Curves associated with spacelike curves on lightlike surfaces in $\mathbb{E}_{1}^{3}$

Abstract

In this paper, we introduce the concept of $k-$directional Darboux curves associated with a given spacelike curve $\alpha $ lying on a lightlike surface in Minkowski $3$-space $\mathbb{E}_{1}^{3}$. These $k-$directional Darboux curves are curves generated by specific vector fields derived from the Darboux frame along the curve. We explore the connections between these associated curves and their curvatures, and we also propose a new method for classifying special curves, such as helices and slant helices, using these ideas. In addition, we give the related examples and their graphics.