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The snip shows a part of the method shown by my teacher, to describe the reflection formula for spherical surface. However I do not understand how the relation of AB and BI (which I have highlighted) is coming. Please explain.

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The equations comes form the law of cosines, but we will see there is an error in the equations, as can be easily observed with dimensional analysis.

The law of cosines says that if you have a triangle with three vertices $A$, $B$ and $C$, and if the lengths of the sides connecting the vertices are $AB$, $BC$ and $CA$, and if the angles at each vertex are denoted by $\theta_A$, $\theta_B$, and $\theta_C$, then these quantities are related by $AB^2 = BC^2 + CA^2 - 2*BC*CA*\cos(\theta_{C})$.

If we apply this formula to the picture, except where $C$ from the formula is $R$ in the picture, we get $AB^2 = BR^2 + RA^2 - 2*BR*RA*\cos(\theta_{R})$. Now the length $BR$ is $r$, the radius of the spherical mirror. Also $\theta_{R}$ and $\theta$ make a straight line so $\theta_R = 180^\circ - \theta$. Plugging these in we get $AB^2 = r^2 + RA^2 - 2*r*RA*\cos(180^\circ - \theta)$. This matches the formula you have except the one you gave is incorrectly missing a factor of $r$ multiplying the cosine.

I will leave it as an exercise to find how to apply the law of cosines to the second triangle to produce the given equation. Think about which of the three angles you should use in the law of cosines. Also notice you will find a similar error in the second formula.

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Well it seems that everyone who tries to teach geometrical optics, as far as ray tracing goes, seems to feel they have to create their own procedures, ignoring the fact that several hundred years of prior work, has established efficient ways to do it. You state the problem is to establish the law of reflection from Fermat's principle. So that can be established (from Fermat's principle as asked), without ANY reference whatsoever to the curvature of the surface; which is totally irrelevant to the problem, since the reflection takes place at the single point of ray incidence, on the surface.

So to move on from there to establish the relationship between your u, v, and r parameters, for your spherical mirror, useful closed form expressions can only be developed (for elementary studies), by assuming that the point B is very close to the axis, so that (theta) and the other two incident and reflected angles, are small enough so that the sin is equal to the angle (in radians). This is the so-called "paraxial region." Then NO trigonometry is required. B can be described by its height (h) from the axis.

In formal ray tracing for optical design, there is no difference between reflection and refraction. If reflection is required, and the incident ray is in a medium of index (n), then the reflected ray is in a medium of index (-n). Distances following reflection are reversed in sign.

The usual sign convention for "left to right" tracing, is that distances to the left of the surface are negative; those to the right are positive. If the center of curvature is to the left of the surface, then the surface radius is negative.

So in your diagram, u, v, and r, are ALL negative. a common convention uses unprimed parameters for quantities before refraction (reflection), and primed parameters, for those after refraction.

So your u, v, r would be l, l', r, and the indices would be n and n'.

Then the standard paraxial refraction formula is:-

    n'/l' = n / l + (n'-n) / r    

So for your spherical mirror case, we would put n = 1 and n' = -1 , giving:-

    -1 / l' = 1 / l + (-1 -1) / r   

or to put it in your variable names;

    -1/v = 1/u + (-1-1)/r

I'll let you finish that off.

For imaging small objects of height (h) or (h') and with marginal rays at angles of (u) and (u'), it can be shown, that the quantity (nhu) is invariant under all optical transformations. It is known as the Lagrange Invariant. The corresponding form outside the small angle paraxial region, says NHsin(U) is invariant. This is the optical sine theorem.

This formalism is NOT new. It was first taught by A. E. Conrady in 1926, an is readily available today as a Dover Press paperback, in two volumes. If you want to learn optical design; buy it.

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