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To show that there is only one linear fractional transformation (also known as a Möbius transformation) that maps three distinct points \( z_1, z_2, z_3 \) in the extended complex plane (Riemann sphere) to three specified distinct points \( w_1, w_2, w_3 \) in the extended \( w \)-plane, we proceed as follows: ### Step 1: General form of the linear fractional transformation A linear fractional transformation has the general form: \[ f(z) = \frac{az + b}{cz + d} \] where \( a, b, c, d \) are complex _________ _____ _______ ________ _________.
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