Direction Curves Associated with Darboux Vectors Fields and Their Characterizations

In this paper, we consider the Darboux frame of a curve $\alpha$ lying on an arbitrary regular surface and we use its unit osculator Darboux vector ${\overline{D}}_o$ , unit rectifying Darboux vector ${\overline{D}}_r$ , and unit normal Darboux vector ${\overline{D}}_n$ to define some direction curves such as ${\overline{D}}_o$ -direction curve, ${\overline{D}}_r$ -direction curve, and ${\overline{D}}_n$ -direction curve, respectively. We prove some relationships between $\alpha$ and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants ${\delta }_o$ , ${\delta }_r$ , and ${\delta }_n$ are, respectively, zero. Finally, we enrich our study by giving some examples.