Earth Curvature Calculator

This calculator relies on the spherical geometry of the Earth, treating the planet as a smooth sphere with an average volumetric radius ($R$) of approximately 6,371,000 meters. To find the distance to the geometric horizon ($d_h$) from an elevated observer, we use the Pythagorean theorem applied to a right-angled triangle formed by the Earth's center, the observer's position, and the tangent point on the horizon:

distance_horizon = sqrt((R + height)^2 - R^2)

When calculating how much of a distant object is obscured by the curvature ($obscured\_part$), the tool compares the straight-line tangent drop against the curved geodetic drop over a given ground distance, factoring in standard atmospheric refraction to deliver real-world accuracy.

Imagine you are standing at an observation deck with your eyes exactly 10 meters above sea level. You want to know how far away the horizon is and how much of a distant boat's hull is hidden if the boat is 25,000 meters (25 km) away.

Step 1: Calculate the horizon distance. Using the Earth radius $R = 6,371,000$ m and height $h = 10$ m:
distance_horizon = sqrt((6,371,000 + 10)^2 - 6,371,000^2)
This evaluates to approximately 11,288 meters (about 11.29 km).

Step 2: Determine the obscured height at 25 km. Since your horizon is at 11.29 km, a target located at 25 km extends well beyond your horizon. Using the geometry of drop-off beyond the tangent point, the calculated obscured part shows that roughly 26.4 meters of the lower structure of the distant object is hidden from your direct line of sight.

Keep these guidelines in mind when working with curvature and horizon calculations:

• Atmospheric Refraction: Light bends slightly downward due to atmospheric density gradients. Standard calculations often apply a refraction coefficient (multiplying Earth's radius by ~4/3), which makes the horizon appear slightly farther away than pure geometric models predict.
• Elevation Reference: Always measure your height relative to the local water or ground datum, not sea level if you are standing on an inland hill or mountain.
• Precision Matters: For short distances under one kilometer, Earth's curvature is negligible, and flat-earth approximations work fine. Curvature calculations become critical only over longer spans.