三次DP曲线定义区间的扩展及其形状优化

为构造一种带形状参数的三次DP曲线，解决DP曲线在给定控制顶点时不具有形状修改功能的缺陷。将传统的三次DP基函数的定义区间由0,1]推广为0,α]，重新参数化后得到一组新的含参扩展基，分析扩展基的性质并将其与固定的控制顶点进行线性组合，构造了三次α-DP曲线。讨论了曲线的性质与形状，分析了形状参数的几何意义，并给出了曲线光滑拼接的条件：当满足一定条件时，曲线可达到G¹或G²连续。同时，运用张量积方法将三次α-DP曲线推广到曲面。实例表明，三次α-DP曲线曲面不仅继承了传统DP曲线曲面的优良性质，而且具有形状可调性。最后给出了3种形状参数的选取方案以及相应实例。

Interval extension of the cubic DP curve and its shape optimization.

In order that the cubic DP curve could have the shape-adjustable, a type of cubic DP curve with a shape parameter is constructed. This paper extends the definition interval of the the traditional cubic DP basis function from 0,1] to 0,α]. After reparameterization, a new set of extended basis functions with a parameter is proposed. The properties of the new basis functions are analyzed. Based on linear combination method,this paper also discusses the properties and the shape of the cubic DP curve and analyses the geometric meaning of shape parameters. Furthermore,G¹ and G² continuous conditions of the curve are given when certain conditions are met. Using the tensor product method, the curve is generalized to construct the cubic DP surface with a parameter. The new curve and surface not only inherits the excellent properties of the traditional DP curve and surface, also can adjust its shape under the fixed control points. Finally, three selection schemes of the shape parameters and the corresponding application examples are given.