Planar coordinate systems

There's supposed to be a big yard sale at Center Street and Palm Avenue, and you and a friend are trying to find the intersection on a map. You spot it first and circle it with a pen. With that stroke, you have just created a locational reference system on a plane surface. Well, maybe "system" is too fine a word, since your circle only helps you find this particular intersection.

Suppose, instead, you take a ruler and pencil, divide the map into squares, and label each square. Then you could write down the names of all the streets in each square and alphabetize your list. To find the intersection of Center and Palm, you would look up Center Street and see what square (or squares) it lies in. You'd do the same for Palm Avenue. The square they have in common is the part of the map where you'll find the intersection. Since this method will work for any street or intersection on the map, it really is a locational reference system.

Note that it doesn't matter what projection the map is in, what its scale is, or what part of the earth it covers. A locational reference system can be applied to any map because the grid of squares has nothing to do with the specific properties of the map. It's just a way to pigeonhole locations to make them easier to find. (The size of the squares is up to you. For that matter, you could use something other than squares, if you really wanted to.)

Is this a coordinate system? It's debatable. Suppose your grid has 36 squares and you label each square with a number from 1 to 36. Where exactly are the coordinates? But suppose instead you number both rows and columns from 1 to 6. In this case, each square is defined in terms of a pair of numbers, like (3, 4) or (6, 1) and it sounds like you do have coordinates, even if they're not very precise.

Planar coordinate systems, however, are usually understood to be systems that assign location references to individual points, not just to areas, and that support analytic geometry (which means you can use them to calculate distances and directions between points). These systems are based on Cartesian coordinates, invented by the French mathematician and philosopher Rene Descartes. You'll look at these systems next.

Left: A street map with three point locations. Center: A grid (locational reference system) in which points can be located somewhat precisely within squares. Right: A coordinate system in which every point has unique coordinate values, as long as you subdivide the numbers on each axis.