On the kinematic geometry of relative screw motions

We present an exact formulation for the central problem of screw theory, namely, the determination of the principal screws of given relative screw motions. Using the Study's dual-line coordinates, a method for the kinematic synthesis of spatial gears with skew axes is introduced. This approach allows showing that the principal screws of the system can be determined via the extreme values of the pitch. The well known equation of Plucker's conoid associated with the three-systems has been derived, and it is shown that the principal axes of it are two screw axes located at its center and at right angles. More specifically, these two axes and the common normal to all screw axes form a reference frame that simplifies the expression of Plucker's colloid. Finally, geometric conditions are discussed for some of the special screw systems.