带形状参数的n次Bézier曲线

The Nth-Degree Bézier Curve with Shape Parameters

提出了一类新的n+1(n≥1)次带参数的多项式调配函数,n次Bernstein基函数Bi,n(t)是它的一特例.由给出的多项式调配函数,建立了带形状参数的分段多项式曲线生成方法.研究了调配函数及其所生成曲线的性质.其调配函数具有递推性、规范性和非负性;所生成曲线具有如端点性质、对称性、凸包性、几何不变性等与Bézier曲线的类似性质.研究结果表明:在控制多边形不变的情况下,可以通过改变形状参数的值来调整曲线的形状.

In this paper,a new class of polynomial blending functions of degree n+1(n≥1)with parameters are presented,and the Bi,n(t),known as the Bernstein basis function of degree n,is a special case of it.Based on the given polynomial blending functions,a method for generating piecewise polynomial curves with shape parameters is established.Then,the properties of blending functions and the curves generated by blending functions are studied,showing that the blending functions have the properties of recursion,normalization and nonnegativity,and that the generated curves have such properties as endpoints,symmetry,convex hull and geometric invariance,which are similar to the Bézier curves.The results indicate that the shape of the curve can be adjusted by changing the value of the shape parameter under the condition that the control polygon is constant.