控制边长成等比的一类三次空间Bézier曲线的挠率分析

Analysis on the torsion of a kind of cubic-spatial Bézier curves with geometric proportion control polygon sides

本文针对控制边长成等比、相邻控制边的夹角相等的三次空间Bézier曲线,根据仿射不变性建立空间直角坐标系,得到曲线挠率的表达式.然后通过换元简化计算,得到挠率导数的表达式.再结合控制多边形的边角几何特征,利用Descartes符号法则分析挠率导数为零时根的分布情况.从而给出了这类三次空间Bézier曲线挠率单调递增和有极小值的参数域刻画.最后给出了几个典型数值实例.

This paper focuses on a kind of cubic spatial Bézier curves with geometric proportion control polygon sides and equal angles between adjacent control polygons.And a rectangular space coordinate system is established according to the affine invariance to obtain the expression of curve deflection rate.In order tosimplify calculation,the corresponding expression for the derivative of this torsion is derived by changing variables.Based on geometric features of the control polygon,the distribution of roots where torsion derivativ eequals zero is analyzed by using the Descartes sign rule.Thus the parametric domains,where the torsion of the corresponding Bézier curve is monotonic increasing or reaches its minimum value,are described.Finally,several typical numerical examples are given.