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 p + ε q where p and q 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.
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.
W. R. Hamilton introduced quaternions in 1843, and by 1873 W. K. Clifford obtained a broad generalization of these numbers that he called biquaternions, which is an example of what is now called a Clifford algebra.
In 1898 Alexander McAulay used Ω with Ω2 = 0 to generate the dual quaternion algebra. However, his terminology of "octonions" did not stick as today's octonions are another algebra.
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