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Math Help - Euclidean Isometries - Help Please!

  1. #1
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    Exclamation Euclidean Isometries - Help Please!

    Need examples of Euclidean isometries f and g such that f\circ{g} g\circ{f}.
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  2. #2
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    Quote Originally Posted by ReneePatt View Post
    Need examples of Euclidean isometries f and g such that f\circ{g} g\circ{f}.
    There are so many examples. For instance, look at what happens when you compose two rotations with different centers. Or two symmetries with respect to non-orthogonal lines.
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  3. #3
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    Quote Originally Posted by Laurent View Post
    There are so many examples. For instance, look at what happens when you compose two rotations with different centers. Or two symmetries with respect to non-orthogonal lines.

    That's my problem. I have no idea how to do this and the book doesn't tell me. I've been searching the internet for examples so that I can better understand. I'm a distant learning student so I have to pretty much teach myself all this stuff.
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  4. #4
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    let A be the matrix [cos90 -sin90|sin90 cos90] a rotation of 90. let X=(x,y) vector.
    let f(X)=AX so the action of f on X rotates it by 90.
    and let g be a rotation with respect to the point (1,0) not the origin.
    so what g does is
    g(X)=A(x-1,y)+(1,0) (i.e. take 1 away from x then rotate w.r.t origin then replace x)

    so (fog)(X)=f(g(x))=A(A(x-1,y)+(1,0))
    and (gof)(X)=g(f(x))=A(AX-(1,0))+(1,0)

    picking point (2,1)
    fog(X)=(-1,0)
    gof(X)=(-1,-2)
    so check by putting values for the vector X and seeing these two arent always equal.
    ((x-1,y) and similar expressions are vectors)
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