On the linear line complex and helicoidal vector field associated with homologous lines of a finite displacement

In finite kinematics, the LLC associated with homologous lines of a finite displacement is lesser known because its nullplanes were not identified, unlike the LLC associated with homologous points. This paper investigates the geometry of the LLC of homologous lines and unveils the nullplanes that constitute the LLC. This paper also demonstrates the existence of a helicoidal vector field corresponding to the LLC of homologous lines. The pitch of the helicoidal vector field (and the LLC) is the translation divided by the sine of the rotation of the screw displacement. Furthermore, the LLC of homologous lines degenerates into the LLC of instantaneous motion when the two positions become infinitesimally separated.