Derivation of Quaternion Formulae
02 Mar 2012, 19:16 UTC
To show that the inverse of a unit quaternion (r,V) is (r,-V) we do:
(r,V) * (r,-V) gives
r' = r*r + V.V = 1
V' = -r.V + r.V + V^V = (0,0,0)
To rotate a vector X by a quaternion we work as follows:
position quat = (0,X)
resulting position quat = (r,V) * (0,X) * (r,-V)
= (0,X')
Working through using the multiplication formula:
(r',X') = (-V.X,rX + V^X)*(r,-V)
r' = -rV.X - (rX + V^X).-V
X' = -V.X(-V) + r(rX + V^X) + (rX + V^X)^(-V)
r' = -rV.X + rX.V = 0
X' = rrX + (V.X)V + r(V^X) + V^(rX + V^X)
We assume the following vector identity:
V^(V^X) = V(V.X) - X(V.V)
This gives us:
X' = rrX + (V.X)V + r(V^X) + rV^X + (V.X)V - (V.V)X
= (rr - V.V)X + 2(V.X)V + 2r(V^X)
Therefore, vector X rotated by quaternion (r,V) is given as follows:
X' = (r.r - V.V)X + 2(V.X)V + 2r(V^X)