Kinematic differential geometry of a rigid body in spatial motion—II. A new adjoint approach and instantaneous properties of a line trajectory in spatial kinematics☆

Based on differential geometry, a new approach of a space curve adjoint to a ruled surface is presented in the paper. The geometrical properties of a point trajectory in spatial motion are extensively researched by means of the new adjoint approach. The invariants of axodes are derived and their kinematic meanings are revealed, and then, the invariants of a point trajectory generated by spatial motion are represented by that of axodes, which leads to the acceleration center, Bresse hyperboloid, geodesic inflection surface and inflection curve in the moving body being readily located. Meanwhile, the curvature theories or the geodesic Euler-Savary analogue and the Euler-Savary analogue are set up. Finally, the cubic of stationary curvature and Burmester's point in the moving body are concisely derived for spherical motion. The novel approach adopted in this paper will be developed in our continued papers. The instantaneous properties of a line trajectory will be discussed, and the characteristic lines in the moving body will be revealed in the consecutive papers. All of our three papers will lay the ground work for a unified theory of spatial kinematic geometry based on the axodes.