LEARN NC

10 New dimensions

All of the maps we’ve looked at so far assume a two-dimensional surface. But we know that the earth’s surface is more complicated than the surface of a true sphere — it has mountains and valleys. Suppose we wanted to add a third dimension to our map to show elevation?

Representing the third dimension

Let’s go back, briefly, to the mathematical concept of mapping. The earth’s surface can be represented in two dimensions, as we’ve seen. A cylindrical projection, like Mercator, renders it in a Cartesian coordinate system in which latitude and longitude are straight lines that intersect at right angles. Other projections use more complicated coordinate systems, in which axes are curved and warped.

Figure 11-1. A two-dimensional Cartesian coordinate system.

two-dimensional cartesian coordinate space

If you want to add a third dimension to a graph, it gets complicated. Students in calculus and analytic geometry often find themselves drawing coordinate systems like this, in which the z-axis is meant to look as if it’s going off into the page:

Figure 11-2. A three-dimensional Cartesian coordinate system.

three-dimensional Cartesian coordinate space

But that’s hard to draw without a computer, and it’s hard to read accurately. The problem is that our representation — our map — is still only two-dimensional. Adding artistic perspective, imagining things going into the page, helps us visualize, say, a cube:

Figure 11-3. A three-dimensional cube represented in two dimensions.

cube

…but isn’t much good for making accurate, readable maps. If we want to clearly represent a third dimension on a two-dimensional map, we have to get creative.