Earth Curvature Calculator

Earth Curvature Calculator: Instantly calculate the horizon distance, geometric drop, and hidden height of targets blocked by the Earth’s curve. Use the inputs below to adjust for observer height, distance, and atmospheric refraction.

Observer Height (Eye Level)

Distance to Target

Target Height (Optional)

Refraction (k)

Approximate k from weather:

Calculating…

Hidden Height
–
Obscured part of target

Horizon Dist

–

From observer

Geometric Drop

–

From tangent line

Dip Angle

–

Degrees

*Diagram is schematic (vertical axis exaggerated) to show geometry.

How to use the Earth Curvature Calculator

Enter your Observer Height (distance from sea level to your eyes). Enter the Distance to the object you are viewing. Optionally, enter the Target Height to see how much of the object remains visible.

The calculator automatically adjusts for Atmospheric Refraction (bending of light), which allows you to see slightly further over the curve than geometry alone would predict. You can adjust this in “Advanced Settings”.

Formulas Used

1. Earth Radius & Refraction

Standard Earth Radius ($R$) is approx 6,371 km. Light bends in the atmosphere, effectively increasing this radius. We use the effective radius ($R_e$):

$$R_e = \frac{R}{1 – k}$$

Where $k$ is the refraction coefficient (Default 0.1667).

2. Distance to Horizon

The distance to the geometric horizon ($d_h$) from an observer height ($h$):

$$d_h = \sqrt{2 R_e h + h^2} \approx \sqrt{2 R_e h}$$

3. Hidden Height

If the total distance ($d$) is greater than the horizon distance ($d_h$), the remaining distance ($d_x = d – d_h$) curves downwards. The hidden amount ($h_x$):

$$h_x = R_e (\sec(\frac{d_x}{R_e}) – 1)$$

Or using the standard parabolic approximation: $$h_x \approx \frac{d_x^2}{2 R_e}$$

4. Geometric Drop

The vertical drop from a tangent line at the observer’s position:

$$Drop = R_e (1 – \cos(\frac{d}{R_e}))$$

Frequently Asked Questions (FAQ)

Why can I see objects that should be hidden?

This is usually due to Refraction. When the air near the water is cooler than the air above (temperature inversion), light bends more strongly around the curve (looming), making objects appear higher. Set “Refraction” to “Strong Inversion” to simulate this.

What is the “Earth Bulge”?

The Earth bulge is the height of the curve at the midpoint between two points. It represents the “hill” of water obstructing the line of sight.