Dual Quaternion

Dual Quaternion

In mathematics and mechanics, the set of dual quaternions is a Clifford algebra that can be used to represent spatial rigid body displacements. A dual quaternion is an ordered pair of quaternions  = (A, B) and therefore is constructed from eight real parameters. Because rigid body displacements are defined by six parameters, dual quaternion parameters include two algebraic constraints.

In ring theory, dual quaternions are a ring constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form q = q₀ + ε q_ε, where q₀ and q_ε are ordinary quaternions and ε is the dual unit (εε = 0) and commutes with every element of the algebra. Unlike quaternions they do not form a division ring. It can be given the Clifford algebra classifications Cℓ_0,2,1(R) ≅ Cℓ0_0,3,1(R).

Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy), and in applications to 3D computer graphics, robotics and computer vision.