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Math Help - describe the matrix transformation

  1. #1
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    Cool describe the matrix transformation

    Hi folks,

    SO, here is the problem:
    A reflection in the line y = x - 1 is followed by an anticlockwise rotation of  90^o about the point (-1,1). Show that the resultant transformation has an invariant line, and give the equation of this line. Describe the resultant transformation in relation to this line.

    For the first transformation I get :

    \overline{x'} = \left( \begin{array}{cc} 1 & 2 \\ 0 & -1 \end{array} \right). \overline{x}

    For the second the rotation about the origin O is:

    \overline{x'} = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0  \end{array} \right). \overline{x}

    and about (-1,1) is:

    \overline{x'} = <br />
\left( \begin{array}{cc} 0 & -1 \\ 1 & 0  \end{array} \right)<br />
\left( \begin{array}{c} x \\ y \end{array} \right) + <br />
\left( \begin{array}{c} 0 \\ 2 \end{array} \right)

    The resultant of a process P that transforms x to x' (i.e. x' = Px) and is followed by a process Q that transforms x' to x'' (i.e. x'' = Qx') is the resultant process x'' = QPx. So:

    \overline{x''} = <br />
\left( \begin{array}{cc} 0 & -1 \\ 1 & 0  \end{array} \right)<br />
\left( \begin{array}{cc} 1 & 2 \\ 0 & -1  \end{array} \right)<br />
\left( \begin{array}{c} x \\ y \end{array} \right) + <br />
\left( \begin{array}{c} 0 \\ 2 \end{array} \right)

    \overline{x''} = <br />
\left( \begin{array}{cc} 0 & 1 \\ 1 & 2  \end{array} \right)<br />
\left( \begin{array}{c} x \\ y \end{array} \right) + <br />
\left( \begin{array}{c} 0 \\ 2 \end{array} \right)

    The result of this transformation on the unit square OABC is as follows:
    O(0,0) goes to O(0,2)
    A(1,0) A'(0,3)
    B(0,2) B'(1,4)
    C(1,1) C'(1,5)

    I have attached a rough sketch. It is not what I was hoping for. The answer is that the transform is a glide reflection in the line x = 1/2 with ((1/2,0) going to (1/2, 3). My transformation doesn't even preserve a closed shape. If you ignore the sequence of the letters OABC it looks like a shear along x = 1/2. Can anyone see what I am doing wrong?
    Attached Thumbnails Attached Thumbnails describe the matrix transformation-m6e16.jpg  
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  2. #2
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    Grandad's Avatar
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    Hello s_ingram
    Quote Originally Posted by s_ingram View Post
    Hi folks,

    SO, here is the problem:
    A reflection in the line y = x - 1 is followed by an anticlockwise rotation of  90^o about the point (-1,1). Show that the resultant transformation has an invariant line, and give the equation of this line. Describe the resultant transformation in relation to this line.

    For the first transformation I get :

    \overline{x'} = \left( \begin{array}{cc} 1 & 2 \\ 0 & -1 \end{array} \right). \overline{x}

    For the second the rotation about the origin O is:

    \overline{x'} = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0  \end{array} \right). \overline{x}

    and about (-1,1) is:

    \overline{x'} = <br />
\left( \begin{array}{cc} 0 & -1 \\ 1 & 0  \end{array} \right)<br />
\left( \begin{array}{c} x \\ y \end{array} \right) + <br />
\left( \begin{array}{c} 0 \\ 2 \end{array} \right)

    The resultant of a process P that transforms x to x' (i.e. x' = Px) and is followed by a process Q that transforms x' to x'' (i.e. x'' = Qx') is the resultant process x'' = QPx. So:

    \overline{x''} = <br />
\left( \begin{array}{cc} 0 & -1 \\ 1 & 0  \end{array} \right)<br />
\left( \begin{array}{cc} 1 & 2 \\ 0 & -1  \end{array} \right)<br />
\left( \begin{array}{c} x \\ y \end{array} \right) + <br />
\left( \begin{array}{c} 0 \\ 2 \end{array} \right)

    \overline{x''} = <br />
\left( \begin{array}{cc} 0 & 1 \\ 1 & 2  \end{array} \right)<br />
\left( \begin{array}{c} x \\ y \end{array} \right) + <br />
\left( \begin{array}{c} 0 \\ 2 \end{array} \right)

    The result of this transformation on the unit square OABC is as follows:
    O(0,0) goes to O(0,2)
    A(1,0) A'(0,3)
    B(0,2) B'(1,4)
    C(1,1) C'(1,5)

    I have attached a rough sketch. It is not what I was hoping for. The answer is that the transform is a glide reflection in the line x = 1/2 with ((1/2,0) going to (1/2, 3). My transformation doesn't even preserve a closed shape. If you ignore the sequence of the letters OABC it looks like a shear along x = 1/2. Can anyone see what I am doing wrong?
    Thanks for showing us your full working. I think it's pretty well correct, except for the first transformation matrix. This doesn't represent the reflection, does it? Don't forget that a transformation represented by a single 2 x 2 matrix will always leave the origin fixed; and you want (0, 0) \to (1, -1).

    You have to reflect first in a parallel line through the origin (that's y = x) and then translate so that the origin moves to the required image. So that's:
    \overline{x'} = \left( \begin{array}{cc} 0 & 1 \\ 1 &   0\end{array} \right). \overline{x}+\left( \begin{array}{c} 1 \\ -1 \end{array} \right)
    Do you want to try again?

    Grandad
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  3. #3
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    May 2009
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    Hi Grandad,

    Thanks very much Grandad! Yes, I was worried about the very point you have made - how to represent a reflection about a line that doesn't go through the origin. I tried it again, using your suggestion and it comes out perfectly! Great!
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