the lofting method for NURBS surface fitting

For reverse engineering, nonuniform rational B-spline surfaces are fitted to measured coordinates of points. Two methods are often applied. The first one consists of simultaneously estimating the unknown control points of the surface. The second method is a special case of the lofting or skinning method where cross-sectional curves are not interpolated but fitted to the measurements, thus obtaining the control points. This approach is considered to give an approximate solution of the first one. However, it is shown here by an analytical proof and confirmed by a numerical example that both methods give identical results. Since the computational complexity of the first approach is the square of the second one, the simultaneous estimation of a large number of control points should be avoided.