Higher-Dimensional Geometry and Singularity Structure of Osculating Type-II Ruled Surfaces in Lorentzian Spaces

In Minkowski 3-space, we establish a geometric framework to osculate Type-II ruled surfaces by utilizing the Type-II Bishop frame in ( E 1 3 ) . Our analysis extends to higher-order singularities such as butterflies and pyramids, including explicit singularity loci. We also compare Type-II Bishop frames with rotation-minimizing frames using timelike base curves and spacelike normals. With RK4 integration, we develop a robust computational model for Weingarten surfaces and subclasses with constant curvature. The theoretical foundation for Type-II Bishop frames is extended to higher-dimensional Minkowski spaces E 1 n for n > 3 through generalized Frenet-type equations and curvature functions. We determine exact stability conditions under perturbations of Bishop curvature using advanced singularity theory. The numerical implementations of our methods, including geometric modeling and relativistic geometry, demonstrate their effectiveness in both theoretical and applied contexts.