Flat vs. Round Earth Calculator

Earth curvature analyzer - visibility of objects, horizon and model comparison

This add-on is a tool forgeometric analysis of visibility on large distances. You enter the height of the observer (e.g. eye height), the height of the target (e.g. lamppost, skyscraper, mast), distance and - optionally - advanced parameters: Earth's radius and refractive index. The calculator answers questions such as:"should the object be visible?", "how many meters are hidden below the horizon?"and"what is the elevation angle of the target in the spherical and flat models?".

parameter in the calculator. What exactly will you calculate in this analyzer?

How to use the calculator - step by step

1. Set the observer height (h1)- usually eye height above ground/water level. If you are standing on the beach, enter e.g. 1.7 m. If you are on a cliff or terrace, enter the real height (e.g. 30 m).
2. Enter the height of the target (h2)- e.g. the height of a lighthouse, building, mast, windmill, or an interesting point on the object (e.g. the upper edge).
3. Enter the distance– in meters or kilometers. In practice, it works best when the distance is measured with a map/GPS/rangefinder.
4. (Optional)Earth's radius (R)- the default can be approximately 6,371,000 m. Changing R makes sense for teaching experiments or local approximations.
5. (Optional)Refraction (k)- if you want to account for the typical "bending" of rays in the atmosphere. Increasing k usually "helps visibility" (the horizon recedes), but in real conditions it varies.
6. ClickCALCULATE GEOMETRYand check: visibility status, percentage of the visible object, how many meters are "hidden" and angle comparison.

What do the results mean? Simple explanations of

metrics Visibility status and "Object % visible"

"Visible" means that in a spherical model (taking into account the set refraction), the line of sight can "hit" the object, and part of it should protrude above the horizon. “Invisible” suggests that the object is entirely hidden behind the curve at a given viewing height and distance.

Visible %is simplified information: what part of the target's height remains above the horizon "aperture". If you see, for example, 60%, it means that the lower 40% of the object's height is geometrically obscured.

"Hidden behind the curve (m)"

This is a value in meters that indicates how much of the object's height (from the bottom) is below the line of sight in the sphere model. When the number is positive, part of the object should be obscured. When it is 0 (or very close to 0), the object is "on the border" of visibility - then small changes in refraction, waves, temperature or height of the site can make the image "sometimes there, sometimes disappears".

Drop

"Drop" is a popular intuitive measure: how much lower the Earth's surface is at a given distance, compared to "butting" the tangent line at the point of observation. This is not a "magic" value from the Internet - it is ordinary circle geometry. For short distances the drop is small, but increases with distance (approximately proportional to the square of the distance).

Your horizon

The distance to the horizon depends on the observer's height. The higher you are, the further the viewing geometry "reaches". Therefore, climbing a dune, cliff, observation tower or building floor often radically changes what can be seen over the water or flat terrain.

Refraction (k) – why is it important?

The atmosphere is not uniform. When air has different temperatures and densities at different altitudes, light rays are bent. In practice, this means that the "line of sight" is not always perfectly straight and the horizon may seem further away. Thekparameter in the calculator allows you to simulate this effect.

• k = 0→ no refraction (pure geometry).
• k > 0→ the rays "bend" downwards, which usually increases the expected visibility.
• Variable conditions→ in real observations k is not constant; mirages may appear during temperature inversion.

Spherical vs flat model – what does the elevation comparison mean?

In the comparison section you get two angles:elevation in the spherical modelandelevation in the flat model. This is the angle at which you must "raise" (or "lower") your line of sight to bring the target into line with your sight line. In the flat model, the geometry is simpler: there is no curvature, so the elevation is mainly due to the difference in height and distance. In the spherical model, there is the fact that the surface "escapes" and the line tangent to the surface has a different course.

“You must be at a height of X m to see the entire target” – how to read it?

This insight tells us how much observer height (h1) is needed for the entire object (from the base to the top) to be above the "aperture" of curvature. This is useful in planning observations: instead of guessing whether it is worth climbing the tower, you can assess whether 10 m is enough or whether you need 60 m.

Most common errors in input data (and how to avoid them)

• Confusing eye height with ground height- if you are standing on a cliff 30 m above the water, h1 is ~31.7 m, not 1.7 m.
• Target height "from sea level" vs "from the base"- calculator needs target height above the local reference level.
• Map distance vs straight line– surface distance is usually used. The calculator also shows the chord as a curiosity.
• Failure to take into account intermediate obstacles- an embankment, trees and buildings can obscure an object faster than the curvature.
• Forced refraction– k is not a knob for adjusting the result to expectations; changes with weather and conditions.

FAQ

Summary

The Earth's curvature analyzer allows you to easily check what the geometry implies: how much of the object should be hidden behind the curvature, how far the horizon is, what the "drop" is at a given distance and how refraction can change the conclusions. It is a great tool for planning observations, verifying scenarios and educational model comparison.