Three-dimensional NURBS surface estimated by lofting method

Abstract

For reverse engineering, nonuniform rational B-spline (NURBS) surfaces expressed by the tensor product are fitted to measured coordinates of points. To estimate the unknown control points, the lofting or skinning method by cross-sectional curve fits leads to efficient computations. Its numerical complexity for estimating k2 control points is O(k3), while simultaneously estimating the control points possesses a complexity of O(k6). Both methods give identical results. The lofting method is generalized here from a two-dimensional surface represented by the tensor product to a three-dimensional one. Such a surface is needed for a deformation analysis or for solving dynamical problems of reverse engineering, where surfaces change with time. It is shown that the numerical complexity to estimate k3 control points for a three-dimensional surface is only O(k4). It is also shown by an analytical proof and confirmed by a numerical example that the lofting method for estimating the control points and their simultaneous estimation give identical results. The numerical complexity increases from O(k4) for the lofting method to O(k9) for the simultaneous estimation of k3 control points. Thus, the lofting method leads to an efficient way of estimating three-dimensional NURBS surfaces for time-depending problems.