Rhumb line

In navigation, a rhumb line (or loxodrome) is a line crossing all meridians of longitude at the same angle, i.e. a path derived from a defined initial bearing. That is, upon taking an initial bearing, one proceeds along the same bearing, without changing the direction as measured relative to true north.

Usage

The name is derived from Old French: "rumb", a line on the chart which intersects all meridians at the same angle.

Early navigators in the time before the invention of the chronometer used rhumb-line courses on long ocean passages, because the ship's latitude could be established accurately by sightings of the Sun or stars but there was no accurate way to determine the longitude. The ship would sail North or South until the latitude of the destination was reached, and the ship would then sail East or West along the rhumb-line, maintaining a constant latitude and recording regular estimates of the distance sailed until evidence of land was sighted .

General and mathematical description

A rhumb line can be contrasted with a great circle, which is the shortest distance between two points on the surface of a sphere, but whose bearing is non-constant. If you were to drive a car along a great circle you would hold the steering wheel in the centre, but to follow a rhumb line you would have to turn the wheel, turning it more sharply as the poles are approached. In other words, a great circle is locally "straight" with zero geodesic curvature, whereas a rhumb line has non-zero geodesic curvature.

Meridians of longitude and parallels of latitude provide special cases of the rhumb line, where their angles of intersection are respectively 0° and 90°. On a North-South passage the rhumb-line course coincides with a great circle, as it does on an East-West passage along the equator.

On a Mercator projection map, a rhumb line is a straight line; a rhumb line can be drawn on such a map between any two points on Earth without going off the edge of the map. But theoretically a loxodrome can extend beyond the right edge of the map, where it then continues at the left edge with the same slope (assuming that the map covers exactly 360 degrees of longitude).

Rhumb lines which cut meridians at oblique angles are loxodromic curves which spiral towards the poles

Mathematical derivation

Finding the loxodromes between two given points can be done graphically on a Mercator map, or by solving a nonlinear system of two equations in the two unknowns tan(α) and λ₀. There are infinitely many solutions; the shortest one is that which covers the actual longitude difference, i.e. does not make extra revolutions, and does not go "the wrong way around".

The distance between two points, measured along a loxodrome, is simply the absolute value of the secant of the bearing (azimuth) times the north-south distance (except for circles of latitude for which the distance becomes infinite).

Etymology and historical description

The 1878 edition ofThe Globe Encyclopaedia of Universal Information describes loxodrome lines as:

See also

References