Dual Quaternion - Formulas

Formulas

In order to describe operations with dual quaternions, it is helpful to first consider quaternions.

A quaternion is a linear combinations 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, that is A = a₀ + A, where a₀ is a real number and A = A₁ i + A₂ j + A₃ k is a three dimensional vector. The vector dot and cross operations can now be used to define the quaternion product of A = a₀ + A and C = c₀ + C as

A dual quaternion is usually described as a quaternion with dual numbers as coefficients. A dual number is an ordered pair â = ( a, b ). Two dual numbers add componentwise and multiply by the rule â ĉ = ( a, b ) ( c, d ) = (a c, a d + b c). Dual numbers are often written in the form â = a + εb, where ε is the dual unit that commutes with i, j, k and has the property ε2 = 0.

The result is that a dual quaternion is the ordered pair of quaternions  = ( A, B ). 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, Â = â₀ + A, where â₀ = ( a, b ) and A = ( A, B ) is the dual vector that defines a screw. This notation allows us to write the product of two dual quaternions as