形状可调的5次组合样条及其参数选择

目的 为了克服3次参数B样条在形状调整与局部性方面的不足，提出带参数的5次多项式组合样条。方法 首先构造一组带参数的5次多项式基函数；然后采用与3次B样条曲线相同的组合方式定义带参数的5次多项式组合样条曲线，并讨论基于能量优化法的5次组合样条曲线参数最佳取值问题；最后定义相应的组合样条曲面，并研究利用粒子群算法求解曲面的最佳参数取值。结果 5次组合样条不仅继承了3次B样条的诸多性质，而且还比3次B样条具有更强的局部性及形状可调性。由于5次组合样条仍为多项式模型，因此方程结构相对较为简单，符合实际工程的需要。利用能量优化法可获得光顺的5次组合样条曲线与曲面。结论 所提出5次多项式组合样条克服了3次参数B样条在形状调整与局部性方面的不足，是一种实用的自由曲线曲面造型方法。

Quintic composite spline with adjustable shape and parameter selection

Objective The quintic polynomial composite spline with parameters is presented in this study to overcome the disadvantages of the cubic parametric B-spline in shape adjustment and local aspects. Method First, a class of quintic polynomial basis functions with parameters is constructed. The quintic polynomial composite spline curves with parameters are then defined according to the same composite way with the cubic B-spline curves. The optimal parameter value of the quintic composite spline curves based on the energy optimization method is discussed. Finally, the corresponding composite spline surfaces are defined, and the problem of the optimal parameter values of the surfaces utilizing the particle swarm algorithm is studied. Result The quintic composite spline not only inherits most properties of the cubic B-spline, but also has stronger local and shape properties than the cubic B-spline. Given that the quintic composite spline is still a polynomial model, the equation structure is relatively simple, which is more in line with the requirements of the actual projects. The smooth quintic composite spline curves and surfaces can be obtained by the energy optimization method. Conclusion The quintic composite spline overcomes the disadvantages of the cubic parametric B-spline in shape adjustment and local aspects, which is a practical method for free curve and surface modeling.