Polynomial Accelerated Iterative Approximation for Higher Order and Rational Bézier Curves

Two accelerated iterative methods for curves approximation are presented in this paper. These presented methods are used to reduce the degree of Bézier curves and approximate rational Bézier curves by polynomials. By employing the preconditioned progressive iterative approximation (PPIA), we approximate the points sampled from target curves, and generate polynomial approximations. The equi-parametric and adaptive sampling methods are introduced. Both of them are well performed in degree reduction of Bézier curves and polynomial approximation of rational Bézier curves. Due to the effectiveness of the preconditioned technique, the accuracy and efficiency of approximation are improved significantly. More importantly, we can obtain the approximation within an user-defined error bound. Numerical examples demonstrate the effectiveness of our methods.