How to use this calculator
- Enter or choose points (latitude, longitude).
- The midpoint (lat, lon) updates as you type. Use Copy result or Share to keep it.
Method
For each point: x = cos φ cos λ, y = cos φ sin λ, z = sin φ; sum the vectors; φₘ = atan2(Z, √(X² + Y²)), λₘ = atan2(Y, X)
- φ, λ
- latitude and longitude
- X, Y, Z
- sums of the unit vectors
- φₘ, λₘ
- midpoint latitude and longitude
Worked example
- London → (0.62241, −0.00139, 0.78269); New York → (0.20885, −0.72865, 0.65227).
- Sum: X = 0.83127, Y = −0.73004, Z = 1.43496.
- Latitude = atan2(Z, √(X² + Y²)) ≈ 52.36844°; longitude = atan2(Y, X) ≈ −41.29031°.
- Each city is half the 5,570.23 km great-circle distance away: 2,785.115 km.
The great-circle halfway point between London and New York is about 52.3684°N, 41.2903°W, in the North Atlantic south of Greenland, about 2,785 km from each city.
What the result means
For two points the result is exactly halfway along the great circle. For several points it is the centre of their unit vectors projected back to the surface, which is a good “central meeting point” but not the point that minimises total travel distance.
Assumptions and limits
- Spherical Earth with mean radius 6,371.0088 km for distances.
- All points have equal weight.
- Undefined when the points balance out exactly (for example two antipodal points).
- For a flat x–y midpoint use the Distance Calculator instead.
Questions
Why is the midpoint of London and New York so far north?
The shortest route between them curves north on a flat map, so its halfway point lies north of both cities’ latitudes.
Can I use it for a plain x, y midpoint?
No; that is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2), which the Distance Calculator gives. This tool is for points on the globe.