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In ring theory, dual quaternions are a non-commutative ring constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician A variety of dualities in mathematics are listed at Duality (mathematics. In Mathematics, the real numbers may be described informally in several different ways A dual quaternion can be represented in the form q = q0 + ε qε, where q0 and qε are ordinary quaternions and ε is the dual unit (εε = 0).

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 and robotics. Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data A robot is a mechanical or Virtual Artificial agent In practice it is usually an electro-mechanical system which by its appearance or movements

In 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. Eduard Study ( March 23, 1862 &ndash January 6, 1930) was a German Mathematician known for work on Invariant theory In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive Three-dimensional space is a geometric model of the physical Universe in which we live He futher 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. Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, The biquaternions are the numbers w + xi + yj + zk \ \! where w x y and z are complex numbers and the elements of {1 i j k} multiply as in the Quaternion group

Eponyms

Since both Eduard Study and William Kingdon Clifford used, and wrote upon, the dual quaternions, at times authors refer to dual biquaternions as “Study biquaternions” or “Clifford biquaternions”. Eduard Study ( March 23, 1862 &ndash January 6, 1930) was a German Mathematician known for work on Invariant theory William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and The latter eponym has also been used to refer to split-biquaternions. In Mathematics, a split-biquaternion is a member of the Clifford algebra C �( R) Read the article by Joe Rooney linked below for view of a supporter of W. K. Clifford’s claim. Since the claims of Clifford and Study are in contention, it is convenient to use the current designation dual quaternion to avoid conflict.

References

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