Rhumb line

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, i.e. a path with constant bearing as measured relative to true or magnetic north.

The effect of following a rhumb line course on the surface of a globe was first discussed by the Portuguese mathematician Pedro Nunes in 1537, in his Treatise in Defense of the Marine Chart, with further mathematical development by Thomas Harriot in the 1590s.

A rhumb line can be contrasted with a great circle, which is the path of shortest distance between two points on the surface of a sphere. On a great circle, the bearing to the destination point does not remain constant. If one were to drive a car along a great circle one would hold the steering wheel fixed, but to follow a rhumb line one 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. On a Mercator projection the North and South poles occur at infinity and are therefore never shown. However the full loxodrome on an infinitely high map would consist of infinitely many line segments between the two edges. On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the North (or South) Pole.

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