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Suppose $M$ is a hyperbolic Moebius transformation with fixed points at $(0, 0), (1, 0)$ which, when applied to the complex $(x_0, y_0)$, yields the result $(x_1, y_1)$.

How do I solve for $M$ given $x_0, y_0, x_1$, and $y_1$?

2 Answers 2

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The transformation we are looking for is $$\frac{w}{w-1}=k~\frac{z}{z-1}$$ with $0

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Hint: You can take your matrix representation of the Möbius transformation to be an element of $PSL(2,\mathbb{R})$, or your transformation to be $T(z)= \frac{az+b}{cz+d}$ with $ad-bc=1$. Then with the conditions $T(0)=0, T(1)=1 $ and $ T(z_0)=z_1$ you should be able to solve for $a, b , c$ and $d$.

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    P.S. I'm assuming T is an isometry of $\mathbb{H}$, the upper half plane.2012-06-02