Generating Functions and Phase Portraits of Slant Helices in Euclidean 3-Space

In this paper, we study slant helices in Euclidean $3$-space by means of a generating function representation. We show that the curvature and torsion of a slant helix can be written in terms of a single differentiable function, and conversely, that every slant helix locally admits such a representation. This formulation gives a simple description of the phase portrait in the curvature--torsion plane and allows several classical subclasses of slant helices to be treated in a unified way. In particular, rectifying curves, Salkowski curves, anti-Salkowski curves and spherical slant helices arise from specific choices of the generating function. We also introduce an associated slant helix construction preserving the same generating-function structure. Furthermore, oscillatory generating functions are considered, leading to periodic, bounded and asymptotically constant-slope families of slant helices. Finally, connections with persistent rigid-body motions and curvature-dependent energy models are discussed.