Overview

Quaternions, called hyper complex numbers, are generalization of complex numbers, and were originally conceived by R.W. Hamilton in 1843 [1]. A complex number is given by

where a and b are both real numbers and i is the imaginary number which is given by

As evident, a complex number is the sum of two elements one real and one imaginary. Complex numbers may be thought of as vectors in a two dimensional space with two element basis { e ₀ , e ₁}, where e ₀=1 and e ₁= i. As such, c in Eq. (B.1) can be represented by

A quaternion, made of four elements, is a generalization of a complex number. Similar to a complex vectors, they can be represented in a four dimensional space using four element basis { e ₀, e ₁, e ₂, e ₃} defined by

Hamilton s original description of this basis was given by [2]

From which we could deduce that

With the basis{ e ₀, e ₁, e ₂, e ₃} a quaternion Q is represented by

where ( q ₀, q ₁, q ₂, q ₃) are scalars. When we substitute for the elements of the basis from Eq. (B.4) in the above equation, the quaternion becomes

Even though Eq. (B.8) is the formal representation of a quaternion, we could present it as a four dimensional vector for computational purposes. For derivation...