Great-circle distance and course (spherical)

Model: Central angle σ = atan2(√((cos φ₂ sin Δλ)² + (cos φ₁ sin φ₂ − sin φ₁ cos φ₂ cos Δλ)²), sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ), distance R·σ; initial course atan2(sin Δλ cos φ₂, cos φ₁ sin φ₂ − sin φ₁ cos φ₂ cos Δλ), and the final course from the reverse course plus 180°. Compared with Karney (2013) on WGS 84

Show your work

1. Central angle

   σ = atan2(√(…), sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ)

   from 40.6413°, -73.7781° to 51.47°, -0.4543° = 0.869566994 rad

2. Distance

   $$R\times \sigma$$

   6,371,008.771 m × 0.869566994 = 5,540.019 km

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Exact on the sphere, at every distance (the atan2 form keeps full precision near 0 and 180°). On the Earth the sphere is off by up to about 0.5% in distance and a few tenths of a degree in course; both are reported

When to use this: Use this when you need the great-circle answer specifically: checking a figure from navigation software, a spreadsheet or a textbook, all of which use the spherical formulas. It reports the ellipsoidal distance beside its own and the gap between them, so you can see whether a discrepancy you are chasing is the sphere or a mistake.

Limitations: The sphere is the approximation and it is deliberate. Against the ellipsoid the distance is out by up to about half a percent, which on a transatlantic route is some fifteen kilometres, and the course by a few tenths of a degree; both are reported rather than described. For the real distance use the geodesic tool. Antipodal points have no unique great circle and coincident points no course.

Worked example: JFK to Heathrow on the mean-radius sphere. Source: Worked from the spherical formulas; the distance equals the haversine scenario's 5,540,019 m. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.

Review: Not yet independently reviewed by a geodesist.

Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.1, core 0.1.0. See this tool in the verification report.

Changes

• 2026-09-23, fixed: Short distances lost their meaning: a 0.859 m geodesic showed as 0.001 km. Distances in the navigation tools (version 1.0.1) still show to the meter in kilometers, but a short one now keeps three significant digits, down to the millimeter, so it shows as 0.000859 km. Differences such as the rhumb line's extra distance keep fixed decimals. Values are unchanged. Changelog

Checked against: 21 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

• Geodetic Reference System 1980 (mean radius R1 = (2a + b)/3), Moritz, H., International Association of Geodesy, Journal of Geodesy 74(1). pp. 128-133.
• Algorithms for geodesics, Karney, C. F. F., Journal of Geodesy, Vol. 87, No. 1. pp. 43-55.

Terms

IUGG: International Union of Geodesy and Geophysics

The scientific union that adopted GRS 80 and the mean Earth radius R1. Source: Geodetic Reference System 1980

R1: IUGG mean Earth radius

The radius of a sphere the size of the Earth on average, (2a + b) ÷ 3 = 6,371,008.771 m for WGS 84. Spherical formulas use it when they treat the Earth as a ball. Source: Geodetic Reference System 1980