Formula Used
The calculator first adjusts Earth radius for refraction.
The effective radius is Re = R / (1 - k).
Here, R is Earth radius and k is the refraction coefficient.
The fast drop estimate is drop = d² / (2Re).
This works well for moderate distances.
The exact tangent drop uses drop = Re × (sec(d / Re) - 1).
The sagitta bulge is Re × (1 - cos(d / Re)).
Observer horizon distance uses surface horizon = Re × acos(Re / (Re + h)).
Hidden height is solved from the tangent line that leaves the observer and just touches the refracted Earth curve.
How to Use This Calculator
- Enter the surface distance from the observer to the object.
- Enter observer height above the local surface.
- Enter target height when checking visibility.
- Keep the default Earth radius unless you need a custom planet model.
- Set refraction to 0 for pure geometry or 0.13 for standard air.
- Choose an output unit and decimal precision.
- Press the calculate button. The result appears above the form.
- Use the CSV or PDF buttons to save the result.
Example Data Table
Examples use Earth radius 6371 km, refraction coefficient 0.13, and zero observer height.
| Surface Distance |
Approximate Drop |
Exact Tangent Drop |
Drop Angle |
| 1 km |
0.068 m |
0.068 m |
0.0039° |
| 5 km |
1.707 m |
1.707 m |
0.0196° |
| 10 km |
6.828 m |
6.828 m |
0.0391° |
| 20 km |
27.311 m |
27.311 m |
0.0782° |
Understanding Vertical Horizon Drop
What the Result Means
Vertical horizon drop describes how far a curved surface falls below a local tangent line.
The idea is useful for surveying, radio planning, photography, marine viewing, and educational checks.
A short distance creates a small drop.
A long distance creates a much larger drop because distance is squared in the common estimate.
Why Refraction Matters
Air bends light slightly downward in many normal conditions.
This makes the apparent Earth curve look flatter.
The calculator handles that effect with an effective radius.
A coefficient of zero gives pure geometry.
A coefficient near 0.13 gives a common standard atmosphere estimate.
Real weather can still change the view.
Observer Height and Hidden Height
Observer height changes what can be seen.
A person standing higher has a longer horizon distance.
A camera on a tower sees farther than a camera near water level.
Hidden height estimates how tall a target must be before its top rises above the horizon line.
This is helpful when checking distant buildings, hills, ships, lights, or towers.
Exact and Approximate Methods
The approximate equation is quick and easy.
It is reliable for many local field cases.
The exact tangent equation is better when distance grows.
The calculator shows both results, so you can compare them.
It also reports sagitta bulge, drop angle, dip angle, and visibility allowance.
These values help you understand the geometry from several useful angles.
Best Use Cases
Use this tool before field observation, line-of-sight planning, drone work, telescope setup, or classroom demonstrations.
Keep input units consistent, or let the unit selectors convert them.
Treat the answer as a planning estimate.
Local terrain, waves, temperature layers, pressure, and elevation datum can affect real observations.
For engineering work, combine this result with measured elevation data.
Frequently Asked Questions
1. What is vertical horizon drop?
It is the amount a curved surface falls below a straight tangent line over a chosen distance.
It is often used for Earth curvature checks.
2. Why are approximate and exact drops different?
The approximate method uses a short-distance simplification.
The exact method uses trigonometry.
Their difference grows as distance becomes larger.
3. What refraction value should I use?
Use 0 for pure geometry.
Use 0.13 for common standard atmosphere estimates.
Real conditions can vary during the day.
4. What does hidden height mean?
Hidden height is the target height needed to rise above the horizon line from the observer position.
5. Does observer height affect the drop value?
Observer height does not change the surface drop itself.
It changes horizon distance, dip angle, and how much of a target is hidden.
6. Can I use this for radio links?
Yes, it helps with basic line-of-sight planning.
Add terrain, antenna height, Fresnel zone, and local obstruction checks for real design.
7. Why is Earth radius editable?
Editable radius supports custom geodetic assumptions, educational comparisons, and non-Earth examples.
The default mean radius suits general use.
8. Are CSV and PDF files generated automatically?
Yes. After calculating, use the export buttons above the form.
They save the current result shown on the page.