Question
Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
Answer :
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Let the linear fractional transformation (LFT) be [ w = f(z) = \frac{az + b}{cz + d}, \quad ad - bc \neq 0. ] We are given three distinct points (z_1, z_2, z_3) in the extended (z)-plane and three distinct points (w_1, w_2, w_3) in the extended (w)-plane. We need to show there is only one LFT mapping (z_i \mapsto w_i) for (i = 1,2,3). --- Step 1: Use the cross-ratio property A fundamental property of LFTs is that they preserve the cross-ratio. That is, for any four points (z_1, z_2, z_3, z), [ [w, w_1, w_2, w_3] = [z, z_1, z_2, z_3], ] where the cross-ratio is [ [z, z_1, z_2, z_3] = \frac{(z - z_1)(z_3 - z_2)}{(z - z_2)(z_3 - z_1)}. ] For our problem, if (f(z)) maps (z_i \mapsto w_i), then for any (z) in the extended plane, [ \frac{(w - w_1)(w_3 - w_2)}{(w - w_2)(w_3 - w_1)} = \frac{(z -
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