Bearing Formula
Calculate the compass direction from one point to another using coordinates.
Essential for navigation and mapping.
The Formula
θ = atan2(sin(Δλ) × cos(φ₂), cos(φ₁) × sin(φ₂) - sin(φ₁) × cos(φ₂) × cos(Δλ))
The bearing formula gives the initial compass direction from one point to another on Earth. Bearing is measured clockwise from north: 0° = North, 90° = East, 180° = South, 270° = West.
Variables
| Symbol | Meaning |
|---|---|
| θ | Bearing (degrees, 0-360) |
| φ₁, φ₂ | Latitude of start and end points (radians) |
| Δλ | Difference in longitude (radians) |
| atan2 | Two-argument arctangent function |
Example 1
Bearing from New York (40.71°N, 74.01°W) to London (51.51°N, 0.13°W)
Δλ = 73.88° east
Using the formula with radians:
Bearing ≈ 51.2° (roughly northeast)
Example 2
Bearing from Los Angeles (34.05°N, 118.24°W) to Tokyo (35.68°N, 139.65°E)
Δλ = 360 - (118.24 + 139.65) = 102.11° (going west across the Pacific)
Using the formula:
Bearing ≈ 305° (roughly northwest — across the Pacific)
When to Use It
Use the bearing formula when:
- Navigating between two geographic coordinates
- Building mapping and direction applications
- Planning sailing, flying, or hiking routes
- Determining which compass direction to travel