Great circles and flight paths

It is an often repeated cliché that the easiest way to become a millionaire is to start out a billionaire then go into the airline business. It is a tough and competitive business which means that we can be fairly certain that planes do fly the shortest possible way between point A and B. Sure, there are many complicating factors but let’s leave them at the moment.

It is an amusing experience to take a globe or Google Earth and find out how an airplane should fly between two points to spend the smallest amount of fuel possible. In order to do that, an airplane should follow a great circle. What is this? It is an imaginary line on the surface of the Earth which is an intersection of the surface and a geometrical plane which is determined by three points: departure and arrival airfields and the center of the Earth.

When we open Google Earth and measure the distance between two points, this distance is measured exactly along a great circle. There is nothing very special about that if these points are relatively close to each other (few thousand kilometers) or if they are roughly on the same meridian (all meridians are great circles). However, things get somewhat surprising when these conditions are not met.

The result is surprising because our Earth is round but we prefer flat maps for practical reasons. Unfortunately, there are no ways to project a spherical surface onto flat map without generating distortions. It is a matter of choice which distortions and where we prefer but we can not go without them. We should keep this in mind or otherwise we may live in illusions which have not much to do with the real world.

This post is motivated by my upcoming trip to Hawaii. I live in Estonia which is a small country in northern Europe. Now let’s imagine how should an airplane fly in order to cover the smallest possible distance. Is it something like this:

http://picasaweb.google.com/107509377372007544953/Coll#5853054459213578562
Actually no, this is not the shortest distance between Estonia and Hawaii. We should not believe this map because it is a two-dimensional representation of a three-dimensional object. Hence, it is unavoidably faulty and limited. In reality, the shortest flight route between Estonia and Hawaii goes almost straight over the north pole:

http://picasaweb.google.com/107509377372007544953/Coll#5853054494721766546
Although there are few countries that have more northerly position than Estonia and Hawaii is not too far from the equator but still the shortest way to Hawaii from Estonia is to head straight towards the north pole. This world is sometimes really weird. Or I should say “round”?

I said in the previous post that after the trip Hawaii would be the most distant place on Earth where I have ever been. This is really nothing to brag about because Hawaii is only 11,000 kilometers away from my home. The circumference of the Earth is 40,000 kilometers. We should travel approximately 20,000 kilometers and then we can say that it really is not possible to go any further, unless we are leaving the Earth entirely.

But let’s come back to the real world. It would be very nice to fly straight from Estonia to Hawaii but this is not to be. It is odd and very unfortunate. Both are fine places for living but surprisingly small number of people are apparently sharing my view. I have to first go to places where more people live and where there is a real market for airlines. In my case, instead of heading north, I will go south. I will go to Istanbul in Turkey from where my plane should fly to Los Angeles. The distance between these two cities is also 11,000 kilometers and the flight will last 14 hours. I already feel that these are going to be frustratingly long 14 hours. Anyway, here is how the route from Istanbul to Los Angeles should theoretically look like:

http://picasaweb.google.com/107509377372007544953/Coll#5853054498486073266
I know what great circles are and how bizarre the shortest route on globe may look. However, I still find it unexpected that one should fly north of Iceland over the Greenland ice shield in order to fly along the shortest route from Istanbul to Los Angeles.

I am not sure that this is the actual route because there are other factors that should be taken into account. Jet streams, for example, are powerful winds high in the atmosphere which airlines try to use when flying in the same direction with wind and avoid when flying in the opposite direction. However, one thing is clear. The third image is much better representation of the actual flight route than the first one.