Singularities of Timelike Bishop Developable Ruled Surfaces in Minkowski 3-Space

This paper investigates timelike developable ruled surfaces generated by the Bishop frame along unit-speed timelike curves in Minkowski 3-space E13. The Bishop frame formulation remains well defined at curvature-zero points where the Serret–Frenet frame may fail to be defined. The singular sets of the resulting developable surfaces are determined explicitly in terms of the Bishop curvature functions ϵ1 and ϵ2. The frontality conditions are identified, and necessary and sufficient conditions for cuspidal edge and swallowtail singularities are obtained under the nondegeneracy condition ϵ1ϵ2≠0. Using B-height functions and versal unfoldings, these singularities are further described through the orders of vanishing of the corresponding height functions and their discriminant sets. A constant normal-offset family is also investigated, and its singular locus along with its cuspidal edge and swallowtail criteria are expressed in terms of both Bishop curvature functions. Cylindrical, conical, and tangential configurations are characterized through the Bishop curvature ϵ1. Examples illustrate the necessity of the frontality condition, demonstrating cuspidal edge and genuine swallowtail singularities together with cylindrical, conical, and tangential configurations. The results refine and extend earlier Bishop frame treatments of timelike developable surfaces by emphasizing the explicit frontality conditions and the singular geometry of the constant normal-offset family.