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Proving the following:



First a few definitions:

(x_{1},y_{1}) = point a.
It is in medium A

(x_{2},y_{2}) = point c.
It is in medium C

(x ,y_{0}) = point b. It is where
Medium A intersects medium C

* Medium A is not necessarily different than
medium C

q_{1}
= Angle between line (a,b) relative to normal

q_{2}
= Angle between line (b,c) relative to normal


Now to Prove:

To prove this, we have to first
show that to travel from a to b is a straight line,
however simple this
might seem, it is quite a lengthy formal proof, so lets just assume that the
quickest way to get from point a to b is a straight line. Then we will
show that the quickest way to get from a to c via b is Snell's Law.

1) The Total Time to get to c
is:


_{2) Which does not help
much, so lets expand this into an integral to get something more exact.}


_{3) Where S is the distance
and V the velocity, and obviously, distance over velocity is time.
Then:}


_{4) Now lets get the change
in distance in terms of the points a, b and c:}


_{5) Since we want the
minimum time, we have to take the derivative of t as a function of x and set
it equal to 0:}


_{6) Then we bring the term
multiplied by 1 over V2 to the other side to make it easier to work with:}


_{7) This may look
complicated, but with some simple trigonometry, the big mess in the brackets
simplify:}



_{8) Very nice, lets plug
this back into part 6 and see what happens:}


_{9) How convenient, now lets
multiply both sides by the speed of light, it will not change the result
since both side are being multiplied by the same thing, we are using the
speed of light constant for ease of use:}


_{10)Lets see what we get
with speed of light over velocity. It is the }_{indices}_{
of refraction:}



_{Now plug this back into 9:}


_{Voila, Snell's Law!}
