From Fermat's Principle to Snell's Law

Fermat's principle of least time postulates that light takes the fastest route between two points.

When light is traveling at an interface between two mediums, the speed light travels changes based on the material.

(Light moves at a constant speed within the same medium no matter how fast the source of light is moving relative to an observer. A beam of light from a flashlight moving towards you has the same speed of a beam of light from a flashlight moving away. That's Einstein's special relativity. What I mean by the 'speed of light changing' is that the constant speed of light is slower in a dense material than it is in a vacuum.)

A beam of light that enters a dense material will bend to travel a shorter distance through the slower region of dense material. When it goes from a dense region to a lighter region, why does it bend in the opposite direction? I don't think Snell's law really explains that with a physical rationale... it's just like the symmetry of the mathematics setup by the light to dense case implies that the opposite would happen, say for example like if you could travel that beam of light backwards.

But the principle of least time, kind of... kind of... has a reasoning to explain why light bends outward from dense to light. It says that given any two points A and B, light travels the path that takes the least time. So if we pre-select points A and B on opposite sides of a interface between two materials, then we can figure out the fastest route to navigate between them using Google Maps.

This is really a cheat, because, well... how do we know when we start at A we will end up at B ahead of time? We don't, we are just asking: "if there is a beam of light that goes from A to B, what would be the path it takes?" We discover at what angle to the interface does that light travel. We say, that if this light that traveled from A to B obeyed this law, then all light beams also obey this law. Then we can make a forward prediction to figure out where we would end up, if we all we knew was we started from A at an angle to the interface.

The derivation from Wikipedia is straightforward and common place. So of course I could figure it out on my own, right? Nah, I looked that shit up after I spent an hour diagramming the interface and attempting to brute force labeling everything and joining labels with basic kinematics.

The idea is straightforward. Write an expression for the time it takes the light to travel from A to B, in terms of the speed of light in each region and the distance traveled. Then take the derivative of that expression in terms of a parameter that controls at which angles theta1, theta2 does the light cross the interface. Set that derivative equal to zero to find the set of angles that minimizes the time.

I got lost in all the lengths, couldn't figure out which one to unify the expression. I thought I had to write time in terms of the angles theta1, theta2 before and after light crosses the interface, then take a partial derivative in terms of each theta.

My unsuccessful kinematics attempt. Do not copy this!

There's a more on-target variable to vary. It's the position y1 at which the light crosses the interface. This is a single variable, that can easily differentiate the expression with, and it fully describes the angles theta1, and theta2, since the distances x1, x2 along the component parallel to the light's motion are fixed by the geometry of the setup. Likewise, the total vertical distance y (= y1 + y2) is fixed, so that the crossing point y1 also describes the length y2 the light travels after the interface.

Once you figure out to use the crossing point as the independent variable, the rest follows just using Pythagorean Pythagoras and SOCATOAn SOCATOA: basic trigonometry. Yeah you need to remember the chain rule in order to take the derivative, but that's a first year calculus level of being careful.

So here's my write-up of the Wikipedia derivation, from Fermat's Principle to Snell's Law.
You should refer to Wikipedia instead of me, but this blog is my refrigerator so I will put my homework on it!