A New Link to Helices in Euclidean $3$-Space

In this paper, we introduce a novel approach for obtaining the parametric expression and description of a general helix, slant helix, and Darboux helix. The new method involves projecting $\alpha $ onto a plane passing through $\alpha \left( 0\right) $ and orthogonal to the unit axis vector $U$ in order to determine the position vector of the general helix $\alpha $. The position vector of the helix with the plane curve $\gamma $ and its axis $U$ is then established. Additionally, a relation between the curvatures of $\alpha $ and $\gamma $ is presented. The proposed technique is then applied to derive the parametric representation of a slant helix and Darboux helix, followed by the provision of various examples obtained through the application of this methodology.