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    Mobius Transformations for Circles

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    Prove: For any given circles R and R' in C_oo, there is a mobius transformation T such that T(r)=R'. Further, we can specify that T take any 3 points on R onto any 3 points of R'. If we do specify Tz_j for j=2,3,4 (distinct z_j in R), then T is unique.

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    Thank you for clarifications.
    Here is for reference the text of the theorem as you wrote it in the message:

    Theorem: Suppose R and R' are 2 circles in C_oo. So, we know that there is a Mobius map T such that T(R) = R' (basically circles go to circles). If a,b,c are any 3 points on R and we can specify T(a), T(b), T(c), then such a T is uniquely determined. (Conway says that this proof is trivial, and we may use the fact that M-maps take circles onto circles).

    The proof can perhaps be indeed regarded as trivial when you are aware of its two key elements:

    A: If we take an infinite straight line as a case of a circle with infinite radius, then is can be proved that ANY Mobius transformation maps ANY ...

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