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Thread: Plane Transformation

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

    Plane Transformation

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

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

    is in fact the reflection $\displaystyle \rho_m$

    Note that $\displaystyle \rho_m$ is the reflection through line $\displaystyle m$

    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 $\displaystyle [r, \alpha]$.

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

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

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

    so $\displaystyle x' = r \cos (2\theta - \alpha)$
    = r (\cos (2 \theta) \cos (\alpha) + \sin (2 \theta) \sin (\alpha))$
    = r \cos (\alpha) \cos (2 \theta) + r \sin (\alpha) \sin (2 \theta)$
    = x \cos (2 \theta) + y \sin (2 \theta)$

    y' can be expanded in the same manner.
    Last edited by badgerigar; Mar 5th 2008 at 04:53 PM. Reason: fixed latex error
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