On a Set of Displacements in Space

Abstract

In this paper we study sets of screw displacements in three-dimensional space when the displacements are not uniquely specified. The two cases considered are displacements which transform a row of points onto a congruent row of points, and displacements which transform a line onto a line. In the first case the locus of all screw axes is shown to be a regulus which is either cubic (a cylindroid) or quadratic (a hyperbolic paraboloid), depending upon, respectively, if the displacement is direct or indirect (a reflection). In the second case the locus of screw axes is shown to be a line congruence of order 3 and class 1.