Research On Constructing And Control Of Bivariate Rational Interpolation Surface

Spline rational interpolation is a useful and powerful tool in computer aided geometric design. In recent years, the rational spline with parameters has received attention. Because of many good properties of the rational spline, it has been applied widely in shape design of industrial products. The theories of rational interplation curves are almost perfect. But some questions should be further studied. The theories of rational interplation surfaces are very difficult to study. Nevertheless , The rational interpolation surface is more useful than the rational interplation curve in parctical design. The rational interplation curves and the rational interplation surfaces have been studied in this thesis. There are three parts are consisted of this thesis. The first part is made of the chapter 1. The second part is made of the chapter 2 and chapter 3. The third part is made of the chapter 4 , 5 , 6 , 7 , 8 and chapter 9. The thesis is organized as follows:There is a short intruduce about CAGD. The transition is from CAGD to the rational interpolation curves and surfaces.The approximation of the derivatives of interpolating function has been studied which is a rational cubic spline with linear denominator. The optimal error coefficient is derived. The precise of the derivatives of interpolation function at the knots are given.A weighted rational cubic spline interpolation has been constructed using two kinds of rational cubic spline with quadratic denominator. The constrain of the interpolation curves to be the given region and the approximation of the interpolation function have been studied.A kind of bivariate rational interpolation which was constructed by Duan has been intruduced. The properties of the interpolation function are given. The convexity control of the interpolation surfaces has been studied. The sufficient and necessary conditions for the interpolation surfaces to be convex are derived.The property of a bivariate rational interpolation with symmetric interpolation data is studied. The relationship between parameters and interpolation surfaces is derived.The bounded property and point control of a bivariate rational interpolating surface has been studied. The limitary of the interpolation function is proved. The approximation expression of the interpolation is given.A bivariate rational interpolation is given which is constructed using both function values and partial derivatives. The properties and bases of the interpolation been studied. The concept of integral weights coefficients of the interpolation is given.The constrain of the interpolation surfaces to be the given region plane has been studied. A sufficient condition of constraint of bivariate rational Interpolation surface is given.