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Math Help - Transformation geometry

  1. #1
    Newbie pozmans's Avatar
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    Transformation geometry

    Hi everyone. I'm confused with determining the general rule when a figure has been rotated NOT at 90, 180, 270, 360 degrees.

    For example: In triangle ABC: A(4;4), B(4;1) C(2;0) has changed, C'(-1.5;1.5)... A'(-5,6;0) and B'(-3,6;3) NB - (not sure about A' and B' they approximations)

    So how would i determine the general rule. How do i go about this? I know you use the formula:

    (x;y) → (xCosθ - ySinθ; yCosθ + xSinθ)
    Can i get some step by step help please... thnx
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by pozmans View Post
    Hi everyone. I'm confused with determining the general rule when a figure has been rotated NOT at 90, 180, 270, 360 degrees.

    For example: In triangle ABC: A(4;4), B(4;1) C(2;0) has changed, C'(-1.5;1.5)... A'(-5,6;0) and B'(-3,6;3) NB - (not sure about A' and B' they approximations)

    So how would i determine the general rule. How do i go about this? I know you use the formula:

    (x;y) → (xCosθ - ySinθ; yCosθ + xSinθ)
    Can i get some step by step help please... thnx
    But the transformation from ABC to A'B'C' is not a rotation
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  3. #3
    Newbie pozmans's Avatar
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    What kind of transformation has occured then?
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  4. #4
    Moo
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    Well, I drew the points. And I drew the triangles.
    And I don't see any simple transformation from that...
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  5. #5
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    Quote Originally Posted by pozmans View Post
    What kind of transformation has occured then?
    To me, it looks approximately like a rotation around (0,0), with angle 135(= \frac{3\pi}{4} rad).

    Your formula (x;y) → (xCosθ - ySinθ; yCosθ + xSinθ) is correct.

    You can prove it in various ways. For instance, you may know that you can write x=r\cos\alpha and y=r\sin\alpha for some r>0 and \alpha\in\mathbb{R} (this is polar coordinates). Then (x,y) is rotated into (r\cos(\alpha+\theta),r\sin(\alpha+\theta)). If you expand the sin and cos using the usual formulas, you'll get the expected formula.
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