Exponential and Cayley Maps for Dual Quaternions

Abstract

In this work various maps between the space of twists and the space of finite screws are studied. Dual quaternions can be used to represent rigid-body motions, both finite screw motions and infinitesimal motions, called twists. The finite screws are elements of the group of rigid-body motions while the twists are elements of the Lie algebra of this group. The group of rigid-body displacements are represented by dual quaternions satisfying a simple relation in the algebra. The space of group elements can be though of as a six-dimensional quadric in seven-dimensional projective space, this quadric is known as the Study quadric. The twists are represented by pure dual quaternions which satisfy a degree 4 polynomial relation. This means that analytic maps between the Lie algebra and its Lie group can be written as a cubic polynomials. In order to find these polynomials a system of mutually annihilating idempotents and nilpotents is introduced. This system also helps find relations for the inverse maps. The geometry of these maps is also briefly studied. In particular, the image of a line of twists through the origin (a screw) is found. These turn out to be various rational curves in the Study quadric, a conic, twisted cubic and rational quartic for the maps under consideration.