We are given the three-dimensional cartesian coordinates of a point $A$, a point $B$ and a point $C$. The distance from $A$ to $B$ is the same as the distance from $A$ to $C$ ($|\vec{AB}| = |\vec{AC}|)$.
The goal is to compute the Euler angle triplet ($\phi$, $\theta$, $\psi$) which rotates the vector $\vec{AB}$ such that it aligns with $\vec{AC}$. These Euler angles describe a rotation around the $z$-axis followed by the $y$-axis and then finally the $x$-axis.
One approach would be to compute a rotation matrix $R$ that encompasses this rotation, and then compute the Euler angles by using $R$. This last step is described here: http://eecs.qmul.ac.uk/~gslabaugh/publications/euler.pdf and here https://web.mit.edu/2.05/www/Handout/HO2.PDF.
Then the question remains: how we do compute this general rotation matrix $R$ by only using the cartesian coordinates of $A$, $B$ and $C$?
If we had the Euler angles, we could compute $R = R_z(\phi)R_y(\theta)R_x(\psi)$.
The question is similar to this one: Reaching a point B in Cartesian coordinate via Euler angles knows its initial point A Euler angles and Cartesian coordinates. However, I do not understand how the answer of the question leads to the general rotation matrix $R$. Instead, it describes how one can compute rotation matrices which map the axis unit vectors $\hat{x}$, $\hat{y}$, $\hat{z}$ onto the normalised vector $\hat{AB}$.
Any help will be greatly appreciated!
Best Answer
Project $\vec{AB}$ onto the $x,y$ plane. Find the angle the projected vector makes with the $x$ axis.
Construct a vector in the $x,y$ plane of the same length as the projected vector, but whose $y$ component is the same as the $y$ component of $\vec{AC}.$ For example, if the length of the projected vector is $r_1$ and the $y$ component of $\vec{AC}$ is $y_2,$ then a vector of length $r_1$ at angle $\arcsin(y_2/r_1)$ with the $x$ axis will do.
The $z$ rotation angle will be the difference between the angles these two vectors make with the $x$ axis. You will need to pay attention to the signs of angles in order to find a rotation angle of the correct magnitude and direction.
After performing the $z$ rotation on $\vec{AB},$ you will have a vector $v$ with the same $y$ coordinate as $\vec{AC}.$ Project both vectors onto the $x,z$ plane, measure their angles with the $x$ axis, and determine the rotation angle around the $y$ axis that rotates $v$ onto $\vec{AC}.$
You do not need a rotation around the $x$ axis.