Dual quaternion

In mathematics, the dual quaternions are an algebra isomorphic to a Clifford algebra of a degenerate quadratic space. 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. 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. 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. In Russia, Aleksandr Kotelnikov developed dual vectors and dual quaternions for use in the study of mechanics. In 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. He further developed the idea in Geometrie der Dynamen in 1901. B. L. van der Waerden called the structure 'Study biquaternions', one of three eight-dimensional algebras referred to as biquaternions. In order to describe operations with dual quaternions, it is helpful to first consider quaternions.