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Math Help - find mobius transformation that maps 3 points

  1. #1
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    find mobius transformation that maps 3 points

    Hi guys,

    Here's the question: find the Mobius transformation mapping 0,1,infinity to 1, 1+i, i respectively.



    I know can find the transformation mapping 1, 1+i, i to 0,1,infinity using:

    f(z) = [(z - z1) / (z - z3)] * [(z2 - z3) / (z2 - z1)]

    so, if z1 = 1, z2 = 1+i, z3 = i, I get

    f(z) = (z - 1) / (iz + 1) = T1

    Next, hmm... if I could invert this transformation, I should then get what I want?
    Does this make any sense?
    I'm going to assume it does, for a moment. Then:

    T1^(-1) = (w + 1) / (1 - iw) = T2


    Ok, now, T2(0), T2(1), T2(infinity) do not give me the expected result.
    Something is wrong. Where is the flaw?

    Thank you!



    EDIT: Actually, I think I may be onto something..

    T2(0) = 1
    T2(1) = 1+i

    But I'm unable to work out T2(infinity). What do I do with this infinity?
    Last edited by director; January 30th 2013 at 06:27 PM.
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  2. #2
    MHF Contributor

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    Re: find mobius transformation that maps 3 points

    for a mobius transformation:

    f(z) = \frac{az+b}{cz+d}

    if c \neq 0, then f(\infty) = \frac{a}{c}
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  3. #3
    Junior Member
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    Re: find mobius transformation that maps 3 points

    That was the missing piece of the puzzle, thanks!!
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