C^3连续的七次PH样条闭曲线插值

C^3 spline interpolation by pythagorean hodograph closed curves of degree seven

针对C3连续的七次PH样条闭曲线的构造问题,提出一种复数域内的七次PH样条曲线的新的计算方法.利用七次PH样条曲线的特殊性质以及各段插值曲线之间的C3连续性,通过在复平面内构造满足平方性质的七次PH样条插值曲线,将C3连续的七次PH样条闭曲线的构造问题转变为复数域内的二次复方程组的求解问题.考虑到二次复方程组的解不具有唯一性,提出变步长的同伦算法.通过动态地调整同伦步长的大小,可以得到二次复方程组的所有解.结果表明,该算法不仅克服传统的同伦算法中解的丢失问题,而且得到所有满足条件的C3连续的七次PH样条闭曲线.

A novel method, based on the expression of PH spline curve of degree seven within complex field, was presented in order to construct the C^3 spline interpolation by PH closed curves of degree seven. Due to the particular properties of PH curves of degree seven and the C^3 continuity of interpolation curves, the problem of constructing interpolated spline curve was transformed into finding the solution of quadratic equations related with complex variables, by constructing PH spline curves of degree seven which satisfied perfect square expression within complex field. Considering that the solution of quadratic equations was not unique, an adaptive homotopy method was proposed. By modifying the homotopy step dynamically, all solutions of quadratic equations were obtained. The simulation results showed that this algorithm not only overcame the loss of solutions in traditional homotopy method, but also obtained all C^3 spline curves by PH closed curves of degree seven satisfying the given conditions.