曲线的主法线曲面

在三维欧氏空间中,作为特殊曲线,Mannheim曲线、Bertrand曲线以及一般螺线具有良好的几何和代数性质.讨论了三维欧氏空间中特殊曲线的主法线曲面.根据渐近曲线的方程,具体给出主法线曲面的一族非直线的渐近曲线.再根据平均曲率、高斯曲率及主曲率函数,能得到曲线的主法线曲面的极小轨迹、常高斯曲率曲线及两个主曲率函数之比为常数的曲线.还给出曲面上测地线和腰曲线的性质.

Principal Normal Surfaces of Curves

Discusses algebraically and geometrically the properties of the ruled surfaces in the Euclidean three-space,especially,the principal normal surfaces on which there are some special curves,such as Mannheim curves,Bertrand curves and general helices.According to the equation of asymptotic curves,a family of nonlinear asymptotic curves are given to the principal normal surface.Then,based on the mean curvature,Gaussian curvature and the function of principal curvature,the minimal loci of the principal normal surfaces of curves,the curves with constant Gaussian curvature and the two curves of which the ratio of their functions is constant can all be obtained,as well as the properties of geodesic curve and lumbar curve.