四维欧氏空间中曲线的运动不变特征

ON Motional Invariants of Curves in Euclidean4-Space

举例证实了经典的Frenet公式所定义的曲线的三个曲率不能唯一确定曲线到一个运动;借助于四维空间中三个向量的向量积运算在正则曲线上构造右手系Frenet标架并重新定义曲线的第三曲率;据此证明四维空间中的运动保持曲线的曲率和挠率不变,但第三曲率当运动含有反射时会改变符号,并证明结论的逆也成立。

In this paper, an example is given to substantiate that the functions k1 （s） k2 （s） k3 （s）, which are defined by traditional Frenet formulae, cannot uniquely determine a curve up to a motion. Secondly, the vector product of three vectors in Euclidean 4-space is introduced for mending above-mentioned conclusion. Then it is constructed that the right-hand frame at the normal point of a curve and obtain the new definition of the third curvature for curves in Euclidean 4-space. Finally, it is proved that, under a motion of 4-space, the curvature and torsion of a curve are not changed, but the third curvature can be changed by multiplied -1 when the motion containing reflection, and that the converse of this conclusion also holds.