# Geometric / Intuitive construction of the rotation axis of a 3D rotation matrix?

I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix.

To put the problem in more familiar terms, let's assume you have the canonical 3D orthonormal frame of reference (i,j,k) and another 3D orthornormal frame of reference (x,y,z). Then, the Matrix M whose rows are respectively the vectors x y and z is a rotation matrix.

I do know how to calculate the rotation axis of that matrix: it's the eigenvector for the eigenvalue 1 and can be found via solving the 3x3 linear system M.v = v for v.

However, solving that system gives me precious little geometric intuition about where that axis is and why. I would love to pick you guys collective brains to see if there is an intuitive, hopefully simple, construction of the rotation axis from the 6 vectors (i,j,k) and (x,y,z).

Edit:

We're talking about 3x3 rotations in R3 vector space here, not affine matrices.

• I have some experience with rotation matrices, so you have 2 coordinate systems and the rotation matrix that describes the rotation to go from $(x,y,z)$ to $(i,j,k)$. And what is it exactly what you need now? The rotation angles of the rotation? – Ruts Oct 30 '15 at 18:15
• @Ruts : No, I know how to derive both the axis (solving the system Mv=v) and the angle (the trace of the matrix is IIRC 1+2*cos(angle)). What I am looking for is an intuitive, geometric construction of the axis. That is: if I give you two coordinate systems in a 3D drawing, can you sort of figure out "with your hands" where the rotation axis of the matrix that transforms from one frame to the other is likely to be, – blondiepassesby Oct 30 '15 at 18:18
• My apologies if I still don't get you. So $(x,y,z)$ and $(i,j,k)$ don't have the same origin. So in order to create $(i,j,k)$ out of $(x,y,z)$ by a rotation you need to rotate $(x,y,z)$ from another coordinate system, let's say $(m,n,o)$. Is this what you mean? – Ruts Oct 30 '15 at 18:37
• If you want geometric intuition switch to quaternions for spatial rotations. – user137731 Oct 30 '15 at 18:49
• @Bye_World : quaternions don't help at all for this particular problem. Rotations become easy once you're in quaternion space, but this problem is basically how you get there in the first place : a quaternion is precisely the rotation axis with a norm more or less proportional to the angle. Intuiting the rotation axis from the matrix is what I'm after here. – blondiepassesby Oct 30 '15 at 19:20