I have 2 points in 3d space, p1 and p2, and need to rotate a long object (long on x-axis) along its 3 axis (first x, then y and finally z) for it to be aligned to the p1-p2 line. Knowing both p1 and p2 coordinates, and the object's centre and rotation axis being exactly between p1 and p2, what would be rx, ry and rz ?
The rotation axis is a world axis and does not rotate with the object. Rotation applies first rx then ry then rz.
Initial situation:
Desired situation:
Given the world space joint axes are the ones being rotated around, you are basically solving $$M_1M_2M_3 v_1 = v_2$$ where $v_1$ is the position of whatever before rotation and $v_2$ the position after rotation and:
$$v_1 = \begin{bmatrix}{v_1}_x\\{v_1}_y\\{v_1}_z\end{bmatrix} \text{ and } v_2 = \begin{bmatrix}{v_2}_x\\{v_2}_y\\{v_2}_z\end{bmatrix}$$ and the matrices for rotation: $$M_1 =\begin{bmatrix}a_1&b_1&0\\-b_1&a_1&0\\0&0&1\end{bmatrix},M_2 =\begin{bmatrix}a_2&0&b_2\\0&1&0\\-b_2&0&a_2\end{bmatrix},M_3 =\begin{bmatrix}1&0&0\\0&a_3&b_3\\0&-b_3&a_3\end{bmatrix}$$ and $${a_i}^2 + {b_i}^2 = 1, i \in\{1,2,3\}$$
I am kind of sure this does not have any unique solution.
say the ortsvektors for $$P_2 = [3,5,7]^T\\ P_1 = [1,2,3]^T$$ then the $v_2$ to aim for should be $$v_2 = P_2-P_1 = [3-1,5-2,7-3]^T = [2,3,4]^T$$