Map Projections: Navigators and Radio Operators

Map Projections

Projections for Navigators and Radio Operators

Besides their pedagogical value, many map projections are invaluable for specialized professionals. For instance, a common problem is finding the shortest route across the Earth surface between two points. Such path is always part of a geodesic or great circle on the globe surface. The geodesic is used by ship and aircraft navigators attempting to minimize distances, while radio operators with directional antennae look for a bearing yielding the strongest signal.

The Cylindrical Conformal Projection

Mercator's Projection

Conventional (equatorial) Mercator map; graticule spacing 10°; map arbitrarily clipped at parallels 85°N and 85° S

The great Flemish cartographer Gerhard Kremer became famous with the Latinized name Gerardus Mercator. A revolutionary invention, the cylindrical projection bearing his name has a remarkable property: any straight line between two points is a loxodrome, or line of constant course on the sphere. In the common equatorial aspect, the Mercator loxodrome bears the same angle from all meridians. In other words, if one draws a straight line connecting a journey's starting and ending points on a Mercator map, that line's slope yields the journey direction, and keeping a constant bearing is enough to get to one's destination.

The only conformal cylindrical projection, Mercator's device was a boon to navigators from the 16th-century until the present, despite suffering from extreme distortion near the poles: Antarctica is enormously stretched, and Greenland is rendered about nine times larger than actual size. Indeed, stretching grows steadily towards the top and bottom of the map (in the equatorial form, in higher latitudes; the poles would be actually placed infinitely far away). Like all conformal projections, Mercator's was not intended for worldwide wall maps.

Although important, a Mercator map is not the only one used by navigators, as the loxodrome does not usually coincides with the geodesic, except in short travels.

This projection was possibly first used by Etzlaub ca. 1511; however, it was for sure only widely known after Mercator's atlas of 1569. Mercator probably defined the graticule by geometric construction; E.Wright formally presented equations in 1599.

Transverse Mercator Projection

Transverse Mercator map, central meridian 30°W

More commonly applied to large-scale maps, the transverse aspect preserves every property of Mercator's projection, but since meridians are not straight lines, it is better suited for topography than navigation.

Equatorial, transverse and oblique maps offer the same distortion pattern.

The transverse aspect, with equations for the spherical case, was presented by Lambert in his seminal paper (1772). The ellipsoidal case was developed, among others, by the great mathematician Carl Gauss (ca. 1822) and by Louis Krüger (ca. 1912); it is frequently called the Gauss conformal or Gauss-Krüger projection.

The UTM Grid

The best known use of the transverse Mercator projection is the specialized form called Universal Transverse Mercator (UTM) projection system.

The UTM defines a grid covering the world between parallels 84°N and 80°S. The grid is divided in sixty narrow zones, each centered on a meridian. Zones are identified by consecutive numbers, increasing from west to east (the first zone, immediately east of the 180° meridian, is numbered 1; zone 31 lies just east of the Greenwich meridian). A set of parallels divides the grid in rows, labeled by letters from C to X (I and O are not used) starting south. Therefore each zone comprises 20 quadrangles, identified by a number-letter pair. Quadrangles are in turn further subdivided in squares 100 km-wide, identified by double letter combinations.

Although no Mercator map is created by a perspective process, the cylinder is a useful visualization aid. The blue strip is zone 13 of the UTM grid; it is part of a cylindrical slice, approximating a spherical lune 6° wide at the equator and clipped by the 84°N and 80°S parallels.

Each zone is separately projected using the ellipsoidal form of the transverse Mercator projection with a secant case: scale of the central meridian is reduced by 0.04%, so two lines about 1°37" east and west of it have true scale. The UTM grid was designed for large-scale topographic mapping in separate sheets, not for whole world maps. In particular, sheets from different zones don't juxtapose exactly.

Within each quadrangle, any point may be located by its distance in meters, east from the central meridian and north from the Equator. The central meridian's coordinate is always 500,000; the Equator's coordinate is designated 0 for quadrangles in the northern hemisphere, and 10,000,000 for quadrangles in the southern hemisphere. Since the distance from poles to Equator is approximately 10,000 km, such offset origins ensure coordinates (called false eastings and false northings) are always positive.

The UTM grid is fairly regular, with a few exceptions:

the polar cap south of 80°S is mapped by the ellipsoidal case of the azimuthal stereographic projection, and comprises semicircular "zones" A (west of Greenwich meridian) and B
likewise, the cap north of 84°N is covered by an azimuthal stereographic projection comprising "zones" Y (west of Greenwich) and Z
all quadrangles span 8° in the south-north direction, except those in row X, which ranges from 72°N to 84°N
all quadrangles span 6° from west to east, except a few at rows X and V immediately east of the prime meridian

The original UTM system was adopted by the U.S. Army in 1949, and variations afterwards by several agencies throughout the world. Despite the name, it is not actually "universal" in the sense that each grid may be based on a different datum, so sheets from different grid sets may or may not be compatible. UTM maps are of course conformal, and distance and area distortion are limited by the large scale of individual sheets.

Azimuthal Equidistant Projection

Part of the azimuthal family and commonly used in the polar aspect, the azimuthal equidistant projection is invaluable for drawing some geodesics as straight lines. It also shows distances in true scale over the Earth surface (the captain of a nuclear submarine could use this projection to check which cities lie inside its destructive range).

Azimuthal equidistant map centered on 47°W 23°S

This azimuthal equidistant map is centered around Campinas, Brazil. This kind of map requires a fairly dense geographical database as its periphery is strongly stretched.

Unfortunately, such properties are valid for the central point only, and each azimuthal equidistant map must be tailor-made for that specific location.
On the other hand, a gnomonic map does not preserve distances and can not show the whole Earth, but it does map all geodesics to straight lines. In other words, there is no optimal single map which can show the shortest path between any two points on Earth.

As a rule, azimuthal projections make straightforward finding true directions from a single point: a radio operator whose hardware is stationed at the map's center could point its antenna for maximum gain towards anywhere on Earth.

Since the equidistant azimuthal preserves radial distances from the central point, here we immediately see that a shortest hypothetical flight from Campinas to Central Australia passes directly south over the pole.