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Math Help - Developing transformation matrix

  1. #1
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    Developing transformation matrix

    I have a question with regards to develop a transformation matrix
    between 2 sets of data.

    set 1:
    A = 53.45 , 54.5
    B = 45 , 51.95
    C = 46.4 , 45.75


    Set 2
    A = -4.1041 , 1.0457
    B = 4.7046 , 0.812
    C = 5.0025 , -5.5006

    By using the transformation matrix derived using the 3 different coordinates found in set 1, i should be able to map point D from (55.9 , 48.25 ) to (-4.8006 , -5.6333) at a certainly of accuracy. The points A,B,C should be mapped from SET 1 to SET 2 exactly with point D transformed at a certain accuracy.

    the problem here is, i am not able to determine the sequence of transformations that took place i.e. translation or rotation or scaling etc.

    thank you so much if there is any kind soul to help me to give me some hints to slve it. thanks!
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  2. #2
    MHF Contributor

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    So you are looking for a matrix \begin{bmatrix}a & b \\ c & d\end{bmatrix} so that
    \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}53.45 \\ 54.5\end{bmatrix}= \begin{bmatrix}-4.1041 \\ 1.0457\end{bmatrix}

    \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}45 \\ 51.95\end{bmatrix}= \begin{bmatrix}4.7046 \\ 0.812 \end{bmatrix}
    and
    \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}46.4 \\ 45.75\end{bmatrix}= \begin{bmatrix}5.0025 \\ -5.5006\end{bmatrix}?

    You may not be able to do that. A linear transformation in two dimensions is determined by two points, not three. If you multiply those out, each matrix equation will give two numerical equations so you will have 6 equations in the four unknowns, a, b, c, and d. Again, there may not be a solution.
    Last edited by HallsofIvy; February 16th 2011 at 05:59 AM.
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  3. #3
    A Plied Mathematician
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    Again, there may not be a solution.
    There isn't. I already checked. However, if you increase the dimensions of the problem, you might be able to get the result. If you look here, that's precisely what they do for 3 dimensions: increase the size of the operator matrix to 4 x 4, and include a "1" as the last component for the vectors. So maybe that's what's required here: to do translations and rotations and reflections, etc., you might need an extra dimension.
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