One Local Fairing Method Of B-spline Curves Based On Weighted Progressive Iterative Approximation

The fairing problem of curves and surfaces has been the focus of the Computer AidedGeometric Design (CAGD for short) in recent years. Besides, B-spline method haspowerful functions in designing and expressing free form curves and surfaces, which isone of the most popular methods applied to shape description. Therefore, the fairingproblem of B-spline curves and surfaces became an important topic of CAGD. Thereare two methods which are used to fair B-spline curves: the global fairing method andthe local fairing method. The energy method is the typical one among the global fairing.However, the resulting curve of this approach is different from the initial curve in alarge scale and approximates to a straight line.In recent years, the Progressive Iterative Approximation(PIA for short) has achievedextensive attentions. A sequence of curves or surfaces are generated by adjusting thecontrol points iteratively. The resulting curves or surfaces will converge to the initialcontrol points as the times of iteration increases. The PIA algorithm has manyapplications in Computer Aided Geometric Design and has been extended to severalother methods.In view of the above two aspects, a new local method, which is based on theweighted progressive iterative approximation (WPIA for short) algorithm, is used to fairplanar B-spline curve. The method is based on weighted progressive iterativeapproximation (WPIA for short) and consists of following steps: finding the bad pointwhich needs to fair, deleting the bad point, re-inserting a new data point to keep thestructure of the curve and applying WPIA method with the new set of the data points toobtain the faired curve.The resulting curves of the proposed method have less strain energies. And they arenot very different from the initial ones. Numerical examples are provided todemonstrate the effectiveness of the method.