As we continue the "Why Study Math"series of articles, here we look at the conic section called the hyperbola. The hyperbola is obtained by intersecting the double-napped cone (see the other articles in this series on this point) with a plane so that both parts of the cone are cut. Those familiar with the parabola might note that this curve almost looks like two parabolas pasted back to back with a space in between them. Mathematically, the hyperbola is not a parabola, although these two conic sections have a similar outward appearance. The hyperbola is the least known of the four conic sections. It is also the most difficult curve to derive algebraically. Probably for this reason, students who study the conic sections, like the hyperbola the least. However, when students see the reason we study this curve, their attitude changes significantly. For this reason, we will now examine some of those applications connected to the hyperbola. Everyone at one time or another has thrown a pebble into a still pond. Picture throwing not one but two pebbles into this pond at the same time. The outward concentric circles that form intersect each other at points which trace out the curve known as the hyperbola. This application is used in radar tracking stations. LORAN, the terrestrial navigation system, uses low frequency radio transmitters to locate objects. Objects are located by sending out sound signals from two sources to a receiving station, such as one found on a boat or plane. The constant time difference between the signals from the two stations is represented by a hyperbola. As we discussed with the applications of the ellipse, most celestial bodies follow elliptical orbits. In the case of comets, however, a hyperbolic path is followed as they shoot through space. The hyperbola is also the shadow cast on a wall by a lamp with a cylindrical shade. And for something a little more earthy, the shape of that horse saddle you get on to ride forms an interesting solid curve called a hyperbolic paraboloid. So you see, the conic sections" ?even the hyperbola" ?might be closer than you think. |
