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Math Help - Transformation help

  1. #1
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    Transformation help

    The question:
    Show that the order in which transformations is important by first sketching the result when the triangle A(1,0) B(1,1) C(0,1) is rotated by 45 degrees about the origin and then translated by (1,0), and then sketch the result when it is first translated then rotated.

    I'm not sure how to determine its position after rotation. Do I have to use a transformation matrix? If so, how would I apply it to this question? Thanks.
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  2. #2
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    I presume you are doing a positive rotation which is anticlockwise. You can measure 45 degrees with a pertractor and then draw a faint line. Then a compass can be used to find the new coordinates.
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  3. #3
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    Ok, thanks. So there's no need to do it accurately with algebra?
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  4. #4
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    You need to create a matrix to multiply. Use the matrix where the top row is cosx -sinx and the bottom row is sinx cosx.
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  5. #5
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    Well, we can't say what your teacher wants!

    A rotation about the origin of \theta degrees corresponds to a matrix multiplication with matrix \begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}. That is the same as saying x'= x cos(\theta)- y sin(\theta), y'= x sin(\theta)+ y'cos(\theta)

    Here, \theta= 45 so x'= x\frac{\sqrt{2}}{2}- y\frac{\sqrt{2}}{2}, y'= x\frac{\sqrt{2}}{2}+ y\frac{\sqrt{2}}{2}

    That is, (1, 0) is mapped into \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right), (1, 1) is mapped into \left( 0, \sqrt{2}\right) and (0, 1) is mapped into \left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right).

    You should be able to see how that matches your graph. Now, "translate by (1, 0)" by adding 1 to each x coordinate.

    It should be easy to see why moving the object away from the origin before rotating is different from rotating and then moving the object.
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  6. #6
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    Ahh, thank you. So basically I apply the rotation matrix on each of the co-ordinates?
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  7. #7
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    Quote Originally Posted by Glitch View Post
    The question:
    Show that the order in which transformations is important by first sketching the result when the triangle A(1,0) B(1,1) C(0,1) is rotated by 45 degrees about the origin and then translated by (1,0), and then sketch the result when it is first translated then rotated.

    I'm not sure how to determine its position after rotation. Do I have to use a transformation matrix? If so, how would I apply it to this question? Thanks.
    I've attached a sketch:

    1. The thick outlined triangle is the original.

    2. In red: First rotation, then translation.

    3. In blue: First translation, then rotation.
    Attached Thumbnails Attached Thumbnails Transformation help-rotundtranslat.png  
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  8. #8
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    Thanks earboth, that looks like what I got using the protractor/compass method.
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