of a dual quaternion is given by taking the dual number conjugate, given by The quaternion and dual number conjugates can be combined into a third form of conjugate given by In the context of dual quaternions, the term "conjugate" can be used to mean the quaternion conjugate, dual number conjugate, or both. Norm The norm of a dual quaternion is computed using the conjugate to compute . This is a dual number called the magnitude of the dual quaternion. Dual quaternions with are unit dual quaternions. Dual quaternions of magnitude 1 are used to represent spatial Euclidean displacements. Notice that the requirement that , introduces two algebraic constraints on the components of Â, that is Inverse If is a dual quaternion, and p is not zero, then the inverse dual quaternion is given by p−1 (1 − ε q p−1). Thus the elements of the subspace do not have inverses. This subspace is called an ideal in ring theory. It happens to be the unique maximal ideal of the ring of dual numbers. The group of units of the dual number ring then consists of numbers not in the ideal. The dual numbers form a local ring since there is a unique maximal ideal. The group of units is a Lie group and can be studied using the exponential mapping. Dual quaternions have been used to exhibit transformations in the Euclidean group. A typical element can be written as a screw transformation. Dual quaternions and spatial displacements A benefit of the dual quaternion formulation of the composition of two spatial displacements DB = ([RB], b) and DA = ([RA],a) is that the resulting dual quaternion yields directly the screw axis and dual angle of the composite displacement DC = DBDA. 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 DC = DBDA. 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