Three independent derivations of how much of a distant object drops below the horizon on a sphere. Pick a method, plug in numbers.
Horizon distance approximation:
| Locations (A–B) | Observer height (km) | Observed distance (km) | Hidden height, geometric (km) | Hidden height, optical (km) |
|---|---|---|---|---|
| Lanzarote (Uga) – Tenerife (Teide) | 0.4 | 300 | 4.2 | 3.37 |
| Barcelona – Mallorca | 0.5 | 200 | 3.5 | 2.81 |
| Valencia – Ibiza | 0.3 | 150 | 2.8 | 2.24 |
| Madrid – Toledo | 0.2 | 100 | 1.5 | 1.20 |
Optical hidden height uses the same standard-atmosphere refraction model as the Earth Horizon Simulator (sea-level pressure/temperature, standard optical lapse rate): coefficient k ≈ 0.198, effective Earth radius R′ = R/(1 − k) ≈ 7,947 km. Scaled here from the geometric figure by R/R′ ≈ 0.802, since these rows are illustrative real-world examples rather than direct outputs of the calculators above. Refraction bends light around the curve, so less of the target is actually hidden than the pure-geometric figure.