Map Projections: Useful Properties

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Map Projections

Useful Map Properties

Matching Projection to Job

No map projection is perfect for every task. One must carefully weigh pros and cons and how they affect the intended map's purpose before choosing its projection. The next sections outline desirable properties of a map, mentioning how projections can be used or misused.

Like the real Earth, this globe's north-south axis is tilted at 23°27'30" from the orbital plane.

For any map, the most important parameters of accuracy can be expressed as:

can distances be accurately measured?
how easy is obtaining the shortest path between two points?
are directions preserved?
are shapes preserved?
are area ratios preserved?
which regions suffer from the most, and which kind of, distortion?

Globe Properties

Unfortunately, only a globe offers such properties for any points and regions. Since crafting a globe is only a matter of reducing dimensions (no projection is involved), every surface feature can be reproduced with precision limited only by practical size, with no loss of shape or distance ratios. As a bonus, a globe is a truly three-dimensional body whose surface can be embossed in order to present major terrain features. But globes suffer from many disadvantages, being:

bulky and fragile, clumsy to transport and store;
expensive to produce, especially at larger sizes, thus impractical for showing fine details;
difficult to look straight at every point, therefore
cumbersome for taking measurements or setting directions;
able to show a single hemisphere at a time;
completely unfeasible for widespread reproduction by printed or electronic media

Changing minor map properties

Several maps using the Mercator projection show how fundamental projection properties are not affected by changes in minor features like scale, aspect, and choice of mapped area. What is constant in all Mercator maps is how scale quickly changes the farther one gets from a reference line (which may or may not be horizontal), but remains constant along a direction parallel to that line; this ensures all shapes are locally preserved, in detriment of area ratios.

1 The Mercator projection is frequently (and usually inappropriately) employed in world maps. Any Mercator map must be arbitrarily clipped, in this normal equatorial aspect at its top and bottom; a complete map would be infinitely tall. The cropped, transverse and oblique maps on the right are much more legitimate uses for this projection.

Equatorial Mercator map emphasizing the Pacific

2 Translating the central meridian is a common and trivial modification strongly dependent on the map's theme. Here a translation is used to avoid interrupting the Pacific Ocean. The distortion pattern remains the same as above, with areas drastically enlarged at the top and bottom.

Equatorial Mercator map emphasizing Australia

3 Departing from tradition, some maps half-jokingly show Australia and other meridional nations "on top of the world". Here, a bit more of Antarctica was arbitrarily cropped off. Of course, neither projection nor aspect differ from the above maps.
Mercator's projection is often naïvely accused of privileging the northern hemisphere—even "lowering" the Equator to aggrandize First World nations—but an uncropped equatorial Mercator map is perfectly symmetrical around the Equator; it's the map's editor who decides whether a large portion of Antarctic ice is relevant enough to show

Equatorial Mercator map emphasizing Africa

4 Scaling—in this case up by 100%—and cropping the map are common editorial decisions which change no projection properties.

5 Still a Mercator map, but in a transverse aspect scaled up 25% and rotated 90°. The graticule seems different and the poles can be visible, however the distortion pattern remains: scale is constant vertically instead of horizontally, growing left and right, not up and down. The central meridian, instead of the Equator, is perfectly represented, while Africa, bent strongly enough to seem split in two, is barely recognizable.

6 This oblique Mercator map (scaled up 25% and rotated) lays more of the Americas in the central band of reduced scale distortion. The Equator is no longer straight.

Exploiting Map Properties

So, flat map projections are usually more important and useful than globes, despite their shortcomings. In particular, no flat map can be simultaneously conformal (shape-preserving) and equal-area (area-preserving) in every point.

However, a reasonably small spherical patch can be approximated by a flat sheet with acceptable distortion. In most projections, at least one specific region—usually the center of the map—suffers from little or no distortion. If the represented region is small enough (and if necessary suitably translated in an oblique map), the projection choice may be of little importance.

On the other hand, the fact that no projection can faithfully portray the whole Earth should not lead to a pessimistic view, since distorting the planet on purpose makes possible—unlike with a globe—uncovering important facts and presenting at a glance relationships normally obscure. Skillfully used, distortion is a powerful visual tool; this becomes explicit in a kind of pseudoprojection called cartogram, where the place a point is drawn depends not only on its location on Earth, but also on attributes of the mapped region, like a county's population or a country's economic yield. In 1934 Erwin Raisz presented cartograms using simple rectangles with areas proportional to the attribute of interest but no relation to the original shapes. Waldo Tobler developed the modern concept of cartogram, which instead uses distorted but actually recognizable shapes; the amount of repeated calculations involved makes electronic computers indispensable tools.

No projection is intrinsically good or bad, and a projection suitable for a particular problem might well be useless or misleading if applied elsewhere.

Major and Minor Properties

For any projection, its "major" properties—concerning whether and how well distances, areas and angles are preserved—are largely independent of changes in scale, aspect and the choice of mapped area (this last detail is strongly associated with the aspect, selection of central meridian, and any eventual post-projection rotation and cropping), even though the graticule and other shapes may appear radically different. Therefore, the same projection may be the source of many maps, often only superficially unrelated.

Sometimes, for historical or convenience reasons, particular uses of a single projection are known by distinctive names. E.g., the Gauss-Krüger projection is a transverse case of the ellipsoidal Mercator projection; the Briesemeister and Nordic projections are oblique (with or without rescaling) aspects of the Hammer projection; and many rescaled versions of Lambert's equal-area cylindrical projection have been proposed. Other examples abound for interrupted, averaged and composite designs.

Identically marked graticules in orthographic (representing the original globe) and van der Grinten III projections at same scale.

The projection known as Van der Grinten's third violates all five graticule properties:

purple lines are very stretched near the poles; green lines are longer farther from the vertical axis (the constant vertical spacing between parallels is deceptive)
red lines are also slightly stretched closer to the map edges
the blue cells are also enlarged near the edges; this is more obvious at high latitudes
of all meridians, only the central remains straight
parallels and meridians cross at right angles only at the Equator and central meridian; elsewhere, there's shearing, symmetrical around the vertical axis: angles are compressed in opposite directions east and west of the central meridian

The Graticule as a Guide to Distortion

Especially for a map in the normal aspect, a quick visual inspection of its graticule provides obvious clues of whether its projection preserves features. For instance, if the coordinate grid is uniformly laid (say, one line every ten degrees),

along any meridian, the distance on the map between parallels should be constant
along any single parallel, the distance on the map between meridians should be constant; for different parallels, should decrease to zero towards the poles
therefore, any two grid "cells" bounded by the same two parallels should enclose the same area

Also,

the Equator and all meridians should be straight unbroken lines, since they don't change direction on the Earth's surface
any meridian should cross all parallels at right angles

Again, for any particular projection, violation of any or all these properties doesn't necessarily make it poorly designed or useless; rather, it suggests (and constrains) both the range of applications for which it is suitable and, for each application, regions on the map where distortion is significant.

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