ON THE CROSS RATIO OF DUAL QUATERNIONS IN THE CONTEXT OF KINEMATICS

In general, the notion of a cross ratio can algebraically be defined for local alternative rings, see [2]. Recently, its kinematic interpretation in case of the non-commutative ring H of dual quaternions was proposed in [8]. Further applications and suggestions in the context of chain geometry can be found in [4]. This paper aims at the notion of the cross ratio for from a more geometric point of view. In the context of its kinematic interpretation as spatial displacements, it is defined for quadruples of points on a projective line over H. At first, we consider algebraic properties of the non-commutative algebra (H, +, .). Afterwards, a projective line over H is constructed and the cross ratio is introduced as an invariant under the action of the projective linear transformation group PGL(2,H). Furthermore, an example of a line-symmetric motions is studied in detail. It is well-known that the cross ratio of four complex numbers is real if the four associated points are either collinear or concyclic. Therefore, the question for a real cross ratio of dual quaternions arises.