# Proof of image formation property of spherical mirrors

• I

## Main Question or Discussion Point

Hi. I'm trying to proof the image formation property of a concave spherical mirror. I know you can do this easily with a particular choice of rays (namely one that hits the vertex and one that passes through the center of the sphere) but I would like to show that a generic ray yields the same result because

1) Any two non-parallel rays will cross paths sonner or later so to use only two rays gives you a result but not a satisfactory proof.

2) The rays chosen for this easy proof pass through special points of the problem's geometry and could be a particular result.

I know there is a purely geometrical approach to this general proof I'm seeking but since I want to keep consistency with the work I've already done, I'm doing it with Analytical Geometry.

The Mirror is given by the equation

$y=R-R\sqrt{1-\left(\fracx}R}\right)^2$,

where $R$ is the radius of the mirror and its axis is parallel to the y-axis. The object in front of the mirror has coordinates $(x_{1},y_{1})$.

The ray reflected at the mirror's vertex has a line equation given by

$y=-\fracy_{1}}x_{1}}\,$

and the one passing through the mirror's center has a line equation

$y=R+\fracy_{1}-R}x_{1}}\,x$.

These rays cross paths at $x_{2}=\fracR}R-2y_{1}}\,x_{1$ and showing that

$y_{2}=-\fracR}R-2y_{1}}\,y_{1$,

we obtain the known amplification factor $x_{2}=-\fracy_{2}}y_{1}}\,x_{1$.

Now... let's consider a ray that hits the mirror at a point $(x_{0},y_{0})$. When this ray is reflected it will have a line equation

$y=y_{0}+\tan\left(2\arctan\left(-\fracR}x_{0}}\sqrt{1-\left(\fracx_{0}}R}\right)^2}\right)-\arctan\left(\fracy_{1}-y_{0}}x_{1}-x_{0}}\right)\right)(x-x_{0}$.

Under the paraxial approximation (which allows us to consider $x_{0}\ll R$) and using the property of arctan addition we can rewrite it as

$y=y_{0}+\tan\left(\arctan\left(\frac2x_{0}R}R^2-x^2_{0}}\right)-\arctan\left(\fracy_{1}-y_{0}}x_{1}-x_{0}}\right)\right)(x-x_{0}$.

I know we can simplify it a bit more by using the $\tan(a+b)$ formula but the way I kept it above is easier to contemplate. I have tested all these rays with mathematica and they converge to the same point so I know everything up to here is ok. My problem is:

If I equal this last ray to one of the others (the one that reflects at the mirror's vertex, for instance) I'll obtain the x at which they cross paths as a function of $x_{0}$ and $y_{0}$. Since all the rays converge to the same point (given the paraxial approximation) I would not expect them to depend upon the point at which they touch the mirror.

What am I missing here? There is some simplification I did not do? Could you lend me a hand?

Thank you very much.