Great-circle destination (spherical)
Where a great circle from a start point on a course takes you after a distance on a sphere, and how far that is from the ellipsoidal answer.
Go the other way: Great-circle distance and course (spherical) →
Planning and education aid. Not for primary navigation. Full disclaimer
Works on a sphere, not on the ellipsoid the Earth actually is. Use the geodesic tools for anything you act on: they are exact on WGS 84, and this tool reports how far the sphere puts it wrong for your inputs. Karney (2013), the geodesic algorithm the geodesic tools use.
The great circle ends at 46.0180795°, -64.8412441°, 2,073 m from where the ellipsoid puts it.
- Destination longitude
- Final course
- Distance from the ellipsoidal destination
Provenance
- Computed by
- navigation.geodesic.spherical-direct 1.0.0, core 0.1.0
- Model
- φ₂ = asin(sin φ₁ cos δ + cos φ₁ sin δ cos θ), λ₂ = λ₁ + atan2(sin θ sin δ cos φ₁, cos δ − sin φ₁ sin φ₂), with δ = distance ÷ R; the final course from the reverse course plus 180°. The ellipsoidal destination is Karney's direct on WGS 84
- Accuracy
- Exact on the sphere; the offset shows how far the sphere lands from the ellipsoidal destination
- Notes
- None
- Cites
- Moritz, H., International Association of Geodesy, Geodetic Reference System 1980 (mean radius R1 = (2a + b)/3); Karney, C. F. F., Journal of Geodesy, Algorithms for geodesics
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: φ₂ = asin(sin φ₁ cos δ + cos φ₁ sin δ cos θ), λ₂ = λ₁ + atan2(sin θ sin δ cos φ₁, cos δ − sin φ₁ sin φ₂), with δ = distance ÷ R; the final course from the reverse course plus 180°. The ellipsoidal destination is Karney's direct on WGS 84
Show your work
Central angle
δ = distance ÷ R1,000,000 m ÷ 6,371,008.771 m= 0.156961014 radDestination latitude
φ₂ = asin(sin φ₁ cos δ + cos φ₁ sin δ cos θ)φ₁ = 40°, θ = 45°= 46.0180795°
The same steps an agent gets from the MCP server with explain: true.
Accuracy: Exact on the sphere; the offset shows how far the sphere lands from the ellipsoidal destination
When to use this: Use this to reproduce where a great-circle calculation says you end up: from a point, on a course, for a distance, on a sphere. The companion to the spherical inverse, and for the same reason -- matching what other software produces rather than bettering it.
Limitations: The sphere, again deliberately. The `offset` says how far this destination is from the ellipsoidal one for the same course and distance, and it grows with distance rather than staying fixed. Going more than half a circumference wraps round the far side, which is correct and rarely what was meant. For the real destination use the geodesic direct tool.
Worked example: 1,000 km on course 045° from 40° N, 74° W. Source: Worked from the spherical formulas on R1, with the ellipsoidal end from the geodesic direct. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.
You enter
- Course
- 45 deg
- Distance
- 1000 km
- Start latitude
- 40 deg
- Start longitude
- -74 deg
You get
- Destination latitude
- 46.0180795°
- Destination longitude
- -64.8412441°
- Final course
- 51.2630499°
- Distance from the ellipsoidal destination
- 2,073 m
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 1.0.0, core 0.1.0. See this tool in the verification report.
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