Great Circle Calculation Formula

Great Circle Formula:

\[ d = R \times \theta \text{ where } \theta = \arccos(\sin(lat1) \times \sin(lat2) + \cos(lat1) \times \cos(lat2) \times \cos(\Delta lon)) \]

Latitude 1:

degrees

Longitude 1:

degrees

Latitude 2:

degrees

Longitude 2:

degrees

Earth Radius:

Great Circle Distance:

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1. What is the Great Circle Formula?

The Great Circle Formula calculates the shortest distance between two points on the surface of a sphere (like Earth). This distance is called the great-circle distance or orthodromic distance.

2. How Does the Calculator Work?

The calculator uses the Great Circle formula:

\[ d = R \times \theta \text{ where } \theta = \arccos(\sin(lat1) \times \sin(lat2) + \cos(lat1) \times \cos(lat2) \times \cos(\Delta lon)) \]

Where:

\( d \) — Distance between points (same units as R)
\( R \) — Radius of the sphere (Earth's radius ≈ 6371 km)
\( lat1, lat2 \) — Latitude of points 1 and 2 (in radians)
\( \Delta lon \) — Difference in longitude between two points (in radians)
\( \theta \) — Central angle between the points (in radians)

Explanation: The formula calculates the central angle between two points using spherical trigonometry, then multiplies by the sphere's radius to get the distance.

3. Importance of Great Circle Calculations

Details: Great circle distances are essential for navigation, aviation, and geography. They provide the shortest route between two points on a globe, which appears as a curve when projected onto a 2D map.

4. Using the Calculator

Tips: Enter coordinates in decimal degrees (positive for North/East, negative for South/West). The default Earth radius is 6371 km (mean radius). For more precision, use 6378.137 km (equatorial) or 6356.752 km (polar).

5. Frequently Asked Questions (FAQ)

Q1: How accurate is this calculation?
A: The formula is mathematically exact for a perfect sphere. Earth's oblateness causes minor errors (typically <0.5%).

Q2: What's the difference between great circle and rhumb line?
A: Great circle is shortest path; rhumb line maintains constant bearing but is longer except along meridians or equator.

Q3: Can I use this for other planets?
A: Yes, just input the appropriate radius for the celestial body.

Q4: Why does the shortest path appear curved on maps?
A: Map projections distort great circles (except Mercator's rhumb lines). The shortest path is only straight on a globe.

Q5: What's the maximum possible great circle distance on Earth?
A: Approximately 20,037 km (half Earth's circumference) between antipodal points.