parabola v. catenary
Friday, March 5, 2010 
A blue parabola and a red catenary. Image courtesy of Ask Dr. Math.
Parabola
A parabola is a curve formed by the intersection of a circular cone and a plane parallel to the side of the cone. It can also be defined as the set of points that are the same distance from a line and a point that isn’t on the line. The graph of a quadratic equation of the form 
generates a parabola.
Many physical relationships can be formally described by quadratic equations: When air resistance isn’t taken into account, thrown objects follow a parabolic path under the influence of gravity. In addition, parabolic reflectors are used in radio and optical telescopes, microphones, satellite dishes, and car headlights.
Catenary
Stan Wagon riding his square-wheeled trike on an inverted catenary path. Image courtesy of Science News. Catenary curves can look nearly identical to parabolas, but they are defined differently. A catenary is the curve formed by a string suspended from its ends. The general equation for a catenary is
where cosh is the hyperbolic cosine function.
An upside-down catenary is the ideal shape for an arch that only supports its own weight because the arch’s structural elements will experience pure compression, without bending or shear forces. In applications where an arch has to support a large amount of weight, a parabolic curve is preferable. A similar situation applies when the curve opens upwards. Simple suspension footbridges with curved decks are catenaries, while larger suspension bridges with horizontal roadways have parabolic cables.