The most natural way of plotting radar data is by using polar coordinates. Recall from the last blog that the radar signal gives us two pieces of information about an object that it detects: its distance (which we will call ) and the angle a line drawn to it makes with a fixed reference direction -- in our example east (or rightward). This angle we called the azimuth (from an Arabic word meaning way or direction) or azimuth angle; we will denote it, as is customary, by the Greek letter (theta). Here is a picture

If we set up a usual coordinate system (the kind you use for plotting points in high school algebra), with the -axis pointing "north" (upwards) and the -axis pointing "east" (rightwards), our object at point will then have coordinates . From elementary right-angle trigonometry:So, from the distance and azimuth information that radar gives us, we can find the position of an object in the "-coordinate system". This can then be transferred to a map (with north-south coordinates) to find the position of the object at . If the distance is fairly large, one also has to take into account the curvature of the earth. This requires using spherical trigonometry, which I mentioned a few weeks ago in the blog about sundials.

The air traffic controllers at airports need to know more from the radar, however: they need to know the height or altitude of the planes in the area. To find this we need to move up to 3 dimensions. The radar still measures the distance and the azimuth angle, but it must also be able to scan upward and measure the angle that a line drawn to an object makes with the horizon. This is called the elevation angle. We are now working in a 3-dimensional grid, with a -axis perpendicular to the earth, pointing straight upward. Here is a representation of this:

Notice that in this coordinate system it is customary to denote the distance of the object at the point (determined by radar) using the Greek letter (rho). From elementary trigonometry again, we see that the actual height of the object above the ground (the vertical dotted line) is the -coordinate which is given by

Designing a radar antenna so that it scans in three dimensions is tricky. Antennas themselves are traditionally "cup" or "U" shaped. More precisely, they are parabolic (shaped like a parabola) in cross-section. This is because of the reflective properties of parabolas --- often proved in calculus courses. There is a certain point inside a parabola called the focus. If a source of radiation is place at the focus, the rays emanating from it hit the parabola and are reflected out of the parabola as parallel rays. Conversely, radiation entering a parabolic reflector (antenna) is reflected in such a way that it "focuses" at the ... focus! Just what you need to make this set-up work efficiently. Here's a sketch:

This view is looking down on the antenna, which rotates at the same time it is scanning up and down. This is accomplished with complicated electronics, designed with the aid of a lot of mathematics!

Some of the newer, more modern antennas can actually be made flat (non-parabolic), and the effect of the parabolic shape can be simulated using special mathematical algorithms on a high-speed computer, which processes the incoming and outgoing data as it is being received/sent.