广义Frenet标架系统及其应用

Generalized Frenet frame and its applications

研究了参数曲线上的一类仿射标架。该标架以参数曲线的一、二阶导矢及其叉积作为标架向量,在弧长参数形式下演变为经典Frenet标架,是广泛意义上的Frenet标架。该标架系统可建立两条同源曲线几何不变量之间的解析关系,用于变形过程几何结构改变的分析。文章推导了广义Frenet标架下的Frenet公式并借助一个实例,推导出了曲线变形前后曲率关系的一个解析公式,该解析式可在有限元分析等工程计算中实现对曲率的精确计算。

An investigation is made on an affine frame of parametric curves, The frame uses the derivatives and their cross product of the first and second order as its basis vectors, can be evolved to be classical Frenet frame in form of arc-length parameter, is a generalized Frenet frame. The frame with its formula can set up analytic relationships among the geometric invariants of two homologous curves, and can be used to analyze the deformation process of geometric structures. A generalized Frenet formula is derived in accordance with the generalized Frenet frame and a fonnula is also demonstrated via an application sample to show analytic relationships between curvatures of a deforming curve before and after a deformation. The formula can be used for precise computation of curvatures in engineering calculation such as finite element method and so on.