Whenever one plans moving from a point on Earth to another far enough to be out of sight, or on the sea or above the clouds where no paths are marked, some very related questions arise:

- what's the initial
**heading**, i.e., direction to turn, for establishing a path with a "straight" course? - if no course corrections are taken, which points will be visited along the way?

Directions on a surface are stated as
a **bearing**, i.e., the angle from a reference
line. On Earth, this line is the meridian through the current
location, and the bearing is usually measured in degrees, from
0° northwards and increasing clockwise to 360°
northwards again — on Earth this angle is more commonly known as the
**azimuth**. The direction of the current meridian can be
obtained from an ordinary compass, whose needle is always aligned
with the magnetic north-south direction, but one must account for
*magnetic declination*, the deviation of magnetic north
from true geographic north. Magnetic declination is not uniform
throughout the world and slowly changes with time (in millions
of years, the magnetic "poles" have even switched hemispheres
several times), therefore nautical charts must be periodically
updated with reference correction marks for declination.
Gyrocompasses and astrocompasses are immune to magnetic
declination; on the other hand, a gyrocompass depends on a power
supply to keep its wheel spinning, while operating an
astrocompass requires a precise clock and up-to-date astronomical tables.

If its angle remains unchanged from *every* current
meridian, a path is
a **loxodrome**
or **rhumb line**, a line of constant bearing. The
concept was, perhaps after suggestions by Martim Afonso de
Sousa, invented by the Portuguese scholar Pedro Nunes (or NuÃ±ez)
ca. 1533, although the precise mathematical details were
understood only much later.

A parallel crosses all meridians at straight angles, thus all
parallels are closed loxodromes in the east-west direction. All
meridians are obviously trivial loxodromes in the north-south
direction. For all other directions the loxodrome is an open
(i.e., with two distinct ends) three-dimensional curve known as
a *spherical helix* or *loxodromic spiral*: each
end reaches a pole after an infinite number of tighter and
tighter turns.

Two points not on the same parallel or meridian can be connected by an infinite number of loxodromes, but one is almost always interested on the shortest, "steeper" one, which crosses less than half of the meridians; the other rhumb lines do one or more additional turns around the Earth.

Clearly rhumb lines are not in general the most direct route to either pole. In fact, the Equator and the meridians alone are not only loxodromes but also great circles, containing the geodesic lines which are the shortest path between two points. On the other hand, in the absence of reference points the loxodrome is the easiest path to follow: once the proper bearing is established, it is enough keeping it true — i.e., the compass's needle straight aligned — to be sure to reach the destination. Reality is a bit more complicated for aquatic and aerial vehicles: crosswinds, waves and streams cause the bearing to deviate from the heading, unless compensated.

How do projections determine how the loxodrome is drawn on flat maps? The azimuthal orthographic projection clearly shows the spherical helix's shape, but it is useless for measuring bearings, unless the starting point lies at the center of projection. On the polar aspect of an azimuthal stereographic map, the loxodrome appears as a logarithmic spiral; this is a direct consequence of this projection's conformality: the logarithmic spiral is a plane curve which intersects all of its radii at the same angle; it is also self-similar, looking the same no matter how magnified.

Most other projections are poorly suited for presenting or
calculating loxodromes, which are mapped to complex curves. The
exception is Gerhardus Mercator's most famous conformal projection:
in the equatorial aspect all meridians are vertical lines,
and *all* loxodromes are straight lines. Consequently,
in order to determine the bearing between two points it is
enough connecting them on an equatorial Mercator map and
measuring its angle, or *slope*, from the vertical: a straightedge
and a protractor are sufficient.

How does Mercator's design compare with another common cylindrical projection, the equidistant cylindrical? In the latter, parallels are equally spaced and the loxodrome is curved towards the Equator. Mercator progressively spaced apart the parallels, in proportion to the trigonometric secant of the latitude; the exact mathematics was not available at the time, so Mercator probably resorted to geometric approximations. Unfortunately, since the secant is infinite at the poles, these lie at infinity and cannot be included in real maps: to do so while keeping correct angles would join meridians at two points, which is impossible in a cylindrical map.

In fact, parallel spacing is so exaggerated at higher latitudes that equatorial Mercator maps are often clipped about 70-80° north and south. This is not too relevant for navigation (near the polar caps, magnetic declination is too significant anyway, so in these comparatively small areas other tools are employed), and Mercator's projection has faithfully served route planners for centuries.

Another noteworthy projection,
the loximuthal,
devised independently by K.Siemon and W.Tobler,
shows *all* loxodromes intersecting at a special point on
the central meridian as straight lines with constant
scale *and* direction. Unfortunately, like
the azimuthal
equidistant and unlike the universal equatorial Mercator,
loximuthal maps must be tailor-made for each point of interest.

Copyright © 1996, 1997, 2008 Carlos A. Furuti