Curvature results
These values update after each calculation and appear above the input form.
Enter calculator values
Use the fields below to compare exact curvature, the common square-law estimate, and an optional refraction-adjusted model.
Plotly graph
The chart compares exact drop, the square-law estimate, and the refraction-adjusted drop across the selected range.
Formula used in this calculator
1) Exact drop below the tangent
For surface distance s and Earth radius R, the central angle is θ = s / R.
Exact drop = R × (1 − cos θ)
2) Common square-law approximation
For shorter distances, curvature is often estimated by the second-order approximation below.
Approximate drop ≈ s² / (2R)
3) Refraction-adjusted effective radius
Atmospheric refraction bends light downward and reduces apparent curvature. The calculator uses an effective radius.
R_eff = R / (1 − k), then Apparent drop = R_eff × (1 − cos(s / R_eff))
4) Horizon distance from eye height
Ground horizon distance and straight-line horizon distance are both reported for observer and target heights.
Ground horizon = R × arccos(R / (R + h)) and Line-of-sight horizon = √((R + h)² − R²)
How to use this calculator
- Enter the surface distance and choose the matching distance unit.
- Pick a radius preset or enter a custom radius with its unit.
- Set a refraction coefficient if you want an apparent-curvature estimate.
- Add observer and target heights to estimate horizon reach and visibility.
- Choose the main output unit plus graph range and graph step size.
- Press Calculate curvature to update the result panel, graph, CSV export, and PDF export.
Example data table
Example values below assume a mean Earth radius of 6,371 kilometers and a refraction coefficient of 0.13.
| Distance (mi) | Exact drop (ft) | Apparent drop (ft) | Square-law estimate (ft) | Notes |
|---|---|---|---|---|
| 1 | 0.6669 | 0.5802 | 0.6669 | Near the familiar eight-inch rule. |
| 2 | 2.6675 | 2.3207 | 2.6675 | Drop scales with distance squared. |
| 5 | 16.6719 | 14.5046 | 16.6719 | Refraction noticeably reduces apparent drop. |
| 10 | 66.6876 | 58.0182 | 66.6876 | Longer ranges magnify small model differences. |
Frequently asked questions
1) What does “curvature per mile” actually mean?
It usually refers to the drop below a starting tangent after traveling a chosen distance. The drop is not linear. It grows roughly with the square of distance, which is why the familiar rule uses miles squared rather than miles alone.
2) Why does the calculator show exact and approximate drop?
The exact model comes from circle geometry. The approximate model uses a small-angle shortcut, which is extremely close at short ranges. Showing both lets you see when the shortcut is sufficient and when exact geometry is better.
3) What is the “eight inches per mile squared” rule?
It is a rounded version of the approximation s² / (2R) when distance is in miles and drop is converted to inches. It is convenient for estimates, but exact geometry is cleaner for detailed comparisons.
4) Why include atmospheric refraction?
Light bends in the atmosphere, especially across long paths near the surface. That bending makes Earth appear slightly less curved than pure geometry predicts. The refraction option helps compare geometric drop with an apparent, observation-style estimate.
5) What do observer and target heights change?
They do not change Earth’s geometry. Instead, they affect horizon reach and whether a distant target may sit beyond the geometric horizon. Taller observer or target heights usually increase the distance over which the target remains visible.
6) Does this calculator work for kilometers too?
Yes. You can enter distance in miles, kilometers, meters, feet, or nautical miles. The calculator converts everything internally to meters, then reports results in your chosen output unit for consistent comparisons.
7) Why does the graph matter?
The graph makes the square-law growth obvious. It also helps you compare exact, approximate, and refraction-adjusted values across a full range instead of at one point. That is useful when you are testing multiple viewing distances.
8) Can this prove what a camera will see?
Not by itself. Real viewing conditions also depend on lens properties, elevation changes, weather layers, waves, haze, and obstacles. This page gives a geometry-based baseline, then adds a simple refraction model for comparison.