Bézier曲线曲面正则性的判别条件

Conditions for Determining the Regularity of Bézier Curve and Surface

正则性是参数曲线曲面的重要代数性质,是由参数曲线曲面的参数化决定的.在计算机辅助制造过程中,要求所处理的参数曲线曲面是正则的,前提是计算机辅助设计得到的参数曲线曲面是正则曲线曲面.然而,直接按照正则参数曲线曲面的定义,采用解方程或方程组的方法来判断曲线曲面是否正则,其计算相当复杂,实际上也是行不通的.通过将Bézier曲线曲面的导矢曲线(法矢曲面)的参数表示转换为隐式表示,得到了一个判断Bézier曲线曲面正则性的简单而实用的充分条件.

Regularity is an important algebraic property of parametric curve and surface, which depends on the parameterization of parametric curve and surface. In computer-aided manufacturing, the processed parametric curve and surface should be regular, so the parametric curve and surface generated by computer-aided design should be regular first. However, the computation of determining the regularity of parametric curve and surface by solving equation or system of equations induced by the definition of regularity is considerably complex, artd is actually infeasible. In this paper, by transforming the parametric representations of derivative vector curve （of Bézier curve） and normal vector surface （of Bézier surface） to their implicit representations, a simple and practical sufficient condition for determining the regularity of Bézier curve and surface is presented.