How to use the rhumb line distance and bearing calculator
- Enter the latitude and longitude of the start, in decimal degrees. North and east are positive, south and west negative.
- Enter the latitude and longitude of the end.
- Choose kilometers, miles or nautical miles.
- Read the rhumb line distance, the constant compass bearing, and the great circle distance for comparison.
Formula
Stretched latitude: psi = ln(tan(pi / 4 + latitude / 2))
Bearing = atan2(dLon, dPsi), with dLon taken the short way around the globe
q = dLat / dPsi (the cosine of the latitude when both latitudes are equal)
Distance = R x sqrt(dLat^2 + q^2 dLon^2), with R = 6371.0088 km, the mean radius of the Earth
A rhumb line cuts every meridian at the same angle, so it appears as a straight line on a Mercator map.
London to New York and two shorter trips
From London (51.5074, -0.1278) to New York (40.7128, -74.006) the rhumb line is 5,794.129 km on a constant bearing of 258.04 degrees (west-southwest). The great circle is 5,570.23 km, so the rhumb line is 223.9 km, or 4.02%, longer. Along the equator through 90 degrees of longitude both are 10,007.56 km and the bearing is due east.
From Sydney to Auckland the rhumb line is 1,166.876 nautical miles on a bearing of 98.82 degrees, 2.782 nautical miles more than the great circle, and across the date line from 170 degrees east to 170 degrees west at 35 degrees north it is 1,821.714 km due east.
Rhumb lines and great circles
The shortest path between two places on a sphere is a great circle, but to follow it you must keep changing your compass heading. A rhumb line is a steady heading: easy to steer and to plot on a Mercator chart, but a little longer unless you travel along the equator or a meridian. Ships and light aircraft often sail rhumb lines for short and medium trips, and use great circles for long ocean crossings.
The difference grows with distance and with latitude. For the shortest route, use the great circle distance calculator; for the accurate distance on an ellipsoid, the Vincenty distance calculator. To find where a bearing and distance take you, see the destination point calculator.
Reading the numbers
The bearing is measured clockwise from true north, from 0 to 360 degrees, and the compass point beside it is the nearest of 16. Compass readings differ from true bearings by the local magnetic variation; the compass variation calculator handles that. To convert a bearing to other formats, use the bearing and azimuth converter.
The Earth is treated as a sphere with a radius of 6371.0088 km, so results can differ from an ellipsoidal model by up to about half a percent. The page does not accept the exact poles, where all rhumb lines spiral and the formula breaks down.
Assumptions and limits
- The Earth is a sphere with a mean radius of 6371.0088 km.
- Longitude differences over 180 degrees are taken the short way across the date line.
- Latitudes of exactly 90 or -90 degrees are not accepted.
- Miles and nautical miles use 0.621371192 and 0.539956803 per kilometer.
Frequently asked questions
What is a rhumb line?
A path of constant compass bearing. It crosses every meridian at the same angle.
Is a rhumb line the shortest route?
No. The great circle is the shortest, except along the equator and meridians, where both are the same.
How do I enter west and south?
Use negative numbers: for example, 74.006 degrees west is -74.006.
Why does the page show the great circle too?
So you can see how much longer the steady-bearing route is.