UNIFYING THE GEOMETRY OF FINITE DISPLACEMENT SCREWS AND ORTHOGONAL MATRIX TRANSFORMATIONS

If the end-locations of a rigid-body displacement P-->Q are represente d by the 3 x 3 orthogonal dual number matrices (P) over cap and (Q) ov er cap, and if the displacement is then re-specified in terms of an or thogonal matrix (H) over cap(\(H) over cap\ = 1) of the same form such that (H) over cap(2)$ = (Q) over cap (P) over cap(T)$, then the skew- symmetric matrix (H) over cap - (H) over cap(T)$ is found to define a finite displacement screw (S) over cap which characterises the displac ement. This screw, of pitch P and sited in the line (s) over cap of Ch asles's axis for the displacement, has the form (S) over cap = 2 <(the ta)over cap>(s) over cap = 2 sin theta (1 + epsilon P)(s) over cap, P = sigma/tan theta, in which the distance 2 sigma and the angle 2 theta are the components of the displacement as measured along and about th at axis. It is a corollary of this finding that finite displacements o f a given pitch P are characterised by a locus which is not a helix bu t, rather, the hyperbolic paraboloid sigma = P tan theta. The derivati on and key properties of the characterising screw (S) over cap are dis cussed in this paper. In particular it is shown, in the limit situatio n of an infinitesimal displacement, that the characterising screw deri ved here is identical with the screw of an infinitesimal or instantane ous twist as defined by other authors. (C) 1997 Elsevier Science Ltd.