Research On Rational Surfaces With Parameters

Curves and surfaces modeling is an important task of CAGD. The construction,display and analysis of curves and surfaces under the computer system is the most basicreserch. And spline interpolation is one of the most powerful tools in curves and surfacesmodeling. It is urgent to solve the problem that how the rational spline curve withshape parameters can be controlled flexibly, as the interpolation conditions aredetermined.Rational interpolation spline which is the the combination of the polynomialfunction and rational approximation is a simple and useful solution for the problem. Theshape of curves and surfaces can be controlled effectively by changing the values ofparameters under the condition that other conditions are determined.In this paper，we have made deep researches on rational surfaces with shape parameters.And the paper is composed of five parts.The first part is the introduction, which is an overview of background and history. Andthe structure of this paper is given.The second part briefly introduces three kinds of curves with parameters. Then rationalsurfaces with shape parameters based on function values, rational Hermite surface with shapeparameters and several Coons surfaces are described.The third part presents two methods of the construction of the blending function. Thefirst one is constructed by using the endpoint properties of blending function. The secondmethod is that is converted into mathematical transformationThe fouth part is the key part of the paper. Based on the rational cubic curve, twosets of rational blending function is constructed.And the blending function have goodproperties.Then the rational surface with shape parameters is defined. Thesurface not only inherits the most good properties of bicubic Coons surface, butalso can adjust the concavity and convexity of the shape by changing the values ofparameters withfixed boundary conditions. Finally numerical examples show theeffective of surface modeling.The fifth part summaries the paper and give some future work prospects.