loxodrome

Let be the longitude of a point on the loxodrome.

Let a loxodrome pass through the point whose longitude and latitude are both 0; call this the "central point".

Each loxodrome spirals infinitely often around each pole.

The later invention of logarithms allowed Leibniz to establish algebraic equations for the loxodrome.

However the full loxodrome on an infinitely high map would consist of infinitely many line segments between the two edges.

Using only the unique shortest loxodrome from the central point to each point p gives only one copy, occupying a sort of oval.

A loxodrome on the surface of the earth is a curve of constant bearing: it meets every parallel of latitude at the same angle.

On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the North (or South) Pole.

The 1878 edition of"The Globe Encyclopaedia of Universal Information" describes loxodrome lines as:

Suppose one starts at the central point and travels a certain distance in a certain direction along this loxodrome and arrives at geographic location .

The angle that the loxodrome subtends relative to the lines of longitude (i.e. its slope, the "tightness" of the spiral) is the argument of "k".

He was the first to propose the idea of a loxodrome and was also the inventor of several measuring devices, including the nonius, named after his Latin surname.

The pole-to-pole length of a loxodrome is (assuming a perfect sphere) the length of the meridian divided by the cosine of the bearing away from true north.

In navigation, a rhumb line (or loxodrome) is a line crossing all meridians of longitude at the same angle, i.e. a path derived from a defined initial bearing.

But theoretically a loxodrome can extend beyond the right edge of the map, where it then continues at the left edge with the same slope (assuming that the map covers exactly 360 degrees of longitude).

He was the first to understand why a ship maintaining a steady course would not travel along a great circle, the shortest path between two points on Earth, but would instead follow a spiral course, called a loxodrome.

Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes.

Portuguese mathematician and cosmographer Pedro Nunes (1502-1578), who first described the loxodrome and its use in marine navigation, and suggested the construction of a nautical atlas composed of several large-scale sheets in the cylindrical equidistant projection as a way to minimize distortion of directions.

A spherical spiral (rhumb line or loxodrome, left picture) is the curve on a sphere traced by a ship traveling from one pole to the other while keeping a fixed angle (unequal to 0 and to 90 ) with respect to the meridians of longitude, i.e. keeping the same bearing.