| As an important tool of curve and surface modeling in CAGD, rational spline interpolation method is widely used in geometric modeling. Compared with the traditional method of polynomial splines, rational method is very flexible and is easy to implement regional control. In recent years, rational interpolation spline methods, as an important branch of numerical approximation theory, have been concerned by many researchers. The main work of this paper includes the following two parts:In the first part, rational quartic spline interpolation based on function values and derivative values is constructed with the quadratic denominator. The C2 continuity property, shape preserving properties and local regional control of the spline function are analyzed. And the point control method of rational interpolation curve is extended to the point control under monotonicity-preserving conditions. On the other hand, rational quartic spline interpolation only based on function values is given with the quadratic denominator. The C2 continuity property, shape preserving properties and local constraint control of the spline function are studied. For the end subintervals, point control of rational quartic interpolation curves are discussed.Rational interpolation spline method is extended to bivariate rational interpolation spline surfaces in the second part. An weighted bivariate cubic rational interpolation spline surfaces are constructed with the quadratic denominator. The integral property, bounded property and error of the spline function are analyzed. The spline has a group of symmetrical basis functions. The effect of the shape of the spline surfaces is discussed based on the various parameters. The local constraint control method of the spline surfaces are studied. On the other hand, in order to reduce the computational complexity, we construct a class of rational weighted linear interpolation surface based on four points, and study its interpolation properties. |
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