Loxodrome

A rhumb line (or loxodrome) is a path of constant bearing on a spherical (or elliptical) object. They are a traditional part of the theory of navigation.

If you follow a given (magnetic-deviation compensated) compass-bearing on Earth, you will be following a rhumb line, which spirals from one pole to the other. Near the poles, they are close to being logarithmic spirals, so they wind round each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a rhumb line is (assuming a perfect sphere) the length of the meridian divided by the sine of the bearing away from true north.

Rhumb lines are not defined at the poles: it is hard to go south-east from the North Pole and even harder to go north-west.

Contrast with: great circle, small circle.

On a Mercator projection map, a loxodrome is a straight line. On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the North (or South) pole.

On a sphere which has coordinates φ (azimuth) and θ (latitude), the equation of a loxodrome is

<math> \phi = a \ln \left| \sec \theta + \tan \theta \right| <math>

or, equivalently,

<math> \phi = a \ln \left| \tan \left( {\theta \over 2} + {\pi \over 4} \right) \right| <math>

where a depends on the bearing.

The word "loxodrome" comes from Greek loxos : oblique + dromos : running (from dramein : to run).

Old maps do not have grids composed of lines of latitude and longitude but instead have rhumb lines which are: directly towards the North, at a right angle from the North, or at some angle from the North which is some simple rational fraction of a right angle. These rhumb lines would be drawn so that they would converge at certain points of the map: lines going in every direction would converge at each of these points. See compass rose.