In this paper, we investigate the parametric representation for a family of surfaces through a given geodesic curve G₃ . We provide necessary and sufficient conditions for this curve to be an isogeodesic curve on the parametric surfaces using Frenet frame in Galilean space. Also, for the sake of visualizing of this study, we plot an example for this surfaces family.

In this study, we examine some properties of Salkowski curves in $\mathbb{E}^{3}$. We then make sense of the angle $(nt)$ in the parametric equation of the Salkowski curves. We provide the relationship between this angle and the angle between the binormal vector and the Darboux vector of the Salkowski curves. Through this angle, we obtain the unit vector in the direction of the Darboux vector of the curve. Finally, we calculate the modified orthogonal frames with both the curvature and the torsion and give the relationships between the Frenet frame and the modified orthogonal frames of the curve.

We investigate null curves of the AW(k)-type (1 ≤ k ≤ 3) in the 3-dimensional Lorentzian space, L ³ , and give curvature conditions of these curves by using the Cartan frame. Moreover, we study harmonic curvatures of curves of AW(k)-type and show that if the α Frenet curve is of type AW(1), then α is a null helix.