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In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be 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 A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.
In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation have six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebraic constraints are used in this application.
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 , and in applications to 3D computer graphics, robotics and computer vision.