Research On Rational Cubic Splines And Interpolation Of Space Closed Curves

Spline function is a useful and powerful tool for designing a curve or surface. Rational spline function, as the combination of spline function and rational approximation, is not only an important means of rational approximation, but also a natural generalization of polynomial spline. It integrates their advantages, and is more flexible and more common. In recent years, rational interpolation splines with parameters, especially rational cubic splines, have received more and more attention because of its advantage in local adjustment.In this thesis, some rational cubic splines, including their construction, error estimation, and the approximation properties of the derivative, etc, are introduced firstly, which are cubic Hermite spline with linear denominators, rational cubic spline based on function values and rational cubic spline based on average difference quotient. Based on these, a "true" rational cubic spline with linear denominators is constructed. This rational cubic spline can be C~2 continuous. Its approximation error is analyzed and error estimation expression is given.And then, the interpolation of a kind of space closed curves under cylindrical coordinate system is studied. The policy is to transform the interpolation problem of space closed curves into the one of plane curves by rational cubic spline with linear denominators through expanding circular cylindrical surface. The space curves finally obtained are shown to be curvature continuous. The error estimation of this method is analyzed, and numerical example shows that the effect is good.