All Topics  
Rhumb line

 
Rhumb Line

 
   Email Print
   Bookmark   Link
 

 

Rhumb line
 
 
  Topics Rhumb line Topic Home

In navigation
Navigation

Navigation is the process of reading, and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks....
, a rhumb line (or loxodrome) is a line crossing all meridians
Meridian (geography)

A meridian is an imaginary arc on the Earth's surface from the North Pole to the South Pole that connects all locations running along it with a given longitude....
 at the same angle, i.e. a path of constant bearing
Bearing (navigation)

In marine navigation, a bearing is the direction of one object in relation to another object, the other object usually being one's own vessel....
. Unlike a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
 route (for which bearing is not constant), following a rhumb line requires turning the vehicle more and more sharply while approaching the poles.






Discussion
Ask a question about 'Rhumb line'
Start a new discussion about 'Rhumb line'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Three views of a of pole-to-pole loxodrome.
In navigation
Navigation

Navigation is the process of reading, and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks....
, a rhumb line (or loxodrome) is a line crossing all meridians
Meridian (geography)

A meridian is an imaginary arc on the Earth's surface from the North Pole to the South Pole that connects all locations running along it with a given longitude....
 at the same angle, i.e. a path of constant bearing
Bearing (navigation)

In marine navigation, a bearing is the direction of one object in relation to another object, the other object usually being one's own vessel....
. Unlike a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
 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
Portuguese people

The Portuguese people are the ethnic group or nation native to the country of Portugal, in the west of the Iberian peninsula of Southern Europe-Western Europe Europe....
 mathematician
Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
 Pedro Nunes
Pedro Nunes

Pedro Nunes , was a Portugal mathematics, cosmographer, and professor, born from a New Christian family.Nunes, considered to be one of the greatest mathematicians of his time, is best known for his contributions in the technical field of navigation, which was crucial to the Portuguese Portugal in the period of discoveries....
 in 1537, in his Treatise in Defense of the Marine Chart, with further mathematical development by Thomas Harriot
Thomas Harriot

Thomas Harriot was an English astronomy, mathematician, ethnographer, and translator. Some sources give his surname as Harriott or Hariot or Heriot. He is sometimes credited with the introduction of the potato to Great Britain and Ireland....
 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
Geographical pole

A geographical pole , is either of two points on the surface of a spinning planet or other spinning body, at 90 degrees from its equator, at one of the two points where the Axis of rotation around which the body spins meets the surface of the body....
 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 spiral
Logarithmic spiral

A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Ren? Descartes and later extensively investigated by Jakob Bernoulli, who called it Spira mirabilis, "the marvelous spiral"....
s (on a stereographic projection
Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....
 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
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
) the length of the meridian
Meridian (geography)

A meridian is an imaginary arc on the Earth's surface from the North Pole to the South Pole that connects all locations running along it with a given longitude....
 divided by the cosine of the bearing away from true north.

Rhumb lines are not defined at the poles.

On a Mercator projection
Mercator projection

The Mercator projection is a Map projection#Triangular presented by the Flemish people geographer and cartographer Gerardus Mercator, in 1569....
 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
Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....
 map, a loxodrome is an equiangular spiral whose center is the North (or South) pole.

Mathematical Derivation


Let ß be the constant bearing
Bearing (navigation)

In marine navigation, a bearing is the direction of one object in relation to another object, the other object usually being one's own vessel....
 from true north of the loxodrome and be the longitude where the loxodrome passes the equator. Let be the longitude of a point on the loxodrome. Under the Mercator projection
Mercator projection

The Mercator projection is a Map projection#Triangular presented by the Flemish people geographer and cartographer Gerardus Mercator, in 1569....
 the loxodrome will be a straight line with slope . For a point with latitude and longitude the position in the Mercator projection can be expressed as Then the latitude of the point will be or in terms of the Gudermannian function
Gudermannian function

The Gudermannian function, named after Christoph Gudermann , relates the circular trigonometric function and hyperbolic trigonometric functions without using complex numbers....
 gd In cartesian coordinates this can be simplified to

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(a) and ?0. 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
Secant

Secant is a term in mathematics. It comes from the Latin secare . It can refer to:* a secant line, in geometry* the Trigonometric functions#Reciprocal functions, reciprocal to the cosine....
 of the bearing (azimuth) times the north-south distance (except for circles of latitude).

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 ??µß??
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
.

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
Compass rose

For Compass Airlines, an Airline in the US using the Callsign "Compass Rose," See Compass Airlines A compass rose is a figure displaying the Orientation of the Cardinal directions, north, south, east and west on a map or nautical chart....
.

There are some Muslim groups in North America that take the rhumb line to Mecca (southeastwards) as their qibla
Qibla

Qiblah is an Arabic language word for the direction that should be faced when a Muslim prayer during Salah. Most mosques contain a mihrab in a wall that indicates the qiblah....
 (praying direction) instead of the traditional rule of the shortest path, which would give Northeast.

See also

  • great circle
    Great circle

    A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
  • small circle
    Small circle

    A small circle of a sphere is the circle constructed by a plane crossing the sphere not in its center. Small circles always have smaller diameters than the sphere itself ....


External links

  • at MathPages