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Thread: Compostion of a reflection and a translation

  1. #1
    Feb 2008

    Compostion of a reflection and a translation

    I am revising for an exam and working through exercise questions for which I have been given the solutions - however there is one solution I really don't understand!

    We are given S a rotation about the origin anti-clockwise by theta and T a translation by b = (b1,b2).
    Want to find the centre of rotation T(S(x)) and the angle and prove our answer.

    The solution says that T(S(x))=Ax+b =A(x-u)+u where
    u= (1/2(b1) - 1/2(b2) cot(theta/2) , 1/2(b1) cot(theta/2) + 1/2(b2) )
    Apparently in order to find u I only need to solve T(x) = x and repeatedly use the identity (1-cos(theta))/sin(theta) = tan(theta/2).

    As far as I can see, T(x)=x only occurs when b=0 as otherwise the point would be translated somewhere else and T(x) would not equal x. I really have no idea how to use this or how they got u!

    Any help would be very much appreciated - this is an old assignment not current assessed work and I am merely using it to help me revise.

    Thanks very much,
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  2. #2
    Member alunw's Avatar
    May 2009

    Typo in question?

    The question must have meant you to solve T(S(x)) = x which should be simple enough.
    It T is a translation then T(x) <> x for any x.
    A rotation has one fixed point (or 2 if you count infinity as a point), so if you find the unique finite point where T(S(x)) you have found the centre of the rotation T(S)). Since the centre of a rotation is a fixed point.
    The composition of a translation and a rotation is always another rotation (in either order)

    The combination of rotation about two different points is also usually a rotation and again has one fixed point. In the exceptional cases where the equation has no solution you would find that it was a translation instead.
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