Constructing $k$-Slant Curves in Three Dimensional Euclidean Spaces

Helices and constant procession curves are special examples of slant curves. However, there is no example of a $k$-slant curve for a positive integer $k\geq 2$ in three dimensional Euclidean spaces. Furthermore, the position vector of a $k$-slant curve for a positive integer $k\geq 2$ has not been known thus far. In this paper, we propose a method for constructing $k$-slant curves in three dimensional Euclidean spaces. We then show that spherical $k$-slant curves and $N_{k}$-constant procession curves can be derived from circles, for $k \in \mathbb{N}$, the set of all nonnegative integers. In addition, we provide a new proof of the spherical curve characterization and define a curve in the sphere called a spherical prime curve. Afterward, we apply $k$-slant curves to magnetic curves. Finally, we discuss the need for further research.