Quaternion multiplication

Quaternion definition ：
$$\mathbf{q}=q_w+q_{x}i+q_{y}j+q_{z}k=\left[\begin{array}{c}{q}_w\\ q_x\\ q_y\\ q_z\end{array}\right]$$
stay Hamilton In representation ：
$$\begin{gathered} i^2=j^2=k^2=ijk=-1 \\ ij=-ji=k,jk=-kj=i,ki=-ik=j \end{gathered}$$
stay JPL In representation ,$-ij=ji=k$,Eigen3 Is used in Hamilton notation , So the following formulas are used Hamilton notation .

Multiply quaternions ：
$$\mathbf{p}\otimes \mathbf{q}=\left[\begin{array}{c}{p}_w{q}_w-p_x{q}_x-p_y{q}_y-p_z{q}_z\\ p_w{q}_x+p_x{q}_w+p_y{q}_z-p_z{q}_y\\ p_w{q}_y-p_x{q}_z+p_y{q}_w+p_z{q}_x\\ p_w{q}_z+p_x{q}_y-p_y{q}_x+p_z{q}_w\end{array}\right]$$

Four yuan number $\mathbf{q}$ For vectors $\mathbf{v}$ Rotate , First, write the vector in quaternion form （ The real part is 0）：
$$\mathbf{p}=[0,x,y,z{]}^T=[0,\mathbf{v}{]}^T$$
And then calculate ：
$${\mathbf{p}}^{\prime }={\mathbf{q}\mathbf{p}\mathbf{q}}^{-1}$$
Can verify ${\mathbf{p}}^{\prime }$ So is the real part 0, Take the imaginary part as the vector after rotation ${\mathbf{v}}^{\prime }$:
$${\mathbf{v}}^{\prime }=\left[\begin{array}{c}(q_w^2+q_x^2-q_y^2-q_z^2)x+2(q_x{q}_y-q_w{q}_z)y+2(q_w{q}_y+q_x{q}_z)z\\ 2(q_w{q}_z+q_x{q}_y)x+(q_w^2-q_x^2+q_y^2-q_z^2)y+2(q_y{q}_z-q_w{q}_x)z\\ 2(q_x{q}_z-q_w{q}_y)x+2(q_w{q}_x+q_y{q}_z)y+(q_w^2-q_x^2-q_y^2+q_z^2)z\end{array}\right]$$