On the one-parameter dual spherical motions

Based on E. Study's dual line coordinates, explicit expressions were developed for the differential properties of one-parameter dual spherical motions that are coordinate systems independent. This eliminates the need of laborious coordinate transformations necessary in the determination of the canonical systems. With the proposed technique, the Disteli formulae of the axodes were derived and the connections between kinematic geometry of a line trajectory were researched. Meanwhile, a ruled analogy of the curvature circle of a curve in planar kinematics was extended into general spatial kinematics. Then a lot of efforts have been directed to establish dual versions of the Euler–Savary equations of a line trajectory resulting in two different expressions depending on the axodes. As a result, theoretical expressions for the direction of a line congruence were revealed. Finally, an example illustrating the application of the obtained formulae was introduced.