Earth Curvature Calculator
Calculate surface curvature drop, refracted curvature drop, horizon distances, dip angle, and maximum line-of-sight distance based on Earth radius, observer height, target height, viewing distance, and atmospheric refraction. Earth Curvature Calculator helps surveyors, photographers, engineers, educators, and outdoor observers understand long-distance visibility and the effects of Earth's curvature.
Highlights
- Estimate Earth's curvature drop over any viewing distance.
- Adjust calculations for none, standard, or strong atmospheric refraction.
- Calculate observer and target horizon distances.
- Find maximum line-of-sight distance between two elevated points.
- Determine horizon dip angle and target visibility status.
Table of Contents
How Earth Curvature Calculator Works?
The calculator models Earth as a sphere with the entered Earth Radius R. It uses the observer height, target height, viewing distance, and selected atmospheric-refraction condition to estimate how the surface falls away from a straight line of sight.
For a viewing distance d, the geometric curvature drop below a tangent line can be represented by the exact spherical expression:
For distances that are small compared with the Earth radius, the calculator may use the common approximation:
Atmospheric refraction changes the apparent curvature by making light follow a slightly curved path. The selected condition is represented by an effective radius Reff. No refraction uses the geometric radius, while standard and strong refraction use progressively stronger effective-radius adjustments.
The horizon distance for an observer at height h is obtained from the right triangle formed by the planet's center, the observer, and the tangent point:
The same relationship is applied to the target height. The maximum line-of-sight distance is the sum of the observer and target horizon distances, using the effective radius associated with the selected atmospheric condition.
The horizon dip angle is the angle below a level horizontal line at which the observer sees the horizon:
Finally, the target is marked visible when the entered viewing distance does not exceed the calculated maximum line-of-sight distance. Otherwise, the geometric horizon lies between the observer and target.
| Output | Calculation basis |
|---|---|
| Geometric Curvature Drop | Planetary radius and viewing distance without refraction |
| Refraction-Adjusted Drop | Viewing distance and the effective radius for the selected condition |
| Observer Horizon Distance | Observer height and planetary radius |
| Target Horizon Distance | Target height and planetary radius |
| Maximum Line-of-Sight Distance | Sum of observer and target horizon distances |
| Horizon Dip Angle | Observer height and planetary radius |
The calculator accepts the Earth radius in m, km, mi, or ft; heights in mm, cm, m, km, in, ft, yd, or mi; and viewing distance in m, km, mi, ft, yd, or nmi. Results are available in the supported distance units and in degrees or radians for the dip angle.
Practical Use Cases
Surveyors can estimate how much distant ground should appear below a straight tangent line and compare the result with elevation measurements, sight-line planning, or preliminary long-distance observations.
Photographers and videographers can estimate whether a distant shoreline, building, or landscape feature should be geometrically visible from a chosen camera height and how atmospheric refraction may alter its apparent position.
Engineers can use the horizon and maximum line-of-sight outputs for early-stage checks involving long-distance optical links, elevated structures, coastal visibility, and other projects where planetary curvature is relevant.
Educators and students can vary Earth radius, observer height, target height, distance, and refraction condition to demonstrate tangent geometry, spherical surfaces, inverse-square-style distance effects in the approximation, and horizon formation.
Outdoor observers can estimate the maximum geometric distance to a horizon from a hill, tower, aircraft, or elevated viewpoint and determine whether a distant object is within the combined observer-target visibility limit.
Accuracy and Limitations
The calculator is mathematically accurate for an ideal spherical planet and the selected refraction model, subject to normal numerical rounding. It assumes a smooth surface, a straight or effectively modeled optical path, and heights measured above the planetary surface; it does not account for mountains, buildings, waves, local terrain, observer or target size, lens elevation, haze, mirages, temperature inversions, humidity gradients, or changing atmospheric layers. The standard and strong refraction options are simplified atmospheric categories rather than real-time meteorological predictions, so actual visibility can differ substantially, especially near coastlines and during unusual weather. The visibility status is a geometric line-of-sight test and does not guarantee that the target is bright, unobstructed, or resolvable by the eye or an optical instrument.
Quick Examples
Example #1: Sea-Level Observer
Estimate curvature and horizon visibility for a person viewing a target across a long, flat surface.
Example #2: Mountain-to-Mountain View
Assess the maximum line-of-sight distance between two elevated locations while accounting for strong atmospheric refraction.
Example #3: Coastal Survey
Calculate curvature drop and horizon distances for a coastal survey conducted in nautical miles with no atmospheric-refraction adjustment.
