Research On The Shape-preserving Rational Interpolation Spline

In recent years, following the development of modern industrial and manufacture, Computer Aided Geometric Design (CAGD) becomes a new cross-cover subject. It is one of the most basic research which studies curve(surface) modeling by the method of interpolation and approximation. As the combination of the polynomial spline function and the rational approximation, the rational spline interpolation is a powerful tool for shape preserving and localization. Especially the rational spline curves and srufaces with parameters are received wide attention. One important problem is the construction of shape preserving surface. For the given data, the surface should have the same geometric character. For example, positivity preserving, convexity preserving and monotony preserving. In this paper, a class of shape-preserving rational interpolation splines with parameters is derived.The first chapter is the introduction, it is an overview of background and significance of the rational spline interpolation with parameters, and the concept and organizational structure of this paper.The second chapter describes several shape-preserving rational cubic interpolation spline curves. Several interpolation curves with different denominator and several rational cubic curves only based on function values are first introduced. Then the quadratic rational spline curve and quartic rational spline curve are introduced.The third chapter introduces several shape-preserving rational interpolation spline surfaces method. First some cubic Hermite interpolation surfaces are introduced. Then the interpolation surfaces only based on function values are given. Finally biquartic rational spline interpolation surface is introduced.Chapter IV are the researches of this paper. We get a bicubic Hermite interpolation spline surface.The error estimation and a positivity preserving method are given. The positivity preserving sufficient condition of the rational spline is given by using the corresponding condition of the bicubic Bézīer surface. It can be achieved by adjusting the directional derivatives of the surface.Chapter V summary the paper and give some future work prospects.