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Math Help - Rotation Transformation

  1. #1
    Senior Member bugatti79's Avatar
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    Rotation Transformation

    Folks,

    I am struggling to understand how the attached expressions were obtained for the simple rotation of 2 axes normal to each other by an angle \theta

    This is different to what is shown the wiki link Rotation matrix - Wikipedia, the free encyclopedia (See in 'two dimensions')

    Ie, x'= x\cos \theta - y \sin \theta and y'=x \sin \theta + y \cos \theta


    Any clues?

    Regards
    bugatti
    Attached Thumbnails Attached Thumbnails Rotation Transformation-imag0126.jpg  
    Last edited by bugatti79; October 16th 2012 at 11:29 AM.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Re: Rotation Transformation

    Quote Originally Posted by bugatti79 View Post
    Folks,

    I am struggling to understand how the attached expressions were obtained for the simple rotation of 2 axes normal to each other by an angle \theta

    This is different to what is shown the wiki link Rotation matrix - Wikipedia, the free encyclopedia (See in 'two dimensions')

    Ie, x'= x\cos \theta - y \sin \theta and y'=x \sin \theta + y \cos \theta


    Any clues?

    Regards
    bugatti
    Here is a picture

    Rotation Transformation-capture.png

    Note that

    T(e_1)=\cos(\theta)e_1+\sin(\theta)e_1

    and

    T(e_2)=-\sin(\theta)e_1+\cos(\theta)e_2

    So the general transformation is

    A=\begin{bmatrix}\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}

    Now for any arbitary x and y you get

    A \begin{bmatrix} x \\ y \end{bmatrix} =\begin{bmatrix}\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

    \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \cos(\theta)-y \sin(\theta) \\ x \sin(\theta)+y \cos(\theta) \end{bmatrix}
    Thanks from bugatti79
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  3. #3
    Senior Member bugatti79's Avatar
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    Re: Rotation Transformation

    Thank you sir, just what I was looking for.

    Regards
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  4. #4
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    Re: Rotation Transformation

    I wrote this out for someone last month, if you want to take a look:

    Understanding how to express basis with trig functions
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