Research On Rational Interpolation Spline Function

Curves and surfaces modeling is an important task Computer-Aided Geometric Design(CAGD for short). It is a focus for the researchers that how to adjust the shape of the rational curves and surfaces flexibly. It is effective to settle the problem for us to construct the rational spline function with free parameters. The surface should have the same geometric character with the given data are not changing, for example, convexity-preserving, positivityPreserving monotonicity-preserving etc. A rational interpolation spline function is constructed frequently. With the conditions of the interpolation are not changed by adjusting the values of parameters, we can control the shape of rational curves and surfaces flexibly.In this paper, we have researched the shape-preserving deeply on positivity-preserving of rational curve and convexity-preserving of rational surface with shape parameters. There are four chapters in this paper.The first chapter is the introduction. In this part, which is introduced of the background and history of the CAGD and rational curve with parameters and coons surface. Besides the major research and the structure of this paper are given.Chapter II presents a rational cubic function with one parameter, which is 1C. In the case of interpolation conditions determined by constraint parameter, the shape of the curve can be changed. Based on a rational cubic spline function with free parameters, the necessary and sufficient condition is given to maintain spline positive. The spline is positive by changing parameters. And a numerical example is given.In the chapter III, the conditions for interpolating surfaces on the free parameters to be convex are derived. First of all, a rational cubic spline function with free parameters is extended to rational bi-cubic partially blended function. These rational patches inherent the good features to coons patches. The convexity-preserving is discussed, and given the sufficient and necessary conditions of the rational spline function. The conditions of boundary curves to be convex are extended to the convexity conditions of the rational spline function. It can be controlled the convexity of the surfaces by adjusting the shape parameters with the data is not changed. Examples are given to show the effectiveness of the methods.Chapter IV is the summary of the paper and presents some future work and the development prospect.