Example preset
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Inputs
Common settings (earth radius/refraction coefficient)
“Tangential drop” is the difference between the tangent at the starting point and the ground surface (the familiar “8 inches per mile squared” discussion). “Sagitta” is the maximum midpoint difference between the arc and the chord.
Results
| Target horizon distance (ground distance) | — |
|---|---|
| Effective earth radius R_eff | — |
| Observer's straight line distance (tangential length) | — |
| Straight line distance of object (tangential length) | — |
| Determination of comparative distance | — |
| Visible margin distance (based on comparison distance) | — |
| required target height at that distance | — |
| Hidden height (approximate) |
| Approximate d²/(2R) | — |
|---|---|
| central angle θ | — |
| distance covered by target | — |
|---|---|
| Effective earth radius R_eff | — |
Schematic diagram
A schematic diagram will be displayed as you input.
Assumptions & limits
- The calculation is an approximation of a spherical (effective Earth radius) model. Terrain, obstacles, and uneven sea surfaces are not handled.
- The refraction coefficient k varies with the atmosphere. Default is k=0 (no refraction).
- At high altitude conditions (approximately over 20 km), the reliability of the refraction approximation with k constant decreases.
- Horizontal distance is mainly displayed as ground distance (arc length), and straight line distance is a reference value.
- At long distances, model simplification (no consideration of terrain or weather) has a greater influence than approximation.
How to choose the right horizon or curvature mode
Start with Horizon distance if your main question is “How far can I see from this height?” or “Can two points see each other?” Switch to Curvature drop when you need the Earth-curvature drop over a known distance, and use Required height when you know the distance and need to estimate the extra target height required for visibility.
Observer height vs target height
For horizon distance, h1 is the observer height and h2 is the target height. Set h2 to zero for a sea-level target, or enter both heights to estimate mutual visibility. If you also enter a comparison distance dc, the page shows an approximate hidden height at that distance.
What the refraction coefficient k means
The coefficient k adjusts the effective Earth radius to approximate atmospheric refraction. Larger k values usually make the visible horizon slightly farther away, but the effect depends on weather and temperature layers. Treat k as a planning assumption, not a guaranteed field observation.
Ground distance vs straight-line distance
This calculator mainly reports ground distance along the Earth surface. A straight-line value is shown only as a reference. For practical horizon questions, the ground distance is usually the more useful number.
Limits to keep in mind
Terrain, buildings, waves, haze, and changing atmospheric conditions are not modeled here. If you need a real-world visibility assessment, combine this page with elevation data and on-site conditions.
See also
FAQ
Is there a difference between ground distance (arc length) and straight line distance?
The difference is usually small but not zero. This calculator mainly shows ground distance (arc length).
What is the refractive coefficient k?
k approximates atmospheric refraction by adjusting Earth's effective radius. Weather can change refraction, so treat k as a guide.
What does “8 inches/mile²” correspond to?
It corresponds to tangent drop from the start point. The sagitta at the same distance is about one quarter of that value.
What does "hidden height (approximate)" indicate?
At comparison distance dc, it is the extra target height needed for visibility at observer height h1 compared with the input height h2. Terrain and weather are not included.
Are mountains, buildings, and weather conditions considered?
Not considered. If necessary, combine it with visual analysis using terrain data.
Horizon distance, mutual visibility, and hidden height
Horizon distance from one height
If you only need to know how far one observer can see over a smooth Earth, use the horizon mode with the observer height only. This answers the common search intent “how far can I see from height?”
Mutual visibility between two heights
When both the observer and the target have height above the surface, the calculator combines the two horizons to estimate whether they can see each other across the surface. This is the right mode for ship-to-shore, tower-to-tower, or lookout-to-peak estimates.
Hidden height at a known distance
If you know the comparison distance first, use the horizon mode with dc or switch to Required height. The hidden-height result estimates how much extra target height is needed to clear curvature at that distance under the same k assumption.
Curvature drop vs sagitta
Curvature drop is the tangent-based drop from the start point, while sagitta is the maximum midpoint difference between an arc and a chord. Use drop for the familiar “8 inches per mile squared” style discussion and sagitta when you need the geometric center difference.
What this page does not include
This page does not include terrain screening, buildings, tree cover, or visibility loss from haze. If those factors dominate your case, treat these numbers as a curvature baseline only.
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