欧几里得空间E4中四元Bertrand曲线的特征
Characterizations of the Quaternionic Bertrand Curve in Euclidean Space E4
摘要：
在[18]中，L. R. Pears证明了E-n（n > 3）中的Bertrand曲线是退化曲线。这个结果在[16]中由Matsuda和Yorozu重申。他们证明了E-n（n > 3）中不存在特殊的Bertrand曲线，并在4维欧几里得空间中定义了一类新的Bertrand曲线，称为（1，3）-型Bertrand曲线。在这项研究中，我们定义了一个四元数Bertrand曲线？(4)在欧几里得空间E4中，并研究了两种情况下它性质。在第一种情况下;考虑欧几里得空间E4中的四元数Bertrand曲线，其中r为空间四元数曲线的挠率; K是四元数曲线的主曲率？(4)在另一种情况下，我们证明了在欧几里得空间E4中不存在四元数Bertrand曲线（r = 0）：因此，我们利用文[16]中类似的方法给出了欧几里得空间E4中四元数Bertrand曲线的概念，我们称之为四元数（N-B ~ 2）Bertrand曲线，并给出了这类曲线的一些特征。
英文摘要：
In [18], L. R. Pears proved that Bertrand curves in E-n(n > 3) are degenerate curves. This result restate in [16] by Matsuda and Yorozu. They proved that there is no special Bertrand curves in E-n(n > 3) and they define new kind of Bertrand curves called (1, 3)-type Bertrand curves in 4-dimensional Euclidean space. In this study, we define a quaternionic Bertrand curve ?(4) in Euclidean space E4 and investigate its properties for two cases. In the first case; we consider quaternionic Bertrand curve in the Euclidean space E4 for r-K = 0 where r is the torsion of the spatial quaternionic curve ?; K is the principal curvature of the quaternionic curve ?(4): And then, in the other case, we prove that there is no quaternionic Bertrand curve in the Euclidean space E4 for r - K = 0: So, we give an idea of quaternionic Bertrand curve which we call quaternionic (N - B2) Bertrand curve in the Euclidean space E4 by using the similar method in [16] and we give some characterizations of such curves.
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