Great Circle Routes: Why the Shortest Path Looks Curved
Draw a straight line from New York to Tokyo on a world map and you have drawn the long way round. The shortest route leaves New York heading north-northwest.
The shortest line on a ball
A great circle is any circle drawn on a sphere whose centre is the centre of the sphere itself. Slice an orange straight through the middle, at any angle, and the cut edge is a great circle. On Earth the equator is one, and so is every meridian paired with its opposite meridian.
The shortest path between two points on a sphere always runs along the great circle that passes through both of them. Navigators call that path an orthodrome. Every other circle of latitude, such as the 50th parallel, is a small circle: its centre sits somewhere up the Earth's axis, not at the middle of the planet, and following it is never the shortest way.
Why it looks curved
Most world maps use the Mercator projection or something like it. Mercator has one famous property: a line of constant compass direction is drawn perfectly straight. That line is called a rhumb line, or loxodrome, and it is very convenient to steer, because you hold the same heading the whole way.
But a rhumb line is not the shortest route. To stay on a constant heading it has to cut every meridian at the same angle, and away from the equator that pulls it the long way round. The true shortest route, the great circle, therefore looks bent on the map, bowing towards the nearer pole. The map is not showing the route taking a detour; the map itself is stretched.
Worked example: New York to Tokyo
Both cities sit at about the same latitude, so on a flat map the obvious route runs almost due west across the Pacific.
- Straight line on a Mercator map: hold 268 degrees (just south of west) for about 12,790 km
- Great circle: set off at 333 degrees (north-northwest), cross Canada and Alaska, then curve down towards Japan
- Great-circle length: about 10,850 km
The great circle is roughly 1,940 km, or 15%, shorter than the flat-map line, which is 18% longer than the great circle. That is why flights between them head north over Canada and Alaska instead of straight out over the ocean.
On a great circle your heading keeps changing
Follow a great circle and the compass direction you are travelling in changes all the way. A flight from London to Los Angeles leaves London heading northwest, at about 312 degrees, passes over Greenland and northern Canada, and arrives travelling southwest, at about 214 degrees. It never turns; the direction changes because north itself is in a different place relative to you.
That is also why the direction back is not simply the opposite. From London, Los Angeles lies at 312 degrees; from Los Angeles, London lies at about 34 degrees, not 132.
How much the straight map line adds
Surface distances, rounded. The flatter and more east-west the route, and the further from the equator, the bigger the gap.
| Route | Great circle | Straight line on Mercator | Extra |
|---|---|---|---|
| New York to Tokyo | 10,852 km | 12,792 km | +18% |
| Buenos Aires to Sydney | 11,801 km | 13,827 km | +17% |
| Chicago to Moscow | 7,999 km | 9,219 km | +15% |
| London to Los Angeles | 8,756 km | 9,733 km | +11% |
| Madrid to Beijing | 9,222 km | 10,207 km | +11% |
| New York to Madrid | 5,768 km | 5,939 km | +3% |
Routes that run mostly north-south barely differ, because meridians are great circles themselves.
Sailors used both
Before computers, a constant compass course was far easier to follow than a curve whose heading changes every hour, so ships often simply sailed the rhumb line. On long ocean passages navigators split the difference: they plotted the great circle, broke it into several shorter rhumb-line legs, and changed course at the end of each one. Aircraft flight-management systems still do a refined version of the same thing.
Great circles in GeoAzimuth
Every arrow in the GeoAzimuth game follows a great circle. That is why, from Paris, the arrow to Tokyo points northeast at about 34 degrees rather than east: it points along the shortest route over the globe. When an arrow seems to point the wrong way, picture the curve from this lesson, not a straight line on a flat map.
Check what you learned
Five questions on great circles and rhumb lines. Pass at 70% or better to complete the lesson.
Practice with Games
Ready to test what you learned? Jump into a game and put it into practice.
Great circle routes, in one paragraph
A great circle is a circle on a sphere whose centre is the centre of the sphere, such as the equator or a pair of opposite meridians, and the shortest path between two points on Earth always lies along one. On a Mercator world map a straight line is a rhumb line, or loxodrome, which keeps a constant compass bearing but is longer than the great circle, often by 10 to 18% on long east-west routes. That is why great-circle flight paths look curved on flat maps, bowing towards the nearer pole: New York to Tokyo is about 10,850 km along the great circle, which leaves New York heading north-northwest, against about 12,790 km for the straight map line. Along a great circle the compass heading changes continuously, so the direction back is not simply the opposite of the direction out. Only the equator and the meridians are both great circles and rhumb lines, and only a gnomonic projection draws every great circle as a straight line.
Common questions about great circle routes
Why do planes fly curved routes?
They are not really curved. Planes fly the great circle, the shortest route over the globe, and it only looks curved because flat world maps stretch the Earth. On a globe the same route looks like a straight line pulled tight between the two cities.
What is the difference between a great circle and a rhumb line?
A great circle is the shortest route between two points on a sphere, but its compass heading changes as you go. A rhumb line keeps one compass heading the whole way and is drawn straight on a Mercator map, but it is longer. The two coincide only along the equator and the meridians.
Is the equator a great circle?
Yes. The equator is the only line of latitude that is a great circle, because it is the only one whose centre is the centre of the Earth. Every meridian together with its opposite meridian also forms a great circle. All other parallels are small circles.
How much shorter is a great circle route?
It depends on the route. For routes that run mostly north-south there is almost no difference. For long east-west routes at mid-latitudes the straight map line is often 10 to 18% longer: New York to Tokyo is about 10,850 km on the great circle against about 12,790 km on a constant bearing.
How do I draw a great circle on a map?
On a globe, stretch a piece of string tight between the two places. On paper, use a gnomonic chart, where great circles are straight lines, or compute points along the route with spherical trigonometry, as the interactive map on this page does, and join them up.