A dual quaternion is an expression
 |
where 
and 
are quaternions, the dual unit 
commutes with quaternions, and 
. Addition is componentwise, while multiplication
follows
![[q_r+epsilonq_d][p_r+epsilonp_d]=q_rp_r+epsilon[q_rp_d+q_dp_r].](/images/equations/DualQuaternion/NumberedEquation2.svg) |
The algebra is associative and noncommutative, and it contains both the quaternion algebra and the dual numbers as subalgebras.
Unit dual quaternions, with an identification of opposite signs, represent orientation-preserving rigid motions of three-dimensional Euclidean space. Their multiplication composes the corresponding rotations and translations, which makes dual quaternions useful in kinematics and computer graphics.
