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Math Help - Combine Scaling and Translation Matrices

  1. #1
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    Joined
    Aug 2010
    Posts
    1

    Combine Scaling and Translation Matrices

    Hello all,

    I am revising for a maths resit paper taking place next week and could really do with some help regarding this past paper question.

    Derive a single matrix to undertake the following collective 2D transformations. A Scaling by a factor of 2 in both X and Y directions followed by a translation by 2 in the X direction and 3 in the Y direction.

    Any advice regarding this would be much appreciated as I am pulling out what's left of my hair trying to figure it out.

    I think I need to do this first:

    Scaling by 2 in both X and Y directions =

    X' = Sx * X Y' = Sy * Y

    Translation by 2 in X and 3 in Y directions would follow as =

    X' = X + tx Y' = Y + ty

    Scaling Matrix would be:

    |X'| |2 0| |X|
    |Y'|=|0 2|* |Y|

    Translation Matrix would be:

    |X'| |2 0| |X|
    |Y'|=|0 3|* |Y|

    And now to combine the two matrices in some way?

    Once again, ANY help would be great.

    Cheers
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  2. #2
    Newbie
    Joined
    Aug 2010
    Posts
    9

    Math.

    Your scaling matrix looks good, but your "translation" matrix is actually still a scaling matrix. To translate \begin{pmatrix}<br />
   x \\<br />
   y <br />
\end{pmatrix}<br />
by
    \begin{pmatrix}<br />
   2 \\<br />
   3 <br />
\end{pmatrix}<br />
, simply add the vectors:
    \begin{pmatrix}<br /> <br />
   x \\<br />
   y <br />
\end{pmatrix}<br />
 + \begin{pmatrix}<br /> <br />
   2 \\<br />
   3 <br />
\end{pmatrix} = <br />
\begin{pmatrix}<br /> <br />
   x + 2\\<br />
   y + 3<br />
\end{pmatrix}<br />

    Alright, so given some (x,y) pair, you're going to scale x by 2 and then translate by 2. So
    x' = 2x + 2
    and likewise,
    y' = 2y + 3
    So we know our solution should look something like this, but we don't have the answer in matrix form.

    You're right that we need to combine the two operations--to accomplish this, just do one after the other. First scale, then translate.

    \begin{pmatrix}<br /> <br />
    2 & 0\\<br />
    0 & 2<br />
 \end{pmatrix} \begin{pmatrix}<br /> <br />
   x \\<br />
   y <br />
\end{pmatrix}<br />
 + \begin{pmatrix}<br /> <br />
   2 \\<br />
   3 <br />
\end{pmatrix} = <br />
\begin{pmatrix}<br />
    2x \\<br />
    2y <br />
 \end{pmatrix}<br />
  + \begin{pmatrix}<br /> <br />
    2 \\<br />
    3 <br />
 \end{pmatrix} = \begin{pmatrix}<br />
     2x + 2\\<br />
     2y + 3<br />
  \end{pmatrix}<br />

    And look, this is the same as the equations for x' and y' given above. Hopefully this helps.
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