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.

The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces.

This study presents a new geometric formulation for constructing embankment–type ruled surfaces in the Lorentzian framework of Minkowski 3-space ${\mathrm{E}}_1^3$, by means of the orthogonal modified frame (OMF). The OMF serves as an orthogonal moving frame that is fully compatible with the Minkowski metric, providing a consistent representation of the causal behavior of both spacelike and timelike curves. Within this framework, three distinct surface families are established namely, the OMF embankment surface, the OMF embankment–like surface, and the OMF tubembankment–like surface. Each class is developed together with its explicit parametric expression and associated geometric invariants. The corresponding first and second fundamental forms are derived, from which closed analytical expressions for the Gaussian curvature and mean curvature are obtained. These computations lead to precise differential conditions governing the developability and minimality of the generated surfaces. Representative examples illustrate the smooth curvature distribution and the Lorentzian metric consistency of the OMF–based models, demonstrating clear advantages over traditional Euclidean constructions. Overall, the proposed formulation offers an efficient and unified framework for analyzing and modeling ruled surfaces in Lorentzian geometry, with prospective applications in relativistic motion theory, geometric design, and computer-aided kinematic simulation.