Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$

Abstract

In this study, we define (1,3)-Bertrand-direction curve and (1,3)-Bertrand-donor curve in the 4-dimensional Euclidean space $E^{4}$. We introduce necessary and sufficient conditions for a special Frenet curve to have a (1,3)-Bertrand-direction curve. We introduce the relations between Frenet vectors and curvatures of these direction curves. Furthermore, we investigate whether (1,3)-evolute-donor curves in $E^{4}$ exist and show that there is no (1,3)-evolute-donor curve in $E^{4}$ .