Identifying Locations

Before we jump into the mechanics of the projection process, its important to know how we precisely identify locations on the Earth. To identify any location, you need a reference point to work from. For example, you can roughly locate the city of Chicago by saying Chicago is in Illinois. Here the reference point (the state of Illinois) is quite large, and the directions from the reference point to Chicago are very general (all that is indicated is that Chicago is "in" Illinois). Thus, this isn't a very specific way of locating the city, but at least it gives you a rough idea where Chicago is located.
Figure 1. Four U.S. Cities.

In a perfect world, we'd like to be able to use the same reference point to locate everything. For example, it is perfectly correct to say Chicago is in Illinois and to say that Los Angeles is in California. However, since we're using two different reference points (Illinois for Chicago and California for Los Angeles), these two statements don't tell you anything about the relative locations of Chicago and Los Angeles. Thus, unless you know something about the relative locations of the two reference points, saying Chicago is in Illinois and Los Angeles is in California does not tell you anything about which city is farther north, or which city is closer to Washington, D.C., and so on. However, if we use the same point of reference to locate the two cities -- for example, if we say that Chicago is northeast of St. Louis and Los Angeles is southwest of St. Louis -- we can deduce that Chicago is north of Los Angeles (Because if Los Angeles is southwest of St. Louis, St. Louis must be northeast of Los Angeles. Thus, since St. Louis is northeast of Los Angeles, and Chicago is northeast of St. Louis, Chicago must be north of Los Angeles.)

In the perfect world we'd also like to have a way of identifying locations that is highly precise. So far, we've been very imprecise in our measurements of distances and directions between our reference points and the points we are trying to locate. For example, all we said was that Los Angeles was southwest of St. Louis; we didn't say how far to the southwest Los Angeles was located, and when you think about it, "southwest" is a fairly general direction. We'd like to come up with a location system that is a lot more precise than the general approach we've used so far.

There is no reason why we can't make the technique that we've been using work like the perfect location identification system we've been wishing for. We could pick a common reference point and precisely measure angles and distances from that point. For example, we could say that Los Angeles is 1,591.29 miles from St. Louis at an azimuth of 258.76 degrees from true north (Figure 2).
Figure 2. A spherical coordinate system originating in St. Louis with the base axis extending true north. The coordinates of Los Angeles are displayed.

This is an example of what is called a spherical coordinate system. In mathematics, you'll hear this referred to as a polar coordinate system, but geodetic scientists tend to stay away from that term because its too easy to get confused with the poles of the Earth. I was once at an academic conference in London, England and heard the British folks call this a "radial coordinate system", but to the best of my knowledge, everywhere outside of London, this is called a spherical coordinate system. Spherical coordinate systems require (1) an origin point (in the case of Figure 2, St. Louis is the origin), and (2) a base axis running through the origin from which angles are measured (in the case of Figure 2, this is the line starting in St. Louis and going due north). Once you've defined these two things, you can locate any point by its distance from the origin, and the angle between the base axis and the line connecting the origin to the point you are locating.

We tend to use spherical coordinates fairly frequently in our everyday lives. Its quite common to say things like "X is about 5 miles east of here." This statement is clearly based on a spherical coordinate approach to locating objects -- it involves an origin point ("here"), a base axis and angle ("east" implies an angle of 90 degrees from a base axis running due north), and a distance ("about five miles"). It's the ability of spherical coordinates to work from just about any origin point that make them very attractive for giving general locations in day-to-day use.

The problem with spherical coordinates is that when you try to use them to precisely locate objects, or to compute the distance between two objects, or just about any other even slightly sophisticated task, the angular part of the coordinates almost always requires you to resort to trigonometry (the sine, cosine and tangent of angles -- remember all that stuff?). Trigonometry is not only something that many people find difficult, it is also something that taxes digital computers (most computers can conduct hundreds or even thousands of simple addition, subtraction, multiplication or division operations in the time it takes to conduct one trigonometric operation). This doesn't mean that you can't use spherical coordinates to precisely locate objects; it just means that using them is more difficult than we'd like. Fortunately, there is an alternative coordinate system that is much easier to use.

Instead of using one axis and an origin point, why not simply use two axis that cross each other at right angles. Put the origin point where the axis cross. Figure 3 shows an example of such a pair of axis superimposed on a map. Now every point on the map can be precisely located by its coordinates on these two axis.
Figure 3. A map with a superimposed Cartesian coordinate grid. Note that it is a simple mater to identify the location of any point (in this case, cities within the U.S.) using this grid.

There are two things to notice about the coordinates system shown in Figure 3:

  1. The two axis cross each other at right angles.
  2. The units of measure along each axis are constant, e.g., the distance between 1 and 2 along the horizontal axis is the same as the distance between -2 and -3 along the same axis.
Any set of axis that meets these two conditions creates what is termed a Cartesian coordinate system (named after Rene Descartes, a French mathematician and philosopher from the 17th century). Cartesian coordinates work extremely well on flat surfaces like a map. Cartesian coordinate systems have lots of wonderful properties, the two most important being that (1) you usually don't have to resort to trigonometry to analyze Cartesian coordinates, and (2) all of the geometry that you (hopefully!) learned in high school works with Cartesian coordinates. This includes real basic stuff, like the idea that the shortest distance between two points is a straight line, and slightly more complicated stuff, like the Pythagorean theorem, which you can use to calculate the distance between any two points that have been located using Cartesian coordinates. The type of geometry which works with Cartesian coordinate systems is called Euclidian geometry (after Euclid, the famous 3rd century B.C. Greek mathematician who wrote Elements, the first scholarly description of geometry).

Cartesian coordinate systems work well on Euclidian spaces like flat maps, but unfortunately, the surface of the Earth is curved, not flat. This means the surface of the Earth is a non-Euclidian place. Figure 4 demonstrates this. In Euclidian spaces, the shortest distance between any two points is a straight line. Figure 3 shows the shortest distance between two points on the surface of the Earth -- a curved line (If you don't believe that the line on Figure 4 is curved, hold a straight edge -- a ruler, a piece of paper, or whatever -- up to the computer screen so that the straight edge lines up with the two endpoints of the line shown on the Figure. You'll see that the curved line on the Figure doesn't follow your straight edge). In fact, the shortest path between any two points on the surface of the Earth is called a great circle route, because the shortest path is actually a portion of a circle that goes all the way around the Earth.
Figure 4. The shortest distance between two points on the surface of the Earth is a portion of a Great Circle.

Since the Earth's surface isn't a Euclidian space, a Cartesian coordinate grid applied to the Earth's surface would become very distorted. Just like the shortest path in Figure 4 curves, the axis in a Cartesian coordinate grid applied to the Earth's surface would stretch and curve. Such distorted coordinates would be almost impossible to work with, and not very accurate.

Fortunately, there are such things as non-Cartesian coordinate systems. There are many possible non-Cartesian coordinate systems, but the one that is used for locating points on the Earth's surface is the Latitude/Longitude system. Just like a Cartesian system, the Latitude/Longitude system uses two reference axis, but instead of measuring distances to these axis, we measure angles.

Latitude measures how far north or south of the equator a point is located. The latitude of any point is defined as the angle between two line segments, one connecting the point to the center of the Earth and the other connecting the center of the Earth to the equator (this is a lot less complicated than it sounds; Figure 5 makes this much more clear). Latitudes start at 0 (on the equator) and range from 90 degrees north (at the North Pole) to 90 degrees south (at the South Pole). To differentiate between paired north and south lines of latitude (i.e., differentiating between 30 degrees north latitude and 30 degrees south latitude), you either have to explicitly say "30 degrees north latitude" or "30 degrees south latitude", or sometimes people make all of the latitudes north of the equator positive and all of latitudes south of the equator negative. Thus, 30 degrees north latitude becomes just "30 degree latitude", and 30 degrees south latitude becomes "-30 degrees latitude."
Figure 5. Latitude is computed as the angle between the equator, the center of the Earth, and a point on the surface of the Earth. This example shows rotating such an angle completely around the Earth, thereby tracing a complete line of latitude (in this case, a line of latitude 48 degrees, 0 minutes, 46.04 seconds north). All points on this line share a common latitude. Note how as the angle is rotated, its center never moves from the center of the Earth.

Longitude measures how far east or west of the prime meridian a point is located. The prime meridian is just a line running north to south along the surface of the Earth that connects the North Pole to the South Pole. Exactly where the prime meridian is located is arbitrary; we could pick any north-south line that connects the poles and call it the prime meridian. However, by international convention, the prime meridian is the north-south line that connects the poles and runs through the Royal Observatory in Greenwich, England (There are still a few countries that officially refuse to recognize the line through Greenwich as the prime meridian, notably Israel and China. However, in practice, virtually everyone, including everyday Israeli and Chinese folks, use the Greenwich line as the prime meridian).

The longitude of a point is the angle between the prime meridian, the axis of the Earth, and the point (Figure 6). Note that unlike the angle used to measure latitude, the center of the angle used to measure longitude does not stay fixed at the center of the Earth; it moves up and down the Earth's axis. This makes latitude and longitude very different from one another. Lines of latitude are all parallel to one another, and they start out being very long at the equator (the equator, which is the zero degree line of latitude, runs around the entire circumference of the planet, for a total length of about 25,000 miles), and get progressively shorter as you near the poles (90 degree latitude, either north or south, is a single point and hence has zero length). Lines of latitude also go all the way around the Earth, forming complete circles. On the other hand, lines of longitude are not parallel (they all start at a single point at one pole, reach their maximum separation at the equator, and then converge again at the other pole). Lines of longitude are also all the same length, and they only go halfway around the planet, forming semicircles.
Figure 6. Longitude is computed as the angle between the prime meridian, the axis of the Earth, and a point on the surface of the Earth. This example shows a line of longitude being traced from one pole to the other. This traces a complete line of longitude; all points on this line share a common longitude (in this case, a line of longitude 90 degrees, 0 minutes and 0 seconds west of the prime meridian). Note how as this line of longitude is traced, the center of the angle moves down the axis of the Earth.

Longitude is typically measured in angles starting at zero on the prime meridian and ranges from 180 degrees east (this takes you across Europe, Africa Asia and Australia and ends in the Pacific) to 180 degrees west (this takes you across the Atlantic, over the Americas and ends in the Pacific (Figure 7). Alternatively, some people assign all of the longitudes east of the prime meridian positive longitudes and all longitudes west of the prime meridian negative longitudes. This technique (along with assigning positive latitudes to the northern hemisphere and negative latitudes to the southern hemisphere) is commonly used in GISs and other computer programs.
Figure 7. Latitudes and longitudes (Craster Parabolic Equal Area Projection)

Since latitude and longitude are both angles, they need to be measured using angular units. By convention, latitude and longitude are measured using either decimal degrees or degrees, minutes and seconds. Decimal degrees are obvious; if the angle between a point, the center of the Earth and the equator is twenty five and three-fourths degrees, its latitude is 25.75 degrees. Decimal degrees are commonly used by GIS and other computer software systems, because computers have little difficulty dealing with numbers that include digits to the right of the decimal point.

The degrees, minutes and seconds system of measuring angles dates back to at least the 15th century, and really the only reason it is still used today is because its been around for so long. It's actually quite a simple system: each degree is broken down into sixty minutes, and each minute is broken down into sixty seconds. Thus, the previously mentioned angle of 25.75 degrees translates into 25 degrees, 45 minutes, and 0 seconds. Converting back and forth between decimal degrees and degrees/minutes/seconds is easy; this page discusses how these conversions are conducted.

One last point. Typically, in the degrees/minutes/seconds system, you would write an angle like 40 degrees, 23 minutes, 16.4 seconds as 40° 23' 16.4". Degrees are denoted by the small superscript circle, minutes by a superscript single line, and seconds by the double superscript line. Unfortunately, its hard to create these symbols on a web page, so in this class, I'll frequently use the a degrees, b minutes and c seconds notation.