Great circle vs rhumb line: why the shortest route looks longer
The great circle is the shortest path on a globe, the rhumb line a route of constant bearing. We explain why planes fly along a curve and how to check it.
Two routes between the same two points
Between Warsaw and New York we can compare two routes: a great-circle route and a rhumb line. Their lengths do not depend on the chosen projection. Only the appearance of the lines on the map changes.
A great-circle route follows the shorter arc of a circle whose centre is the centre of the sphere. It is the shortest path over that surface. A rhumb line, also called a loxodrome, maintains a constant angle to geographic north. It is no shorter than the great-circle route. The two routes can coincide along the equator or a meridian.
Which route is shortest on a sphere?
The rhumb line: constant bearing and straight on Mercator
On a standard Mercator map, a rhumb line is straight. Its constant angle to the meridians can be read from the map, which helps with navigation planning. This bearing is measured from geographic north. Simply holding a fixed magnetic compass reading is not enough, because the difference between magnetic and geographic north varies with location.
Not every rhumb line is a spiral. Due east or west it follows a parallel, and due north or south it follows a meridian. A rhumb line with an oblique bearing, extended towards a pole, winds around it in a spiral. Its length difference from the great-circle route depends on both endpoints and the route direction, not just distance from the equator.
The great circle: same path, different appearance
On a Mercator map, a great-circle route is usually curved. Routes along the equator and meridians are straight, however. For Warsaw and New York the great-circle route bends north on this map, but the appearance of an arc does not mean a longer journey.
On the gnomonic projection, every displayed great-circle segment is straight. This map covers less than a hemisphere. The orthographic projection resembles a globe, but only great circles passing through the centre of the view appear straight. On the sphere itself, the route remains an arc. A straight segment between its endpoints would pass through the interior of the sphere.
What is the difference for Warsaw and New York?
For the city points used in our tool, the great-circle route is about 6,854 km and the rhumb line about 7,349 km. That is about 495 km, or 7.2% extra relative to the great-circle route. These results use a sphere with a radius of 6,371 km and coordinates of 52.23°N, 21.01°E for Warsaw and 40.71°N, 74.00°W for New York. They are not distances between particular airports.
This difference does not automatically translate into a set number of minutes or litres of fuel. A real flight plan also considers wind, weather, available airspace and air traffic control requirements.
Test it yourself
Open the great circle and rhumb line comparison for Warsaw and New York. Switch projections and watch both lines. The route lengths stay the same even though their appearance on screen changes. Then choose other cities or your own points on the map and compare the results.
The example distances were calculated using the model in the route lab.