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:
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:
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:
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),
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.