Three-dimensional NURBS surface estimated by lofting method

• Published in: International journal of advanced manufacturing technology Vol. 49; no. 9-12; pp. 1059 - 1068
• Main Author: Koch, Karl-Rudolf
• Format: Journal Article
• Language: English
• Published: London Springer-Verlag 01.08.2010; Springer Nature B.V
• Subjects: CAE) and Design; Complexity; Computer-Aided Engineering (CAD; Deformation analysis; Engineering; Estimation; Industrial and Production Engineering; Lofting; Mathematical analysis; Mechanical Engineering; Media Management; Original Article; Reverse engineering; Software reviews; Tensors; Reverse engineering; Lofting method; Time depending NURBS surface; Simultaneous estimation of control points
• ISSN: 0268-3768; 1433-3015
• DOI: 10.1007/s00170-009-2460-6

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 k 2 control points is O ( k 3 ), while simultaneously estimating the control points possesses a complexity of O ( k 6 ). 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 k 3 control points for a three-dimensional surface is only O ( k 4 ). 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 ( k 4 ) for the lofting method to O ( k 9 ) for the simultaneous estimation of k 3 control points. Thus, the lofting method leads to an efficient way of estimating three-dimensional NURBS surfaces for time-depending problems.