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.

- Cylindrical Projections
- The Mercator Projection (including Gauss)
- The Gall Stereographic (also the Braun)
- The Miller Cylindrical Projection
- The Plate Carrée
- The Lambert Equal-Area Cylindrical Projection

- Azimuthal Projections
- Conic Projections
- Pseudocylindrical and Pseudoconic Projections (the Loximuthal)
- Polyconic Projections
- Conventional Projections
- Other Conformal Projections
- The Lagrange Conformal
- August's Conformal Projection (also the Eisenlohr)
- Guyou's Doubly-Periodic Projection (also some related projections by Adams, and the Pierce Quincunctial)

- Other Equal-Area Projections
- The Hammer-Aitoff Projection (also Aitoff's Equal-Area)

- Miscellaneous Projections
- Winkel's Tripel Projection (also the Aitoff)
- The Aitoff-Wagner Projection (also the Larrivée)
- The Laskowski Tri-Optimal Projection (also Canters' Minimum-Error Polyconic)