Why Are Great Circles the Shortest Flight Path?

Great Circles (Geodesic) vs Small Circles

great circle sphere

Last Updated: February 21, 2018

Why do you fly over Greenland in an airplane flight?

Or why is it that when you see flight paths on a map they always take a curved route between 2 cities?

It’s because planes travel along the true shortest route in a 3-dimensional space.

This route is called a geodesic or great circle route. They are common in navigation, sailing and aviation.

But geodesics can be confusing when you’re looking at a 2-dimensional map as they follow quite the odd flight path. Let’s dig into this concept a bit deeper.

Great circle routes explained

In a flight path from New York to Madrid, if I asked you which line is shorter, you’d say the straight one, right?

great circle flight path

However, a straight line in a 2-dimensional map is not the same as a straight line on a 3-dimensional globe.

This is why flight paths take an arc route between an origin and a destination.

Now here’s how the same flight paths look like on a sphere. Remember that the straight line in the Mercator map above followed the 40° latitude line.

great circle new york madrid

This paints quite the different story, doesn’t it? It’s deceiving to the human eye.

The takeaway is this:

A route that looks longer on the map is because of the distortion created with map projections like the Mercator projection. In navigation, pilots often use great circles (geodesic) as the shortest distance flight.

Great circles vs small circles

Now that you have a visual understanding of great circles. Here’s a definition of what a great circle is:




  • A great circle is a circle on the globe such that the plane passing through the sphere’s center is equal to the circumference of the Earth.
  • Alternatively, a great circle is where the radius is equal to that of the globe when taken at the center of globe representing the shortest distance between two points on the surface of the earth.

In basic terms, imagine you’re cutting into an orange. You can cut them at any angle – north-south, east-west, diagonally. As long as you cut two identical portions, then the circle where cut was made is a great circle.

For example, the equator is a great circle because it’s the maximum possible circle:

Equator

You could also cut it at the north and south pole. This longitudinal line also cuts two equal portions. Any meridian line is a great circle as well.

Longitude

From New York to Madrid, here’s how the plane create two equal segments.

A great circle generates two arcs with shorter one being the shortest path. Here is the shortest path and how the plane is angled to create the shortest path.

great circle new york madrid

How about when you follow along the 40° latitude line? Anywhere that it doesn’t cut two equal pieces is a small circle.

While a rhumb line track is at a constant azimuth, a geodesic line changes direction all the time.

small circle

READ MORE: Rhumb Lines: Setting it Straight with Loxodromes

How geodesics work

Planes travel along the true shortest route in 3-dimensional space.

This route is called a geodesic or great circle.

While map projections distort these routes confusing passengers, the great circle path is the shortest path between two far locations.

This is why pilots fly polar routes saving time and distance. And this is why pilots often fly over Greenland.

13 Comments

  1. Every airline in the world has a business route. This route determines where the aircraft can and cannot fly. For example certain routes over the Atlantic are more costly to make but much faster, the final determination is down to the pilot, in regards to how much fuel he has on board, how delayed the aircraft is, in respect to flying faster than he normally would. This is a very significant thing when determining which transatlantic route to take, because there are at least a dozen of which only 3 of those routes are the fastest and most economical, the rest vary with flight cost.
    In respect to flying over Greenland, there was a very valid reason to do so especially during the winter and early spring months. As the air gets colder, it gets thinner, and the air thins it gets more and more difficult for an aircraft to stay in the sky. This meant under aviation rules, those aircraft with only 2 Jet Engines had to make the shortest distance between land masses, incase an engine stopped working. This way the pilot could make an emergency landing. Aircraft such as the 747 had no such problems, and could easily fly the quickest transatlantic routes. However over the decades Jet Engines have become increasingly reliable, so much so, that they too can now fly over the Arctic to whatever destination.
    There are very few airlines who actually use Arctic routes, because these routes are extremely expensive to use.
    An example of a flight from Helsinki (EHFK; HEL) to Barcelona (LEBL, BCN) does not use a Great Circle nor a Small circle, it uses a direct path from point A to B.

    Small circles are not always used for short distances, infact they are seldom used in Europe. Instead most short distances are literally from A->B.

  2. Let me correct you. As the air gets colder, it gets denser, not thinner. However, as you increase in altitude, the air gets thinner and colder. Pilots will always want to fly higher as the low density of the air reduces drag and thus increases the efficiency of the fuel.

  3. Mr Bir has it right on, as soon as I read it, I questioned it. But we appreciate your explanation of the great circle route. Good article.

  4. J Bir, unlike liquids the viscosity of gases inceases with increasing temperature. This can be explained using the kinetic theory of gases.

  5. I too, like Mr. Bir, noticed that there was something wrong with the explanation regarding the density of air. Yes, as air gets colder it gets denser, that’s why to start a cold gasoline engine you have to activate the ‘choke’ which in turn will supply the fuel-air mixture with an extra amount of fuel in order to compensate the leanness that otherwise would not start a combustion.

  6. Actually, you’re all right, but you’ve each only got two-thirds of the puzzle. There is no direct relationship between temperature and density. There IS a relationship between PRESSURE and temperature and density.

    In aviation, there is a concept called “termperature altitude.” So the actual relationship is between all three of them, but the focus is only on the two. The higher you go, the colder and thinner the air gets because the pressure is dropping. For every 1000 feet you go up, there is an expected drop in temperature and pressure, and this remains constant. So flying through colder air is like flying higher. The colder air makes the pressure go down while the density remains the same. The airplane acts as if the air is less dense. Got it?

  7. Airplanes need air pressure to fly, it doesn’t matter how dense the air is if there’s no pressure, and pressure and altitude have a consistent relationship. Does that make sense?

  8. …and yes, thinner colder air is better for fuel efficiency, but bad for lift. The higher you go, the harder it is to maintain altitude, so like everything else, there is a trade-off. Airplanes are designed to take advatange of the thickness of the air at a certain altitude. That altitude is not fixed relative to the ground, but depends on the weather. So on a hot day, they can fly higher. On a cold day, they.have to fly lower. But there is always an optimum “temperature altitude” for any given plane on any given day. You can’t just keep going up forever. You need pressure under the wings, or you’re flying a brick.

  9. Chris, you are wrong. For air, D=PM/(RT), where D=mass density, P=pressure, M=molecular weight average, R=ideal gas constant, & T=absolute temperature. This is why you need to fill your tires with air in the winter. The temperature drop compresses the air, which means you have to add air to fill in the same amount of relative space.

    There are many other atmospheric factors in play such as humidity, wind speed, and wind direction to consider as well.

  10. You have such great figures. May I use one of them in a presentation to illustrate great circle routes?

  11. Thinner air does not give better engine performance. The reason flying high is more economical compared to flying at low altitude is because the TAS is higher due to density. Thus the amount of fuel burned per nm travel is lower.

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