Direction Curves Associated with Darboux Vectors Fields and Their Characterizations

In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D¯o, unit rectifying Darboux vector D¯r, and unit normal Darboux vector D¯n to define some direction curves such as D¯o‐direction curve, D¯r‐direction curve, and D¯n‐direction curve, respectively. We prove some relationships between α 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 δo, δr, and δn are, respectively, zero. Finally, we enrich our study by giving some examples.