Geometric and singularities insights of swept surfaces via the Bishop frame in Euclidean 3-Space

This study investigates the geometry and singular behavior of swept surfaces generated by the move of Bishop frame along a spatial curve in Euclidean 3-space. The surface is defined as the envelope of a family of unit spheres whose centers trace an axial trajectory, with the contact points forming great circles within a prescribed plane. A parametric representation is established to highlight the dependence of the surface on both the axial and profile curves. Key geometric features including the coefficients of the first fundamental form, unit normal vectors, and curvature characteristics are analyzed, revealing that the profile curves act as planar geodesics and curvature lines. We further examine singularities, offset surfaces, and parabolic curves, deriving conditions for smoothness, geodesicity, and convexity. Special attention is given to the criteria under which the surface becomes developable, particularly when it reduces to a cylinder, cone, or tangent surface. Several illustrative examples, including those arising from mate curves of slant helices and circular trajectories, demonstrate the resulting geometric phenomena.