In this paper, firstly, we define a W -direction curve and W -rectifying curve of a Frenet curve in 3-dimensional Euclidean space E 3 by using the unit Darboux vector field W of the Frenet curve and give some characterizations together with the relationships between the curvatures of each associated curve. We also introduce a V -direction curve, which is associated with a curve lying on an oriented surface in E 3 . Later, some new associated curves of a Frenet curve are defined in E 4 .

In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4-space E 4 . Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E 4 . Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface.

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