Curvature Theory of the Envelop Curve / Surface for a Line in Planar Motion and a Plane in Spatial Motion

The curvature theories for the envelop curve of a line in planar motion and the envelop ruled surface of a plane in spatial motion are extensively researched in the differential geometry language. A line-envelop curve in planar motion is firstly derived by means of the adjoint approach. The higher order curvature theory of the envelop curve reveals a unified form in the infinitesimal and finitely separated positions for a line in planar motion. And then, a plane in spatial motion traces the envelop surface, which is a developable surface and whose invariants are concisely derived. The geodesic curvature of the spherical image curve for the generator’s unit vector is readily derived and compared with that of the unit normal vector of the envelop surface. As a result, the curvature theory for a plane-envelop surface in spatial motion are shown in terms of that of the spherical motion, corresponding to the generator’s unit vector and unit normal vector of the envelop surface. Meanwhile, the instantaneous cubic (cone) of stationary curvature and the direction “Burmester’s line” of the generator of the developable envelop surface are revealed. Therefore, a solid theoretical basis is provided for the synthesis of mechanisms and the machining of surface.