Earth Curve Calculator
Calculate the physical drop, horizon distance, and obscured target height due to the Earth's curve. Incorporates standard atmospheric refraction corrections.
Calculator Settings
Height of your eyes above the surface.
Horizontal distance to the object being viewed.
Height of the target. Set to 0 to find geometric drop only.
Corrects calculations for bending light rays in the atmosphere. Highly recommended for real-world optical measurements.
Hidden Height vs. Distance Chart
The plot above visualizes how much height of a target is obscured relative to how far it moves away from you, matching your current observer height and refraction setting. The flat part represents distances within your direct line of sight (zero hidden height), after which the hidden portion rises quadratically.
Understanding Earth Curve Calculator: The Horizon and Beyond
From our daily perspective on the ground, the Earth appears flat. However, as an oblate spheroid with an average radius of approximately 6,371 kilometers (3,959 miles), the surface curves away beneath us in every direction. This curvature imposes a definite limit on our line of sight, creating the boundary we call the horizon.
When viewing objects across long distances, such as ships at sea, wind turbines, or distant skyscrapers, the Earth's curve acts as a physical barrier. As the target moves further away, its lower portion drops below our sight line. This creates the phenomenon where objects appear to sink bottom-first into the sea, which has been documented by navigators and astronomers since antiquity.
Mathematical Calculations and Geodesic Formulas
Calculating Earth curvature drop and obscured target height requires spherical geometry. Here are the core formulas implemented within this tool:
1. Geometric Horizon Distance
Determines how far an observer at height $h$ can see to the horizon point:
dh = √(2 · R · h + h2)
Where R is the Earth radius and h is the height of the observer.
2. Geodesic Curvature Drop
Determines the drop height from the flat tangent horizontal line:
Drop = R - √(R2 - d2)
Where d is the distance between the observer and the target.
3. Hidden (Obscured) Target Height
When the target distance d is greater than the horizon distance dh, the hidden height is:
Hidden Height = R - √(R2 - (d - dh)2)
This calculates the curvature drop starting from the horizon line of sight boundary, which determines the lower portion hidden from view.
In imperial units, a simplified rule of thumb is often used: 8 inches per mile squared (represented as Drop = 8 × d2). While this approximation is highly accurate for short distances (under 100 miles), it begins to break down at larger ranges and does not account for observer height or atmospheric conditions. This calculator uses exact trigonometric formulas to ensure mathematical rigor.
Understanding Refraction (7/6 R)
In the real world, light does not travel in a perfectly straight line through the air. The Earth's atmosphere is denser at sea level than at higher elevations. This vertical density gradient causes light rays to bend (refract) downward as they travel.
This downward bending allows light to follow the curvature of the Earth slightly, letting observers see "around" the curve. As a result, objects appear higher in the sky, and the horizon appears further away than it would in a pure geometric vacuum.
To model standard atmospheric refraction under typical meteorological conditions, geodesists use an effective Earth radius. The standard factor is 7/6 of the physical radius (often called the 7/6 refraction model, corresponding to a refraction coefficient of $k \approx 0.17$). Applying this correction increases the effective radius from 6,371 km to approximately 7,433 km, which matches observation data collected by surveyors.
Geodesy and Historical Wagers
The debate over Earth's curvature was a subject of high-profile scientific tests in the 19th century. In 1838, Samuel Rowbotham, a proponent of the flat-Earth theory, conducted the famous Bedford Level Experiment along a straight 6-mile stretch of the Old Bedford River in Cambridgeshire, England. He claimed to see a boat with a flag over the full distance without any drop, asserting this proved the Earth flat.
In 1870, Alfred Russel Wallace, co-discoverer of natural selection, accepted a wager from a flat-earth proponent to replicate the experiment with scientific controls. Wallace understood the atmospheric refraction issues Rowbotham ignored. By using a telescope placed at an equal height above the water at both ends, and placing a target pole with marker bands in the middle, Wallace demonstrated that the middle target appeared higher than the sight lines at either end. This confirmed a curvature drop of approximately 5.7 feet, matching geodesic calculations exactly.
Frequently Asked Questions
For an observer standing at sea level with eye height at 1.8 meters (5 feet 11 inches), the horizon is only 4.8 kilometers (3.0 miles) away. If you climb to a viewing deck at 100 meters (328 feet), your line of sight to the horizon increases to approximately 35.7 kilometers (22.2 miles), illustrating how horizon distance increases with height.
The "8 inches per mile squared" rule ($Drop = 8 \times d^2$) is a simplified imperial rule of thumb for estimating physical drop. It is derived from the parabolic approximation of a circle. It is highly accurate for distances under 100 miles. However, it only calculates the geometric drop from a flat horizontal plane, and does not account for the height of the observer or refraction.
Atmospheric refraction depends heavily on temperature gradients. On hot days when air near the ground is warmer than the air above, a reverse density gradient can occur, bending light rays upward. This creates thermal mirages. On very cold days or over cold water (thermal inversion), refraction increases significantly, making objects visible that are geometrically far below the horizon (a phenomenon called 'looming').
Typically, no. At standard commercial flight altitudes of 10,600 meters (35,000 feet), the horizon is curved by only about 0.05 degrees, which is almost imperceptible to the human eye due to the limited field of view of airplane windows. Visible curvature is generally only clear from altitudes above 15,200 meters (50,000 feet) in specialized aircraft.