Example of pole-to-pole loxodrome

In navigation, a rhumb line (or loxodrome) is a line crossing all meridians at the same angle, i.e. a path of constant bearing. Image File history File links Loxodrome Description: The Loxodrome hat im Gegensatz zur archimedischen Kugelspirale einen konstanten Winkel zwischen den Längengraden und den Breitengraden, der hier das Verhältnis von 10 Länge zu 1 Breite hat. ... Image File history File links Loxodrome Description: The Loxodrome hat im Gegensatz zur archimedischen Kugelspirale einen konstanten Winkel zwischen den Längengraden und den Breitengraden, der hier das Verhältnis von 10 Länge zu 1 Breite hat. ... Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ... In navigation, a bearing is the clockwise angle between a reference direction (or a datum line) and the direction to an object. ...

The idea of a loxodrome was invented by a Portuguese Mathematician Pedro Nunes in the 1500s. Pedro Nunes (latin, Petrus Nonius), (1502, Alcácer do Sal – August 11, 1578, Coimbra) was a Portuguese mathematician, maybe born from a New Christian (of Jewish origin) family. ...

If you follow a given (magnetic-deviation compensated) 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 (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. Magnetic deviation is the error induced in a compass by local magnetic fields, which must be allowed for, along with magnetic declination, if accurate bearings are to be calculated. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Stereographic projection of a circle of radius R onto the x axis. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... On the earth, a meridian is a north-south line between the North Pole and the South Pole. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Rhumb lines are not defined at the poles.

Contrast with: great circle, small circle. For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the... A small circle, of a sphere, is the circle constructed by a plane crossing the sphere not in its middle point. ...

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. Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569) The Mercator projection is a cylindrical map projection presented by the German geographer and cartographer Gerardus Mercator, in 1569, in a large planisphere measuring 202 by 124 cm, printed in eighteen separate sheets. ...

On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the North (or South) pole. Stereographic projection of a circle of radius R onto the x axis. ... Logarithmic spiral (pitch 10°) Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ...

On a sphere which has coordinates φ (latitude), λ (longitude) and α (azimuth), the equation of a loxodrome is Latitude, usually denoted symbolically by the Greek letter φ, gives the location of a place on Earth north or south of the Equator. ... Longitude, sometimes denoted by the Greek letter λ, describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... Azimuth is the horizontal component of a direction (compass direction), measured around the horizon toward the East, i. ...

(where is the inverse Gudermannian function

 and is the longitude where the loxodrome passes the equator.)

Or Gudermannian function with its asymptotes y = ±π/2 marked in gray. ...

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). Secant is a term in mathematics. ... The 4 main circles of latitude on Earth A circle of latitude is an imaginary east-west circle on the Earth, that connects all locations with a given 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. A compass rose or wind 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 praying direction instead of the traditional rule of the shortest path that would give Northeast. Facing the Qibla at a prayer in Damascus The geometrical calculation of Qibla Qibla (Arabic: ‎ ) is an Arabic word referring to the direction that should be faced when a Muslim prays. ...

External links

• Loxodromes: A Rhumb Way to Go (PDF)