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DISTANCE / HEIGHT / SIGHT LINE

Earth curvature calculator

Set surface distance, eye height and target height. Compare tangent drop, hidden and visible portions, horizon dip, midpoint bulge and the maximum mutual sight line.

RUN THE NUMBERS

Earth curvature, horizon and target visibility

Set the observer and target heights, their surface separation and an assumed refraction coefficient. Tangent drop and chord bulge use the physical radius; sight-line results use the selected effective radius.

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.

m

Height of the target above the same smooth surface. Compare it with hidden height to find its visible part.

km

Mean Earth radius is 6,371.0084 km. A different radius changes physical drop, horizon and bulge together.

Hidden target height17.54m

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

Visible target height0 m

The entered target above the limiting ray; zero means the whole target is below that smooth-surface sightline.

Geometric tangent-plane drop31.39 m

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

Surface arc to observer’s horizon5.05 km

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

Straight distance to the horizon5.048 km

Tangent length from the eye, distinct from the horizon’s surface arc.

Horizon dip0.0454 °

Depression below local horizontal in the selected effective-radius scenario.

Maximum mutual sight-line range16.336 km

Observer and target horizons combined; terrain and variable atmospheric layers are not entered here.

Midpoint bulge above endpoint chord7.85 m

The physical surface’s maximum separation above the chord joining two surface endpoints.

Solid: effective-radius surface and heights · dashed: limiting sightline · vertical scale expanded
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
  20. Target distance from effective centre6,371,008.4 + 10 = 6,371,018.4
  21. Target tangent triangle ratio6,371,008.4 ÷ 6,371,018.4 = 0.999998
  22. Target horizon angleacos(0.999998) = 0.001772
  23. Target horizon surface arc0.001772 × 6,371,008.4 = 11,288.046834
  24. Sum of observer and target horizon arcs5,048.170652 + 11,288.046834 = 16,336.217486
  25. Maximum mutual sight-line range16,336.217486 ÷ 1,000 = 16.336217
  26. Horizon dip below horizontal0.000792 × 57.29578 = 0.045399
  27. Squared tangent distance factor6,371,010.4 − 6,371,008.4 = 2
  28. Sum of eye and surface radii6,371,010.4 + 6,371,008.4 = 12,742,018.8
  29. Straight tangent length squared2 × 12,742,018.8 = 25,484,037.6
  30. Straight tangent distance√(25,484,037.6) = 5,048.171709
  31. Straight tangent distance in kilometres5,048.171709 ÷ 1,000 = 5.048172
  32. Target height minus hidden height10 − 17.544924 = -7.544924
  33. Visible target heightmax(0, -7.544924) = 0
  34. Half the physical arc angle0.003139 ÷ 2 = 0.00157
  35. Cosine half-anglecos(0.00157) = 0.999999
  36. Midpoint sagitta factor1 − 0.999999 = 0.000001
  37. Physical surface above endpoint chord6,371,008.4 × 0.000001 = 7.84805

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
Compare the historic Bedford Level sight lines