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RHUMB LINE OF THE SPHERE
Curve studied by Pedro
Nunes in 1537, Simon
Stevin in 1608, Maupertuis
in 1744; name given by Snellius
in 1624.
Other names: loxodrome, wind line. 
Writing = longitude, = latitude, where is the constant angle between the curve and the parallels:
Spherical differential equation: . Except the case of the parallels: = constant obtained for = 0: Spherical equation: . where gd^{1} is the inverse of the Gudermann function, defined for by: gd^{1} (x) = ln (tan(x/2 + /4)) = arsinh(tan x) = artanh(sin x) = sign(x).arcosh(sec(x)), (the plot of which is the pseudoelliptic radioid). Cylindrical equation for : (hence the parametrization: ) Radius of curvature: . Radius of torsion: . 
The rhumb lines of the sphere, associated to a given axis, are the curves that form a constant angle with the parallel (or the meridians).
Do not mistake the rhumb lines for the spherical helices, that form a constant angle with the equatorial plane, nor for the clelias.
The rhumb lines correspond to the straight lines in Mercator coordinates ; in other words, on the maps of the Earth that use the Mercator projection, the rhumb lines are represented by straight lines. The angle a that the images of the rhumb lines form on the map with respect to the horizontal is the same as the angle they form on the sphere with respect to the parallels.
If we know the geographic coordinates and of two points, the angle a associated to the shortest rhumb line joining these two points is obtained by the formula: and the length is given by: .
The notion of rhumb line is opposed to that of geodesic, shortest path joining two points on the sphere, which is an arc of a great circle ; by comparison, the length of the geodesic joining the two points above is given by the formula .
The rhumb line (in red) and the geodesic (in blue) joining the point with longitude 15° west and latitude 15° south to the point with longitude 150° west and latitude 60° north.

The same on a map in Mercator projection!!! 
The double lattice of rhumb lines forming an angle of ±45° with the meridians creates an elegant lattice of orthogonal curves on the sphere.
Compare to the lattice of Viviani curves. 


The orthogonal projection of the rhumb line on the equatorial plane is, as the above cylindrical equation shows, the unbounded Poinsot spiral: .
The stereographic projection from the North pole on the equatorial plane is the logarithmic spiral: , which forms the same angle with the radius vector as the rhumb line forms with the meridians (since the stereographic projection is a conformal map).
Rhumb lines as seen by Escher
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© Robert FERRÉOL, Jacques MANDONNET 2018