Converting local rotation to global rotation.

Question

I have a rotation matrix in a local coordinate system and a 4x4 homogeneous matrix. I'm trying to convert the local rotation matrix to the global rotation matrix. I tried to find the dot product between both hoping that would work out but it doesn't. How can I find the rotation in the global coordinate system provided that I have a 4x4 matrix of the global coordinate system?

Answer 1

Apparently this means: (i) $x'_l=H_l\cdot x_l$ is a known matrix that converts in the local coordinate system, (ii) $x'_g=H_g\cdot x_g$ is an unknown matrix that converts in the global coordinate system, and (iii) $x'_g=H_{l\to g}\cdot x'_l$ is a known matrix which converts local to global coordinates. This means $H_{g\to l}=H^{-1}_{l\to g}$ is the known matrix that converts from global to local coordinates. The primes at the coordinates denote the images of the points after rotation/conversion. It follows immediately that $x'_g = H_{l\to g}\cdot H_l\cdot x_l = H_{l\to g}\cdot H_l\cdot H_{g\to l}x_g$. Therefore $H_g=H_{l\to g}H_lH_{g\to l}$ is a triple matrix product of known matrices.