About Rational Curves And Surfaces Of The Polynomial Approximation

Rational curves and rational surfaces, which are a class of important approximation functions, are extensive applied in CAD/CAM. After NURBS being assured as the international criteria, rational functions have become more and more important in CAD. However, due to the complex of computation and the need of the design, sometime we need to use polynomial approximation for a rational curves and surfaces.In theory of approximations, the classic methods of polynomial approximation for rational expression are various interpolations and operator approximations, such as Lagrange interpolation, Hermite interpolation and Bernstein polynomial approximation. These approximation methods converge too slowly or even hard to converge.Base on the practical need, the concepts interval curves and interval surfaces are presented in theory of approximations. Interval curves and interval surfaces are application and extension of interval analysis method in CAGD, which is important tool for error analysis in numerical analysis field. With the deeper research of interval curves and interval surfaces, people begin to approximate curves (surfaces) with interval curves (surfaces).In this paper, we survey the development of polynomial approximation of rational curves (surfaces) and state the knowledge of interval curves (surfaces) in the first chapter. In the second chapter, the works have been done is discussed. In the third chapter, interval polynomial approximation of rational curves is introduced. In the final chapter, we discuss interval polynomial approximation of rational surfaces. Firstly, we briefly state the interval-surface approximation based on the Taylor Expansion, and later, we give out a better interval-surface approximation based on the optimization method, which is also the main work of this paper.