Theorem E2
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Proof Suppose that t is an isometry with t(0) =b and t(1) =g.
Suppose that s maps z to az + b.
Then s(0) = b, s(1) = a + b.
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The idea is to construct a map with the same effect as t on 0 and 1.
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We must check that s is an isometry. This is easy since |s(z) - s(w)| = |(az+b)-(aw+b)| = |a(z-w)| = |z-w|. |
The case where s(z) =az*+b |
A simple calculation shows that r = s-1°t fixes 0 and 1. By Lemma 1, r is the identity or rx (reflection in the x-axis). Hence t = s or s°rx, so has the required form.
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s-1°t is more or less determined. |
euclidean group |