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In mathematics, the dual quaternions are an algebra isomorphic to a Clifford algebra of a degenerate quadratic space.
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 p + ε q, where p and q are ordinary quaternions and ε is the dual unit (which satisfies εε = 0) and commutes with every element of the algebra. Unlike quaternions, they do not form a division ring.
In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions.[1] A dual quaternion is an ordered pair of quaternions  = (A, B), constructed from eight real parameters. Because rigid transformations have six real degrees of freedom, dual quaternions include two algebraic constraints for this application.
