Quaternions and spatial rotation

See quaternions and, if necessary, complex number.

I Introduction Recall the geometric version of the product of two quaternions, q = (a, u) and q' = (a', u'), where a and a' are the real parts, u and u' are the imaginary parts, also seen as vectors of the three dimensional space R³: q.q' = (aa' - u.u', au' + a'u + u×u'). u.u' designates the scalar product, and u×u' the vector product. We will denote by q^* the quaternion conjugate of q: q^* = (a,-u).