四次C-曲线的性质及其应用

Properties and applications of degree 4 C-curve

以1,t,t2,t3,…为基底的Bézier曲线和B样条曲线是构造自由曲线、曲面强有力的工具.但是它们不能精确地表示某些圆锥曲线如圆弧、椭圆等,也不能精确地表示正弦曲线.本文利用一组新的基底sint,cost,t2,t,1,构造了两条新的曲线,这两条曲线依赖于参数α>0.当α→0时极限分别是四次Bézier曲线和四次B样条曲线,称之为四次C-曲线:四次C-Bézier曲线和四次C-B样条曲线.它们具有一般Bézier曲线和B样条曲线的性质:如端点插值,凸包,离散等,还可以精确的表示圆弧、椭圆及正弦曲线.作为应用,文章最后给出了四次C-Bézier曲线表示正弦曲线的条件.

Bézier curves and uniform B\|splines are powerful tools for constructing curves and surfaces of free form. But they can not represent the arcs,cylinders and sine curves.In this paper,with the basis functions 1,t,t\+2,sint,cost, two new curves which depend on a parameter α＞0 are constructed. These two curves have the same properties of Bézier curves and uniform B\|splines such as:terminal property,convex hull property,affine invariance,etc.As one of the applications of these two new curves,the representation of sine curve is given at the end of this paper.