transferring coordinate systems

Hello,

So there is a 'world coordinate system' in normal 3D space 'xyz'.

If I apply a rigid-body transformation to this coordinate system I get a new coordinate system, (A).

By 'rigid-body transformation' I mean first applying a 3X3 matrix R_1 on any point in world coordinate system and then translating it by 3-vector T_1.

Now imagine I apply a different 'rigid-body transformation' on the same global coordinate system , denoted by rotation matrix R_2 and translation vector T_2, to get a new coordinate system (B).

Now my question is what Rotation/Translation will transform coordinate system (A) to (B)?

will it be the following?

Rotation = R_2 X (inverse of R_1)

Translation = T_2 - R_2 X (inverse of R_1) X T_1

There is this software I am using and it needs these rotation and translation matrix/vector. But above values don't work!

Thanks for your help.

Re: transferring coordinate systems

Assuming that R_2, and R_1 are linearly indepdent. If for any $\displaystyle y \in A $ $\displaystyle y = R_1x + T_1 $. Since R_1 is lin independent, we are assured some x(unique) exists in our global coordinate system. So $\displaystyle y - T_1 = R_1x $, again an inverse also exists for $\displaystyle R_1 $, call it $\displaystyle R^{-1}_1 $. So $\displaystyle R^{-1}_1(y - T_1) = R^{-1}_1 R_1 x = x $. So, x is in our global coordinate sys and we know how to get from our global coordinates to B, so $\displaystyle R_2(R^{-1}_1(y - T_1)) + T_2 = R_2x + T_2 $, so to get from A to B, we have $\displaystyle R_2*R^{-1}_1 y - R_2*R^{-1}_1T_2 $ will do the transformation