Polar Geometry

The shortest distance between two points is a straight line. However, if you distort the surface on which the line is drawn, you distort the line.

One of the most popular maps of the Earth is the Equirectangular projection which makes for easy coordinate systems because latitudes and longitudes are all equally spaced throughout the map.

Lines, when placed upon a polar disk which is then distorted into an equirectangular-type map, are altered in a very predictable way. Due to the nature of the polar disk, lines will always tend toward the center of the disk--the upper part of the map.

This can easily be seen in an image such as this:
Here we see three straight lines on a polar disk traced out onto an equirectangular map. Notice how they all curve upward toward the north pole (point A)? (Note: the central vertical line on the rectangular map is the equivalent of the international date line)

Now, if you take a gander at this map, you'll see that airlines don't always do this.
In fact, they often appear to do the exact opposite--bending southward instead of northward.

My question is this: if the Monopole Model of the Flat Earth is correct, why do commercial flights not take the obvious shorter route, opting instead for routes that are much much longer than they need to be?

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Edit:
Here's an animation showing the real flight paths across a polar model:

To iterate the question, why do commercial airliners take these routes when the straight line routes (see the first animation) are vastly shorter?

I also asked this questions in my thread called Flight Times and received no scientific answer.

http://www.theflatearthsociety.org/forum/index.php/topic,59549.0.html

I was hoping Ævan would answer because he apparently has a pilot's licence.

Ævan can you address the points in this thread please?

Please can the FE'ers refrain from unsubstantiated comments like this:

They are simply mistaken.

Why would those pilots take a flat earth route on their international flights if they do not believe that the earth is flat?

Because they observe that flight times are longer when they do not use, for whatever reason,  the RE route.

Why would those pilots take a flat earth route on their international flights if they do not believe that the earth is flat?

I did a deployment last year. It was a cruise to Japan from San Deigo. On the way their there was no rush so we intended to do a couple port visits including one in Hawaii. With Hawaii in between the way their was a nearly straight route if we used a flat map like the one we are used to. On the way back we were in a hurry and intended to make no stops so we did a Great Circle route that almost reached Alaska.

The route back was nearly a week shorter.

Why would those pilots take a flat earth route on their international flights if they do not believe that the earth is flat?

Because the airlines have observed that in real life example that these are the fastest and most fuel efficient routes perhaps??

Can you show why on a supossed flat earth these routes are the fastest and most fuel efficient Tom?

Why would those pilots take a flat earth route on their international flights if they do not believe that the earth is flat?

If a pilot decides to go FE way, then he must fly via North Pole. So:

RE way: Distance between Sydney and Johannesburg = 6,900 miles
FE way: Sydney - North Pole (8,600 miles) + North Pole - Johannesburg (8,000 miles) = 16,900 miles

FE way is almost three times longer.