My apologies in advance: I am not very proficient in the field of mathematics. If my question seems discombobulated, it probably is, and I would greatly appreciate your help cleaning it up! Alright, onward...
I have a few coordinate systems to coordinate. I'm using a camera that detects markers in space and returns their location and attitude (in a vector and rotation matrix). I'm using a machine that moves in a room along standard Cartesian axes that are pre-determined in the room. The camera has its own coordinate system according to which it is detecting the markers.
I have a matrix, $R$ that can take a vector with respect to the room coordinate system and rotate it so that it is now expressed relative to the camera coordinate system.
I take record the position of the markers on the machine at an instant ($T_o$), and would like to report how much the machine has rotated ($D$) with respect to the room's coordinate system given the instantaneous positions of the markers ($T_i$).
yields the rotation in the camera's coordinate system.
seems to give the correct answer, but I don't understand it.
Please let me know the formula for $D$ or, if the formula above is correct, could you give a quick explanation for it?