Direction curves associated with Darboux vectors fields and their characterizations (Q2068268)

Summary: 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.