Complex Numbers

Stereographic Projection

Quaternion Number System

Quaternions are an extension of the complex number system and are defined to be the quotient of two directed lines in a three-dimensional space, or rather, the quotient of two vectors. Multiplication of a quaternion is non-communicative. A quaternions are generally represented in the form:

$$ a + b\ \textbf{i} + c\ \textbf{j} + d\ \textbf{k} $$

Where $a, b, c,\ and\ d\ \in \reals$; and $i, j,\ and\ k$ are basic complex numbers (a.k.a, imaginary numbers).

These imaginary numbers are what allow for the projection of three dimensional rotation from this forth imaginary dimension and are refered as the vector component while the leading real is refered to as the scaler component.

Ther following must be true about the vector compontent

$$ i^2 = j^2 = k^2 = -1 \\ ij = -ji = k \\ ki = -ik = j \\ jk = -kj = i $$

These equations state that the values of the complex numbers will distribute via associativity and anticommutativity in order to allow for quaternion multiplication.

Application

Quaternions are used in pure mathematics but also in applied mathematics where much of their usefulness resides as quaternions can be used in calculations involving three-dimensional rotation in fields such as computer graphics and computer vision . Quaternions can also be used along with other methods of rotation such as Euler angles and rotation matrices or as an alternative to them entirely, depending on the desired application of the rotation.