关于有理B－样条曲线的权因子的研究

An Investigation on the Weights of Rational B-Spline Curves

对有理Ｂ－样条曲线中的权因子作了研究。首先，阐述了权因子的几何意义，证明了权因子是有关四点的交比；分析了权因子的极限性质与零权因子对曲线形状的影响。其次，就曲线上一点在控制点方向与任意方向移动这两种情况，给出了曲线形状调整时的权因子计算公式，从而得到调整后的曲线方程，并给出一个应用算例。再次，提出了有理Ｂ－样条曲线的分解表示方法，即将ｎ次有理Ｂ－样条曲线用ｑ个二次有理Ｂ－样条与（ｎ－２ｑ）个一次有理Ｂ－样条的乘积形式表示，并给出相应的条件。最后讨论了有理Ｂ－样条曲线对控制多边形的逼近问题。

The weights in the Rational B-Spline curves are studied more completely in this paper. First of all, the geometric meaning of the weights is explained, the weights are proved to be of cross-ratios at the associated four points. The effects of the limit properties of weigths and zero-weight on the shape of the curves are analyzed. Nextly,when the shape of curves is varied because of one point of the curve moving along the direction of control point or along any direction,the formulae of calculating weights are given with an applied example. Thirdly,the equations of Rational B-spline curves are dissolved into linear and quadric multiplicators in order to provide a new explanation for Rational B -spline curves with corresponding conditions. Finally the equation of Rational B-spline curves approaching to the control polygon is discussed.