Why the line from Seoul to Paris bends north

Set Seoul as your origin in CouchColumbus and pick Paris, and the line drawn on the map does not run straight. It swells upward, then settles back down into Europe. At first glance it looks like the route calculation is wrong. What is wrong is not the line. It is the map.

The shortest path on Earth is not a straight line on a map

The shortest path between two points follows the surface of the Earth. Slice the sphere with a plane through its centre and you get a circle; the path runs along that circle. This is called a great circle.

The trouble starts when you move that path off a curved surface and onto a flat one. However you unfold it, a sphere cannot be laid flat without distortion. This service uses the equirectangular projection — longitude and latitude used directly as x and y, the simplest approach there is. Easy to build and good for fitting the whole world on one screen, but the higher the latitude, the more it stretches horizontally beyond life size. A path that runs north passes through that stretched region, so on the map it appears to bend upward.

The line is not curved. The map holding the line has been pulled apart at the top.

Seoul to Paris: what it actually passes over

The Seoul–Paris great circle distance this service computes is 8,966km, on a bearing of north-northwest from Seoul. Paris is west of Seoul, and yet the path heads north.

Follow that route and note the country underneath and you get this.

South Korea → North Korea → China → Mongolia → China → Mongolia → Russia → Finland → Sweden → Denmark → Germany → Netherlands → Belgium → France

It does not go through Central Asia or the Middle East. It crosses Siberia. On a flat map that looks like a far longer way round; on a globe it is the shorter one.

New York is more dramatic still. The Seoul–New York great circle is 11,053km, and the route runs South Korea → North Korea → China → Russia → Canada → United States. It does not cross the Pacific head-on; it swings up near the pole. On a flat map it does not look like a shortest path at all.

The line on the map is not a real flight

One thing worth stating plainly: what this service draws is the pure geometric shortest path, not the route an actual flight takes.

Real routes have a lot layered on top. Overflight permissions are required, conflict zones are avoided, and aircraft will deliberately go the long way to ride a jet stream and burn less fuel. The same route shifts a little every time depending on the season and the day's winds. That is also why North Korea appears in the list above — the great circle passes over it, but real airliners route around that airspace.

Which is why the waypoints do not come out strange either

The same calculation picks the waypoints. For each candidate city we take the distance from origin to destination by way of that city, divided by the direct distance — the detour ratio. The closer to 1, the more it sits on the way.

Going from Seoul to Paris, 211 cities come in under 1.4. Fukuoka scores 1.12, Qingdao 1.05. Cities on the far side of the planet, by contrast, shoot up into the dozens and drop out. Which is why no itinerary ever stops in Buenos Aires on the way to Paris.

In one line

The curve on the map is not this service showing off. It is what happens when you lay something round out flat, and we simply did not straighten it back out.