Dual quaternion

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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. A dual quaternion can be represented in the form q = q₀ + ε q_ε, where q₀ and q_ε 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.

In 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. 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.

[edit] 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”. The latter eponym has also been used to refer to split-biquaternions. 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.

[edit] References

• A.T. Yang (1963) Application of quaternion algebra and dual numbers to the analysis of spatial mechanisms, Ph.D thesis, Columbia University.
• A.T. Yang (1974) "Calculus of Screws" in Basic Questions of Design Theory, William R. Spillers, editor,Elsevier, pages 266 to 281.
• J.M. McCarthy (1990) An Introduction to Theoretical Kinematics, pp.62-5, MIT Press [ISBN 0262132524].
• Dual Quaternions for Rigid Transformation Blending
• Joe Rooney William Kingdon Clifford, Department of Design and Innovation, the Open University, London.