 In general, the dual quaternion associated with a spatial displacement "D" = (["A"], d) is constructed from its screw axis "S" = (S, V) and the dual angle ("φ", "d") where "φ" is the rotation about and "d" the slide along this axis, which defines the displacement "D".The associated dual quaternion is given by,

Let the composition of the displacement DB with DA be the displacement "D""C" = "D""B""D""A".The screw axis and dual angle of DC is obtained from the product of the dual quaternions of DA and DB, given by
That is, the composite displacement DC=DBDA has the associated dual quaternion given by

Expand this product in order to obtain

Divide both sides of this equation by the identity
to obtain

This is Rodrigues' formula for the screw axis of a composite displacement defined in terms of the screw axes of the two displacements.