The world we live in...
is a round globe. Small areas on it, of course, are so nearly flat, one cannot tell the difference.
But when one is drawing a map of the whole world, or even a whole country, one has to decide how one is going to represent our spherical world on flat paper. Many different solutions have been developed over the years, and a few of them will be examined on these pages.
Map projections can be grouped together in two basic ways; and a third characteristic, although it divides different way of using the same projection, is sometimes considered important enough that different versions of the same projection varying only in this characteristic are given different names.
The first characteristic is aspect. This identifies the basic layout of the projection. The most important projections are either cylindrical, conic, or azimuthal. A flat piece of paper can, without stretching, be bent into a cone or a cylinder, and in this way, it can touch a globe along an extended line; if left flat, it only touches the globe at a point.
The basic mathematics of obtaining several important properties of maps is different in these aspects.
There are other types of projection than these three basic types. A projection can be pseudocylindrical, which means that although the parallels of latitude are straight lines, the scale along them varies, so the meridians are no longer perpendicular to them, or (necessarily) straight lines. Pseudoconic and even pseudoazimuthal projections are also possible. As well, there is the polyconic projection, and there are many conventional projections.
Projections are also classified in terms of their properties. Specifically, on the basis of two very important properties: whether they are conformal, or equal-area. A conformal projection maintains the shape of small regions, so angles at any point are correct, although sizes will change. An equal-area projection, on the other hand, maintains size at the expense of shape. Maintaining both size and shape, of course, requires a globe.
In general, and this is true for the projections in the three basic aspects of cylindrical, conic, and azimuthal, scale going away from the center of a map increases for a conformal projection, and decreases for an equal-area projection. Most projections that are neither conformal nor equal area have a scale behavior that is somewhere in between. However, two very important azimuthal projections lie outside this range: the gnomonic projection, which can be used to find the great circle path between two places, and whose scale expands more quickly than that of a conformal projection, and the orthographic projection, which looks like a picture of a globe, whose scale shrinks more quickly than that of an equal-area projection.
Finally, there is the case of a projection. Just as you can move a globe you are holding in your hand, so too one can think of the graticule, the lines of latitude and longitude on the globe, as being movable.
So one can move the globe, or move the graticule on the globe, and draw a map of a shifted world: that is, although the usual rules for drawing a projection place the lines of latitude and longitude on it in a given way, one can shift the world under the projection's graticule, and treat the original graticule of the world like the coastlines and borders on the globe, as simply things to be drawn where they happen to be.
There is the conventional (or, in the case of an azimuthal projection, polar) case, where the projection is drawn in the normal and easiest fashion. There is the transverse (or, in the case of an azimuthal projection, equatorial) case, in which the globe has been shifted by 90 degrees before the map is drawn, and there is the oblique case where the globe is shifted by a lesser amount.
Although some azimuthal projections can be drawn in other cases relatively easily, drawing most other map projections in a case different from the conventional one was very difficult before computers came along to draw maps for us. Thus, the transverse case of the Mercator projection is also known as the Gauss Conformal Projection; the transverse case of the Plate Carré projection is known as Cassini's projection.
I've written a little BASIC program that draws maps of the world in the form of .xbm graphic files. I then view them in my web browser, press Alt-Print Screen, paste the result in a paint program, and save the images in a more compact format I present on these web pages.
The data points I use to plot these maps are from GSHHS, a cartographic data set available at the U. S. National Geophysical Data Center. The lowest-resolution version of the data set is used by my BASIC program, as converted to text format by the program provided. Note that the data files at that site, although they bear the ".gz" extension, are NOT compressed, although they are binary. (This may have been done to avoid problems with web server software that did not recognize the .b extension of those files.) The ones with the ".bz2" extension at the Hawaii site are compressed, using the bzip2 program, which is available in a version for conventional PCs, even though it is mostly used under Linux.
Subsequently, I added to the program the ability to use files produced by pscoast within GMT, the Generalized Mapping Toolbox. However, in order to keep the ability to exclude small lakes, I continue to read the coastlines in the original GSHHS format, which includes more information, and is faster for my program to read (which is not a criticism of GMT, as it is intended to work in the world of the UNIX operating system and compatible operating systems, not with QBASIC programs).
The information for the construction of the projections discussed here is, for many of the more common projections, derived from any of a number of standard works, but for some of the more exotic projections, various specialized sources, such as D. H. Mahling's Coordinate Systems and Map Projections, An Album of Map Projections by Voxland and Snyder, Elements of Map Projection by Deetz and Adams, and others, were helpful.
This site, needless to say, does not describe all the map projections in existence. However, even some fairly well-known ones have not been discussed here yet. Although I have encountered formulae for the two-point equidistant projection and even the Chamberlin Trimetric projection, I felt that sort of projection would present some problems in implementation in my very simple BASIC program without a fancy interface. I admit I ignored the Goode Homolosine projection specifically because of a dislike for one feature of it; there is a sudden change in the direction of lines crossing the parallel at which the Sinusoidal and Mollweide projections are joined. On the other hand, while I like the Armadillo projection very much, I haven't seen formulae for the little area with New Zealand on it - and it is a projection in a class by itself, requiring a new chapter. Note, too, that some projections, instead of getting a section to themselves, are merely mentioned (and, usually, illustrated) in passing during the discussion of another projection. These projections are mentioned in the table of contents below so that you can find them as well.