Where two courses cross

The point where two courses from two positions meet, on geodesics or rhumb lines, and how far each has to run to get there.

Planning and education aid. Not for primary navigation. Full disclaimer

Experimental: not yet fully verified. How results are checked

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

145,044.023m

The courses cross 145,044.023 m along the first and 60,092.661 m along the second.

Run from the second position
Crossing latitude
Crossing longitude

Provenance

Computed by: navigation.route.course-intersection 1.0.1, core 0.1.0
Model: Geodesic: Newton's method on the two distances, x1(s1) = x2(s2), using each geodesic's direction at its current end (Karney 2013 direct and inverse), started from the great-circle crossing on the mean sphere; of the two crossings the one ahead of both positions is reported. Rhumb: both are straight lines in Mercator coordinates (longitude, isometric latitude) and meet exactly
Accuracy: The geodesic crossing closes to under a nanometer; the rhumb crossing is exact to floating point
Notes: 1 shown with the answer
Cites: Karney, C. F. F., Journal of Geodesy, Algorithms for geodesics

Something look off?

Your values

Showing an example. Change anything.

First position latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees First position longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees First course True, like 045 deg · plain numbers mean degrees Second position latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees Second position longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees Second course True, like 315 deg · plain numbers mean degrees Ellipsoidoptional wgs84 (default), grs80, clarke1866, intl1924, airy1830, bessel1841, or krassovsky1940

More options 3

Line typeoptional geodesic (the shortest path, the default) or rhumb (constant course) Custom semi-major axisoptional With inverse flattening, like 3396190 m (Mars) · plain numbers mean m Custom inverse flatteningoptional Like 169.894; 0 for a sphere

Run many at once from a CSV

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How we got thisFormula, worked example, sources, and proof

Model: Geodesic: Newton's method on the two distances, x1(s1) = x2(s2), using each geodesic's direction at its current end (Karney 2013 direct and inverse), started from the great-circle crossing on the mean sphere; of the two crossings the one ahead of both positions is reported. Rhumb: both are straight lines in Mercator coordinates (longitude, isometric latitude) and meet exactly

Show your work

Run along the second course
the distance that closes the gap between the two courses
course 135.0° from 43.6600°, -70.2600° = 60,092.661 m
Run along the first course
the matching distance on the first course
course 45.0° from 42.3600°, -71.0000° = 145,044.023 m

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

Accuracy: The geodesic crossing closes to under a nanometer; the rhumb crossing is exact to floating point

Worked example: Two courses across the Gulf of Maine. Source: Worked on the ellipsoid: both runs checked by carrying each course its distance with the geodesic direct problem. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.

You enter

First course: 45 deg
Second course: 135 deg
First position latitude: 42.36 deg
Second position latitude: 43.66 deg
First position longitude: -71 deg
Second position longitude: -70.26 deg

You get

Run from the first position: 145,044.023 m
Run from the second position: 60,092.661 m
Crossing latitude: 43.2763375°
Crossing longitude: -69.7365371°

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: 8 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.

Experimental means this tool has not yet met the stable bar: at least 20 golden vectors, differential tests, and an independent worked example.