Now that we can convert angles (i.e. latitudes and longitudes) to linear distances (i.e. meters or feet), we’re all set to develop our own projection; right! Well, to a certain extent, anyway. Using the material discussed in our last couple of issues, we can develop the spherical form of a recognized projection.

Writing formulas for a projection might be getting ahead of ourselves a bit. However, putting knowledge to useful work is beneficial from an educational standpoint. So, for the purpose of this specific article, let us simply assume that a projection is a set of formulas which convert the angular spherical coordinates of latitude and longitude to the linear cartesian coordinates of X and Y.

To do so efficiently and effectively, we must first adopt some conventions. We will use the convention used on a huge majority of maps: that north is up and east is to the right. Note also, that in the cartesian coordinate system, having a vertical Y axis and a horizontal X axis is also a convention. These conventions are so widely used, it may surprise many Casual Cartographers that they are just conventions. Using these conventions, it is natural to assume that there will be one formula to convert longitude values to appropriate values for X, and a corresponding formula for converting values of latitude to appropriate values for Y. For the simplest projection of them all, this is indeed the case. (But not so for just about all other projections.)

I should point out that we will be using lower case lambda, l , for longitude and lower case phi, f , for latitude. This is quite common in the literature written in English. Other notations that we will use, and which are universally used (in English documents), are given in Table 1. Beyond the values given in the table, there does not seem to be much in the way of standard notation. Note that the use of the R implies a sphere, where the use of a and/or b implies an ellipsoid.

Converting latitude on the sphere to a linear Y value is:

There are several important implications that may not be obvious to the Casual Cartographer. First, the units of the resulting y value are the same as the units of the R value used. Thus, if you use a radius value given in meters, the resulting y value will be in meters. Second, this formula implies that the y value of the equator is zero. Third, converting a latitude in the southern hemisphere, where latitudes are usually considered to be negative, will produce a negative y value. Finally, as we have mentioned before, for this formula to work, the latitude must be expressed in radians (not degrees).

Similarly, for the X axis, we can write:

The same considerations apply. That is, the units of the resulting x value are the same as the units used to express the radius of the sphere; and x will be positive for geographic positions east of the prime meridian and negative for geographic positions west of the prime meridian. The prime meridian is usually Greenwich, but again that is only a convention. And (one last time I promise), the longitude must be expressed in radians, rather than degrees.

The above formulas do indeed provide formulas for the simplest of all projections, the projection known as the Platte Carrée or Simple Cylindrical. Special attention should be given to the case where R is 57.29577951. Perhaps you’ll recognize that this is the reciprocal of the factor required to convert a longitude (or latitude) from degrees to radians. If indeed R is 57.29577951, our formulas simplify to:

One feature of this projection as far as we’ve defined it, is that all points along the Prime Meridian (typically Greenwich) have an X value of zero, the X value assigned to all geography to the east of the Prime Meridian is positive, and all geography to the west is in negative X territory. If you live in the western hemisphere, you may find this inconvenient. If Russian geography is a region of significant interest, you may find having Russia straddling the ±180 degree crack to be rather annoying. That is, the easternmost sliver of Russia ends up on the opposite edge of the map than the rest of Russia. Similarly, the Alaskan Islands of Kiska and Attu end up on the opposite edge of the map from the rest of Alaska. To address issues of this sort, it is customary to introduce the concept of an origin longitude. Technically, this specific longitude becomes the Prime Meridian for specific purpose of making the desired map. We can accomplish this, mathematically, by subtracting this "map specific prime meridian" from the geodetic longitude which is normally referenced to Greenwich. The end result of this is that the origin longitude now produces a zero for x, and all geography to the west of the origin longitude is assigned negative x values. Also, depending upon our choice of origin longitude, Russia is no longer split across the 180 degree longitude crack.

Thus, when actually making a map using the Platte Carrée projection, it is often desirable to choose a longitude to use for the origin longitude. For the reasons related to Russia above, it is often most convenient to choose an origin longitude to be in the center of the geography our map is most concerned with. Thus, this origin longitude thing is also frequently called the central meridian as it is the meridian of the center of the map we will produce.

Given this little extra complication, we arrive at the official formulas for the spherical form of the Platte Carrée projection:

Perhaps the most important part of this discussion is that we have introduced, already, our first couple of map projection parameters. That is, in order to use the formulas of this, the simplest of all map projections, we need two parameters: the radius, and the origin longitude (or if you prefer, the central meridian). In future discussions of map projections, and other transformations, we will get deeply involved in all sorts of projection parameters.

While the mathematics involved may be much more complex, the idea is the same. Map projection parameters are used to generalize the formulas of a projection so that the projection can be most effectively used for more than a single map. In our case above, we suggested using the origin longitude to shift the map so that Russia was no longer cut in two by the 180 degree longitude crack. This will please the folks in Russia, but now maybe the folks in Alaska are unhappy, as Alaska would be cut into two pieces. What’s a Casual Cartographer to do?

Of course, if the purpose of your map is to map Alaska, you’ll probably not worry too much about the Russians. If the purpose of your map is to map Russia, or Asia, you’ll probably have ignore the negative comments from the Alaskans. In any case, as the cartographer, you have to make this choice, and the implementation of the projection formulas provide you with this choice through the means of a projection parameter.

Therefore, in response to a question about what coordinate system is a map based, it is improper to respond Platte Carrée. Platte Carrée does indeed specify the formulas used, but not the parameters used. In response to the question "What coordinate system?" one must provide not only the projection, but also all appropriate parameters.

When you are given a map, or set of coordinates, to process; do not ask "What’s the projection?" You’ll only get a small fragment of the information you need. Rather, in these cases, always ask "What is the coordinate system?"; and don’t be satisfied with a response that does not include some parameters. In future articles of the Casual Cartographer, as we explore some of the different projections, we’ll learn exactly what parameters are required for each individual projection.

Aha! Projections are easy!!! Well maybe not. Remember that we’ve only done the simplest of all projections, and only the spherical form of it. But our discussion this time has introduced a very important concept, one that I’m sure most Casual Cartographers have encountered often. That is the projection parameter. What we’ve seen this time is that a projection is (implemented as) a set of formulas (like those given above). When projection parameters are applied to customize these formulas to a specific region, a specific piece of geography, a specific mapping project, then and only then do you have a coordinate system.

Most U.S. Casual Cartographers deal in well defined coordinate systems such as State Plane Coordinate Systems (SPCS) and Universal Transverse Mercator (UTM) zones. In these cases, the projection parameters are well established and publicized. In the case of State Planes, it literally takes an act of the State Legislature to change them. Thus, the specific projection parameters are directly implied when one specifies a specific zone. However, even in these cases, it is important not to be satisfied with just the name or description of the zone. When given a map or set of coordinates to process, be sure to obtain the datum, and the unit of the coordinate system in addition to the zone. (A coordinate system of the NAD27 State Plane variety can be safely assumed to use the U. S. Survey foot as its unit.)

Several map projections have been developed which are specific to a specific piece of geography, the New Zealand Map Grid is one such system. Therefore, when using this projection, it is unlikely that you will need to specify any parameters. Similarly, the Bipolar Oblique Conic Conformal Projection was originally invented for a specific map, a map of North and South America. Again, when referencing this projection in a GIS type product, it is unlikely that you will be asked for, or need, specific parameters.

There is, of course, another alternative to the Russian problem we discussed above. Could we not simply extend the range of our map beyond +180 and –180 degrees of longitude? That is, instead of limiting longitude to be greater than –180 degrees and less than or equal to +180 degrees as is quite common, why not set limits at, say –210 and +210. This would have the effect of duplicating some geography at two different locations. Many GIS products would frown on this idea as it may require two copies of the same data in a database; never a good idea. However, looking at the formulas for the spherical form of the Platte Carrée projection, I think you can see that there is no reason why the projection formulas would not work as expected. At least for this projection anyway.