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Thread: Plane Transformation

  1. #1
    Junior Member hercules's Avatar
    Sep 2007

    Plane Transformation

    Let \theta be an angle, and let m be the straight line through the origin with inclination \theta to the positive x-axis. Show that the plane transformation Q = f(P) that maps the point P(x,y) to the point Q(x',y') where

    x' = x \cos 2 \theta + y \sin 2 \theta
    y' = x \sin 2 \theta - y \cos 2 \theta

    is in fact the reflection \rho_m

    Note that \rho_m is the reflection through line m

    Thanks guys
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  2. #2
    Senior Member
    Dec 2007
    I really can't remember how I used to do these questions, but here is the way I would do it now. It is likely that I have forgotten a much simpler method.

    Before the transformation, the polar coordinates of a point are [r, \alpha].

    Now we rotate the entire graph clockwise by \theta so that the line we are reflecting over becomes the x axis. [r, \alpha] \to [r, \alpha-\theta].

    To reflect over the x axis, we simply multiply the angle by -1. [r, \alpha - \theta] \to [r, \theta - \alpha].

    Now rotate anticlockwise by \theta to restore the original orientation and the transformation is complete. [r, \theta - \alpha] \to [r, 2 \theta - \alpha]

    so x' = r \cos (2\theta - \alpha)
    <br />
 = r (\cos (2 \theta) \cos (\alpha) + \sin (2 \theta) \sin (\alpha))
    <br />
= r \cos (\alpha) \cos (2 \theta) + r \sin (\alpha) \sin (2 \theta)
    <br />
= x \cos (2 \theta) + y \sin (2 \theta)

    y' can be expanded in the same manner.
    Last edited by badgerigar; Mar 5th 2008 at 05:53 PM. Reason: fixed latex error
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