earth curve calculator

What this earth curve calculator does

This tool estimates how Earth’s curvature affects long-distance visibility. You can enter a distance, your eye height above the surface, the height of a distant object, and an optional atmospheric refraction factor. The calculator then reports:

• Curvature drop over the entered distance
• Your horizon distance
• Target horizon distance
• Combined line-of-sight limit
• Required target height to be just visible
• Estimated hidden height (if any)

Core geometry behind the calculation

Earth is modeled as a sphere with mean radius R = 6,371,000 meters. For a surface distance d, the exact curvature drop from a local tangent is:

drop = R × (1 − cos(d / R))

For short ranges, this closely matches the common approximation:

drop ≈ d² / (2R)

Both are included so you can compare the exact and approximate values.

How visibility and hidden height are estimated

1) Horizon distance from observer height

Given observer height h, the arc distance to the horizon is:

s = R × arccos(R / (R + h))

2) Combined visibility limit

If both observer and target have height, the target can be seen when the total separation is less than or equal to:

observer horizon + target horizon

3) Required target height

When the target is farther than the observer’s horizon, the tool computes the minimum target height needed to become visible. Any shortfall is reported as hidden height.

What atmospheric refraction changes

Light bends slightly downward in the atmosphere, effectively increasing Earth’s optical radius. This is handled by replacing R with an effective radius:

R_eff = R / (1 − k)

With standard conditions (k ≈ 0.13), distant objects may appear slightly higher than they would in a no-refraction vacuum model.

Example use cases

• Photography: Estimate whether a skyline peak should be visible from a distant shoreline.
• Marine navigation: Approximate when a lighthouse top appears above the horizon.
• Survey planning: Evaluate required mast or antenna height for line-of-sight links.
• Education: Explore spherical geometry and the impact of atmospheric refraction.

Common mistakes to avoid

• Mixing units (miles with meters, feet with kilometers) without conversion.
• Assuming flat-Earth distance formulas for long-range visibility.
• Ignoring observer and target heights in visibility checks.
• Using one fixed refraction value for all weather conditions.

Quick interpretation guide

If hidden height is 0

The target top is predicted to be at or above the geometric horizon line under your selected settings.

If hidden height is positive

That amount of the object is below the horizon from your viewpoint. Increasing observer height, target height, or using stronger downward refraction can reduce hidden height.

Final note

This calculator provides a clean geometric baseline and is excellent for planning and estimation. For precision-critical work, combine these results with local terrain, elevation profiles, instrument height offsets, and measured atmospheric conditions.