Why your flight to Tokyo goes over the Arctic

You board in London. You are flying to Tokyo. Tokyo is east of London — obviously east, unambiguously east, about 9,600 km east.

Two hours in, you look at the moving map and the aircraft is over Norway, heading almost due north.

Nothing has gone wrong. The map is the problem.

The flat map is a compromise

Every world map you have ever seen is a lie of some kind, because every one of them takes a sphere and flattens it. There is no way to do that without distorting something — area, shape, distance or direction. You choose which lie you can live with.

The Mercator projection, the one most maps default to, preserves angles. That made it invaluable for navigation by compass: a straight line drawn on a Mercator map is a constant compass bearing, and a ship could simply hold that heading. The price is that it stretches everything away from the equator, catastrophically so near the poles. Greenland looks the size of Africa. Africa is fourteen times larger.

So when you draw London to Tokyo as a straight line on a Mercator map, you are not drawing the shortest route. You are drawing the route that holds one compass bearing — which on a sphere is a longer, curving path called a rhumb line.

The great circle

The actual shortest distance between two points on a sphere lies along a great circle: the circle you get by slicing the sphere through both points and its centre. The equator is a great circle. So is every line of longitude. Lines of latitude, apart from the equator, are not.

Fly the great circle from London to Tokyo and it takes you up over Scandinavia, across the top of Russia, and down through Siberia. On a globe, that path is visibly, undeniably the straight one. On a flat map it looks like a detour over the North Pole.

The saving is not marginal. London to Tokyo along a great circle is roughly 9,580 km. Along a constant-bearing rhumb line it is closer to 11,200 km — an extra 1,600 km, or about two hours of flying and a great deal of fuel.

Why some flights don't take the shortest path

Aircraft do not fly perfect great circles, for reasons that are mostly practical:

• Jet streams. A band of very fast wind sits at cruising altitude in the mid-latitudes. Riding it eastbound can save an hour; fighting it westbound costs one. Transatlantic flights routinely detour hundreds of kilometres north or south to find or avoid it, which is why New York to London is reliably shorter than London to New York.
• ETOPS rules. Twin-engine aircraft must stay within a certified flying time of a suitable diversion airport. Over the North Pacific and the Southern Ocean that constrains the route.
• Airspace. Some is closed, some is expensive, some is at war. Routes bend around all three.
• Air traffic control. The published airways structure is not a smooth field of great circles.

The result is a path that approximates the great circle and deviates from it for good reasons — usually by a few percent.

What this means for a logbook

If you're recording flights, the distance you want is the great circle distance. It's the standard measure, it's reproducible, and it's what every airline and every frequent-flyer programme uses.

It is also not something you should be calculating by hand. The formula — the haversine, if you want to look it up — is straightforward but fiddly, and you need accurate coordinates for both airports.

Orbit does this for you. Enter two airport codes and it computes the great circle distance from an offline database of 8,801 airports, then draws the resulting arc onto a globe you can spin — where, correctly, it looks like a straight line.

That is the part people find surprising. Once your flights are on a sphere instead of a rectangle, the routes stop looking strange. The one to Tokyo goes over the Arctic because that is where the short way is.

A test you can run

Get a globe and a piece of string. Hold one end on London and the other on Tokyo, and pull it taut.

The string goes over the Arctic. It has to. There is no shorter path, and the string has no opinion about map projections.

Related reading: the world's longest non-stop flights and how many times around the Earth you've flown.