(untitled) (European Journal of Pure and Applied Mathematics, 2026)
This research examines directional Smarandache curves derived from spatial polynomial curves within the context of FlC-frames in three-dimensional Euclidean space E3. The authors define four novel categories of directional Smarandache curves: TD2, TD1, D2D1, and TD2D1 based on specific combinations of the FLC frame vectors. For each type, the authors derive the complete Frenet apparatus (tangent, normal, and binormal vectors, curvature, and torsion) and thecorresponding FLC apparatus. The theoretical findings are shown by a detailed analysis of a third-order polynomial curve, in which specific parameterizations for directional Smarandache curves are derived. Graphical comparisons of directional Smarandache curves created from the classical Frenet frame and the novel FLC frame are provided.