New Special Curves Using a Pair of Orthogonal Curves.

In this paper, we introduce a new family of special curves in Euclidean 3-space, called the generalized k−slant helix, which extends the classical notion of the k−slant helix. A particular subclass of this family, referred to as the generalized k−slant helix of type NB (denoted by G-k-SH-NB), is studied in detail. By employing a pair of orthogonal associated curves, we derive the intrinsic equations of this class of curves and obtain the parametric representation of their position vectors with respect to the arc-length parameter. Furthermore, starting from a general helix, we apply this approach to determine the parametric representation of the G-k-SH-NB in terms of the arc-length parameter. Several illustrative examples are presented, for which both the intrinsic equations and the corresponding parametric representations are obtained and graphically visualized.