Earth Curvature Line of Sight Calculator

Calculator Inputs

Observer height

Height above sea level or local ground.

Target height

Set to 0 for sea-level target.

Output distance unit

Affects horizon and LOS outputs.

Earth radius

Mean radius is 6371 km.

Atmospheric refraction

Enable effective radius model

Refraction coefficient (k)

Typical optical value ~0.13. 0.25 gives a 4/3 radius model.

Separation distance (optional)

Used for visibility, drop, and midpoint bulge.

Example Data Table

Observer height (m)	Target height (m)	Refraction k	Observer horizon (km)	Max LOS (km)	Visible at 50 km?
2	0	0.13	5.17	5.17	No
10	30	0.13	11.56	31.58	Maybe
50	50	0.25	29.16	58.32	Yes

Values are illustrative. Use the calculator for exact outputs with your selected units and radius.

Formula Used

Effective Earth radius with refraction

When refraction is enabled, the Earth behaves like it has a larger radius:

R_eff = R / (1 − k)

k is a refraction coefficient. k = 0 means no refraction.

Straight-line distance to the horizon

Using a right triangle from the Earth's center:

d = √(h(2R_eff + h))

Surface distance to the horizon (arc length)

Find the central angle and multiply by the effective radius:

θ = arccos(R_eff / (R_eff + h)), s = R_eff·θ

Curvature drop and midpoint bulge

For surface separation S, let φ = S / R_eff:

drop = R_eff(1 − cos φ), bulge = R_eff(1 − cos(φ/2))

How to Use This Calculator

Enter the observer height and select its unit.
Enter the target height, or set it to zero.
Keep the radius at 6371 km, or customize it.
Enable refraction if you expect standard bending.
Optionally enter separation distance to test visibility.
Choose your preferred output unit and calculate.
Download the results table as CSV or PDF.

Practical Article

1) Why Earth curvature matters for line of sight

Over long distances, the surface bends away from a straight viewing path. That bend creates a geometric horizon even on a clear day. This calculator estimates how far an observer can see, and whether a distant target is hidden by the horizon.

2) Choosing observer and target heights

Heights are the strongest inputs. A small increase near the ground can noticeably extend visibility. For example, raising an observer from 2 m to 10 m increases the horizon by several kilometers. Enter realistic values for eyes, antennas, towers, or mast heads.

3) Mean Earth radius and custom radii

The mean radius is about 6,371 km, which works well for general planning. You can also enter a custom radius if you prefer an equatorial or polar value, or if you are modeling another planetary body. Radius affects every distance and curvature value.

4) Atmospheric refraction and the k factor

Light typically bends slightly downward in the lower atmosphere, extending the apparent horizon. A common engineering shortcut is an effective radius model, where R_eff = R / (1 − k). A typical optical value is k ≈ 0.13, while k = 0.25 corresponds to the classic “4/3 Earth” approximation.

5) Straight-line horizon versus surface horizon

The tool shows both a straight-line distance to the tangent point and a surface (arc) distance. The straight-line number is useful for direct geometry checks, while the surface value is often better for maps, routes, and “distance over ground” comparisons.

6) Maximum visibility between two elevated points

If both observer and target are above the surface, their horizon distances add. This creates a maximum line-of-sight range, assuming a smooth surface and no obstacles. If you enter a separation distance, the calculator tests whether it is within that limit.

7) Curvature drop and midpoint bulge numbers

For a chosen separation S, the central angle is φ = S / R_eff. The “drop” is R_eff(1 − cos φ), which describes how much the surface falls below a local tangent. The midpoint “bulge” is R_eff(1 − cos(φ/2)), a useful estimate when comparing to a straight chord line.

8) Using units and interpreting results in practice

Choose kilometers, miles, nautical miles, or meters for distance outputs. Heights are converted internally to meters for stable math. Remember that terrain, buildings, waves, temperature layers, and instrument height definitions can dominate real-world results. Use this tool as a baseline, then add safety margin for route planning, surveying, and radio or optical links.

FAQs

1) What does “Visible” mean in the results?

It means the entered separation distance is less than or equal to the maximum surface line-of-sight computed from the two heights and the chosen refraction setting.

2) Should height be above sea level or local ground?

Use height above the surface you are looking across. For coastal views, sea level is common. For land views, use local ground elevation for both points when possible.

3) Why do horizon distances differ from other calculators?

Different tools may use different radii, refraction assumptions, and whether distance is along the surface or a straight line. Matching those choices usually brings results into close agreement.

4) What k value should I choose for refraction?

k ≈ 0.13 is a common default for standard conditions. For radio-style “4/3 Earth” modeling, use k = 0.25. For no refraction, turn it off.

5) Does this include hills, buildings, or unusual ducting?

No. This is smooth-surface geometry plus an optional standard refraction shortcut. Real visibility can be blocked or extended by terrain, structures, and strong atmospheric layers.

6) Why are drop and bulge shown in meters?

These are vertical distances, and meters keep them consistent and easy to compare. You can convert to feet by multiplying meters by 3.28084 if needed.

7) Can I use nautical miles for marine planning?

Yes. Select nautical miles for output distances, and enter separation in nautical miles if you like. One nautical mile equals 1,852 meters.