I understand that parabolic mirrors have focal points, i.e., points that incident light rays parallel to the optical axis of the mirror reflect through. But spherical mirrors aren't (exactly) parabolic mirrors. How then does one define the focal point of a spherical mirror?
Do we simply say "it doesn't matter, it's basically a parabolic mirror" or are we more precise than that? For instance, is it best to say that the focal point is halfway between the mirror and the center of curvature?
Related to this, does it make sense to define the center of curvature of a parabolic mirror? If so, why?
Best Answer
It is customary to define the focal point of a spherical mirror as that point on the principal axis where rays which are near to and parallel to the principal axis meet.
The reason for including the words near to can be shown as follows.
In the diagram below there is a concave mirror of radius $R$ and centre $C$.
The principal axis is $CP$ and the incident ray, parallel to the principal axis , is shown as $AB$.
Triangles $ACB$ and $CFB$ are similar so $$\dfrac {AB}{AC}= \dfrac {CB}{CF} \Rightarrow \dfrac {AB}{R}= \dfrac {R}{CF}$$
As the incident ray gets closer to the principal axis $AB \to 2R$ and so $CF \to \dfrac R2$.
So it is really all to do with how close the length of $AB$ to a diameter $2R$.
Using such a construction you could quantify the words near to by deciding how different $CF$ can be from $\dfrac R2$.
A parabola $y=\frac 14 x^2$ (blue) and a circle $(y-2)^2+x^2=4$ (red) are shown in the diagram below.
Curvature of both graphs is the same at point $P$ as is the centre of curvature $C_{\rm circle}$.
Just by eye you can see that the two graphs are "very" similar close to $P$.
All rays parallel to the principal axis arrive at $F$ at the same time because $BF= BD$ and so this is the focal pont of the parabola.
There is however a restriction in that this will only happen if the incoming parallel rays are parallel to the principal axis.
The position of the centre of curvature of the parabola $C_{{\rm parabola},B}$ depends on the position of $B$ and only coincides with that of the circle at position $P$.
The necessary construction to find the centre of curvature os a parabola is shown below and more details can be found here from where the gif image was taken.