On the fourth order finite screw system: general displacement of a point in a rigid body

Abstract

In this paper, the displacement of a point on a rigid body which moves from one spatial position to another is considered by the concept of screw matrix. Since there are infinity(3) possible screws to achieve such displacement, using Parkin's definition (measure) for pitch, one can represent this infinite set of screws as a 4-system of screws. This has been demonstrated by Huang and Roth by the use of screw triangle [1]. In this paper, however, without any assumption for the expression of pitch we ask the question that whether there exist different appropriate measure of pitch other than Parkin's such that all the available finite screws also form a 4-system. In fact by the use of screw matrix concept, a general expression of the pitch is derived so that the screws realize the stated displacement form a 4-system. It is shown that Parkin's pitch, plus an arbitrary constant; is the only appropriate measure of pitch. To arrive at this con elusion, a special coordinate system is selected. This selection of the coordinate system also makes it possible to yield several new geometrical properties of the set of infinity(3) screws.