How Can Fermat's Principle Prove the Second Law of Reflection?

[deltafee]

Hi, I am trying to prove the second law of reflection using fermat's principle and I am not entirely sure how to start it.
By the way the second law of reflection is: The incident ray, reflect ray and normal ray all lie in a single plane.

 

[Simon Bridge]

Fermat's principle - light follows path of least time?
You do it pretty much the same way as you would for the first rule and for Snell's law... fix a point that the incedent ray passes through, and another that the reflected ray passes through, but vary the point of reflection (constrained by the first law).

 

[deltafee]

Yeah I used the three variable Pythagorean Theorem and than took the derivative and than placed values for x and y so I could graph it.

Here's the typed worksheet: https://dl.dropbox.com/u/77575413/F.pdf

on the second page I have the graphs of Time and the derivative of Time and as you can see I don't get a minimum in the derivative of time graph, but I get a minimum on the time graph. So I am really not sure what I did wrong.

Oh by the way just to make it easier to see the graph I left the value of c out from the equation.

 

[Simon Bridge]

It looks like at least the derivative is wrong.
You realize you can check your calculations against the actual answer because you know it already right?

I don't get a minimum in the derivative of time graph, but I get a minimum on the time graph.

example: y=x^2 has a minimum, but the derivative function y'=2x does not have a minimum.

I don't follow what you have done though - i.e.
The diagram at the top of the first page has no labels.

That 1/2c looks a little suspect. Comes from the 2d in the first line - but since there are no labels on the diagram I have no idea if it is OK or not.

I see you have written:$$\frac{1}{2c}\left [ \frac{10+z}{\sqrt{58}+z^2}+\frac{z-6}{\sqrt{106}+(20-z)^2} \right ]$$ for both $T$ and $T^\prime$.
(Last equation page 1, and top pf page 3).

I'm surprised you didn't try for a simpler geometry.