Intuition of fermats principle

[aaaa202]

I have used the attached photo to show that light takes the path described by snells law such that the time it takes from point A to B is minimized. But conceptually there is something wrong for me in this derivation. Because why would the light ray coming in, know that there is a different medium later on? It certainly would suggest that it does in this derivation, because here we vary the length that the light travels horizontally in the first medium with the idea of minimizing its path OVERALL. Am I misunderstanding something?

 

[voko]

Because why would the light ray coming in, know that there is a different medium later on?

Of course the light ray would not know this. It does not even exist to begin with. It is a simplification of the underlying wave propagation. The light wave propagates in all directions, so you could imagine, in a way, that this is how the fastest path is "felt".

 

[aaaa202]

Having read about fermats principle I don't really think I understand it. It says the light will take the path between two points that take the least time. But what are these two points? - the points you aim from and at with your lightray?