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DISTANCE / HEIGHT / SIGHT LINE

Earth curvature calculator

Set surface distance and eye height. The result separates geometric drop from the target height hidden beyond the observer’s horizon.

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Earth curvature and hidden height

Distance sets geometric drop; eye height and optional k change hidden height. Refraction k is assumed, not measured weather.

Unit changes preserve the same physical inputs. The working tape stays in SI units.

km

Longer distances increase curvature drop and, beyond the eye horizon, hidden height.

m

A higher observer reduces hidden height but does not change tangent-plane drop at that distance.

Zero means straight rays. Positive k bends rays toward the surface; this constant-gradient scenario does not model a mirage.

Geometric tangent-plane drop31.39m

A different quantity from hidden height: vertical displacement below the observer’s local tangent plane.

Hidden target height17.54 m

Minimum radial height at the target needed to meet the limiting ray; zero before the observer’s horizon.

Surface arc to observer’s horizon5.05 km

An arc length, unlike the straight tangent length in the first instrument.

Working tape
  1. 1 − refraction coefficient1 − 0 = 1
  2. Effective spherical radius R/(1 − k)6,371,008.4 ÷ 1 = 6,371,008.4
  3. Surface distance in metres20 × 1,000 = 20,000
  4. Distance ÷ effective radius20,000 ÷ 6,371,008.4 = 0.003139
  5. Effective radius + eye height6,371,008.4 + 2 = 6,371,010.4
  6. Radius ÷ raised eye radius6,371,008.4 ÷ 6,371,010.4 = 1
  7. Angle to the eye horizonacos(1) = 0.000792
  8. Target angle beyond the eye horizon0.003139 − 0.000792 = 0.002347
  9. Keep zero before the horizonmax(0, 0.002347) = 0.002347
  10. Cosine of remaining anglecos(0.002347) = 0.999997
  11. Reciprocal cosine1 ÷ 0.999997 = 1.000003
  12. Secant minus one1.000003 − 1 = 0.000003
  13. Minimum target height above surface6,371,008.4 × 0.000003 = 17.544924
  14. Surface distance ÷ physical Earth radius20,000 ÷ 6,371,008.4 = 0.003139
  15. Cosine of full surface anglecos(0.003139) = 0.999995
  16. One minus cosine1 − 0.999995 = 0.000005
  17. Drop below local tangent plane6,371,008.4 × 0.000005 = 31.392179
  18. Effective radius × dip angle6,371,008.4 × 0.000792 = 5,048.170652
  19. Observer surface-arc horizon5,048.170652 ÷ 1,000 = 5.048171

What the results measure

At 20 km, the geometric drop below a local tangent is 31.39 m. Eye height and the refraction assumption affect a different result: hidden target height.

Geometric drop is the Earth’s surface below the horizontal plane tangent at the observer. Changing camera height or k does not change it.

Hidden target height is the minimum height at the distant location that meets the limiting ray from the observer. Eye height and the assumed k can change it.

Horizon distance is surface arc length from the observer to that limiting point. It is not the tangent’s straight-line length.

Read the horizon investigation and its sources
Compare the historic Bedford Level sight lines