Great-circle distance and bearing
The shortest path between two points on a sphere is a great circle, and it is not a straight line on most charts. This gives the distance along that path and the heading to start out on.
Calculator
This is an educational and planning aid. Use current official weather, your aircraft flight manual, and the rules that apply to your flight before you act on any figure here. See the aviation disclaimer.
What the result means
The distance is the shortest route over the surface of the Earth, which is what an aircraft with enough range would actually fly.
The initial bearing is the true bearing at the start. On a great circle the bearing changes continuously, which is why the final bearing on arrival is shown separately and is usually different.
On a short leg the two bearings are nearly identical. On a transatlantic leg they can differ by tens of degrees, which is the whole reason great-circle navigation exists.
When this is useful
- Estimating the length of a route between two airfields when you only have their coordinates.
- Understanding why long haul tracks curve north on a flat map.
- Checking a distance from a flight planning tool against an independent calculation.
How it is calculated
The haversine formula gives the angular separation between two points on a sphere, and multiplying by the radius of the Earth converts that angle into distance. Haversine is used in preference to the simpler spherical law of cosines because it stays accurate for very short distances where the other form loses precision.
The initial bearing comes from a separate spherical trigonometry expression. The final bearing is found by computing the initial bearing of the reverse leg and taking its reciprocal.
a = sin2(dLat/2) + cos(lat1) x cos(lat2) x sin2(dLon/2), distance = 2 x R x atan2(sqrt(a), sqrt(1-a))
Worked example
- From
- 53.4213 N, 6.2701 W (Dublin)
- To
- 51.4700 N, 0.4543 W (London Heathrow)
The great-circle distance is about 243 NM, or 450 km, on an initial true bearing of 117 degrees.
Check: The published great-circle distance between these two airports is about 449 km, which is 242 NM. The calculated figure agrees to well under one percent.
Limitations
- The Earth is modelled as a sphere of mean radius 6,371.0088 km. GPS uses the WGS-84 ellipsoid, and against that the error here stays below about half a percent, which is a few nautical miles on a very long leg.
- Bearings are true, not magnetic. Apply local variation before comparing them with a compass or a runway.
- It is a direct path with no regard for airspace, terrain, airways or diversion airfields.
Common mistakes
- Getting the sign of a coordinate wrong. South latitude and west longitude are both negative, and a dropped minus sign puts the position in the wrong hemisphere.
- Entering degrees, minutes and seconds into a field expecting decimal degrees. 53 degrees 25 minutes is 53.4167, not 53.25.
- Flying the initial bearing the whole way on a long leg, which drifts well off the great circle.
Sources
- Aviation Formulary , Ed Williams
- Calculate distance, bearing and more between latitude/longitude points , Movable Type Scripts, Chris Veness