三维欧氏空间中的球面曲线

Spherical Curves in Euclidean 3-Space

依据经典微分几何空间曲线的基本理论与特征,采用一种新的活动标架——三维欧氏空间中的球面Frenet标架,并利用三维曲线的Frenet标架场,对三维欧式空间中的球面曲线进行研究,得到了在三维空间E^3下的贝特朗、曼海姆及从切等特殊曲线,给出了一个由曲线的曲率与挠率的一阶常微分方程描述的三维欧氏空间中的球面曲线,得出了比对应微分方程阶数更低的条件,且大大简化了计算过程.

Eased on the basic theory and characteristics of space curves in classical differential geometry, a new kind of moving construction the spherical Frenet construction of this kind were finany obtained in 3-D Euclidean space, as well as the 3-D curves Frenet construction field were introduced to inspect the spherical curves in 3-D Euclidean space. Bertrand, Malmheim, rectifying curves, and special curves of this kind were finally obtained in the 3-D E^3 space,giving spherical curves in 3-D Euclidean space which can be described by first-order ordinary differential equations with respect to curve＇s curvature and torsion, and producing a low-level curve representation for the corresponding differential equation. The new characteristics of spherical curves were verified and the calculation process was simplified.