Theorem S2 If t is a direct similarity, then t(z) = az + b, where a, b are in C, and a nonzero. If t is an indirect similarity, then t(z) = az* + b, where a, b are in C, and a nonzero.


Proof From the Corollary to the Similarity Theorem, we know that t can be written as r° s_{k}, where r is an isometry, and s_{k} is the scaling mapping z to kz (k > 0). From Theorem E2, we know that r(z) = gz + b or gz* + b, with g, b complex, g = 1. Thus t has the required form, with a =kg.
Conversely, each mapping of the given form can be written as r°s_{k},

similarity group 