Primary sources · 3
- [1] Spherical trigonometry — Bowditch (American Practical Navigator) — The reference text for great-circle navigation, published by the U.S. National Geospatial-Intelligence Agency · NGA Pub. 9, 2019 edition · 2019 https://msi.nga.mil/Publications/APN
- [2] NGA.STND.0036_1.0.0_WGS84 — Defines the spherical reference radius derived from the WGS-84 ellipsoid · NGA Standard · 2014 https://earth-info.nga.mil/index.php?dir=wgs84&action=wgs84
- [3] ICAO Circular 328 — Modernizing the Global Air Navigation System — Describes Performance-Based Navigation and great-circle routing as the operational standard · ICAO · 2012 https://www.icao.int/publications
A great circle is the shortest path between two points on a sphere — and the closest spherical analogue to the geodesic that aircraft actually fly on the WGS-84 ellipsoid. The difference between the two is small enough that "great-circle distance" is the everyday name for what AirMilesCalc computes, even though the underlying math uses the ellipsoid.
What makes a circle "great"
A great circle is the intersection of a sphere with a plane that passes through the sphere's centre. The equator is a great circle; every meridian is a half great circle joined to its antimeridian. Any line drawn on a sphere that does not pass through the centre's plane is a small circle — a parallel of latitude (other than the equator) is the canonical example.
Why the great-circle path looks curved on a map
Mercator projections preserve angles — they were invented in 1569 for sailors who needed to read a constant compass course as a straight line. The price for that property is dramatic area distortion near the poles, and great-circle routes, which want to head north then turn back south on trans-hemispheric flights, look like dramatic arcs against the projection. On a globe the great-circle route is the straight line.
| Route | Great-circle (km) | Rhumb (km) | Detour |
|---|---|---|---|
| JFK → LHR | 5,555 | 5,597 | +0.8 % |
| LAX → SYD | 12,051 | 12,238 | +1.6 % |
| JFK → HKG | 12,983 | 15,231 | +17.3 % |
| LHR → AKL | 18,330 | 23,000 | +25.5 % |
| LHR → NRT (Tokyo) | 9,564 | 11,180 | +16.9 % |
How great-circle distance is computed
The spherical law of cosines gives a one-line answer: cos σ = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ, after which the surface distance is d = R · σ for some chosen Earth radius R. For small distances the law of cosines suffers floating-point cancellation; the Haversine formula is a numerically stable rearrangement that is preferred in production code. AirMilesCalc uses Haversine only as a fallback — Vincenty's ellipsoidal solution is preferred for everything else.
When great-circle is wrong
The great-circle and Vincenty distances disagree by up to 0.5 % because Earth is not a sphere. On a long polar route — like Helsinki → Tokyo — that error reaches roughly 40 km. For a flight plan, 40 km is enough fuel to matter; for casual reference, it is not. AirMilesCalc always reports the ellipsoidal number, and the "great-circle distance" label is a shorthand for the geodesic on WGS-84.