A curvature-of-the-earth calculator computes horizon distance, drop due to Earth’s curvature, and mutual visibility between two heights over a given surface distance using Earth’s radius.

How to use the Curvature-of-the-Earth Calculator

Understanding how Earth’s curvature affects what you can see is useful for boating, surveying, photography, and education. This calculator gives three practical outputs: the distance to the horizon from any height, the vertical drop of the Earth’s surface over a chosen distance, and whether two observers (or an observer and an object) can see each other over the curve. It also includes a visual Plotly diagram to illustrate the geometry with vertical exaggeration for clarity.

Inputs and controls

Enter the observer height (meters) and — optionally — the target height (meters). You may either enter the straight-line or surface distance between points (km), or leave distance empty to compute the horizon distance from the heights. Use the “Mode” selector to choose between horizon-distance calculations, drop-over-distance, and visibility test. The chart area shows a scaled cross-section of the Earth, the observer, the target, the tangent (observer’s horizontal line of sight), and the connecting sightline. The vertical-exaggeration control increases the visual prominence of small heights so towers, ships, or small hills appear clearly while the underlying math remains exact.

Important formulas used

• Radius of Earth (mean): 6,371,000 meters.
• Horizon distance (straight line): d = sqrt((R+h)^2 – R^2) (meters).
• Horizon arc distance (surface): s = R * arccos(R/(R+h)).
• Drop over a surface distance s (exact): drop = R * (1 – cos(s/R)). For short distances this approximates to drop ≈ s^2 / (2R).
• Visibility rule: If horizon_arc(observer) + horizon_arc(target) ≥ separation_distance then line of sight clears curvature.

Reading results

• Horizon distance: shows both straight-line and arc (surface) distances. Use the arc distance if you care about geographic distance along the Earth’s surface, and the straight-line distance when considering direct line-of-sight range.
• Drop: the vertical amount the Earth’s surface falls below the observer’s tangent after a specified surface distance. This is useful when planning sightlines or predicting how much of a ship or tower is obscured by curvature.
• Visibility: an easy yes/no answer that checks whether the observer can see the target at the entered distance and heights.

Visualization notes

The Plotly diagram scales the huge Earth radius down so you can clearly see heights. A vertical exaggeration control lets you increase how tall objects look relative to Earth’s radius — this makes small hills, towers, or ships visible within the chart while preserving correct geometry calculations (the math uses true values; exaggeration is only for display). The visual representation helps confirm numeric outputs: if the dashed sightline intersects the arc representing the Earth surface, the target is behind the horizon.

Practical examples

1. If you stand 2 m above sea level, your horizon arc distance will be roughly 4.5 km. The calculator shows both this and the straight-line distance.
2. For a 100 km separation, the approximate drop is about (100000^2)/(2*6,371,000) ≈ 784 meters — substantial; tall objects might still be visible if they exceed that drop.
3. A 30 m lighthouse vs a 2 m observer at 20 km separation can usually be seen because the lighthouse’s horizon contribution supplements the observer’s horizon.

Step-by-step walkthrough

1. Choose your units and enter the observer height in meters. If you are using a camera tripod or survey instrument, include the instrument height above ground.
2. Optionally enter the target’s height. If the target is sea-level (like a ship’s deck), use the height of the visible feature.
3. Either type the separation distance in kilometers or leave it blank to compute how far the horizon is from each height. Press “Calculate.”
4. Read the numeric outputs: horizon distances, drop over the entered distance, and a visibility result. Inspect the Plotly diagram — if the sightline intersects the Earth’s arc the target is hidden. Toggle vertical exaggeration to inspect small details.

Use this calculator as an educational and planning tool; for safety-critical decisions combine it with local surveys and professional advice and regulations.

Final notes for WordPress integration

This tool is supplied as a single self-contained HTML/JavaScript block that you can paste into a WordPress Custom HTML block or code widget. It is sized to fit a content column between two sidebars (conservative max-width of 720px and responsive behavior). The background is white to match common content areas and ensure readable contrast. Plotly.js is loaded from the CDN. Because the plot reduces distances to kilometers and exaggerates vertical scale visually, users should rely on the numeric outputs for engineering or legal decisions and use the visual only for illustration or teaching.

Disclaimer

This calculator uses a spherical-Earth model with mean radius 6,371 km and ignores local terrain, atmospheric refraction, and tidal effects. Use results as estimates for planning and educational purposes only, not as precise legal, navigational, or engineering guidance.

Frequently Asked Questions (FAQ)

Q: What’s the difference between straight-line and arc horizon distance?

A: Straight-line is the direct line-of-sight path from eye to horizon; arc distance follows the Earth’s surface. The arc distance is what your map would show.

Q: Can atmospheric refraction change the visible distance?

A: Yes. Typical refraction slightly extends the horizon (about 8% over calm sea conditions). For critical planning include refractive corrections or consult professional survey data.

Q: Why does the chart look exaggerated vertically?

A: Earth’s radius is enormous compared to typical heights. Vertical exaggeration helps you visualize the observer, target, tangent, and sightline while the calculations remain exact.

Q: Is the calculator accurate for hills, mountains, or local terrain?

A: No. The tool assumes open, unobstructed sea-level conditions along a smooth spherical surface. Local topography will change sightlines.

Q: Can this be used for radio or microwave line-of-sight?

A: The curvature geometry applies, but radio propagation often includes additional effects (diffraction, tropospheric bending). For radio planning, use specialized propagation models.