三维欧氏空间中的达布曲线

Darboux curves in 3-dimensional Euclidean space

在三维欧氏空间中,经过空间曲线上的一点,且与该点的达布向量平行的直线称为曲线在此点的达布线。如果一条曲线和另一条曲线的点之间建立这样的一一对应关系,使得在对应点的达布线重合,则这2条曲线被称为达布曲线对,其中一条叫做另一条的达布侣线。主要研究了三维欧氏空间中达布曲线对的一些性质,并得到了如下结论:2条曲线的主法线平行的充分必要条件是它们的达布线平行;2条曲线的主法线平行,则它们在对应点的切向量成固定角;2条曲线为达布曲线对的充分必要条件是在对应点它们的从切平面重合。同时研究了三维欧氏空间中一条空间曲线具有达布侣线时它的曲率和挠率需要满足的关系。

In three dimensional Euclidean space, a straight line passing through a point of a space curve and paralleling to the Darboux vector of this point is called the Darboux line of the curve at this point. If the Darboux lines of two curves are coincident at the corresponding points, the curves are called a Darboux curve pair. Some characteristics for Darboux curve pair in three dimensional Euclidean space would be found in this paper, and we also get following results.. 1）The principal normal lines of two curves are parallel if and only if their Darboux lines are parallel; 2）If the principal normal lines of two curves are parallel, their tangent vector fields make a fixed angle; 3）Two curves are Darboux curve pair if and only if their tangent planes are same at the corresponding points. At the same time, we study the relation between the curvature and torsion of a space curve if it has a Darboux pair in three dimensional Euclidean space.