Why Study Math? - The Hyberbola
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.