In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be 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 , where A and B are ordinary quaternions and ε is the dual unit, which satisfies and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.
In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. 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. Since unit quaternions are subject to two algebraic constraints, unit quaternions are standard to represent rigid transformations.
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. Polynomials with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of mechanical linkages design.
History
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.
Formulas
In order to describe operations with dual quaternions, it is helpful to first consider quaternions.
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 , to obtain , and or . Now because , we see that this product yields , 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 , where a0 is a real number and is a three dimensional vector. The vector dot and cross operations can now be used to define the quaternion product of and as
A dual quaternion is usually described as a quaternion with dual numbers as coefficients. A dual number is an ordered pair . Two dual numbers add componentwise and multiply by the rule . Dual numbers are often written in the form , where ε is the dual unit that commutes with i, j, k and has the property .
The result is that a dual quaternion can be written as an ordered pair of quaternions . Two dual quaternions add componentwise and multiply by the rule,
It is convenient to write a dual quaternion as the sum of a dual scalar and a dual vector, , where and is the dual vector that defines a screw. This notation allows us to write the product of two dual quaternions as
Addition
The addition of dual quaternions is defined componentwise so that given,
and
then
Multiplication
Multiplication of two dual quaternion follows from the multiplication rules for the quaternion units i, j, k and commutative multiplication by the dual unit ε. In particular, given
and
then
Notice that there is no BD term, because the definition of dual numbers requires that .
This gives us the multiplication table (note the multiplication order is row times column):
Conjugate
The conjugate of a dual quaternion is the extension of the conjugate of a quaternion, that is
As with quaternions, the conjugate of the product of dual quaternions, , is the product of their conjugates in reverse order,