Progressive Iterative Approximation Method And Its Application In Approximation Of Rational B-spline Curves

Rational curves and surfaces are widely used in Computer Aided Geometric Design and Manufacture. After NURBS has been defined as a unique mathematical method for the geometry shapes of industrial products, rational functions further laid a dominant position in Computer Aided Design (CAD). However, because of the complexity of calculation and the need for design & data exchange between different systems, polynomial approximation of rational curves and surfaces is often needed.In recent years, the method of progressive iterative approximation (abbr. PIA) has been widely used in the field of CAD. By iterating and adjusting the control points of the mixed curves/surfaces, new curves/surfaces with better approximation effects are obtained. As a new method of data fitting, PIA has good adaptability and stability, and avoids the trouble of solving linear equations in reverse engineering, so it has a good application prospect in the approximation of curves and surfaces.In view of the above two aspects, we present an iteration method for Sample-based polynomial approximation of rational B-spline curves. For a given rational B-spline curve, initial control points are sampled from the curve. Meanwhile, by keeping the node vector unchanged, the control points of the initial polynomial B-spline curve are adjusted by the method of PIA. A set of polynomial B-spline curves with better approximation effects can be obtained gradually. At each iteration, using the error reduction factor decides whether the iteration should be continued or not, until the iteration terminates.Main works are as follows. First of all, the development history of PIA is reviewed and two effective iteration methods are also introduced:weighted progressive iterative approximation (WPIA) and local progressive iterative approximation (LPIA). Examples show that these two methods have faster convergence rate than PIA method, and more flexibly in process; Secondly, the basic properties of B-spline are introduced; also the PIA property of B-spline curves/surfaces is introduced in this section. At last, a sample-based polynomial approximation method of rational B-spline curve is presented. Numerical examples show that the error reduction factor makes the iteration process easy and fast. The approximation error has been optimized after making certain iteration steps.