Unified curvature modeling of surface constrained helices and associated ruled surfaces

This paper introduces a new class of surface constrained curves, termed generalized B-helices, in the Euclidean three-space ${\mathrm{E}}^3$. The analysis is developed through a rotational modification of the classical Darboux frame, called the $\mathbb{B}$-Darboux frame $\{\mathrm{T},{\chi }_1,{\chi }_2\}$. This frame combines the effects of geodesic and normal curvatures into two coupled invariants, referred to as the $\mathbb{B}$-Darboux curvatures, providing a unified description of curve geometry on regular surfaces. Within this framework, three types of generalized $\mathbb{B}$-helices are defined, and their characterization conditions are expressed in terms of these curvature functions. The spherical representations corresponding to the $\mathbb{B}$-Darboux frame vectors are also examined, and explicit relations for their curvature and torsion are derived. It is shown that each type of generalized $\mathbb{B}$-helix generates a circular indicatrix on the unit sphere, establishing a direct geometric link between spatial and spherical curves. To validate the theoretical developments, illustrative examples and graphical simulations of ruled surfaces associated with the frame vectors are presented, confirming the applicability and effectiveness of the proposed model.