How Do Curved Mirrors Form Clear Images?

[dEdt]

My Optics text shows how the light rays reflecting off a spherical mirror appear to have been emitted from a single point, hence producing a well-formed image. The text then says that this is true "for any mirror whose curvature is gentle enough and that is symmetric with respect to rotation about the perpendicular line passing through its center". I was wondering if anyone had a proof for this statement.

 

[Doc Al]

I won't offer a proof, but I will make a comment that might shed some light on what they mean by the statement. One thing to realize is that spherical mirrors do not exactly form sharp images, but parabolic mirrors do. The use of spherical mirrors is just a practical convenience. The distortion due to using a spherical mirror is called spherical aberration. For simple optics, such complications are often ignored. (The so-called paraxial approximation is used.) The flatter the mirror (the less spherical and more parabolic) the less the effect of spherical aberration.

 

[dEdt]

Are parabolas the only mirror shape that leave no aberrations?

 

[Jeff.N]

Are parabolas the only mirror shape that leave no aberrations?

It depends on the curvature of what your reflecting. If you drew a picture on the inside of a large sphere, which has the same center as the smaller spherical mirror, then the spherical mirror will not produce any distortion. Sphereical mirrors produce distortion when the thing you reflect isn't on a sphere with the same center. Conversely, if you had something drawn on the surface of a sphere and reflected it in a plane or parabolic mirror it would have distortions.