[GWM] Quaternions are a generalised complex number which have had a lot of use in scientific analysis of 3 dimensional space, navigation and guidance systems for rockets etc. Invented (or discovered?) by Sir W Hamilton on his way to a meeting, he stopped at the side of a bridge in Dublin and scratched the equation into the stone.  The bridge is now known as Hamilton's Bridge or Brougham Bridge.

Quaternions can be thought of as rotations, and consist of 4 real components. http://mathworld.wolfram.com/Quaternion.html

For comparison with Complex Numbers, they can be written:

 q = q0 + q1.i + q2.j + q3.k

where i,j,k are all square roots of -1 AND all different.  i,j,k obey the following rules:

 i*i = j*j = k*k = ijk = -1
 i*j = k
 j*k = i
 k*i = j

Using these rules 2 quaternions can be multiplied to make a new quaternion which obeys the same rules.  Thus the product of 2 quaternions results in another quaternion - it is a closed system of 'numbers'.  If we want to visualise a quaternion, consider i,j,k to be the X,Y,Z axes of space and the q1,q2,q3 is a position in space (or a vector) and q0 a magnitude associated with that position.

Note that the products of i with j, j with k etc are similar to the right hand rule for cross product of 2 vectors. Thus if a quaternion relates in some way to a physical field, then the product of quaternions can represent such effects as gyroscopic precession, spin precession in atoms etc.

An important class of quaternions is a normalised or unit quaternion (one where q0^2+q1^2+q2^2+q3^2 equals 1); the product of 2 unit quaternions is still a unit quaternion.  If the q0 parameter is regarded as a rotation about the vector (q1,q2,q3) the the product quaternion is equivalent to the total rotation produced by applying the rotation of the first quaternion to an object rotated by the second quaternion. Thus quaternions relate strongly to rotation matrices.  They have the advantage over [Euler Angles] that they do not have singularities and are a more linear rate of change when used to represent the attitude of a vehicle. (Imagine an aeroplane suddenly turning nose upward; as the nose goes through vertical the direction of flight of your aircraft changes from due North to due South in zero seconds, a problem for integrating Euler angles).

Non-unit quaternions can be used to express the distortion of time and space expressed in Einstein's Relativistic relations.

[Category Mathematics]