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Math Help - transferring coordinate systems

  1. #1
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    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.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: transferring coordinate systems

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