Catenary

catenary, in mathematics, a curve that describes the shape of a flexible hanging chain or cable—the name derives from the Latin catenaria (“chain”). Any freely hanging cable or string assumes this shape, also called a chainette, if the body is of uniform mass per unit of length and is acted upon solely by gravity.

Early in the 17th century, the German astronomer Johannes Kepler applied the ellipse to the description of planetary orbits, and the Italian scientist Galileo Galilei employed the parabola to describe projectile motion in the absence of air resistance. Inspired by the great success of conic sections in these settings, Galileo incorrectly believed that a hanging chain would take the shape of a parabola. It was later in the 17th century that the Dutch mathematician Christiaan Huygens showed that the chain curve cannot be given by an algebraic equation (one involving only arithmetic operations together with powers and roots); he also coined the term catenary. In addition to Huygens, the Swiss mathematician Jakob Bernoulli and the German mathematician Gottfried Leibniz contributed to the complete description of the equation of the catenary.

Precisely, the curve in the xy-plane of such a chain suspended from equal heights at its ends and dropping at x = 0 to its lowest height y = a is given by the equation y = (a/2)(e^x/a + e^−x/a). It can also be expressed in terms of the hyperbolic cosine function as y = a cosh(x/a). See the figure.

Although the catenary curve fails to be described by a parabola, it is of interest to note that it is related to a parabola: the curve traced in the plane by the focus of a parabola as it rolls along a straight line is a catenary. The surface of revolution generated when an upward-opening catenary is revolved around the horizontal axis is called a catenoid. The catenoid was discovered in 1744 by the Swiss mathematician Leonhard Euler and it is the only minimal surface, other than the plane, that can be obtained as a surface of revolution.

The catenary and the related hyperbolic functions play roles in other applications. An inverted hanging cable provides the shape for a stable self-standing arch, such as the Gateway Arch located in St. Louis, Missouri. The hyperbolic functions also arise in the description of waveforms, temperature distributions, and the motion of falling bodies subject to air resistance proportional to the square of the speed of the body.