How is the direction of the largest slope between two planes calculated?

Question

I have the following problem. I not only want to calculate the angle between two planes, I would also like to know what's the direction of this angle, basically the yaw angle. So I would set plane A to some angle, then calculate the angle between the direction of plane A and the direction of the highest slope between both planes. I know angle alpha in the image thats no problem

So far I have two planes, defined by their normal vector. Then I can calculate the cross product of these two, which give me the line of intersection. Then I take the normal vector of this line, project it to reference plane A and say plane A has a certain direction vector. Now I calculate the angle between the direction vector of plane A and the projected normal vector of the line of intersection. But then I still don't know if I am going up or down.

I would like to know the angle beta in this picture. Black is a reference vector for the first (reference plane A), which I can assign to this plane, red is the direction of the highest slope. How do I find this vector? Then I could calculate beta.

Maybe there is also another possibility. If I calculate roll and pitch angle (so set the normal component of the reference plane one time to zero for x -> roll angle between both plane, and one time to zero for y - pitch angle) separately, then I could from a combination of these, also calculate the yaw angle between two planes?

Does someone know the solution? I hope I have explained it clearly.

Thanks

Answer 1

If you have the two normal vectors, lets call them $u, v$

then $\frac {u\cdot v}{\|u\|\|v\|} = \cos \theta$

This Theta will be the angle between the normal vectors. But it will also be the angle that you are looking for for the intersection of the planes.