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Innavigation, arhumb line (alsorhumb (/rʌm/) orloxodrome) is anarc crossing allmeridians oflongitude at the sameangle. It is a path of constantazimuth relative totrue north, which can besteered by maintaining acourse of fixedbearing. Whendrift is not a factor, accurate tracking of a rhumb line course is independent ofspeed.
In practical navigation, a distinction is made between thistrue rhumb line and amagnetic rhumb line, with the latter being a path of constant bearing relative tomagnetic north. While a navigator could easily steer a magnetic rhumb line using amagnetic compass, this course would not be true because themagnetic declination—the angle between true andmagnetic 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 theAge of Sail, as the correct declination could only be determined if the vessel'slongitude was accurately known, the central unsolved problem of pre-modern navigation.
Using asextant, under a clear night sky, it is possible to steer relative to a visiblecelestial 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 theCrux constellation (also known as the Southern Cross).
Steering a true rhumb line by compass alone became practical with the invention of the moderngyrocompass, an instrument that determines true north not by magnetism, but by referencing a stable internal vector of its ownangular momentum.
The effect of following a rhumb line course on the surface of a globe was first discussed by thePortuguesemathematicianPedro Nunes in 1537, in hisTreatise in Defense of the Marine Chart, with further mathematical development byThomas Harriot in the 1590s.
A rhumb line can be contrasted with agreat 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 zerogeodesic 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 theequator.
On aMercator 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.[1] On a Mercator projection theNorth Pole andSouth 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 astereographic projection map, a loxodrome is anequiangular spiral whose center is the north or south pole.
All loxodromes spiral from onepole to the other. Near the poles, they are close to beinglogarithmic spirals (which they are exactly on astereographic 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 perfectsphere) is the length of themeridian divided by thecosine of the bearing away from true north. Loxodromes are not defined at the poles.
The wordloxodrome comes fromAncient Greek λοξόςloxós: "oblique" + δρόμοςdrómos: "running" (from δραμεῖνdrameîn: "to run"). The wordrhumb may come fromSpanish orPortugueserumbo/rumo ("course" or "direction") and Greekῥόμβοςrhómbos,[2] fromrhémbein.
The 1878 edition ofThe Globe Encyclopaedia of Universal Information describes aloxodrome line as:[3]
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.[3]
A misunderstanding could arise because the term "rhumb" had no precise meaning when it came into use. It applied equally well to thewindrose lines as it did to loxodromes because the term only applied "locally" and only meant whatever a sailor did in order to sail with constantbearing, with all the imprecision that that implies. Therefore, "rhumb" was applicable to the straight lines onportolans 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:[4]
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'.
For a sphere of radius 1, the azimuthal angleλ, the polar angle−π/2 ≤φ ≤π/2 (defined here to correspond to latitude), andCartesian unit vectorsi,j, andk can be used to write the radius vectorr as
Orthogonal unit vectors in the azimuthal and polar directions of the sphere can be written
which have thescalar products
λ̂ for constantφ traces out a parallel of latitude, whileφ̂ for constantλ traces out a meridian of longitude, and together they generate a plane tangent to the sphere.
The unit vector
has a constant angleβ with the unit vectorφ̂ for anyλ andφ, since their scalar product is
A loxodrome is defined as a curve on the sphere that has a constant angleβ with all meridians of longitude, and therefore must be parallel to the unit vectorβ̂. As a result, a differential lengthds along the loxodrome will produce a differential displacement
where and are theGudermannian function and its inverse, and is theinverse hyperbolic sine.
With this relationship betweenλ andφ, the radius vector becomes a parametric function of one variable, tracing out the loxodrome on the sphere:
where
is theisometric latitude.[5]
In the Rhumb line, as the latitude tends to the poles,φ → ±π/2,sinφ → ±1, the isometric latitudearsinh(tanφ) → ± ∞, and longitudeλ increases without bound, circling the sphere ever so fast in a spiral towards the pole, while tending to a finite total arc length Δs given by
Letλ be the longitude of a point on the sphere, andφ its latitude. Then, if we define the map coordinates of theMercator projection as
a loxodrome with constantbearingβ from true north will be a straight line, since (using the expression in the previous section)
with a slope
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 unknownsm = cotβ andλ0. 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 pointsΔs, measured along a loxodrome, is simply the absolute value of thesecant of the bearing (azimuth) times the north–south distance (except forcircles of latitude for which the distance becomes infinite):
whereR is one of theearth average radii.
Its use in navigation is directly linked to the style, orprojection of certain navigational maps. A rhumb line appears as a straight line on aMercator projection map.[1]
The name is derived from Old French or Spanish respectively:"rumb" or "rumbo", a line on the chart which intersects all meridians at the same angle.[1] On a plane surface this would be the shortest distance between two points. Over the Earth's surface at low latitudes or over short distances it can be used for plotting the course of a vehicle, aircraft or ship.[1] Over longer distances and/or at higher latitudes thegreat circle route is significantly shorter than the rhumb line between the same two points. However the inconvenience of having to continuously change bearings while travelling a great circle route makesrhumb line navigation appealing in certain instances.[1]
The point can be illustrated with an east–west passage over90 degrees of longitude along theequator, for which the great circle and rhumb line distances are the same, at 10,000 kilometres (5,400 nautical miles). At 20 degrees north the great circle distance is 9,254 km (4,997 nmi) while the rhumb line distance is 9,397 km (5,074 nmi), about 1.5% further. But at 60 degrees north the great circle distance is 4,602 km (2,485 nmi) while the rhumb line is 5,000 km (2,700 nmi), a difference of 8.5%. A more extreme case is the air route betweenNew York City andHong Kong, for which the rhumb line path is 18,000 km (9,700 nmi). The great circle route over the North Pole is 13,000 km (7,000 nmi), or5+1⁄2 hours less flying time at a typicalcruising speed.
Some old maps in the Mercator projection have grids composed of lines oflatitude andlongitude but also show rhumb lines which are oriented directly towards 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. Seecompass rose. Such maps would necessarily have been in the Mercator projection therefore not all old maps would have been capable of showing rhumb line markings.
The radial lines on a compass rose are also calledrhumbs. The expression"sailing on a rhumb" was used in the 16th–19th centuries to indicate a particular compass heading.[1]
Early navigators in the time before the invention of themarine 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 (actually aparallel, which is a special case of the rhumb line), maintaining a constant latitude and recording regular estimates of the distance sailed until evidence of land was sighted.[6]
The surface of the Earth can be understood mathematically as aRiemann sphere, that is, as a projection of the sphere to thecomplex plane. In this case, loxodromes can be understood as certain classes ofMöbius transformations.
The formulation above can be easily extended to aspheroid.[7][8][9][10][11][12] The course of the rhumb line is found merely by using the ellipsoidalisometric latitude. In formulas above on this page, substitute theconformal latitude on the ellipsoid for the latitude on the sphere. Similarly, distances are found by multiplying the ellipsoidalmeridian arc length by the secant of the azimuth.
Note: this article incorporates text from the 1878 edition ofThe Globe Encyclopaedia of Universal Information, a work in the public domain
