Isometries in The Complex Plane
In terms of complex numbers, the isometries of the plane either of the form
or of the form
for some complex numbers a and ω with |ω| = 1. This is easy to prove: if a = f(0) and ω = f(1) − f(0) and if one defines
then g is an isometry, g(0) = 0, and g(1) = 1. It is then easy to see that g is either the identity or the conjugation, and the statement being proved follows from this and from the fact that f(z) = a + ωg(z).
This is obviously related to the previous classification of plane isometries, since:
- functions of the type z → a + z are translations;
- functions of the type z → ωz are rotations (when |ω| = 1);
- the conjugation is a reflection.
Read more about this topic: Euclidean Plane Isometry
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