Rhumb line
In
navigation, a rhumb line is a line crossing all meridians at the same angle, i.e. a path of constant bearing.
The idea of a loxodrome was invented by a Portuguese Mathematician Pedro Nunes in the 1500s.
If you follow a given compass-bearing on Earth, you will be following a rhumb line, which spirals from one pole to the other, with the exception of 90 and 270 degrees, lines of constant latitude, e.g. the equator. Near the poles, they are close to being
logarithmic spirals , so they wind round each pole an infinite number of times but reach the pole in a finite distance.
Encyclopedia
In
navigation, a
rhumb line is a line crossing all meridians at the same angle, i.e. a path of constant bearing.
The idea of a loxodrome was invented by a Portuguese Mathematician Pedro Nunes in the 1500s.
If you follow a given compass-bearing on Earth, you will be following a rhumb line, which spirals from one pole to the other, with the exception of 90 and 270 degrees, lines of constant latitude, e.g. the equator. Near the poles, they are close to being
logarithmic spirals , 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 the length of the
meridian divided by the
cosine of the bearing away from true north.
Rhumb lines are not defined at the poles.
Contrast with:
great circle,
small circle.
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 pole.
On a sphere which has coordinates
φ ,
λ and
α , the equation of a loxodrome is
- '
Or
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 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 times the north-south distance .
The word "loxodrome" comes from Greek
loxos : oblique +
dromos : running .
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 as their
praying direction instead of the traditional rule of the shortest path that would give Northeast.
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