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Concept
Dual quaternion
Summary
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 ε2 = 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 has six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebraic constraints are used in this application. Since unit quaternions are subject to two algebraic constraints, unit quaternions are standard to represent rigid transformations.
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. Polynomials with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of mechanical linkages design.
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.
In Russia, Aleksandr Kotelnikov developed dual vectors and dual quaternions for use in the study of mechanics.
Official source
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