On the Finite Screw Cylindroid Represented as a 2-system of Screws

Abstract

The problem of displacing a line with a definite point on it from one spatial position to another is studied by utilizing the concept of screw matrix. It is known that all the available finite twists (screws) associated with this displacement form a ruled surface, the so-called finite screw cylindroid. If the definition of the pitch given by Parkin is used, then the finite screw cylindroid can be regarded as a 2-system of screws. This brings to one's mind the question as to whether there exist different appropriate measures for pitch other than Parkin's under which all the available finite twists form a 2-system. This question is answered in this paper. By deriving a general expression of the pitch for these available finite twists under the said condition, it is shown that Parkin's pitch plus an arbitrary constant is the only possible measure of pitch under which the finite screw cylindroid represents a 2-system of screws. However, since adding a constant to the pitches of all screws of any 2-system still gives a 2-system, constant term may be omitted. It is also shown that the determined 2-system of screws can be described as a linear combination of two special basis screws which are called in this paper the α = 0 and the α = π screws.