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Useful Map Properties: Areas

Are Area Ratios Preserved?

Apart from navigational purposes, for most maps of general interest areal equivalence is the most essential property. A map is an instrument to convey information at a glance, much faster and more concisely than tables or lists of numbers, and visual comparisons often require true area ratios. Important applications include:

The classic comparison: Mercator ⨉ Mollweide
Partial Mercator map
Greenland in Mercator map
Africa in Mercator map

Maps at identical scaling factor using the equatorial aspect of Mercator's conformal projection (above) and Mollweide's equal-area projection (below). Both rows show a cropped small-scale map, Greenland and Africa. The 10°-graticule makes clear how the Mercator projection stretches scales near the poles. On the other hand, it shows no distortion along the Equator, where the Mollweide map visibly exaggerates vertical scale.

Partial Mollweide map
Greenland in Mollweide map
Africa in Mollweide map

The latter issue is quite worrisome as school and college textbooks, to say nothing of newspapers and TV shows, are often careless in their choice of map projections. This is unfortunate, as maps make powerful and long-lasting visual images; a wrongly presented map could permanently distort one's view of the world (however, this very argument has also been used as misleading and myopical propaganda).

As a classical example, Mercator's projection, an extremely powerful tool in the proper context, is frequently — and sadly — used for wall maps and popular illustration, where its conformality is useless and areal ratios misleading.

In the conventional aspect of Mercator's projection, surface stretching increases quickly but continuously to infinity towards the poles. Even Greenland's northern half looks much broader than the southern one — this projection preserves shapes but only locally. In contrast, Mollweide's elliptical projection shows the correct size proportion between all portions of the map.

Relative areas compared, equatorial aspect
Mercator Greenland
and Earth

An equal-area (also called equivalent, equiareal or authalic) projection preserves areal relationships; in other words, given any two regions A and B on the Earth and the corresponding regions A' and B' on an equal-area map, the surface ratios A/A' and B/B' are identical (A and B need not have the same shape; shapes A and A' will probably be different).

An equal-area projection is not necessarily equidistant; in fact, in order to preserve area, at any point the scale distortion in a given direction must be the inverse of the scale distortion in the orthogonal direction. For instance, in the conventional aspect of Mollweide's projection the horizontal scale is slightly too low along the Equator, and slightly too high in the vertical direction: the net effect is making the continents a bit too slender.

Although Mercator's projection makes Africa (one of the largest continents, 29,800,000 km²) and Greenland (the largest island, 2,175,600 km²) apparently similar in size, an area-preserving projection shows their true area ratio — about 13.7 : 1 — much more clearly.

Greenland in recentered Mercator map
Greeland in recentered Mollweide map

At identical scaling factors, Mercator (left) and Mollweide (right) maps recentered on Greenland. The first's area is still exaggerated, but barely noticeably.

Relative areas compared, oblique aspect
Mercator Greenland
Mollweide Greenland

Nevertheless, in life few things come for free, and cartography is no exception. The Mollweide projection, like many equal-area designs, pays a heavy toll in the form of shearing, especially noticeable in high latitudes or far from the central meridian. Although Greenland is not especially affected in the equatorial aspect centered on the Greenwich meridian, it appears much better — and still with correct area — after recentering in an oblique aspect. To be fair, an equivalently recentered Mercator map avoids much of the area stretching of the original, and presents Greenland with better shape than the Mollweide version, which noticeably stretches vertical scale along the projection's central line. Yet another evidence that, whatever strong or weak points of a projection may be, their relevance is definitely influenced by the mapped area of interest.

HomeSite MapShapesMap PropertiesDistortion Pattern - Tissot's Indicatrices — June 16, 2018
Copyright © 1996, 1997 Carlos A. Furuti