Quick background theory:
One of the FE models out there posits the Earth is a non-Euclidean space. There is a lot to the background of this, but the simplest may be as follows: the distance from point A to point B will not be the straight-line distance.
Mathematically, Euclidean space is a set equipped with a metric: coordinates along with a calculation that tells you how far apart two coordinates are. In the Euclidean case, to get from (0,0) to (3,4) you can see it's just the hypotenuse of a right angled triangle with sides 3 and 4. You'd use Pythagoras' theorem.
In a non-Euclidean situation, this wouldn't be the function. In strictly abstract mathematics, d(a,b), the distance from a to b, could be defined as 0 if a=b, and 1 otherwise. That's called a totally disconnected space.
I got bored for a bit today, and figured out a typically workable metric for a flat Earth. It's possible there are calculation errors at some points, I quoted a number of results without proof so I may have overlooked some details, but the gist is unfortunately the following.
There might be ways to simplify, and unfortunately it's not too useful yet as we'd need a map of the Earth with coordinates marked, though I can tell you how to derive that map.
This is not a pitch for an FE model, it's no more than an illustration of how a non-Euclidean map might work.
First off:
The Earth is defined by the x and y axes, with the North Pole (for the sake of tradition) at 0, and the South at infinity. An infinite plane is used so that if you reach the South, you can come out the far side. Once the metric is used, the distance shouldn't actually be infinite. This is just our set.
Longitude is 0 along the line y=0, latitude is zero on the circle of radius 1 centred at the North pole.
Now then, our spectacularly awful looking metric. For points P1=(x1,y1), P2 = (x2,y2):
[jsTex]d(P_1,P_2) = \cos^{-1} \left( \sin \left(\tan^{-1} \left(\frac{x_1^2 + y_1^2 -1}{2x_1} \right) \right) \sin \left(\tan^{-1} \left(\frac{x_2^2 + y_2^2 -1}{2x_2} \right) \right) +\cos \left(\tan^{-1} \left(\frac{x_1^2 + y_1^2 -1}{2x_1} \right) \right) \cos \left(\tan^{-1} \left(\frac{x_2^2 + y_2^2 -1}{2x_2} \right) \right) \cos \left|\tan^{-1} \left( \frac{y_1}{x_1} \right) - \tan^{-1} \left( \frac{y_2}{x_2} \right) \right| \right)[/jsTex]
The code, for those that might want to copy/paste it, is:
d(P_1,P_2) = \cos^{-1} \left( \sin \left(\tan^{-1} \left(\frac{x_1^2 + y_1^2 -1}{2x_1} \right) \right) \sin \left(\tan^{-1} \left(\frac{x_2^2 + y_2^2 -1}{2x_2} \right) \right) +\cos \left(\tan^{-1} \left(\frac{x_1^2 + y_1^2 -1}{2x_1} \right) \right) \cos \left(\tan^{-1} \left(\frac{x_2^2 + y_2^2 -1}{2x_2} \right) \right) \cos \left|\tan^{-1} \left( \frac{y_1}{x_1} \right) - \tan^{-1} \left( \frac{y_2}{x_2} \right) \right| \right)
This is no more than an approximation. It simply comes from mapping the plane in question to a ball (using the stereographic projection), and finding the great circle distance between the resulting points. It's not perfect, as altitude isn't accounted for, and the Earth isn't a perfect sphere, but it's pretty close.
If you wanted to develop a map of the Earth, then you'd perform such a projection of the globe map, and calculate the distances between two points on its surface by means of the above rather than taking the straight-line distance. This seems reasonable, by Davis' model, as he has said the globe is a valid projection of the Earth's surface. This simply gives us a way to view it as a plane, as well as an idea of how distance might be calculated between two points, in this non-Euclidean space.
It's going to be difficult to work with points at and near the South pole, because the coordinates would be infinite and so would blur together, but equally not much information is lost because, viewed externally, the points nearby would seem absurdly stretched out.