Isometries of Unit Circle

My lecturer says that a translation of the unit circle is not an isometry because then it’s no longer a bijection to itself (smth like that).

But if that’s true then how can a translation be an isometry? I’ve attached the example because I can’t understand when a translation is an isometry and when it isn’t.

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Reply 1

RichE

Original post by Y2_UniMaths

An isometry of a set is a distance-preserving map of the set to itself - it's commonly also required to be a bijection.

A translation is an isometry of the plane, but a translation would move the circle to a different circle. You could call that an isometry between the two different circles.

But any isometry of a single circle has to map the circle to the circle, not somewhere else.