An iterative algorithm for polynomial approximation of rational triangular Bézier surfaces

This paper presents the weighted progressive iteration approximation (WPIA) property for the triangular Bernstein basis over a triangle domain with uniform parameters, which is extended from the PIA property for triangular Bernstein basis proposed by Chen and Wang in [J. Chen, G.J. Wang, Progressive-iterative approximation for triangular Bézier surfaces, Computer-Aided Design 43 (2011) 889–895]. We also provide how to choose an optimal value of the weight to own the fastest convergence rate for triangular Bernstein basis. Furthermore, a new and efficient iterative method is proposed for polynomial approximation of rational triangular Bézier surfaces. The algorithm is reiterated until a halting condition about approximation error is satisfied. And the approximation error in Lp-norm (p = 1, 2, ∞) is calculated by the symmetric Gauss Legendre quadrature rule for composite numerical integration over a triangular surface. Finally, several numerical examples are presented to validate the effectiveness of this method.