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I have recently been thinking about some "paradoxes" about mirrors, such as this one .

Yet I still have not found a mathematical (or physical) full description of an ideal mirror (no adsorption, no distortion, etc.).

Let's say that I have a 2D image in front of a mirror: How can I can I describe it and how can I describe its reflection?

I suppose there must be some kind of operator that acts on it and I suppose it is a reflection or rotation of $\pi$ around a given axis; but what axis?

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It is reflection not rotation. And there is no axis for reflection but there is a reflection plane. The operation, in Cartesian coordinates, is (x,y,z) -> (x,y,-z) where z is the axis normal to the mirror plane. The mirror plane is one corresponding to the surface of the plane mirror.

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