Quaternions

The quaternions, devised by Hamilton (1843), generalize complex numbers.

H = {a 1 + b i + c j + d k | a,b,c,d ∈ R},   where i² = j² = k² = ijk = -1,

 

from which it follows that

 

*	1	i	j	k
1	1	i	j	k
i	i	-1	k	-j
j	j	-k	-1	i
k	k	j	-i	-1

(note, non-commutative)

and

(a+bi+cj+dk) (A+Bi+Cj+Dk) = (aA-bB-cC-dD) + (aB+bA+cD-dC)i + (aC-bD+cA+dB)j + (aD+bC-cB+dA)k.

 

Suppose  q = a + bi + cj + dk;  q may also be written (a, b, c, d);  define

conjugate( q ) = q^* = a - bi - cj - dk,

norm( q ) = ||q|| = √( q q^* ) = √( a² + b² + c² + d² ),

inverse( q ) = q^-1 = q^* / ||q||²,

 

e.g.,

 

Note that

(p q)^* = q^* p^*,

||p q|| = ||p|| ||q||,   and

(p q)^-1 = q^-1 p^-1.

Vectors

A 3-D vector, v, can be mapped to, and from, a quaternion with zero real-part:

V2Q(<v_x,v_y,v_z>) = (0,v_x,v_y,v_z).

Rotations

The rotation of a 3-D vector, v, by angle, θ, about the axis specified by the unit vector, n, is

 

rotation(θ, n) =

let q = cos θ/2 + (n_x sin θ/2) i + (n_y sin θ/2) j + (n_z sin θ/2) k

in function( v ) Q2V( q V2Q(v) q^-1 )

 

This gives three degrees of freedom when choosing a rotation: θ, and <n_x, n_y, n_z> subject to n_x² + n_y² + n_z² = 1.

 

e.g.,