Earth Curvature Calculator
Horizon, line of sight and hidden height
The geometric horizon is the point where the observer's line of sight just touches the spherical Earth. This line is a tangent. The higher the observer, the farther away the horizon lies.
Beyond the horizon, the Earth can hide the lower part of an object. The hidden height h1 is measured radially at the target, along the local perpendicular to the sphere's surface. It extends to the visibility threshold. At or before the horizon, h1 is zero.
Blue shows the selected measurement distance d0, the dashed line shows the horizon tangent, and orange shows the hidden height. In straight-line mode, the target's base lies radially below the measurement endpoint Q.
Exact spherical geometry for both distance definitions
R is the Earth's radius, h0 is the observer height and d0 is the input distance. All lengths in the formulas use the same unit; angles are measured in radians.
α = arccos(R / (R + h0))
ℓ = √(h0 · (2R + h0))
α is the horizon angle; ℓ is the straight-line distance from the observer to the horizon.
Straight tangent distance (default)
d0 runs from the observer along the horizon tangent to Q. With q = d0 − ℓ:
d1 = ℓ
h1 = 0 for q ≤ 0
h1 = √(R² + q²) − R for q > 0
Beyond the horizon, Q marks the visibility threshold at the target. Before the horizon, Q is only the measurement endpoint; the hidden height remains zero.
Surface arc
d0 is the arc length along the Earth's surface between the observer's base and the target's base.
d1 = R · α
δ = max(0, d0 / R − α)
h1 = R · (1 / cos(δ) − 1)
Both calculations are exact for their respective distance definitions. The code uses algebraically equivalent, numerically stable expressions, not approximation formulas. The same value of d0 describes different geometric situations in the two modes.
Example: 1 m observer height and an 80 km straight-line distance
For an observer height of 1 m, a straight tangent distance of 80 km and R = 6,371,000 m, the geometrically hidden height is 458.436 m (rounded to three decimal places).
The straight-line distance to the horizon is approximately 3,569.594 m. An object extending radially from the target's base would need to exceed the calculated hidden height for its top to rise above the geometric horizon line. This is a fixed example and does not change your inputs.
Model and limitations
The Earth is modelled as an ideal sphere with a radius of 6,371 km without accounting for atmospheric refraction.
Real observations may differ from the model because of terrain, the Earth's departure from a perfect sphere, atmospheric refraction and visibility conditions, among other factors. The calculated hidden height describes only the geometric situation.
This calculator supports observer heights of 0–500 km and target distances of 0–5,000 km. h1 is a radial height.