Enter Calculator Values
Curvature and Visibility Chart
Example Data Table
| Observer Height | Target Height | Distance | Approx Result |
|---|---|---|---|
| 1.8 m | 30 m | 20 km | Partly visible in normal conditions |
| 10 m | 100 m | 50 km | Often visible, depending on refraction |
| 2 m | 5 m | 25 km | Mostly hidden below the horizon |
| 50 m | 200 m | 80 km | Visible range improves with height |
Formula Used
The calculator uses spherical Earth geometry with an optional refraction correction.
Effective Earth radius: Re = R / (1 - k)
Horizon distance: d = √(2Rh + h²)
Curvature drop: drop = D² / (2R)
Hidden height beyond horizon: hidden = (D - dObserver)² / (2R)
Here, R is Earth radius, h is height, D is viewing distance, and k is refraction factor.
How to Use This Calculator
Enter the observer height first. This may be eye level, camera height, building height, hill height, or aircraft altitude.
Next, enter the target height. This is the visible object, such as a ship, tower, mountain, building, or lighthouse.
Add the viewing distance along Earth’s surface. Select matching units for height, distance, and radius.
Use the default Earth radius for general work. Set refraction to zero for pure geometry. Use 0.13 for a common atmospheric estimate.
Press the calculate button. The result appears above the form and below the header section.
Understanding Earth Curvature and Visibility
Why Height Changes the Visible Horizon
Earth is curved, so the ground slowly bends away from a straight line of sight. A person standing at sea level has a short horizon. A person on a tower sees farther. This happens because extra height raises the line of sight above the curved surface. The calculator estimates that visible range with a spherical Earth model.
What Hidden Height Means
Hidden height is the part of a far object blocked by curvature. For example, the lower section of a ship may vanish first. The mast can remain visible because it is higher. This tool compares the target height with the hidden amount. If the hidden amount is smaller than the target height, part of the target may still be seen.
Why Refraction Matters
Light does not always move in a perfect straight path near Earth. Air density changes can bend light slightly downward. This makes distant objects appear a little higher. The refraction factor adjusts the effective Earth radius. A common value is 0.13, but real conditions vary. Hot ground, cold water, haze, and mirage layers can change the result.
Practical Uses
This calculator is useful for coastal viewing, marine navigation, photography, surveying, astronomy planning, and landscape analysis. It can estimate whether a lighthouse, mountain, building, ship, or antenna should be visible. It also helps compare pure curvature drop with practical viewing height. Results are estimates, not survey-grade measurements. Local terrain, waves, air quality, and optical zoom can affect what is actually seen.
Reading the Result
Start with the observer horizon. Then review the target horizon. Their sum gives the approximate visible range. Compare your distance with that range. Also check hidden height and visible target height. These values explain whether the full object, part of it, or none of it may be visible.
FAQs
1. What does this calculator measure?
It estimates horizon distance, curvature drop, hidden height, and visible target height using observer height, target height, viewing distance, Earth radius, and refraction.
2. Is Earth radius adjustable?
Yes. You can change Earth radius for different models, planets, or local assumptions. The default value is 6371 kilometers.
3. What is hidden height?
Hidden height is the estimated lower part of a distant target blocked by Earth’s curvature beyond the observer’s horizon.
4. What refraction factor should I use?
Use 0 for pure geometry. Use about 0.13 for a common atmospheric estimate. Actual refraction changes with weather and temperature layers.
5. Can this predict ship visibility?
Yes. Enter eye height, ship height, and distance. The calculator estimates whether the ship is visible, partly visible, or hidden.
6. Why does a taller observer see farther?
A taller observer has a higher line of sight. That line touches Earth farther away, so the geometric horizon distance increases.
7. Are results exact?
No. They are useful estimates. Terrain, waves, haze, optical equipment, and atmospheric refraction can make real visibility different.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary of the calculated results.