Rhumb Line Distance Formula
Calculate rhumb line distance for constant-bearing navigation on a Mercator projection.
Used in maritime and aviation navigation.
The Formula
θ = arctan(Δλ / Δψ)
Δψ = ln(tan(π/4 + φ2/2) / tan(π/4 + φ1/2))
A rhumb line (also called a loxodrome) is a path on Earth's surface that crosses every meridian at the same angle. If you set a constant compass bearing and follow it without ever turning, you travel along a rhumb line. On a Mercator projection map, a rhumb line appears as a perfectly straight line, which is why Mercator maps became so popular for maritime navigation after their introduction by Gerardus Mercator in 1569 in Flanders (modern-day Belgium).
The rhumb line distance formula uses the Mercator projection's mathematical properties to calculate the distance between two points. The key variable Δψ is the difference in Mercator-projected latitudes, which stretches the latitude scale to make rhumb lines straight. The bearing θ is calculated from the ratio of longitude difference to projected latitude difference. The actual distance D is then found by dividing the latitude difference Δφ (in radians, multiplied by Earth's radius) by the cosine of the bearing.
A rhumb line is not the shortest path between two points on Earth — that distinction belongs to the great circle. However, a rhumb line is much easier to follow in practice because the navigator maintains a constant compass heading. For short distances, the difference between a rhumb line and a great circle is negligible. For long voyages at high latitudes, the difference can be significant. For example, a transatlantic crossing from New York to London is about 5,570 km via great circle but approximately 5,800 km via rhumb line — roughly 4% longer.
A fascinating mathematical property of the rhumb line is that if the bearing is not exactly north, south, east, or west, the path spirals toward a pole without ever reaching it, creating an infinitely long curve.
Variables
| Symbol | Meaning |
|---|---|
| D | Rhumb line distance (meters or nautical miles) |
| Δφ | Difference in latitude (radians) — φ2 − φ1 |
| Δλ | Difference in longitude (radians) — λ2 − λ1 |
| Δψ | Difference in Mercator-projected latitude |
| θ | Constant bearing angle (radians) — the compass heading to follow |
| φ1, φ2 | Latitudes of the start and end points (radians) |
| R | Earth's radius — approximately 6,371 km or 3,441 nautical miles |
Example 1
A ship sails from Miami (25.76°N, 80.19°W) to Bermuda (32.32°N, 64.78°W). What bearing should the navigator hold?
Convert to radians: φ1 = 0.4495, φ2 = 0.5641, Δφ = 0.1146
Δλ = (−64.78 − (−80.19)) × π/180 = 15.41° = 0.2690 rad
Δψ = ln(tan(π/4 + 0.5641/2) / tan(π/4 + 0.4495/2)) = ln(1.3218/1.2536) ≈ 0.0530
θ = arctan(0.2690 / 0.0530) ≈ arctan(5.076) ≈ 78.9°
The navigator should hold a bearing of approximately 079° (almost due east-northeast)
Example 2
Using the same Miami-to-Bermuda route, what is the rhumb line distance?
D = R × Δφ / cos(θ)
D = 6,371 × 0.1146 / cos(78.9°)
D = 730.1 / 0.1927
D ≈ 1,308 km (or about 706 nautical miles)
When to Use It
Use the rhumb line formula when you need a constant-bearing route that is easy to navigate with a compass.
- Maritime navigation — plotting courses on nautical charts with a constant heading
- Aviation route planning for short to medium distances where simplicity matters
- Comparing rhumb line paths with great circle routes to evaluate distance penalties
- GIS and mapping applications that use Mercator projections
- Understanding why Mercator projection maps are shaped the way they are