Rhumb line

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Three views of a of pole-to-pole loxodrome.

In navigation, a rhumb line (or loxodrome) is a line crossing all meridians at the same angle, i.e. a path of constant bearing. Unlike a great circle route (for which bearing is not constant), following a rhumb line requires turning the vehicle more and more sharply while approaching the poles. 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.

If you follow a given (magnetic-deviation compensated) compass-bearing on Earth, you will be following a rhumb line. All rhumb lines spiral from one pole to the other unless the bearing is 90 or 270 degrees, in which case the loxodrome is a line of constant latitude, such as the equator. Near the poles, they are close to being logarithmic spirals (on a stereographic projection they are exactly, see below), 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 cosine of the bearing away from true north.

Rhumb lines are not defined at the poles.

On a Mercator projection map, a loxodrome is a straight line; beyond the right edge of the map it continues on the left with the same slope. The full loxodrome on the full infinitely high map would consist of infinitely many line segments between these 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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[edit] Mathematical Derivation

Let β be the constant bearing from true north of the loxodrome and \lambda_0\,\! be the longitude where the loxodrome passes the equator. Let \lambda\,\! be the longitude of a point on the loxodrome. Under the Mercator projection the loxodrome will be a straight line

x=\lambda, y = m (\lambda - \lambda_0)\,

with slope m=\cot(\beta)\,\!. For a point with latitude \phi\, and longitude \lambda\,\! the position in the Mercator projection can be expressed as

x= \lambda, y=\tanh^{-1}(\sin \phi).\,\!

Then the latitude of the point will be

\phi=\sin^{-1}(\tanh(m (\lambda-\lambda_0))),\,

or in terms of the Gudermannian function gd \phi=\rm{gd}(m (\lambda-\lambda_0)).\, In cartesian coordinates this can be simplified to

x = r \cos(\lambda) / \cosh(m (\lambda-\lambda_0)),\,
y = r \sin(\lambda) / \cosh(m (\lambda-\lambda_0)),\,
z = r \tanh(m (\lambda-\lambda_0)).\,

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).

[edit] Etymology

The word "loxodrome" comes from Greek loxos : oblique + dromos : running (from dramein : to run). The word "rhumb" may come from Spanish/Portuguese rumbo (course, direction) and Greek ῥόμβος.[1]

[edit] Historical description

The 1878 edition of 'The Globe Encyclopaedia of Universal Information' describes loxodrome lines as:[2]

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.[2]

[edit] Usage

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.

There are some Muslim groups in North America that take the rhumb line to Mecca (southeastwards) as their qibla (praying direction) instead of the traditional rule of the shortest path, which would give Northeast.

[edit] See also

[edit] References

  1. ^ Rhumb at TheFreeDictionary
  2. ^ a b Ross, J.M. (editor) (1878). "The Globe Encyclopaedia of Universal Information", Vol. IV, Edinburgh-Scotland, Thomas C. Jack, Grange Publishing Works, retrieved from Google Books 2009-03-18;

[edit] External links


Note: this article incorporates text from the 1878 edition of The Globe Encyclopaedia of Universal Information, a work in the public domain

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