Mapping the Earth

Curvy flight paths

Did you ever notice that flight paths on world maps are not straight lines? Here is an example from flightradar24.com: Quantas Flight 28 from Santiago, Chile to Sydney, Australia. The picture below is from November 20, 2024.

Apparently curved flight path of a flight from Santiago to Sydney

Notice how the flight curves southward rather than flying directly west. Why go so far off the direct path?.

First, a straight line on a rectangular world map is generally NOT the shortest path. This is because flat maps distort distances on the nearly spherical Earth.

Second, there are many reasons why a flight might detour away from the shortest path. These include avoiding headwinds, bad weather systems, or high mountains. Some detours avoid war zones or politically hostile or legally forbidden air spaces. Planes with a small number of engines take a longer route to keep closer to emergency landing places along the way.

It turns out, however, that the above path is fairly close to the shortest path. We'll return to Quantas 28 later.

How to project a globe onto a flat map?

There are many ways of doing this. In order to visualize the distortions created by projections, let's imagine walking from pole to pole along the prime meridian and spray painting 1000-km-diameter circles at regular intervals. Here's what the globe would look like from far away.

A globe with circles painted at regular intervals on the prime meridian

The picture below shows one way of projecting the globe onto a rectangular world map. First the parallels (latitude circles) are projected onto a cylinder at equal intervals. The equator stays the same size, but as we go toward either pole the parallels need to get larger and larger compared to their sizes on the globe. Then the cylinder is sliced down one of the meridians, say 180W, and unrolled.

How parallels on a globe can be projected at equal intervals onto a flat rectangular map

Here's the map we get, showing how the circles we painted get warped. North-south distances are correctly proportioned, but east-west distances get progressively more stretched as we go toward either pole. Greenland is badly deformed; its northern coast is stretched about three times as much as its southern coast.

When parallels on a globe are projected at equal intervals onto a flat rectangular map, east-west distances are increasingly stretched as one moves toward either pole.

We can address the shape deformation by projecting the parallels onto the cylinder at unequal intervals. If we do this just right, we stretch the north-south distance just as much as the east-west distance so that the circles we painted are still circles. At least in local regions, things maintain their correct shapes. This is called the Mercator projection:

Parallels on a globe are projected at unequal intervals onto a flat rectangular map, so that both east-west and north-south distances are increasingly stretched as one moves toward either pole.

Features are much less misshapen, but we pay a price. Now Greenland has an area about as large as Africa. In reality Africa is about 14 times larger.

The Mollweide projection uses an oval map and projects the circles to keep their areas equal The price is a little bit of north-south stretching at the equator and east-west stretching near the poles. It's an aesthetic compromise.

Parallels on a globe are projected onto a flat oval map, so that circles are distorted slightly but maintain their original areas

There are zillions of possible projections. There's no way of projecting a spherical surface onto a plane without introducing some kind of distortion.

Here's one more. We project the globe onto a flat circular disk centered at the North Pole. Parallels become concentric circles and meridians become spokes. We can do this so that north-south distances, which are close to 69 miles per degree of latitude, are preserved.

Parallels on a globe are projected onto a flat circular disc with the North Pole at its center

Here is the result. East-west distances are increasingly stretched as we go out toward the Antarctic rim. The South Pole, a point on the globe, is now a circle with a circumference of 76,906 miles.

When parallels on a globe are projected onto a flat circular disc with the North Pole at its center, east-west distances are severely stretched as one moves to the periphery

Fly Quantas, but not on a flat Earth!

On a spherical globe, the shortest route between two cities on the equator follows the equator itself. The equator is an example of a great circle, which is a circle that divides the globe into hemispheres. By symmetry, the shortest path between any two cities is the great circle that connects them. The website greatcirclemap.com allows us to visualize the 7000-mile great circle route between Santiago and Sydney. Here is the viewpoint of an observer in the plane of that great circle. You can click on this image and play around with the viewpoint on the website. Quantas 28 has a flight time just under 14 hours, consistent with the 7000-mile distance at typical 500 mph aircraft speeds.

Great circle flight path from Santiago to Sydney

A good way to visualize great circles is to pick two points on a globe and stretch a piece of dental floss as tightly as possible between the two points.

Just for comic relief, suppose that the flat disk is an accurate map of the world. The shortest path from Santiago to Sydney is a straight line. This would be a 16,000-mile, 32-hour journey that passes over the western United States. Bring plenty of snacks and games.

Straight line flight path from Santiago to Sydney on a circular disc projection