Calculate Euler yaw pitch and roll for a 3d point to face another

This may be a really basic question but the typical atan2 method isn't working when I write it in a program.

I need a camera object to face a certain target in 3D space.
I need to calculate the 3 Euler angles (yaw, pitch, roll) for the camera. How can I do this?

• The solution will not be unique since you can always rotate the camera on its axis and still have it pointing at a target. Like choosing portrait vs. landscape in a real camera. – Jyotirmoy Bhattacharya Oct 13 '10 at 23:36
• You're right, but at least the X and Y can be calculated to face the point, if not a proper Z (roll). – Robinicks Oct 14 '10 at 5:06
• Can you explain exactly how your use of two-argument arctangent is failing? It seems to me that any correction to your algorithm would involve the addition or subtraction of some multiple of $\pi/2$. – J. M. is a poor mathematician Oct 20 '10 at 11:07

From looking at your diagram, it appears to me that the transformation takes the unit vector $(i_x,i_y,i_z)$ and rotates it to point to $(f_x,f_y,f_z)$:
$$\begin{pmatrix}c_\Psi&s_\Psi&0\\-s_\Psi&c_\Psi&0\\0&0&1\end{pmatrix} \begin{pmatrix}1&0&0\\0&c_\theta&-s_\theta\\0&s_\theta&c_\theta\end{pmatrix} \begin{pmatrix}c_\phi&-s_\phi&0\\s_\phi&c_\phi&0\\0&0&1\end{pmatrix} \begin{pmatrix}i_x\\i_y\\i_z\end{pmatrix} = \begin{pmatrix}f_x\\f_y\\f_z\end{pmatrix}$$
where $c_\phi = \cos(\phi), s_\phi = \sin(\phi)$, etc.
It isn't hard to solve for $\phi,\theta$ and $\Psi$ if you assume that your camera begins by pointing in a convenient direction such as $(i_x,i_y,i_z) = (0,0,1)$.