带局部形状参数的三次均匀B样条曲线的扩展

带形状参数的B样条曲线的构造已成为计算机辅助几何设计中的热点问题.为了使形状参数具有局部修改功能,给出了两类带局部形状参数的调配函数,它们都是三次均匀B样条基函数的扩展.基于给出的调配函数,定义了两种带局部形状参数的分段多项式曲线.可以通过改变局部形状参数的取值对曲线进行局部调整.调整形状参数可使三次多项式曲线在三次均匀B样条曲线远离控制多边形的一侧摆动,而四次多项式曲线在三次均匀B样条曲线的两侧摆动.最后讨论了它们在曲线设计及曲线插值中的应用.造型实例表明,该类曲线在计算机辅助几何设计中具有重要的应用价值.

Extensions of Uniform Cubic B-Spline Curve with Local Shape Parameters

The construction of B-spline curves with shape parameters has become the hotspot in computer aided geometric design. However, the shape parameters of the curves in previous papers are all global parameters. In order to introduce B-spline curves with local shape parameters, two classes of polynomial blending functions with local shape parameters are presented in this paper. Both of them are extensions of cubic B-spline basic functions. The blending functions have similar properties of classical cubic B-spline basic functions. Based on the given blending functions, a method of generating piecewise polynomial curves with local shape parameters is proposed. By changing the values of the local shape parameters, the shape of the curves can be manipulated locally. The cubic curves can be manipulated to approximate the cubic B-spline curves from their sides away from the control polygons by changing the values of the shape parameter. Similarly, the quartic curves can also be manipulated to approximate the cubic B-spline curves from their both sides by changing the values of the shape parameters. The geometric meanings of the local shape parameters are also discussed. Their applications in curve design and interpolation are also presented. The modeling examples illustrate that these new curves are very valuable for computer aided geometric design.