Highest point of a geodesic (vertex)

The vertex of a geodesic: the northernmost point it reaches, where it runs due east or west, and how far along the line from the start it lies.

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

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.

53.671067616°

The geodesic peaks at 53.671067616°, -22.988028393°.

Vertex longitude
Distance to the vertex
Between the two points
Azimuth at the equator

Provenance

Computed by: navigation.geodesic.vertex 1.0.1, core 0.1.0
Model: Clairaut's relation on the auxiliary sphere: sin α₀ = sin α₁ cos β₁ and σ₁ = atan2(sin β₁, cos α₁ cos β₁); the northern vertex is at σ = 90°, reached by the direct problem in arc length from the start
Accuracy: Exact to rounding: the azimuth at the vertex is 90° to within 1e-9°
Notes: None
Cites: Karney, C. F. F., Journal of Geodesy, Algorithms for geodesics

Something look off?

Your values

Showing an example. Change anything.

Start latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees Start longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees Second point latitudeoptional Or give an azimuth instead; decimal degrees, like 51.47 · plain numbers mean degrees Second point longitudeoptional Decimal degrees, east positive, like -0.4543 · plain numbers mean degrees Ellipsoidoptional wgs84 (default), grs80, clarke1866, intl1924, airy1830, bessel1841, or krassovsky1940

More options 3

Azimuthoptional Instead of a second point, like 51.38 · plain numbers mean degrees 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

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

Model: Clairaut's relation on the auxiliary sphere: sin α₀ = sin α₁ cos β₁ and σ₁ = atan2(sin β₁, cos α₁ cos β₁); the northern vertex is at σ = 90°, reached by the direct problem in arc length from the start

Show your work

Arc from the equator crossing
σ₁ = atan2(sin β₁, cos α₁ cos β₁)
β₁ = 40.546223497°, α₁ = 51.381647858° = 53.88664819°
Vertex latitude
direct problem to σ = 90°
36.11335181° of arc from the start = 53.671067616°

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

Accuracy: Exact to rounding: the azimuth at the vertex is 90° to within 1e-9°

When to use this: Use this to find the highest latitude a great-circle route reaches, and whether it reaches it on the way. A route from New York to London tops out at 53.7 degrees north, well above either end, which is what decides whether it crosses an area of operations, a weather band or an airspace you care about.

Limitations: The vertex need not lie between the two points: when it does not, the route never reaches that latitude, and the `within` field says so -- reading the latitude without it is the mistake this tool invites. Vertices repeat around the globe, and the one reported is the one nearest the start, which may be behind it at a negative distance. A geodesic along the equator has no vertex, and a meridian's vertex is the pole, where longitude names nothing.

Worked example: The JFK to London geodesic. Source: Karney (2013) auxiliary sphere; GeographicLib GeodSolve -a. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.

You enter

Start latitude: 40.6413 deg
Second point latitude: 51.47 deg
Start longitude: -73.7781 deg
Second point longitude: -0.4543 deg

You get

Vertex latitude: 53.671067616°
Vertex longitude: -22.988028393°
Distance to the vertex: 4,014,306.918 m
Between the two points: yes
Azimuth at the equator: 36.420816908°

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

Algorithms for geodesics, Karney, C. F. F., Journal of Geodesy, Vol. 87, No. 1. pp. 43-55.