Compare flat assumptions with curved Earth geometry. Set radius, distance type, and refraction coefficient quickly. See drop, horizon range, and hidden height instantly today.
This tool reports the sagitta (bulge) between a straight chord and a circular arc.
For a circle of radius R and chord length c:
s = R − √(R² − (c/2)²)
If your input distance is a surface (arc) distance L, we convert it to chord length:
c = 2R·sin(L/(2R))
Atmospheric refraction is approximated using an effective radius:
R_eff = R / (1 − k)
Horizon distance from height h uses d = √(2R_eff h + h²) (straight line), with a surface alternative using the central angle.
Standard radius, surface distance, k = 0.13. Drops shown in meters.
| Distance (km) | Drop (no refraction) (m) | Drop (with refraction) (m) |
|---|---|---|
| 1 | 0.01962 | 0.01707 |
| 5 | 0.490504 | 0.426738 |
| 10 | 1.962015 | 1.706953 |
| 20 | 7.84806 | 6.827812 |
| 50 | 49.050322 | 42.673793 |
The primary output is the sagitta, meaning the maximum “bulge” of a circular arc above the straight chord connecting two endpoints. It is a symmetric measure: the arc is compared to a straight line drawn between the endpoints, not a tangent line drawn from one end.
The default spherical radius is 6,371 km. If you choose “Custom radius”, the same equations run with your radius value. Larger radii reduce the sagitta for the same distance; smaller radii increase it.
Many real-world distances (roads, coastlines, GPS tracks) are better treated as surface (arc) length. The tool converts surface length to chord length internally. If you already know the straight distance between endpoints, switch to Chord mode to avoid conversion.
For short ranges, sagitta is well-approximated by s ≈ L²/(8R). Using the standard radius, a surface distance of 10 km produces about 1.962 m sagitta; 20 km produces 7.848 m; and 50 km produces 49.050 m. The calculator uses the full trigonometric form, which stays accurate for longer distances.
Atmospheric refraction is modeled with an effective radius Reff = R/(1 − k). With the common default k = 0.13, the effective radius becomes about 7,323 km, reducing predicted sagitta. For example, at 10 km sagitta becomes about 1.707 m, and at 50 km about 42.674 m.
When you enter an observer height, the calculator estimates horizon range using d = √(2Reffh + h²) for the straight-line distance. The surface alternative uses the corresponding central angle, which is useful when comparing to map-based distances.
With both observer and target heights, the tool checks whether the straight line between the two elevated points passes above the effective surface. A negative clearance means the segment intersects the modeled surface, while a positive clearance indicates an unobstructed geometric line-of-sight under the chosen assumptions.
Some online discussions use a tangent-based “drop from the observer’s tangent,” which is roughly 4× larger than sagitta for small distances. This calculator’s drop is sagitta relative to the endpoint chord, matching the formula shown in the “Formula used” section.
1) Does “Flat” mode prove anything scientifically?
No. It simply sets curvature terms to zero so you can compare outputs. Real observations depend on measurement quality, atmosphere, terrain, and instrument alignment.
2) What distance definition should I pick most of the time?
Use Surface for mapped travel distance or along-the-ground separation. Use Chord when you already have a straight endpoint-to-endpoint distance.
3) What is a typical k value for refraction?
A common engineering approximation is k ≈ 0.13, but real conditions vary widely. Temperature gradients and pressure profiles can change refraction along the path.
4) Why are my results smaller than “8 inches per mile squared”?
That rule of thumb is usually tangent-based. This calculator reports sagitta relative to the chord between endpoints, which is about four times smaller for short ranges.
5) What does negative line-of-sight clearance mean?
It means the modeled straight line between elevated points intersects the effective surface. Increasing heights, reducing distance, or increasing k can change the result.
6) Can I use miles and feet together?
Yes. Distance and height units are independent, and the tool converts everything internally to meters before computing drops, horizons, and clearances.
7) Is the horizon distance affected by refraction here?
Yes. Horizon calculations use the effective radius derived from k, so larger k increases Reff and slightly increases the predicted horizon distance.
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.