Euclidean Plane Isometry - Isometries in The Complex Plane

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.