Importance of Hyperbolas in Life
A hyperbola is the mathematical shape that you obtain when vertically cutting a double cone. Many people
learn about this shape during their algebra courses in high school or college, but it is not obvious why this shape
is important. The hyperbola has a few properties that allow it to play an important role in the real world. Many
fields use hyperbolas in their designs and predictions of phenomena.
Satellites

Satellite systems make heavy use of hyperbolas and hyperbolic functions. When scientists launch a satellite into
space, they must first use mathematical equations to predict its path. Because of the gravity influences of
objects with heavy mass, the path of the satellite is skewed even though it may initially launch in a straight path.
Using hyperbolas, astronomers can predict the path of the satellite to make adjustments so that the satellite gets
to its destination.
Radio

Radio systems’ signals employ hyperbolic functions. One important radio system, LORAN, identified
geographic positions using hyperbolas. Scientists and engineers established radio stations in positions according
to the shape of a hyperbola in order to optimize the area covered by the signals from a station. LORAN allows
people to locate objects over a wide area and played an important role in World War II.
Inverse Relationships

The hyperbola has an important mathematical equation associated with it -- the inverse relation. When an
increase in one trait leads to a decrease in another or vice versa, the relationship can be described by a
hyperbola. Graphing a hyperbola shows this immediately: when the x-value is small, the y-value is large, and
vice versa. Many real-life situations can be described by the hyperbola, including the relationship between the
pressure and volume of a gas.
Lenses and Monitors

Objects designed for use with our eyes make heavy use of hyperbolas. These objects include microscopes,
telescopes and televisions. Before you can see a clear image of something, you need to focus on it. Your eyes
have a natural focus point that does not allow you to see things too far away or close up. To view such things as
planets or bacteria, scientists have designed objects that focus light into a single point. The designs of these use
hyperbolas to reflect light to the focal point. When using a telescope or microscope, you are placing your eye in
a well-planned focal point that allows the light from unseen objects to be focused in a way for you to view
them.
1. Dulles Airport
Dulles Airport, designed by Eero Saarinen, is in the
shape of a hyperbolic paraboloid. The hyperbolic paraboloid is a three-dimensional
curve that is a hyperbola in one cross-section, and a parabola in another cross section.
3. Gear transmission
Two hyperboloids of revolution can provide gear transmission between two skew axes.
The cogs of each gear are a set of generating straight lines.
4. Sonic Boom
In 1953, a pilot flew over an Air Force Base
flying faster than the speed of sound. He damaged every building on the base.
As the plane moves faster than the speed of sound, you get a cone-like wave.
Where the cone intersects the ground, it is an hyperbola.
The sonic boom hits every point on that curve at the same time. No sound is
heard outside the curve. The hyperbola is known as the "Sonic Boom Curve."
In the picture below, the sonic boom is "visible" due to the humidity.
Details about the Photo
Comparing Speeds in Miles per Hour
Human Walking
3 mph
Human Running
34.3 mph
Race Horse
44.9 mph
Cheetah Running
65 mph
Car on Interstate Highway in Colorado
75 mph
Fastest Train
250 mph
Passenger Jetliner
(McDonald Douglas DC-9)
575 mph
Speed of Sound
(At sea level, 59 degrees)
761 mph
Concorde
1,450 mph
Fastest Jet Fighter
4,500 mph
Space Shuttle in Orbit
17,000 mph
5. Cooling Towers of Nuclear Reactors
The hyperboloid is the design standard for all nuclear
cooling towers. It is structurally sound and can be built with straight steel beams.
When designing these cooling towers, engineers are faced with two problems:
(1) the structutre must be able to withstand high winds and
(2) they should be built with as little material as possible.
The hyperbolic form solves both of these problems. For a given diameter
and height of a tower and a given strength, this shape requires less material
than any other form. A 500 foot tower can be made of a reinforced concrete
shell only six or eight inches wide. See the pictures below (this nuclear power
plant is located in Indiana).