New Frameworks for Constructing Special Curves via f-Weighted Direction Curves

Source abstract

In this paper, we investigate three classes of f-weighted direction curves, namely tangent, principal, and binormal direction curves, obtained by integrating positive smooth multiples of the corresponding Frenet vector fields of a generating Frenet curve in Euclidean three-space. We derive curvature and torsion relations for these associated curves and investigate their connections with general helices, slant helices, clad helices, and g-clad helices. In particular, the weighted tangent direction construction preserves certain helix invariants of the generating curve, while suitable choices of the weight function modify its curvature and torsion. We establish conditions on the weight functions under which the resulting curves satisfy the characterizations of rectifying, Mannheim, and Bertrand curves. We also investigate conditions for constructing spherical weighted direction curves and introduce g-weighted normal, rectifying, and osculating direction curves associated with them. Explicit examples illustrate successive weighted constructions, including the generation of a rectifying clad helix and a Salkowski curve. The main contribution is a common construction framework in which the choice of a Frenet direction field and a positive weight function allows several classes of special space curves to be investigated through their geometric characterizations, subject to the required regularity and nonvanishing conditions.