# Dictionary Definition

loxodrome n : 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 line,
rhumb]

# User Contributed Dictionary

## English

### Noun

loxodrome- a line on a surface (such as the Earth) that cuts all meridians at a constant angle (but not a right angle)
- the path followed by a ship or aircraft that maintains a constant course by the compass

### Synonyms

### See also

# Extensive Definition

In navigation, a rhumb line (or
loxodrome) is a line crossing all meridians
at the same angle, i.e. a path of constant bearing.
It is obviously easier to manually steer than the constantly
changing heading of the shorter great circle
route. 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 (\lambda - \lambda_0)\,

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

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

- 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).

The word "loxodrome" comes from Greek loxos :
oblique + dromos : running (from dramein : to run).

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. Jews, who face Jerusalem during
prayer, have traditionally faced directionally toward Israel using a
general plumb line (ie: from North America one would face east).
However there are now some who prefer a Great Circle path

## See also

## References

## External links

loxodrome in Asturian: Loxodrómica

loxodrome in Catalan: Loxodròmia

loxodrome in Czech: Loxodroma

loxodrome in German: Loxodrome

loxodrome in Spanish: Loxodrómica

loxodrome in French: Loxodromie

loxodrome in Italian: Lossodromia

loxodrome in Hebrew: לוקסודרום

loxodrome in Dutch: Loxodroom

loxodrome in Japanese: 等角航路

loxodrome in Norwegian: Loksodrom

loxodrome in Norwegian Nynorsk: Loksodrom

loxodrome in Polish: Loksodroma

loxodrome in Portuguese: Loxodrómia

loxodrome in Russian: Локсодрома

loxodrome in Finnish: Loksodromi

loxodrome in Swedish: Loxodrom