Parabola

Uses

Parabolas show up in a variety of places. The path of a bomb dropped from an airplane is a section of a parabola. The cables of a well-designed suspension bridge follow a parabolic curve. The surface of the water in a bowl that is rotating on a turntable will assume the shape of a parabola rotated around its axis. The area of a circle is a parabolic function of its radius. In fact, the graphs of all polynomial functions y = ax2 +bx + c, of degree two are parabolic in shape.

Perhaps the most interesting application of a parabola is in the design of mirrors for astronomical telescopes. The rays of light from a star, a galaxy, or even such a nearby celestial object as a planet are essentially parallel. The reflective property of a parabola sends a ray that is parallel to the parabola's axis through the focus. Therefore, if one grinds a mirror with its surface in the shape of a parabola rotated around its axis and if one tilts such a mirror so that its axis points at a star, all the light from that star which strikes the mirror will be concentrated at the mirror's focus.

Of course, such a mirror can be pointed at only one star at a time. Even so, the mirror will reflect rays from nearby stars through their own "foci" which are near the real focus. It will bring into focus not only the one star at which it is pointed, but also the stars in the area around the star.

The process can be reversed. If the light source is placed at the focus, instead of concentrating the rays, the Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group.
Figure 4. Illustration by Hans & Cassidy. Courtesy of Gale Group.
reflector will act to send them out in a bundle, parallel to the axis of the reflector. The large searchlights used during air raids in World War II were designed with parabolic reflectors.

Parabolic reflectors are used in other devices as well. Radar antennas, the "dishes" used to pick up satellite television signals, and the reflectors used to concentrate sound from distant sources are all parabolic.

Resources

Books

Ball, W.W. Rouse. A Short Account of the History of Mathematics. London: Sterling Publications, 2002.

Finney, Thomas, Demana, and Waits. Calculus: Graphical, Numerical, Algebraic. Reading, MA: Addison Wesley Publishing Co., 1994.

Hilbert, D., and S. Cohn-Vossen. Geometry and the Imagination. New York: Chelsea Publishing Co. 1952.

Zwikker, C. The Advanced Geometry of Plane Curves and Their Applications. New York: Dover Publications, Inc., 1963.

J. Paul Moulton

KEY TERMS

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Directrix

—The fixed line in the focus directrix definition of a conic section.

Focus

—A point, or one of a pair of points, whose position determines the shape of a conic section.

Parabola

—A set of points which are equidistant from a fixed point and a fixed line.