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.

 

[sardar]

I can provide a response to this content by explaining the concept of curved mirrors and how they form images. The statement in the text is referring to spherical mirrors, which are mirrors with a curved surface that forms a portion of a sphere. These mirrors can be either concave (curving inward) or convex (curving outward).

When light rays hit a spherical mirror, they reflect off the surface and converge or diverge depending on the curvature of the mirror. In the case of a concave mirror, the light rays converge towards a single point known as the focal point. This is the point at which the reflected rays intersect and form a well-defined image. Similarly, in the case of a convex mirror, the light rays diverge away from a single point, also known as the focal point.

The statement in the text is true because of the geometry of a spherical mirror. When the curvature of the mirror is gentle enough, the reflected rays form a cone shape, with the apex at the focal point. This cone of reflected rays creates a clear and sharp image, as all the reflected rays are focused at a single point.

This is also why the mirror must be symmetric with respect to rotation about its center. If the curvature of the mirror is not uniform, or if the mirror is not symmetrical, the reflected rays will not converge or diverge towards a single point, resulting in a distorted or blurred image.

In summary, the statement in the text is true because of the geometric principles of spherical mirrors. The curvature of the mirror and its symmetry play a crucial role in producing a well-formed image by ensuring that the reflected rays converge or diverge towards a single point.