Bertrand-like Curves in Euclidean 3-Space

Abstract

In this paper, the pair of Bertrand-like curves is introduced with a linear dependency between the normal-like vectors in the Frenet-like curve frames of two curves at their corresponding points. The necessary and sufficient conditions to be a Bertrand-like curve are obtained. The main characteristic property of any Bertrand curve is known as the existence of a linear relation between its curvature and torsion. Its analogue is found for a Bertrand-like curve as $\lambda {d_1} - \mu {d_3} = 1$ for non-zero $\lambda ,\,\mu  \in \mathbb{R}$. More clearly, the existence of a linear relation between the Frenet-like curvatures ${d_1}$ and ${d_3}$ of a curve is the necessary and sufficient condition for it to be a Bertand-like curve. We present some characterizations for the conjugate of any Bertrand-like curve. Besides, the relations between the curvatures of each of the pairs are found. An example is presented with a graphic of a Bertrand-like curve pair.