Table of Contents
Earth Curvature
The Earth is approximately spherical with a mean radius of 6,371 km. This curvature means that over long distances, the surface drops away from a flat tangent plane. Understanding Earth's curvature is essential for surveying, navigation, long-distance shooting, bridge and tunnel construction, and telecommunications tower placement. The drop is surprisingly significant even over moderate distances.
For short distances, the curvature is negligible (about 8 cm per km). But over longer distances, the effect becomes substantial: over 10 km, the drop is nearly 8 meters, and over 100 km, the surface has dropped almost 800 meters below a horizontal line. This is why ships disappear hull-first over the horizon and why tall buildings are visible from much farther away than ground-level features.
Formula
Where R is Earth's radius (6,371 km) and d is the horizontal distance. The approximation works well for distances much smaller than Earth's radius. For 1 km: drop = 0.0784 m. A common rule of thumb: drop in feet approximately equals 0.667 times the distance in miles squared.
Curvature Drop Table
| Distance | Drop |
|---|---|
| 1 km | 0.078 m (3.1 in) |
| 5 km | 1.96 m (6.4 ft) |
| 10 km | 7.85 m (25.7 ft) |
| 50 km | 196 m (643 ft) |
| 100 km | 785 m (2,575 ft) |
Applications
- Surveying: correcting level measurements over long distances
- Bridge construction: accounting for curvature in span design
- Telecommunications: line-of-sight tower placement
- Long-range ballistics: trajectory correction for curvature
FAQ
Is the drop formula exact?
The approximation d²/(2R) is excellent for distances up to hundreds of kilometers. For extreme distances, the exact formula using the Pythagorean theorem should be used. At 100 km, the approximation error is only about 0.01%, well within practical requirements for almost all engineering applications.
How does refraction affect observations?
Atmospheric refraction bends light downward, partially compensating for curvature. Standard refraction extends the visible range by about 8%, or equivalently reduces the apparent curvature. Surveyors use a refraction coefficient of about 0.13, meaning the apparent drop is about 87% of the geometric drop under normal atmospheric conditions.
Why can I sometimes see objects beyond the geometric horizon?
Superior mirages caused by temperature inversions can bend light around the curvature more than normal refraction. In extreme cases, objects well beyond the geometric horizon become visible. This phenomenon is called looming and is common in polar regions and over cold ocean surfaces where temperature inversions are frequent and strong.