Model long range sightlines across curved terrain. Switch units, adjust Earth radius, and refraction factor. See clearance instantly, then download a report anytime here.
| Scenario | Observer height | Target height | Distance | k | What to expect |
|---|---|---|---|---|---|
| Two people on flat shore | 2 m | 2 m | 10 km | 0.13 | Small drop; borderline clearance. |
| Observer on tower | 30 m | 2 m | 25 km | 0.13 | Often visible; check minimum clearance. |
| Boat to lighthouse | 5 m | 40 m | 35 km | 1/7 | Usually visible if atmosphere is stable. |
| No refraction geometry | 10 m | 10 m | 40 km | 0 | More blockage; larger hidden height. |
Use the calculator to reproduce these cases and export your results.
The calculator models Earth as a sphere and uses an effective radius to approximate atmospheric refraction: Reff = R / (1 − k).
Line of sight is tested by sampling the straight ray between observer and target, then subtracting Earth curvature via the drop term to find the minimum clearance.
Line of sight over long distances is limited by Earth’s curvature. This tool compares an observer height, a target height, and a surface distance to estimate whether a straight viewing ray clears the curved surface. It is useful for towers, coastal viewing, and antennas.
For a spherical Earth, the surface falls away from the observer’s tangent line. The exact drop at distance x uses drop = R_eff − √(R_eff² − x²). For short ranges, the approximation drop ≈ x²/(2R_eff) is close. As reference, at 10 km drop is about 7.8 m when k=0.
Light bends slightly downward in typical atmospheres, effectively increasing the Earth radius. The calculator models this with R_eff = R/(1 − k). Common engineering values are k = 0.13 (standard) and k ≈ 1/7 for stable layers. Temperature inversions can raise k, while turbulent air can reduce it toward zero.
Each height has a horizon. With refraction included, surface horizon distance is d = R_eff · arccos(R_eff/(R_eff + h)). For example, an eye height of 2 m gives roughly 5 km to the horizon, while 30 m reaches near 20 km under typical k. Two endpoints can often see each other at about the sum of their horizons.
Even if the endpoints seem high enough, the lowest clearance can occur between them. The calculator samples points along the route and reports the minimum clearance and where it happens, helping you spot the tightest part of the sightline. A negative minimum indicates how much of the target is hidden below the curve.
Real views often include dunes, buildings, trees, or ridges. Enter an obstacle height and its distance from the observer to test if that object intersects the viewing ray. This is helpful for microwave links and line-of-sight surveying where one hill can dominate the result.
Heights may be entered in meters or feet, while distance can be kilometers, miles, or nautical miles. Use along-surface distance (map distance) for best consistency. If you know local Earth radius, or want to test sensitivity, you can override the default 6371 km. Keep inputs realistic: extreme k values can mislead.
'Clear line of sight' means the sampled clearance stays non-negative. If blocked, 'hidden height' tells how much additional target height is needed at minimum. Use the required-height outputs to plan towers, antennas, or viewing platforms with margin. For critical work, add clearance for terrain and refractive variability.
Use along-surface (map) distance between the points. For typical ranges, the difference from straight-line distance is small, but surface distance matches the curvature model and keeps results consistent with horizon formulas.
k is a refraction factor that increases the effective Earth radius. Higher k means light bends more downward, so the horizon appears farther. Use k=0 for purely geometric checks, and k≈0.13 for typical conditions.
The horizon-sum is a quick screening rule. The ray between endpoints can still dip closest to the surface somewhere in between. The calculator reports the minimum clearance location to show where blockage actually occurs.
It is good when distance is much smaller than the Earth radius. At tens of kilometers it is usually close, but the exact formula is safer, especially for long links or when you compare different refraction settings.
Hidden height is how much of the target would be below the curved surface along the tightest point of the path. If it is nonzero, increasing target or observer height by at least that amount can restore line of sight.
Enter the obstacle height above the surface and its distance from the observer. The calculator checks the viewing ray clearance at that point and reports whether the obstacle blocks the path.
Yes. Line-of-sight links are sensitive to curvature and refraction. Use conservative assumptions, add extra clearance, and consider terrain data and Fresnel zone requirements for engineering-grade link planning.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.