text: The screw axis appears in the dual quaternion formulation of a spatial displacement . The dual quaternion is constructed from the dual vector  defining the screw axis and the dual angle , where φ is the rotation about and d the slide along this axis, which defines the displacement D to obtain,
  A spatial displacement of points q represented as a vector quaternion can be defined using quaternions as the mapping
  where d is translation vector quaternion and S is a unit quaternion, also called a versor, given by
  that defines a rotation by 2θ around an axis S.
  In the proper Euclidean group E(3) a rotation may be conjugated with a translation to move it to a parallel rotation axis. Such a conjugation, using quaternion homographies, produces the appropriate screw axis to express the given spatial displacement as a screw displacement, in accord with Chasles’ theorem.