Research On The Properties And Point Control Of Bivariate Rational Interpolating Surface

Spline function is a very important tool for designing curves or surface, it has been widely applied in Computer Aided Geometric Design and engineering mathematics. Rational spline function, as the combination of spline function and rational approximation, integrates both the advantages of them, and is more flexible and more common. In recent years, rational spline functions with parameters, especially rational cubic splines, have aroused more and more attention because of their advantages in local control.In this thesis, we introduce several typical rational interpolating spline functions with parameters firstly, describe the construction of a bivariate bi-quadratic rational interpolation surface which is only based on function values and discuss the smoothing condition of this interpolation surface. The expressions and properties of the interpolation basis function are given next. Under the condition of equal knot spacing, we give the matrix representation of the bivariate rational interpolation, from which we can deduce that the interpolation is symmetric with regards to interpolation data and has uniqueness independent of the interpolating direction.Then we discuss the bounded property and approximation properties of the bivariate rational interpolating surface and present a "center point -mean value" control method which can locaOlyl adjust the surface, by selecting the appropriate parameters. The interpolating function can be changed under the condition that the interpolating data remain the same, so the shape of the interpolating surface can be modified. A numerical example shows the effective at last. Interestingly, though four parameters are selected in the thesis, they can be changed into two parameters in special case because the function values are only related to the ratio of each pair of parameters. Thus, the expression is more concise and the calculation is more convenient. It is more convenient and more flexible for us to control and modify the surface. The flexibility of designing and constructing surface is increased, and the effect is good.