Rhumb Line Bearing Calculator
A rhumb line, or loxodrome, is the route you follow by holding one constant compass heading. It crosses every meridian at the same angle, which makes it easy to steer though slightly longer than the great-circle path. This calculator takes two points in decimal degrees and returns the constant rhumb line bearing (clockwise from true north) and the rhumb line distance, using the inverse Gudermannian latitude stretch on a sphere of mean radius 6,371 kilometers. The longitude difference is taken the shorter way around the globe.
Rhumb line formula
dPsi = ln( tan(pi/4 + lat2/2) / tan(pi/4 + lat1/2) )
Adjust dLon to the shorter way (abs <= pi)
Bearing = atan2(dLon, dPsi), normalized to 0..360
q = (lat2 - lat1) / dPsi, or cos(lat1) if dPsi is tiny
Distance = R * sqrt(dLat^2 + (q * dLon)^2), R = 6,371 km
The dPsi term is the difference in stretched (Mercator) latitude, which keeps the bearing constant. One nautical mile is 1.852 kilometers.
Rhumb line context
- A rhumb line holds a single compass heading the whole way.
- It is longer than the great-circle route but simpler to steer.
- The longitude difference is taken the shorter way around the globe.
- North and east are positive decimal degrees.
- Distance is reported in kilometers and nautical miles (1 nm = 1.852 km).
Rhumb line: frequently asked questions
What is a rhumb line?
A rhumb line (or loxodrome) is a path of constant compass bearing. Unlike a great circle, it crosses every meridian at the same angle, so you can follow it by holding a single heading. It is longer than the great-circle route but simpler to steer.
How is the rhumb line bearing calculated?
It uses the difference in the inverse Gudermannian (stretched) latitudes. The bearing is the arctangent of the longitude difference divided by the difference of the natural logs of tan(pi/4 plus lat/2). The longitude difference is adjusted to take the shorter way around if it exceeds 180 degrees.
How is rhumb line distance found?
The distance equals the Earth radius times the square root of (latitude difference squared plus (cosine of the projected latitude times longitude difference) squared), all in radians. When the two latitudes are equal, the projected cosine is used directly to avoid division by zero.
When would I use a rhumb line instead of a great circle?
Rhumb lines are convenient for short to medium voyages and for traditional chart navigation because the heading never changes. Great circles are shorter over long distances but require continual course adjustment. Many passages combine both.
What coordinate convention is used?
Decimal degrees with north and east positive, south and west negative. The result bearing is given clockwise from true north in the 0 to 360 degree range, and the distance is reported in kilometers and nautical miles.
Official sources
- NOAA National Geodetic Survey: National Geodetic Survey.
- NIST DLMF: Digital Library of Mathematical Functions.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.