# Rhumb line

Rhumb line

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. At lower latitudes, however, a loxodrome may be easier to follow than a great circle. The effect of following a rhumb line course on the surface of a globe was first discussed by the Portuguese mathematician Pedro Nunes in the 1530s, 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.

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 :$x=lambda, y = m \left(lambda - lambda_0\right),$with slope . For a point with latitude $phi,$ and longitude $lambda,!$ the position in the Mercator projection can be expressed as :$x= lambda, y= anh^\left\{-1\right\}\left(sin phi\right).,!$Then the latitude of the point will be:$phi=sin^\left\{-1\right\}\left( anh\left(m \left(lambda-lambda_0\right)\right)\right),,$or in terms of the Gudermannian function "gd" $phi= m\left\{gd\right\}\left(m \left(lambda-lambda_0\right)\right).,$In cartesian coordinates this can be simplified to:$x = r cos\left(lambda\right) / cosh\left(m \left(lambda-lambda_0\right)\right),,$:$y = r sin\left(lambda\right) / cosh\left(m \left(lambda-lambda_0\right)\right),,$:$z = r anh\left(m \left(lambda-lambda_0\right)\right).,$

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).

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". [" [http://www.thefreedictionary.com/rhumb Rhumb] " at TheFreeDictionary]

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
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.

ee also

*great circle
*small circle

References

* [http://www.cwru.edu/artsci/math/alexander/mathmag349-356.pdf Loxodromes: A Rhumb Way to Go (PDF)]
* [http://www.mathpages.com/home/kmath502/kmath502.htm Constant Headings and Rhumb Lines] at MathPages

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### Look at other dictionaries:

• rhumb line — Rhumb Rhumb, n. [F. rumb, Sp. rumbo, or Pg. rumbo, rumo, probably fr. Gr. ??? a magic wheel, a whirling motion, hence applied to a point of the compass. See {Rhomb}.] (Navigation) A line which crosses successive meridians at a constant angle;… …   The Collaborative International Dictionary of English

• rhumb line — n. the course of a ship that keeps a constant compass direction, represented on a map, chart, or globe by a line that cuts across all meridians at the same angle …   English World dictionary

• rhumb line — rhumb′ line n. navig. the path of a ship that maintains a constant compass direction • Etymology: 1660–70 …   From formal English to slang

• rhumb line — noun a line on a sphere that cuts all meridians at the same angle; the path taken by a ship or plane that maintains a constant compass direction • Syn: ↑rhumb, ↑loxodrome • Hypernyms: ↑line * * * ˈrəm|līn noun : a line on the surface of the earth …   Useful english dictionary

• rhumb line — A line on the surface of the earth that cuts all meridians at the same angle. It appears as a curved line on the surface of a sphere. Only one such line may be drawn through any two points. Although this is not the shortest distance, the… …   Aviation dictionary

• Rhumb Line Resort — (Kennebunkport,США) Категория отеля: 3 звездочный отель Адрес: 41 Turbat s Creek …   Каталог отелей

• rhumb line — a curve on the surface of a sphere that cuts all meridians at the same angle. It is the path taken by a vessel or aircraft that maintains a constant compass direction. Also called loxodrome, rhumb. [1660 70] * * * …   Universalium

• rhumb line — noun Etymology: Spanish rumbo Date: 1669 a line on the surface of the earth that follows a single compass bearing and makes equal oblique angles with all meridians called also loxodrome …   New Collegiate Dictionary

• rhumb line — noun a) A line that cuts all meridians at the same angle. b) The path of a vessel that maintains a constant compass direction. Syn: loxodrome …   Wiktionary

• Rhumb line — Die Loxodrome von A nach B schneidet alle Meridiane im konstante Winkel η Die Loxodrome (gr. loxos „schief“, dromos „Lauf“) ist eine Kurve auf einer Kugeloberfläche, die immer unter dem gleichen Winkel die Meridiane im …   Deutsch Wikipedia