# Difference between two rigid body transformations

Hello,

I would like to express the error between two rigid body transformations in terms of *translation* and *rotation*. What is the correct way to do this? I already know that you can determine the translational part (the norm of the 3x1 vector in the upper right corner) and rotational part (see angle theta here) of a single 4x4 matrix. But what kind of matrix should I use here?

Obviously, you have to consider some kind of "relative" transformation between the two original transformations. Most of the time, I see people using something like

or`Inverse[T1]*T2`

,`Inverse[T2]*T1`

which both lead to the same translation and angle. But what about the following?

or`T2*Inverse[T1]`

`T1*Inverse[T2]`

Can somebody explain the difference between these two possibilities (maybe even geometrically)? Is there any reference for this kind of topic?

Thanks in advance!