Ruled Surfaces Generated by the Frenet Vectors of the Original Curve Along Its Parallel Curve

Source abstract

This paper introduces four families of ruled surfaces generated by the tangent, principal normal, binormal, and general Frenet vector fields of an original space curve by taking its principal normal parallel curve as the base curve. Although ruled surfaces associated with moving frames have been extensively studied, a unified differential geometric treatment of ruled surfaces generated from the Frenet vectors of an original curve along its parallel curve has not been presented. To address this gap, explicit parametrizations of the proposed ruled surface families are established, and their differential geometric properties are investigated by deriving closed-form expressions for the first, second, and third fundamental forms, Gaussian curvature, mean curvature, distribution parameter, and striction curve. All geometric invariants are expressed explicitly in terms of the curvature, torsion, and offset parameter of the original curve, yielding a unified analytical characterization of the constructed surfaces. Two representative examples, namely a circular helix and a polynomial space curve, are presented to verify the theoretical developments and to demonstrate the applicability of the proposed framework to both constant-curvature and variable-curvature curves. The obtained results provide a systematic approach for studying ruled surfaces associated with parallel curves and establish a foundation for further investigations involving alternative moving frame systems and non-Euclidean ambient spaces.