Dual quaternion

In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions.

[1] 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.

[2] 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.

[7] Polynomials with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of mechanical linkages design.

[3] In 1898 Alexander McAulay used Ω with Ω2 = 0 to generate the dual quaternion algebra.

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

[17] A quaternion is a linear combination of the basis elements 1, i, j, and k. Hamilton's product rule for i, j, and k is often written as Compute i ( i j k ) = −j k = −i, to obtain j k = i, and ( i j k ) k = −i j = −k or i j = k. Now because j ( j k ) = j i = −k, we see that this product yields i j = −j i, which links quaternions to the properties of determinants.

A convenient way to work with the quaternion product is to write a quaternion as the sum of a scalar and a vector (strictly speaking a bivector), that is A = a0 + A, where a0 is a real number and A = A1 i + A2 j + A3 k is a three dimensional vector.

Two dual numbers add componentwise and multiply by the rule â ĉ = ( a, b ) ( c, d ) = (a c, a d + b c).

In particular, given and then Notice that there is no BD term, because the definition of dual numbers requires that ε2 = 0.

Dual quaternions of magnitude 1 are used to represent spatial Euclidean displacements.

It happens to be the unique maximal ideal of the ring of dual numbers.

The dual numbers form a local ring since there is a unique maximal ideal.

Dual quaternions have been used to exhibit transformations in the Euclidean group.

A benefit of the dual quaternion formulation of the composition of two spatial displacements DB = ([RB], b) and DA = ([RA], a) is that the resulting dual quaternion yields directly the screw axis and dual angle of the composite displacement DC = DBDA.

The screw axis and dual angle of DC is obtained from the product of the dual quaternions of DA and DB, given by That is, the composite displacement DC=DBDA has the associated dual quaternion given by Expand this product in order to obtain Divide both sides of this equation by the identity to obtain This is Rodrigues' formula for the screw axis of a composite displacement defined in terms of the screw axes of the two displacements.

[18] The three screw axes A, B, and C form a spatial triangle and the dual angles at these vertices between the common normals that form the sides of this triangle are directly related to the dual angles of the three spatial displacements.

Assemble the components of Ĉ into the eight dimensional array Ĉ = (C1, C2, C3, c0, D1, D2, D3, d0), then ÂĈ is given by the 8x8 matrix product As we saw for quaternions, the product ÂĈ can be viewed as the operation of Ĉ on the coordinate vector Â, which means ÂĈ can also be formulated as, The dual quaternion of a displacement D = ([A], d) can be constructed from the quaternion S = cos(φ/2) + sin(φ/2)S that defines the rotation [A] and the vector quaternion constructed from the translation vector d, given by D = d1i + d2j + d3k.

Using this notation, the dual quaternion for the displacement D = ([A], d) is given by Let the Plücker coordinates of a line in the direction x through a point p in a moving body and its coordinates in the fixed frame which is in the direction X through the point P be given by, Then the dual quaternion of the displacement of this body transforms Plücker coordinates in the moving frame to Plücker coordinates in the fixed frame by the formula[clarification needed] Using the matrix form of the dual quaternion product this becomes, This calculation is easily managed using matrix operations.

It might be helpful, especially in rigid body motion, to represent unit dual quaternions as homogeneous matrices.

The displacement part can be written as The dual-quaternion equivalent of a 3D-vector is and its transformation by

is given by[19] These dual quaternions (or actually their transformations on 3D-vectors) can be represented by the homogeneous transformation matrix where the 3×3 orthogonal matrix is given by For the 3D-vector the transformation by T is given by Besides being the tensor product of two Clifford algebras, the quaternions and the dual numbers, the dual quaternions have two other formulations in terms of Clifford algebras.

First, dual quaternions are isomorphic to the Clifford algebra generated by 3 anticommuting elements

, then the relations defining the dual quaternions are implied by these and vice versa.

Second, the dual quaternions are isomorphic to the even part of the Clifford algebra generated by 4 anticommuting elements

Since both Eduard Study and William Kingdon Clifford used and wrote about dual quaternions, at times authors refer to dual quaternions as "Study biquaternions" or "Clifford biquaternions".

Read the article by Joe Rooney linked below for view of a supporter of W.K.

Since the claims of Clifford and Study are in contention, it is convenient to use the current designation dual quaternion to avoid conflict.