Rhumb line Totally Explained

Everything about Loxodrome totally explained




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, for example 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'll 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're close to being logarithmic spirals (on a stereographic projection they're 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 ť

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

with slope

m=cot(eta),!

. For a point with latitude

phi,

and longitude

lambda,!

the position in the Mercator projection can be expressed as ť

x= lambda, y=	anh^(m (lambda-lambda_0)).,

In cartesian coordinates this can be simplified to ť

x = r cos(lambda) / cosh(m (lambda-lambda_0)),,

y = r sin(lambda) / cosh(m (lambda-lambda_0)),,

z = r 	anh(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, for example doesn't make extra revolutions, and doesn't 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 don't 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'd 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

Further Information

Get more info on 'Loxodrome'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://rhumb_line.totallyexplained.com">Rhumb line Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.