Study Of Adaptive Cubic Spline Interpolation Approximation Algorithm

In science computations, spline interpolation plays an important role in lots of aspects which are involved in function approximation, multi-statistics, system control and Computer Aided Geometry Design. Therefore, the study concerning on theory and applying of spline interpolation has been paid more attention. In the field of Computer Numerical Control (CNC), a general numerical control system just contains straight line and circle arc interpolation methods. For a non-circle curve, we must firstly use straight line or circle arc to fit it, then we use straight line and circle arc interpolation methods to machine the fit curve. Because of the high precision required, these two methods will generate large NC programs. It is hard to improve the quality at the same time. Recently, spline interpolation method is popular in research. Another shortcoming in NC is the weakness of adaptive. So, present an adaptive spline interpolation approximation algorithm that fit for the special requirement is very important.In this thesis, we study the problem of adaptive spline interpolation approximation algorithm. Assuming that the expression of the original function (or curve) is known, we use cubic spline function (or curve) to interpolate approximation it. We concentrate on the distribution of the spline knots, which influence the approximation error heavily. Firstly, we translate the problem of knots distribution to a triangular matrix creation. Therefore, the study of the knots distribution becomes the study of the matrix. Then, on the base of transformation, we present an adaptive algorithm to find the ideal knots distribution. The numerical experiments imply that different curves with different knots distribution and for the same curve the knot distribution is spare in the flat department and dense in the department whose curvature is big. These prove the property of adaptive of our algorithm. Our algorithm meets the tolerance requirement and gives as little knots as possible in the sense of probability. And the knot distribution tends to uniform at the same time.The thesis is organized as follows:In chapter 3, in the case of the original function is explicit expression, we present an adaptive spline interpolation algorithm in which cubic natural spline function is used to approximation the given function. The basic idea of the algorithm and the content of the algorithm are introduced. Some numerical examples are provided in the end. In chapter 4, in the case of the original curve is a parameter curve, we use the basic mind of the algorithm and the content of the algorithm which are introduced in chapter 3 to find the knot distribution of the parametric cubic B-spline curve.In chapter 5, on the base of chapter 4, consider the bound derivative of the curve, so the approximation curve preserves shape well.At last we conclude the thesis and analysis the problem that we interested in for the further study.