# Combining rotations about different coordinate systems

I'm working on what I imagine is a common problem, but am struggling to find high quality resources. It is easiest to describe with an example:

Let's say I have a series of frames and coordinate system for a problem looking out the window of an airplane:

• Global cartesian coordinate system $$[x, y, z]_G$$.
• Local North-East-Down (NED) frame at some location in the sky $$[x, y, z]_{N}$$
• Body coordinate system of the airplane body $$[x, y, z]_P$$. This is described by Euler angle rotations (heading, pitch, roll) about the NED axes.
• Coordinate system of the view looking out the airplane window $$[x,y,z]_V$$. This is described by Euler angle rotations about the plane body axis.

I can easily compute the rotation matrix of each step ($$R_{NG},\ R_{PN},\ R_{VP}$$). But what I want is the rotation matrix $$R_{VG}$$ which describes the rotation of the view out the airplane window frame with respect to the global frame.

As I understand it, these are essentially intrinsic rotations. Instead of simply chaining the rotation matrices together, I need to 'project' each rotation matrix to a different coordinate system. e.g.:

$$R_{PG} = R_{NG}\ R_{PN}\ R_{NG}^{-1}$$ $$R_{VG} = R_{PG}\ R_{VP}\ R_{PG}^{-1}$$

I found this solution in the bottom of a Wikipedia article, and indeed seems to give me the correct answer compared to an example problem.

However, I'm concerned about my lack of understanding here, and lack of any other robust resource. My questions are:

1. Is this correct?
2. What is the technical name for this operation? A rotation projection?
3. What is the best reading (book, website etc.) to understand this better, including any limitations.

It seems like the $$R$$ matrices are actually orthogonal matrices? If so, computing their inverse is very simple, it's just a transposition.
It's also likely that the determinant of your matrices is $$1$$, it it was $$-1$$ then it is a reflexion (which would be quite surprising for your problem). If that's the case then the rotation matrices are part of the $$SO(3)$$ group.
I am not aware of the exact term for this "rotation" but in that case, the matrices will be "congruent", with the further restriction that the rotation matrices are in $$SO(3)$$.
• Thanks. Yes, you are correct - all $R$ are orthogonal with $det(R)=1$. I will look into literature on the $SO(3)$ group. Jul 15, 2023 at 0:42