Greenwood press

18
CONIC SECTIONS
Cutting a cone to make 
Parabolic reflector used in a 
A parabolic reflector 
a parabola.
microwave antenna (Corel)
concentrates energy.
Various representations of the conic section called a “parabola.”

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parabolic shape is ideal for car headlights. It would also be ideal for television
tubes were consumers not so demanding that picture screens be rectangular.
Television tube manufacturers have to do some clever engineering to maximize
the benefits of parabolic reflectors and still provide rectangular screens.
If the arrows are reversed in the right-hand drawing, then the parabolic reflec-
tor accumulates and concentrates energy from outside sources. For example, the
dot for the energy source might represent a pipe containing water. Then the para-
bolic reflector can concentrate the sun’s rays to heat the water as part of a solar
heating system. Pipes in highly polished parabolic troughs can focus enough sun-
light to heat an enclosed fluid as high as 750°F or turn water to steam. Hand-held
parabolic reflectors that were invented for spying are available for sport and hobby
activities such as bird watching. The parabolic reflector picks up weak sounds,
such as distant bird calls, and focuses them on a microphone at the focal point.
Sometimes a diffuse view is important. Since they can provide almost 360°
views, hyperbolic mirrors are used for security surveillance in buildings. The
reflection in hyperbolic mirrors is from the convex side, rather than the concave
side used for parabolic mirrors. This is what makes exterior mirrors on the pas-
senger sides of cars show wider views and justify the warning, “Objects may be
closer than they appear.”
Parabolas appear in science and engineering. A hard-hit baseball flies off the
bat in a parabolic path. The large cables strung between towers of a suspension
bridge, such as the Golden Gate Bridge in San Francisco, form a parabola. Con-
nection to the roadway of the bridge is important in shaping the large cables to
parabolic shape. A telephone wire that curves because of its own weight is not a
parabola, but is a catenary. If a heavy liquid like mercury is placed in a large can,
and the can is spun, the surface of the liquid will form a paraboloid (every verti-
cal cross section through the center of the can is a parabola). Parabolas are used
in design and medical applications to determine smooth curves from three spec-
ified points in a solid or the image of a solid, such as the points provided in a
medical CAT scan.
Ellipses are a oval conic section that look like squashed circles. They have
two foci that act as centers of the ellipse. Hitting a ball from one focus on an
elliptical pool table will result in a carom from the side of the table that sends the
ball to the other focus. Rooms that have elliptical ceilings or shapes will reflect
the sound of a pin dropping at one focal point so that it is audible many yards
away at the other focal point. The Mormon Tabernacle in Salt Lake City and Stat-
uary Hall in the U.S. Capitol in Washington, D.C., are two rooms that have
remarkable acoustics because of their elliptical shapes.
Ellipses are an outcome of some common architectural techniques. The
Romans invented the Groin Vault, the joining of two identical barrel (cylindrical)
vaults over a square plan. The intersection of the vaults form ellipses that go
diagonally to the corners of the square. Although the groin vault is common in
ancient and medieval buildings, it is also found in modern structures such as the
terminal building at the St. Louis Airport.