Question is related about reflection definition because I don't understand in what context they say that 'mirror' is

[..] isometries have a set of fixed points (the "mirror") that is an affine subspace [..]

I wish to build a reflection (an isometry) using a a set of fixed points ('mirror') but I don't understand well if I need to start from affine subspace of vector space or from affine subspace of affine space.
Can you provide me an example to build a reflection from a generic fixed points ?

So I wish to understand better also if there is a morphism or some algebraic structure that link directly an affine subspace of vector space with affine subspace of affine space.

  • $\begingroup$ I think you are overthinking the description. When you look in the mirror, things on one side of the mirror appear as if they were on the other side (so they seem to have "moved"), but the surface of the mirror itself appears to remain fixed ("unmoved"). Such is the nature of a reflection transformation on an affine space. $\endgroup$ – hardmath Apr 19 '18 at 12:54
  • $\begingroup$ Ok, Mirror doesn't reflect itself, for this reason it is works like an affine subspace. But if I want to apply a reflection transformation to reflect also mirror itself what should I need consider ? $\endgroup$ – Jacky Ned Apr 19 '18 at 21:33
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    $\begingroup$ If your affine transformation doesn't fix a nontrivial affine subspace then we don't call it a reflection. It might be some other kind of isometry, but not a refection. $\endgroup$ – hardmath Apr 19 '18 at 21:44

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