Earth Curvature Calculator
ConvertEverything runs locally in your browser — nothing is uploaded
Two horizons have to be kept apart. The geometric horizon is `sqrt(2Rh + h^2)`, the one with no atmosphere; the horizon with refraction is about 6 to 8 percent farther, because air density falls with height and light bends downwards, revealing ground beyond the straight line. Most arguments about seeing the far shore from the sea, or a building from a summit, come down to that 8 percent, so both figures are given. ⚠️ The drop formula `h = d^2/(2R)` is an approximation: it expands the arc as a parabola, is very accurate while d is far smaller than R (centimetres at 10 km) and loses accuracy as d grows. The radius is the mean radius, 6371 km, written as a named constant; the equatorial or polar radius would give different numbers. Everything runs in your browser.
Features
- Geometric horizon (`sqrt(2Rh + h^2)`) and the horizon with refraction given separately
- Refraction uses a factor of 1.08, an empirical 8 percent that varies with temperature and pressure
- The drop uses `h = d^2/(2R)` and is labelled an approximation that drifts at long distances
- Line of sight between two points: the sum of both horizons, geometric and refracted
- The height a target is hidden by, plus a visible or hidden verdict
- Three modes: horizon distance, hidden height, and line of sight
- The Earth radius is a named constant, 6371 km, the mean radius
- Everything runs locally and nothing is uploaded
How to use
- Pick a mode (horizon, hidden height or line of sight)
- Enter the observer height; add a target height to check visibility
- Enter the distance to see the drop and the verdict
- Copy any value with its own button
FAQ
- How much do the two horizons differ?
- About 6 to 8 percent. The geometric horizon assumes light travels in a straight line; in reality air density falls with height and light bends downwards, so a little more is visible. The tool gives both (multiplying by 1.08 for refraction), because arguments about just being able to see something usually come down to that 8 percent.
- How far can h = d^2/(2R) be used?
- It expands the arc as a parabola and assumes d is far smaller than R. At 10 km the error is centimetres, at 100 km it becomes noticeable, and beyond a few hundred kilometres it should be replaced by spherical trigonometry. The tool labels it an approximation for that reason.
- Why this radius of 6371 km?
- It is the mean radius of the Earth (IUGG, 6371.0088 km rounded). The equatorial radius of 6378 km and the polar radius of 6357 km would give different answers. Writing it as a named constant makes it verifiable which one is in use rather than scattering the number through the formulas.
- Why is a target sometimes visible and sometimes hidden?
- Because the curve keeps dropping the ground beyond the horizon. The observer first sees their own horizon, and every further x kilometres drops the ground by x^2/(2R) more; only the part of the target above that drop is visible. Change the distance, eye height or target height and the verdict changes.
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