1) Input coordinates
Les Sables-d’Olonne
φ₁ = latitude of start point (north positive)
λ₁ = longitude of start point (east positive)
Saint-François (Guadeloupe)
φ₂ = latitude of end point
λ₂ = longitude of end point
$$L = \lambda_2 - \lambda_1$$
L = longitude difference between the two points (initial estimate)
3) Reduced latitudes
$$\tan U_i = (1-f)\tan\varphi_i$$
U = reduced latitude (latitude on a reference sphere)
φ = geographic latitude
Used to simplify ellipsoid geometry
$$U_1,\ U_2$$
U₁ = reduced latitude of start point
U₂ = reduced latitude of end point
4) Iterative solution for longitude λ
$$\lambda_0 = L$$
λ = longitude difference on auxiliary sphere (iterated value)
$$
\sin\sigma =
\sqrt{(\cos U_2 \sin\lambda)^2 +
(\cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos\lambda)^2}
$$
σ = angular distance between points on the sphere
sinσ = sine of that distance
$$
\cos\sigma =
\sin U_1 \sin U_2 +
\cos U_1 \cos U_2 \cos\lambda
$$
cosσ = cosine of angular distance
$$\sigma = \operatorname{atan2}(\sin\sigma,\cos\sigma)$$
σ = spherical arc length (radians)
$$
\sin\alpha =
\frac{\cos U_1 \cos U_2 \sin\lambda}{\sin\sigma}
$$
α = azimuth of geodesic at equator crossing
sinα = how tilted the path is east-west
$$\cos^2\alpha = 1 - \sin^2\alpha$$
cos²α = squared north-south component of the path
$$
\cos2\sigma_m =
\cos\sigma -
\frac{2\sin U_1 \sin U_2}{\cos^2\alpha}
$$
σₘ = midpoint of geodesic on the sphere
cos2σₘ = used for ellipsoid corrections
| Iter | λ | Δλ | sinσ | cosσ |
σ | sinα | cos²α | cos2σₘ |
| 0 | -1.038143289 | — | — | — | — | — | — | — |
| 1 | -1.040417114 | -2.27e-3 | 0.83951 | 0.54331 | 0.99322 | -0.67245 | 0.54739 | -0.21772 |
| 2 | -1.040421414 | -4.30e-6 | 0.84310 | 0.53790 | 1.00175 | -0.67690 | 0.54204 | -0.21005 |
| 3 | -1.040421422 | -8.13e-9 | 0.84355 | 0.53705 | 1.00387 | -0.67722 | 0.54138 | -0.20935 |
| 4 | -1.040421422 | -1.5e-11 | 0.84355 | 0.53705 | 1.00387 | -0.67722 | 0.54138 | -0.20935 |
| 5 | -1.040421422 | -2.9e-14 | 0.843553 | 0.537046 | 1.003866 | -0.677215 | 0.541379 | -0.209354 |
Δλ = change between iterations → convergence when very small
5) Distance computation
$$u^2 = \cos^2\alpha \frac{a^2-b^2}{b^2}$$
u² = ellipsoid shape factor
$$A,\ B$$
A = scale factor
B = correction factor
$$\Delta\sigma$$
Correction to spherical distance
$$s = bA(\sigma - \Delta\sigma)$$
s = true geodesic distance on ellipsoid (meters)