三维欧氏空间中主法线曲面的结构函数

Structure Functions on Normal Ruled Surface in 3-D Euclidean Space

在三维欧氏空间中,主法线曲面作为特殊的非可展直纹面具有良好的代数和几何性质.运用微分几何的方法研究主法线曲面的结构函数.根据三维欧氏空间中不可展直纹面的定义和标准方程,给出曲线的主法线曲面的定义和标准方程.从主法线曲面的定义和标准方程出发,得到主法线曲面的结构函数之间满足的关系,以及曲线的主法线曲面的结构函数、准线和腰曲线三者之间的联系.讨论Mannheim曲线和一般螺线的主法线曲面,得到Mannheim曲线的主法线曲面是其侣线的副法线曲面,一般螺线的主法线曲面是正螺面.

As special non-developable ruled surface,the normal ruled surface has good algebraic and geometric properties. Using the classical methods of differential geometry,the structure functions of the normal ruled surface in 3-D Euclidean Space are studied. According to the definition and standard equation of non-developable ruled surface in 3-D Euclidean Space,the definition and standard equation are given to the normal ruled surface. Based on the definition and standard equation of the normal ruled surface,the deep relation of the structure functions is obtained. Then some conclusions about the directrix,the striction line and the structure functions are obtained. By discussing the normal ruled surfaces of general helices and Mannheim curves in 3-D Euclidean Space,conclusions can be drawn that the normal ruled surfaces of general helices are positive spiral surfaces and the normal ruled surfaces of Mannheim curves are binormal ruled surfaces of their Mannheim partner curves.