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

Proof From the Corollary to the Similarity Theorem, we know that t can be written as r° sk, where r is an isometry, and sk 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°sk, where k = |a| and r is an isometry (since |a/|a|| = 1).

similarity group