TITLE: Projecting an angle from one plane to another plane.
QUESTION [5 upvotes]: Consider I have two intersecting planes with an angle ($\theta$). I have two intersecting vectors ($\vec a$ and $\vec b$) on one of the planes that make an angle ($\gamma$). If I project these two vectors onto the other plane, what the projected angle ($\alpha$) between the projected vectors ($\vec a \prime$ and $\vec b \prime$) would be (as a function of the other angles)?
To make it simple, consider: 

Vectors intersect on the hinge of the two planes.
Vectors are symmetric to the perpendicular line to the hinge.

Regards,

REPLY [8 votes]: This is how I understand the problem in a Cartesian coordinate system:

And this is how I solved it (note that two angles are renamed for this via θ → α, α → β:

The following equations are valid:
\begin{align}
R_x = R_{xy} \cos \alpha =  R_{xz} \cos \beta \\
R_y = R_{xy} \sin \alpha \\
R_z = R_{xz} \sin \beta =  R \sin \gamma \\
R_{xy} = R \cos \gamma
\end{align}
Putting them all together results in:
\begin{align}
\tan \gamma = \cos \alpha * \tan \beta
\end{align}
So in the notation of the original question (with the angles restated as in the OP) the solution is:
\begin{align}
\tan \gamma = \cos \theta * \tan \alpha
\end{align}
I had a similar problem and thought it might be useful for others, as the equation looks way less daunting and is easily solved for any of the 3 angles.