Vincenty Inverse Method — Complete Numerical Example

1) Input coordinates

Start: Les Sables-d’Olonne
φ₁ = 46.494953° → 0.81149001541 rad
λ₁ = −1.792091° → −0.03128378623 rad

End: Saint-François (Guadeloupe)
φ₂ = 16.252360° → 0.28365719322 rad
λ₂ = −61.273320° → −1.06942707541 rad

$$L = \lambda_2 - \lambda_1 = -1.03814328918$$
L = initial longitude difference (radians)

2) WGS-84 ellipsoid

$$a = 6378137\ \text{m},\quad f = \frac{1}{298.257223563},\quad b = 6356752.314245\ \text{m}$$
a = equatorial radius, b = polar radius, f = flattening

3) Reduced latitudes

$$\tan U_i = (1-f)\tan\varphi_i$$
U = latitude on auxiliary sphere (accounts for flattening)
$$U_1 = 0.80981293556,\quad U_2 = 0.28275610843$$

4) Iterative solution for λ (full convergence table)

$$\lambda_0 = L = -1.0381432891827342$$
Initial guess: longitude difference on the auxiliary sphere
Iter λ (rad) Δλ (rad) sin σ cos σ σ (rad) sin α cos² α cos 2σₘ
0 -1.0381432891827342 — — — — — — —
1 -1.0404171135171536 -2.2738243344e-03 0.839510 0.543310 0.993219 -0.672447 0.547387 -0.217723
2 -1.0404214142043005 -4.3006871469e-06 0.843102 0.537903 1.001753 -0.676899 0.542038 -0.210051
3 -1.0404214223337993 -8.1294952e-09 0.843553 0.537045 1.003865 -0.677215 0.541379 -0.209354
4 -1.0404214223491663 -1.5367000e-11 0.843553 0.537045 1.003866 -0.677215 0.541379 -0.209354
5 -1.0404214223491954 -2.9087843e-14 0.8435532581 0.5370455295 1.00386554952 -0.67721538895 0.54137931697 -0.20935377160
Δλ = λₙ − λₙ₋₁. Convergence criterion: |Δλ| < 10⁻¹² rad (achieved at iteration 5).

5) Distance computation

$$ u^2 = \cos^2\alpha \frac{a^2 - b^2}{b^2} = 0.00364862414 $$
$$ A = 1.00091153296,\quad B = 0.00091049548 $$
$$ \Delta\sigma = -0.00016088012 $$
$$ s = bA(\sigma - \Delta\sigma) $$
$$ s = 6\,388\,165.05\ \text{m} = 6\,388.17\ \text{km} $$
$$ s_{NM} = \frac{s}{1852} = 3\,449.33\ \text{NM} $$

6) Bearings

$$ \alpha_1 = \operatorname{atan2} (\cos U_2 \sin\lambda,\; \cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos\lambda) $$
$$ \alpha_2 = \operatorname{atan2} (\cos U_1 \sin\lambda,\; -\sin U_1 \cos U_2 + \cos U_1 \sin U_2 \cos\lambda) $$
$$ \alpha_1 = 259.11^\circ,\quad \alpha_2 = 224.85^\circ $$
Initial course ≈ west-southwest; final course at destination

Final result