Hermite Interpolation And Optimization Of Non-uniform Cubic B-spline Curve Based On PIA

B-spline is widely used in the field of computer aided geometric design.Pro-gressive iteration approximation(PIA)is an important iterative method in data interpolation.Hermite interpolation is a common interpolation method.Therefore,it is of great significance to realize the interpolation of non-uniform cubic B-spline curves by Hermite interpolation method based on PIA.This paper proposes a Hermite interpolation algorithm for non-uniform cubic B-spline curves based on PIA.By using this algorithm the B-spline curve has the less number of iterations under the condition of satisfying the interpolation error accuracy.Firstly,the given data points are taken as the initial control points,and the node sequence is obtained by the cumulative chord length method.The control points are updated by constructing the error vector,and the non-uniform cubic B-spline curve for interpolating G~0datas are generated iteratively.Secondly,when the given data points and the unit tangential direction need to be interpolated,a new node is inserted into each node interval by dichotomy to obtain a new control vertex and iteratively generate a cubic B-spline curve of interpolating G~1datas.Finally,when the given data points,unit tangent and curvature need to be interpolated,the ternary subdivision method is used to insert two nodes in each node interval,update the control vertices,and then iterate to obtain the B-spline curve of interpolating G~2datas.The results of numerical examples show that compared with the uniform cubic B-spline curve interpolation algorithm,when the distance between adjacent data points changes more,under the same error conditions the convergence speed of the algorithm is faster and the number of iterations is less.In order to minimize the energy when the curve satisfies the given interpolation error,the energy term is added on the basis of the above algorithm,and the Hermite interpolation algorithm of non-uniform cubic B-spline curve with energy term is proposed.Firstly,a B-spline curve satisfying the prescribed error is obtained by Hermite interpolation algorithm of non-uniform cubic B-spline curve based on PIA,and the curve is taken as the initial interpolation curve.An energy term is added,and the initial value of the energy coefficient is the energy coefficient proposed by Floater,and by the PIA-based Hermite interpolation algorithm with energy term obtain a new interpolation curve.Finally,according to the prescribed error,the energy coefficient is continuously adjusted by the dichotomy,and then the Hermite interpolation method after adding the energy term is used to obtain the updated curve.The energy coefficient is continuously updated and iterated,and finally the B-spline curve with the energy coefficient as large as possible and satisfying the interpolation error condition is found.The results of numerical examples show algorithm of this paper can reduce the energy of the curve.