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 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.

[edit] Formulas

[edit] Addition/Subtraction

Addition of dual quaternions is a simple addition of its coefficients.

[edit] Multiplication

Multiplication of dual quaternions follows the rules of its components - the dual numbers and the quaternions.

$Q_1 = r_1 + \varepsilon d_1$

$Q_2 = r_2 + \varepsilon d_2$

$Q_1 * Q_2 = r_1 * r_2 + \varepsilon (r_1 * d_2 + d_1 * r_2)\,\!$

Note that there is no $d 1 * d 2$ , as the definition of dual numbers say that $\varepsilon ^2 = 0$ .

This gives us the multiplication table

$Q_1 * Q_2\,\!$	$Q_2.1\,\!$	$Q_2.i\,\!$	$Q_2.j\,\!$	$Q_2.k\,\!$	$Q_2.\varepsilon$	$Q_2.\varepsilon i$	$Q_2.\varepsilon j$	$Q_2.\varepsilon k$
$Q_1.1\,\!$	1	i	j	k	$\varepsilon$	$\varepsilon i$	$\varepsilon j$	$\varepsilon k$
$Q_1.i\,\!$	i	-1	k	-j	$\varepsilon i$	$-\varepsilon$	$\varepsilon k$	$-\varepsilon j$
$Q_1.j\,\!$	j	-k	-1	i	$\varepsilon j$	$-\varepsilon k$	$-\varepsilon$	$\varepsilon i$
$Q_1.k\,\!$	k	j	-i	-1	$\varepsilon k$	$\varepsilon j$	$-\varepsilon i$	$-\varepsilon$
$Q_1.\varepsilon$	$\varepsilon$	$\varepsilon i$	$\varepsilon j$	$\varepsilon k$	0	0	0	0
$Q_1.\varepsilon i$	$\varepsilon i$	$-\varepsilon$	$\varepsilon k$	$-\varepsilon j$	0	0	0	0
$Q_1.\varepsilon j$	$\varepsilon j$	$-\varepsilon k$	$-\varepsilon$	$\varepsilon i$	0	0	0	0
$Q_1.\varepsilon k$	$\varepsilon k$	$\varepsilon j$	$-\varepsilon i$	$-\varepsilon$	0	0	0	0

[edit] Conjugate

A dual quaternion has three different definitions of conjugate, which can be expressed as follows. Where $q$ is the dual quaternion, $r$ is a the 'real' quaternion part, and $d$ is the dual part.

$q^\dagger = r^* + \varepsilon d^*$

$q_\varepsilon = r - \varepsilon d$

$q^\dagger_\varepsilon = r^* - \varepsilon d^*$

[edit] Inverse

Just as with normal quaternions, the inverse of a dual quaternion is defined as

$\hat Q^{-1} = {\hat Q^\dagger \over {\hat Q^2}}.$

[edit] Norm

The norm of a dual quaternion can be written as

$\|\hat Q\| = \sqrt{\hat Q\hat Q^\dagger} = \sqrt{\hat Q^\dagger \hat Q}.$

[edit] Dual quaternions for rigid motion representation

When using dual quaternions for rigid motion representation, a somewhat different definition is used. Specifically, the $\varepsilon$ dual unit is defined to anticommute with the quaternion elements i, j, and k. This gives the following multiplication table:

$Q_1 * Q_2\,\!$	$Q_2.1\,\!$	$Q_2.i\,\!$	$Q_2.j\,\!$	$Q_2.k\,\!$	$Q_2.\varepsilon$	$Q_2.\varepsilon i$	$Q_2.\varepsilon j$	$Q_2.\varepsilon k$
$Q_1.1\,\!$	1	i	j	k	$\varepsilon$	$\varepsilon i$	$\varepsilon j$	$\varepsilon k$
$Q_1.i\,\!$	i	-1	k	-j	$\varepsilon i$	$-\varepsilon$	$-\varepsilon k$	$\varepsilon j$
$Q_1.j\,\!$	j	-k	-1	i	$\varepsilon j$	$\varepsilon k$	$-\varepsilon$	$-\varepsilon i$
$Q_1.k\,\!$	k	j	-i	-1	$\varepsilon k$	$-\varepsilon j$	$\varepsilon i$	$-\varepsilon$
$Q_1.\varepsilon$	$\varepsilon$	$-\varepsilon i$	$-\varepsilon j$	$-\varepsilon k$	0	0	0	0
$Q_1.\varepsilon i$	$\varepsilon i$	$\varepsilon$	$-\varepsilon k$	$\varepsilon j$	0	0	0	0
$Q_1.\varepsilon j$	$\varepsilon j$	$\varepsilon k$	$\varepsilon$	$-\varepsilon i$	0	0	0	0
$Q_1.\varepsilon k$	$\varepsilon k$	$-\varepsilon j$	$\varepsilon i$	$\varepsilon$	0	0	0	0

The motions described by dual quaternions of this form, are described in detail under screw theory.

[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.
Martin Baker www.euclidianspace.com

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Dual quaternion

From Wikipedia, the free encyclopedia

Contents

[edit] Formulas

[edit] Addition/Subtraction

[edit] Multiplication

[edit] Conjugate

[edit] Inverse

[edit] Norm

[edit] Dual quaternions for rigid motion representation

[edit] Eponyms

[edit] References

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