Earth Curvature Calculator with Refraction

Example data table

These examples use a mean Earth radius and standard refraction. Real visibility can vary with temperature gradients, pressure, and terrain.

Formula used

This calculator uses the effective Earth radius method to model refraction: R_eff = R / (1 − k). Curvature drop over distance D is approximated by: drop ≈ D² / (2R) (no refraction) and drop_ref ≈ D² / (2R_eff) (with refraction).

The midpoint bulge (sagitta of the chord) is approximated by: bulge_mid ≈ D² / (8R_eff).

Refraction-adjusted horizon distance for height h is: d_h ≈ √(2R_effh). If D ≤ d_h1 + d_h2, the target is geometrically visible.

How to use this calculator

1. Enter the surface distance between observer and target.
2. Set observer and target heights above the local surface.
3. Choose a refraction coefficient, or keep k = 0.13.
4. Select an Earth radius preset, or enter a custom radius.
5. Pick output units, then press Calculate to view results.
6. Use CSV or PDF buttons to export the report.

Article

1) What “curvature drop” means

Over a surface distance D, a straight sightline and a curved Earth separate by a small amount. For small angles, the drop is approximated by drop ≈ D²/(2R). Using the mean radius R ≈ 6,371,000 m, the no‑refraction drop is about 0.785 m at 5 km, 3.14 m at 10 km, and 12.56 m at 20 km.

2) Why refraction changes the picture

Air density normally decreases with height, bending light slightly downward and extending visibility. This calculator uses the effective radius method: R_eff = R/(1−k). With a standard value k ≈ 0.13, the apparent curvature drop becomes about 87% of the no‑refraction value, because drop_ref = (1−k)·drop.

3) Typical k values and when they shift

Near‑standard conditions often sit around k = 0.10 to 0.20. Strong temperature inversions over cold water can push refraction higher, sometimes producing looming or mirages. Unstable hot ground can reduce k, shortening the visible range and increasing apparent drop in daytime heat.

4) Horizon distance from height

The refraction‑adjusted horizon distance for height h is d_h ≈ √(2R_effh). For h = 2 m and k = 0.13, the horizon is roughly 5.35 km. For h = 10 m, it grows to about 11.95 km. Add observer and target horizons to estimate maximum line‑of‑sight.

5) Midpoint bulge for long spans

If you stretch a straight chord between two equal‑height points, the Earth rises under the chord. The midpoint bulge is approximated by bulge ≈ D²/(8R_eff). With k = 0.13, bulge is about 0.214 m at 5 km and 0.857 m at 10 km. This is useful for bridge, laser, and long‑level checks.

6) Visibility and hidden height

The tool checks whether D ≤ d_h1 + d_h2. If not, it estimates the required target height to be seen, given the observer height. The difference between required height and actual height is reported as hidden height, an easy way to gauge how much of a distant object would be below the skyline.

7) Why Earth radius presets matter

Earth is slightly flattened, so radius depends on latitude. The equatorial radius is about 6,378,137 m, while the polar radius is about 6,356,752 m. Over a 50 km span, that difference can shift predicted drop by several centimeters, which matters for precision surveying.

8) Practical tips for better inputs

Use shoreline‑to‑shoreline distances rather than map straight‑line distances through hills. Keep heights referenced to the same local surface, not sea level, unless the terrain is flat. When conditions are unknown, compute with multiple k values (0.07, 0.13, 0.20) to bracket outcomes.

FAQs

1) What does k represent?

k is a refraction coefficient that approximates how the atmosphere bends light. Higher k means stronger downward bending, a larger effective Earth radius, and longer visibility.

2) Why is k often set to 0.13?

0.13 is a common “standard atmosphere” approximation used in surveying and radio horizon estimates. Real conditions can differ, so testing a range is recommended.

3) Does this include terrain and obstacles?

No. It is a geometric model. Mountains, buildings, trees, and haze can block sight even when the geometric visibility test says “visible.”

4) Is the curvature drop formula exact?

It is a small‑angle approximation that works well for common distances. For very long distances, a full spherical geometry model gives slightly different results.

5) Why can distant objects look taller than expected?

Strong inversions can increase refraction, producing looming and mirage effects. In those cases, the effective k may be much higher than 0.13.

6) Which radius preset should I choose?

Use mean radius for general work. Use equatorial or polar if you want to bracket extremes. Use custom when a local radius or planet size is known.

7) Can I use this for radio or microwave links?

Yes for first‑pass planning. Radio engineers often use an effective radius approach too, but also add Fresnel zone clearance, diffraction, and surface roughness models.