The quaternion representation of 3-D angular position

q = sin(a/2) n

where

n is the axis of rotation from center to the current position

a is the amplitude of the rotation

q is 3-D: torsional (out of screen), vertical & horizontal (in the plane of the screen)

The advantage of this representation is that all q's will lie on a plane (if Listing's law is obeyed).

That is Listing's law can simply be stated as q₁ = 0.