In the context of using unit quaternions to represent 3D rotations, I'm aware that the inverse is the conjugate.
So if I have a unit quaternion Q = {qw, qx, qy, qz} (where qw is the scalar part), the conjugate Q' = {qw, -qx, -qy, -qz}.
Can't I simply use Q' = {-qw, qx, qy, qz} as the conjugate instead? Isn't it identical?
Best Answer
If you think about quaternions and rotations, $$q=\left(\cos\frac\theta 2,\sin\frac\theta 2 e_x,\sin\frac\theta 2 e_y,\sin\frac\theta 2 e_z\right)$$ Here $(e_x,e_y,e_z)$ represents the rotation axis. An inverse rotation can be obtained by either switching $(e_x,e_y,e_z)$ to $(-e_x,-e_y,-e_z)$, or switching $\theta$ to $-\theta$. Both have the effect of $$q^*=q^{-1}=\left(\cos\frac\theta 2,-\sin\frac\theta 2 e_x,-\sin\frac\theta 2 e_y,-\sin\frac\theta 2 e_z\right)$$