Point part way along a great circle
The point a given fraction of the way between two points along the great circle on a sphere, beside the same fraction along the ellipsoidal geodesic.
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 point is at 47.5784923°, -59.3177261°, 2,054 m from the same fraction along the ellipsoidal geodesic.
- Longitude
- Ellipsoidal latitude
- Ellipsoidal longitude
- Distance between the two
Provenance
- Computed by
- navigation.geodesic.intermediate-point 1.0.0, core 0.1.0
- Model
- Spherical linear interpolation of the unit vectors: A = sin((1 − f)δ) ÷ sin δ, B = sin(fδ) ÷ sin δ, p = A·p₁ + B·p₂, with δ the central angle; the ellipsoidal point is f × s12 along the Karney geodesic on WGS 84
- Accuracy
- Exact on the sphere; the offset shows the approximation. Undefined for antipodal points, where every great circle joins them
- Notes
- None
- Cites
- Karney, C. F. F., Journal of Geodesy, Algorithms for geodesics
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: Spherical linear interpolation of the unit vectors: A = sin((1 − f)δ) ÷ sin δ, B = sin(fδ) ÷ sin δ, p = A·p₁ + B·p₂, with δ the central angle; the ellipsoidal point is f × s12 along the Karney geodesic on WGS 84
Show your work
Central angle
δ between the two pointsfrom 40.6413°, -73.7781° to 51.47°, -0.4543°= 0.869566994 radInterpolated latitude
atan2(z, √(x² + y²)) of A·p₁ + B·p₂f = 0.25= 47.5784923°
The same steps an agent gets from the MCP server with explain: true.
Accuracy: Exact on the sphere; the offset shows the approximation. Undefined for antipodal points, where every great circle joins them
When to use this: Use this for a point a given fraction of the way along a route -- a waypoint every tenth, a reporting point, a place to draw a label. It returns both the spherical answer most code produces and the ellipsoidal one that is actually right, with the distance between them, so you can see what the approximation costs on this route.
Limitations: The two answers differ, and on a long route they differ by kilometres: the spherical one is exact on a sphere and the Earth is not one. Use the ellipsoidal pair unless you are reproducing someone else's spherical figure. Antipodal points have no unique path between them -- every great circle joins them -- so there is no intermediate point to give. And a fraction of the distance is not a fraction of the flight time, which depends on the wind.
Worked example: A quarter of the way from JFK to Heathrow. Source: Worked by spherical interpolation, with the ellipsoidal point from the geodesic direct at a quarter of the distance. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.
You enter
- Fraction of the way
- 0.25
- Start latitude
- 40.6413 deg
- End latitude
- 51.47 deg
- Start longitude
- -73.7781 deg
- End longitude
- -0.4543 deg
You get
- Latitude
- 47.5784923°
- Longitude
- -59.3177261°
- Ellipsoidal latitude
- 47.5963373°
- Ellipsoidal longitude
- -59.3106731°
- Distance between the two
- 2,054 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: 48 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.
Sources
- Algorithms for geodesics, Karney, C. F. F., Journal of Geodesy, Vol. 87, No. 1. pp. 43-55.