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Projections for Navigators and Radio Operators

Normal Mercator map

Conventional (equatorial) Mercator map arbitrarily clipped between parallels 85°N and 85°S

Two classic map projections, one cylindrical, one azimuthal, deserve descriptions separate from their groups. While the Mercator and azimuthal equidistant projections are essential for specialized cartographic applications, the Internet has spread the former's visibility and audience much beyond its intended goal.

The Cylindrical Conformal Projection

Mercator's Projection

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 a straight line is drawn on an equatorial Mercator map connecting a journey's starting and ending points, that line's slope yields the journey's unchanging direction; keeping a constant bearing is enough to arrive at the destination.

Transverse Mercator map

Transverse Mercator map, central meridian 30°W. This projection is not intended for small-scale or whole world maps, as the scale and areal distortion is obvious even moderately far from the central meridian. On the other hand, it is usually the best choice for large-scale precision mapping, rivaled only by Lambert's conformal conic.

This projection is almost always presented in a tangent case, with the Equator as a standard parallel free of distortion. When using a secant case, two parallels symmetrically opposite the Equator become standard lines; the resulting map is nearly identical after a change in aspect ratio, much like variations of the equal-area cylindrical projection. In this context, the words “tangent” and “secant” are only conceptual, since the Mercator projection is not defined by a perspective process on a developing cylinder.

The only conformal cylindrical projection, Mercator's device was a boon to navigators from the 16th-century until the present, despite suffering from extreme area distortion near the poles: in order to keep shapes undistorted, 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). Mercator maps seldom extend above the 80°N parallel or below 75°S.

Both this apparent shift of the Equator southwards and the areal exaggeration of intermediate latitudes, which mostly coincide with developed nations, have repeatedly incited disapproval about its supposed bias against the Third World (it was even claimed to aid racial discrimination by promoting a supposed superiority of Europe, the U.S.S.R. or the United States); misguided or naïve controversies and proposals to fix the wrong problem include the “Peters” projection, which (like many others) preserves areas, but strongly distorts shapes and has no especially interesting property to compensate for. Although historically several maps have been enlisted in political propaganda, this is no fault of the projections themselves: like all conformal projections, Mercator's was never intended for world wall maps. Nevertheless, it was once common in textbooks.

More recently, the spherical (but using an ellipsoidal datum, as a result not exactly conformal) case was chosen for the world view of Google Maps, clipped between the 85°3′4″ latitudes, which yield a square map, convenient for efficient storage and retrieval. This variant is usually referred to as the Web Mercator.

The Web Mercator projection: not a bad choice at all

All around the Internet, countless voices have expressed from puzzlement to indignation about the choice of the Web Mercator projection by Google Maps and similar Internet services like Bing Maps. Although shallow reasoning and political correctness seem to have tinged most criticism, technical issues and practical benefits justify that decision.

Why an equatorial cylindrical projection?

Tiles organized by scaling factor, or zoom levels.

An extremely simplified overview of the process which converts cartographic assets into mapped tiles deliverable to users.

Tiles organized by scaling factor, or zoom levels.

An extremely simplified overview of the process which converts cartographic assets into mapped tiles deliverable to users.

Unlike a standalone navigation GPS device, which stores maps internally, millions of simultaneous users of Google Maps and its peers receive them across the Internet. Therefore, given the raw cartographical assets — from satellite imagery to aerophotogrammetry data to geocoordinated information collected by official and private agencies — as much as possible of the mapmaking process, including perspective correction (pictures taken from air and space are essentially tilted perspective projections which must be reprojected to a common format) and cartographic projection, is carried offline and stored as tiles in huge computer servers. Later, users are served tiles, commonly overlaid with dynamic data which is optional (like markers and routes), localized (street and place names) or frequently updated (such as traffic conditions).

Tiles are precalculated images which can be juxtaposed seamlessly, generated in multiple sets of scaling factors, commonly referred to as zoom levels. In a simple scheme, information in each tile of level n is also represented in four tiles of level n + 1; if all tiles have identical size, each level has twice the linear resolution of the previous one (more sophisticated schemes allow finer-grained scale changes). Stacked zoom levels comprise an inverted square-based pyramid, where one goes up only as necessary to get enough detail; it is essentially analogous to a quadtree, a classic data structure appropriate for two-dimensional information which may be sparse — for instance, low-priority, seldom visited or featureless areas like open ocean may be absent from higher levels but are still visible in lower resolution (missing pixels are eventually scaled up to match those of higher-level neighbor tiles). Thus it is important for all levels to share the same projection.

Tiles organized by scaling factor, or zoom levels.

Mapped tiles as sets organized by scaling factor, or “zoom level”. Uniformly-shaped tiles are an efficient way to deal with variable-detail information because not all tiles in a view necessarily belong to the same level — for instance, in level 2, since the tile west of South America is mostly an uninteresting ocean, its data is likely to be omitted in higher levels: level 2's version is simply scaled up, trading detail for bandwidth.

Tiles organized by scaling factor, or zoom levels.

Mapped tiles as sets organized by scaling factor, or “zoom level”. Uniformly-shaped tiles are an efficient way to deal with variable-detail information because not all tiles in a view necessarily belong to the same level — for instance, in level 2, since the tile west of South America is mostly an uninteresting ocean, its data is likely to be omitted in higher levels: level 2's version is simply scaled up, trading detail for bandwidth.

Because each user should be sent only the minimum data to assemble his or her area of interest, equatorial cylindrical projections are almost ideally suited for tiling: as the user pans and zooms a view of the world, tiles slide across the screen (a very fast operation once downloaded) and replace one another smoothly; because tile edges are aligned with cardinal directions, crossing beyond the left and right borders of the map is trivial, and it's fairly simple both detecting which tile to load next once the view nears a tile's edge and determining which Earth feature corresponds to a pixel clicked by the user. Other classes of projections would impose either nonuniform tile shapes or irregular map boundaries, making navigation slower or awkward.

Why Mercator's?

Mercator's areal exaggeration far from a central line is, or should be, a well-known side effect of conformality. However, most users of Google Maps are probably less interested in statistical comparisons — where equivalence is paramount — than in urban navigation where the Web Mercator's near-conformality is essential. Any equatorial cylindrical projection must cope with east-west scale, compressed between the standard parallels, inflated outside them. What happens in the north-south direction?

Parallels on a sphere Parallels on three cylindrical maps

On a sphere, the Equator is about 26.5% longer than the parallel at San Francisco, California. On any equatorial cylindrical map, all parallels have the same length, therefore if the Equator has correct scale, horizontal scale across San Francisco is 26.5% too large. Cylindrical projections (from left to right, Mercator, Plate Carrée and Gall-Peters) are fundamentally defined by how vertical scale deals with this distortion.

San Francisco, California; Mercator map San Francisco, California; Mercator map

Because the Web Mercator projection adjusts the vertical scale to exactly match the horizontal one, a small portion of San Francisco, California shows correct angles at street intersections and building corners. Original “satellite” and “roadmap” data by Google, north at the top.

San Francisco, California; Mercator map San Francisco, California; Mercator map

The same place on a Plate Carrée projection: at this latitude, vertical scale is smaller than the horizontal. Shape deformation and the sun's position contribute to the illusion of an aerial view tilted almost upside-down.

San Francisco, California; Mercator map San Francisco, California; Mercator map

The same place mapped by the Gall-Peters projection, evincing shape deformation in the perpendicular direction.

Suppose a Web Mercator map of a portion of San Francisco, California. At this latitude of 37°47′N, cylindrical projections with a standard Equator exaggerate the horizontal scale to about 126.5% of actual. Mercator's projection stretches the vertical scale by exactly the same amount, greatly expanding areas but preserving local shapes: buildings and crossroads duly reproduce the original 90° angles.

If the same spot is represented with identical horizontal scale by the Plate Carrée, the most common particular case of the cylindrical equidistant projection, the map is equidistant along all meridians; in other words, vertical scale is true and constant. Here, it is only 79.04% of the horizontal, distorting building plans and the street grid.

Vertical/horizontal scale for selected cylindrical projections

The ratio vertical/horizontal scale as a function of latitude, for selected equatorial cylindrical projections. For each map, minimizing distortion by choosing an appropriate projection entails assessing not only the standard parallels but also the range of scale variation over the regions of interest. For the Web Mercator, 90° is a theoretical maximum. All projections based on a sphere, when used with "real-world" data, suffer from additional deviations from the 100% ideal.

Conversely, in the same spot mapped by the Gall-Peters, one of several variations of Lambert's equal-area cylindrical projection, the vertical scale is 24.93% too large, and angles are deformed in the perpendicular direction.

Clearly, the outcome of any such comparisons depends on the latitude: the Plate Carrée projection is free of distortion only at the Equator, while the Gall-Peters is correct only along its standard parallels 45°N and S (it's nearly optimal, e.g., for Ottawa and Montreal, Canada, and Turin, Italy) — like Gall's isographic cylindrical, another case of the equidistant cylindrical. Also, in places where streets are aligned with the cardinal directions, any cylindrical equatorial projection would correctly show 90° intersections, though city blocks would still be squashed or stretched, except at standard parallels. But the Web Mercator projection does present all but correct shapes regardless of orientation, almost everywhere.

Plotting the ratio of scales in the north-south and west-east direction provides an insight of how shape distortion changes with latitude, and can assist selecting a projection given the regions to be mapped and the set tolerance for distortion. For cylindrical maps, such a chart conveys essentially the same information as angular deformation patterns.

Although fundamental, a Mercator map is not the only one used by navigators, as the loxodrome does not usually coincide with the geodesic, except in short travels. The geodesic may be plotted on a gnomonic map, and later transferred to a Mercator map and split in loxodromes piecewise.

This projection was possibly first used in the equatorial aspect by Etzlaub ca. 1511; however, it was for sure only widely known after Mercator's atlas of 1569. Since a rigorous underlying mathematical theory was not available at the time, Mercator probably defined the graticule by geometric approximation; E.Wright formally presented equations in 1599.

Transverse Mercator Projections

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 large-scale topographic maps than navigation. It does map a single central meridian (two when secant) with no distortion.

Oblique Mercator map

No aspect of the Mercator projection is appropriate for global or small-scale maps. Finding an oblique aspect which shows all major landmasses with no interruptions is a simple exercise for the spherical case — visually interesting but of little practical value; ellipsoidal versions for small regions pose a much more useful and difficult problem.

As usual, equatorial, transverse and oblique versions of Mercator's projection offer exactly 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.

Oblique Mercator Maps

A mathematical challenge, oblique aspects of the Mercator projection in the ellipsoidal case have attracted professional interest for large-scale local or regional maps. Several approaches have been suggested, usually based on intermediate projection surfaces (if successive conformal mappings are applied in sequence, the end result remains conformal) and differing in details like the range of scale distortion; in general the constant scale along parallel lines of the spherical version is not retained.

Jean Laborde's version (1926), applied to Madagascar, first transformed the ellipsoid into a conformal sphere by appropriately shifting parallels, then used an ordinary transverse Mercator projection, followed by a rotation to align Madagascar's longest dimension with what would be the map's central meridian.

Martin Hotine's better-known method, sometimes called the Hotine projection, was first employed (ca. 1946) for Southeast Asia, then other areas including in the United States. Instead of a sphere, its intermediate surface is an aposphere, a parametric surface of constant total curvature.

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 unused, avoiding confusion with numbers) 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.

An UTM zone

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. This is not the shape of UTM maps; rather, each UTM zone is covered by many partly juxtaposed rectangular large-scale sheets.

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 world or regional maps. In particular, sheets from different zones don't juxtapose exactly.

Within each quadrangle, any point may be located by two distances in meters: the easting, east from the central meridian and the northing, 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 (the false eastings and false northings) are always positive.

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

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 map
Azimuthal orthographic map

First, oblique azimuthal equidistant map centered on Campinas, Brazil. An extra superimposed green graticule helps determining direction and distance from Campinas to every other point on Earth. For instance, it shows Central Australia to be as far as India, and can be reached by a shortest route directly south over Antarctica. The second map, an azimuthal orthographic one, is easier to visualize but not so useful for measurements.

The Azimuthal Equidistant Projection

A common mapping task 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 azimuthal equidistant projection is trivially easy to draw in the polar aspect and, like all azimuthal designs, it features some special properties for the central point alone: all straight lines touching it are geodesics, and the angle between any two of those lines is the same as on Earth. Hence, oblique azimuthal equidistant maps must be tailor-made for each specific location.

The projection's modern name is due to Antonio Cagnoli, who reinvented it in 1799; earlier it had been mentioned by J.Lambert in his seminal paper of 1772, Guillaume Postel (1581), who is often credited as the original author, and Glareanus (ca. 1510), among others.

As a rule, azimuthal projections make straightforward finding true directions from a single point (the reference straight line is, of course, the local meridian): a short wave radio operator whose hardware is stationed at the map's center can use it in order to orient its antenna for maximum gain towards anywhere on Earth.

Furthermore, since of all azimuthal projections the equidistant alone preserves radial distances from the central point, the operator may estimate how much power is required for ensuring stable communication. Likewise, the captain of a nuclear submarine could use this projection to check which cities lie inside its destructive range. Other projections simply are not appropriate.

HomeSite MapPremodern ProjectionsMap Projections - ContentsApplications — August  5, 2018
Copyright © 1996, 1997, 2008 Carlos A. Furuti