On the Regulus Associated With the General Displacement of a Line

In the two position theory of finite kinematics, we are concerned with not only the displacement of a rigid body, but also with the displacement of a certain element of the body. This paper deals with the displacement of a line and reveals the regulus that characterizes such a displacement. Residing on a special hyperbolic paraboloid, the regulus is obtained by the intersection of three linear line complexes corresponding to a specific set of basis screws of a 3-system. The degeneration of the regulus when two positions of a line intersect is also discussed. In this paper, the regulus of intersection is obtained geometrically as well as analytically. The discovery of the regulus lays a geometric foundation for dealing with line-based problems in computational kinematics and computational line geometry.