On the third-order finite screw system: general displacement of a line

Abstract

The problem of displacing a line, being a part of a rigid body, from one spatial position to another is studied by using the concept of screw matrix. It is known that there are ∞2 possible screws to realize such displacement; based on the definition of pitch given by Parkin, any one of the ∞2 screws cannot be expressed as the linear combination of three basis screws. In this paper an attempt is made towards a search of a new definition (expression) for pitch under which the set of ∞2 screws form a 3-system. It is shown that such expression for pitch exists and contains six arbitrary constants. By assigning some values to these constants two different but meaningful and valid expressions of pitch are obtained. These expressions are modified to include terms corresponding to Parkin's definition of pitches. Explicit analytical expression of the 3-system is also discussed.