On-the-linear-line-complex-and-helicoidal-vector-field-associated-with-homologous-lines-of-a-finite-displacement_2008_Mechanism-and-Machine-Theory.pdf

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and
Machine Theory
Mechanism and Machine Theory 43 (2008) 138–146
ate/mechmt
On the linear plex and helicoidal vector field
associated with homologous lines of a finite displacement
Chintien Huang a,*, Wuchang Kuo a, Bahram Ravani b
a Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan
b Department of Mechanical and Aeronautical Engineering, University of California at Davis, Davis, CA, USA
Received 18 October 2006; accepted 13 March 2007
Available online 10 May 2007
Abstract
In finite kinematics, the LLC associated with homologous lines of a finite displacement is lesser known because its null-
planes 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 e infinitesimally separated.
Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction
A linear plex (LLC) contains a set of lines whose Plu¨cker