The Research Of Application On Shape-Preserving Rational Interpolation

Interpolation is the foundation to study the topics of differential equations, function approximation, numerical integration, numerical differentiation etc. Interpolation has a long history, from the production practice, for example, quadratic interpolation and linear interpolation has been successfully applied to calendar study more than a thousand years by our scientists, but by the time limitations of the theoretical knowledge, it’s application is restricted. Interpolation of theoretical knowledge has been further improved with the generation of the calculus, it is life has become increasingly widespread in the daily. Especially the arrival of industrial era and the information age, and the actual needs of the mechanical design, marine, aviation, and other practical problems.With the rapid development of spline interpolation the in recent decades particularly, the breadth and depth of the interpolation application board to a new level.This paper describes the the rational spline development process and the theoretical background. Based on this rational spline curves and surfaces containing shape parameter and Constructing Shape Preserving is analyzed. This work has the following parts:1. Introducing the CAGD’s development process and the research status and background theory of Constructing Shape Preserving rational interpolation.2. Introducing the basic knowledge of linear rational interpolation spline for the following three chapters as the theory foreshadowing.3.Introducing the conformal cubic rational spline and equidistant curve generation algorithm, discussing yasumasa monotonicity and convexity sufficient condition, then the establishment of the spline model and a few scattered points gets con-esponding smooth curve and equidistant curve finally, a numerical example illustrates the timeliness of the algorithm.4. Introducing mixed rational interpolation based on function values, combining3/1rational interpolation function and standard cubic Hermite interpolation to process tensor product, and using difference quotient instead of the parameter derivative at the interpolation node, constructing a bivariate blending rational difference format,at last through data instances explainning it’s flexibility and effectiveness in computer-aided design.5. Introducing the rational interpolation spline surfaces on a rectangular grid, constructing a Yasumasa rational interpolation spline surfaces based on a given set of positive data points.