Calculate The Initial Bearing Between Two Points in Degrees

What is initial bearing?

Initial bearing, also known as forward azimuth, is the angle measured clockwise from a north-based reference line (true north) to the line connecting two points. It represents the direction you would need to face to travel from the starting point to the destination point.

This concept is fundamental in navigation, geography, and surveying. Understanding initial bearing helps in plotting courses, determining flight paths, and creating accurate maps. The bearing is typically expressed in degrees, with 0° representing true north, 90° representing east, 180° representing south, and 270° representing west.

How to calculate initial bearing

Calculating the initial bearing between two points requires knowing their geographic coordinates (latitude and longitude) and using spherical trigonometry. The process involves converting the coordinates to a Cartesian coordinate system, calculating the differences in coordinates, and then determining the angle using trigonometric functions.

The calculation is based on the spherical Earth model, which assumes the Earth is a perfect sphere. This approximation works well for most practical purposes, especially over short to medium distances.

The formula

The initial bearing θ (theta) between two points with coordinates (lat1, lon1) and (lat2, lon2) can be calculated using the following formula:

The result is in radians, which should be converted to degrees by multiplying by 180/π. The bearing is then normalized to the range 0° to 360°.

Worked example

Let's calculate the initial bearing from New York City (40.7128° N, 74.0060° W) to Los Angeles (34.0522° N, 118.2437° W).

1. Convert coordinates to radians:
   • lat1 = 40.7128° × π/180 ≈ 0.7107 radians
   • lon1 = -74.0060° × π/180 ≈ -1.2915 radians
   • lat2 = 34.0522° × π/180 ≈ 0.5936 radians
   • lon2 = -118.2437° × π/180 ≈ -2.0626 radians
2. Calculate Δlon = lon2 - lon1 ≈ -2.0626 - (-1.2915) ≈ -0.7711 radians
3. Apply the formula:
   • Numerator = sin(-0.7711) × cos(0.5936) ≈ -0.6986 × 0.8387 ≈ -0.5856
   • Denominator = cos(0.7107) × sin(0.5936) - sin(0.7107) × cos(0.5936) × cos(-0.7711) ≈ 0.7547 × 0.5571 - 0.6561 × 0.8387 × 0.7374 ≈ 0.4216 - 0.4119 ≈ 0.0097
   • θ ≈ atan2(-0.5856, 0.0097) ≈ -0.5736 radians
4. Convert to degrees: θ ≈ -0.5736 × 180/π ≈ -32.88°
5. Normalize to 0-360°: 360° - 32.88° ≈ 327.12°

The initial bearing from New York City to Los Angeles is approximately 327.12°, which is about 33° west of north.