In navigation, a rhumb line (also rhumb () or loxodrome) is an arc crossing all meridians of longitude at the same angle. It is a path of constant azimuth relative to true north, which can be steered by maintaining a course of fixed bearing. When drift is not a factor, accurate tracking of a rhumb line course is independent of speed. In practical navigation, a distinction is made between this true rhumb line and a magnetic rhumb line, with the latter being a path of constant bearing relative to magnetic north. While a navigator could easily steer a magnetic rhumb line using a magnetic compass, this course would not be true because the magnetic declination—the angle between true and magnetic north—varies across the Earth's surface. To follow a true rhumb line, using a magnetic compass, a navigator must continuously adjust magnetic heading to correct for the changing declination. This was a significant challenge during the Age of Sail, as the correct declination could only be determined if the vessel's longitude was accurately known, the central unsolved problem of pre-modern navigation. Using a sextant, under a clear night sky, it is possible to steer relative to a visible celestial pole star. The magnetic poles are not fixed in location. In the northern hemisphere, Polaris has served as a close approximation to true north for much of recent history. In the southern hemisphere, there is no such star, and navigators have relied on more complex methods, such as inferring the location of the southern celestial pole by reference to the Crux constellation (also known as the Southern Cross). Steering a true rhumb line by compass alone became practical with the invention of the modern gyrocompass, an instrument that determines true north not by magnetism, but by referencing a stable internal vector of its own angular momentum.
Introduction
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, any 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 it can extend beyond a side edge of the map, where it then continues from the opposite edge at the same slope and latitude it departed at (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 Pole and South Pole 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. All loxodromes spiral from one pole to the other. Near the poles, they are close to being logarithmic spirals (which they are exactly on a stereographic projection, see below), so they wind around each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome (assuming a perfect sphere) is the length of the meridian divided by the cosine of the bearing away from true north. Loxodromes are not defined at the poles.
Etymology and historical description The word loxodrome comes from Ancient Greek λοξός loxós: "oblique" + δρόμος drómos: "running" (from δραμεῖν drameîn: "to run"). The word rhumb may come from Spanish or Portuguese rumbo/rumo ("course" or "direction") and Greek ῥόμβος rhómbos, from rhémbein. The 1878 edition of The Globe Encyclopaedia of Universal Information describes a loxodrome line as:
Loxodrom′ic Line is a curve which cuts every member of a system of lines of curvature of a given surface at the same angle. A ship sailing towards the same point of the compass describes such a line which cuts all the meridians at the same angle. In Mercator's Projection (q.v.) the Loxodromic lines are evidently straight. A misunderstanding could arise because the term "rhumb" had no precise meaning when it came into use. It applied equally well to the windrose lines as it did to loxodromes because the term only applied "locally" and only meant whatever a sailor did in order to sail with constant bearing, with all the imprecision that that implies. Therefore, "rhumb" was applicable to the straight lines on portolans when portolans were in use, as well as always applicable to straight lines on Mercator charts. For short distances portolan "rhumbs" do not meaningfully differ from Mercator rhumbs, but these days "rhumb" is synonymous with the mathematically precise "loxodrome" because it has been made synonymous retrospectively. As Leo Bagrow states:
the word ('Rhumbline') is wrongly applied to the sea-charts of this period, since a loxodrome gives an accurate course only when the chart is drawn on a suitable projection. Cartometric investigation has revealed that no projection was used in the early charts, for which we therefore retain the name 'portolan'.
Mathematical description For a sphere of radius 1, the azimuthal angle λ, the polar angle −π/2 ≤ φ ≤ π/2 (defined here to correspond to latitude), and Cartesian unit vectors i, j, and k can be used to write the radius vector r as
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