On the Regulus Associated With the General Displacement of a Line and Its Application in Determining Displacement Screws

Abstract

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 unveils the regulus that corresponds to such a displacement. The regulus is then used as a basic entity to determine the displacements of a rigid body from line specifications. 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 three-system. When determining the displacements of a rigid body from line specifications, a displacement screw is obtained by fitting a linear line complex to two or more line reguli. When two exact pairs of homologous lines are specified, we obtain a unique linear line complex, which determines the corresponding displacement screw. When more than two pairs of homologous lines with measurement errors are specified, it becomes a redundantly specified problem, and a linear line complex that has the best fit to more than two reguli is determined.