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.

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?

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

Same projection,
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.

On a technical note, large-scale Mercator maps in the ellipsoidal case pose considerable implementation challenges and are not as easily changed as suggested above; nonetheless, this neither concerns the map's user nor affect the arguments above.

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's primitive
cartograms presented abstract rectangles with areas proportional to the
attribute of interest but bearing no resemblance to
real-world shapes. The prolific 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.

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.

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 one 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

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

- 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.

www.progonos.com/furuti/MapProj/Normal/CartProp/cartProp.html — June 16, 2018

Copyright © 1996, 1997 Carlos A. Furuti