Catenary

The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force. The word catenary is derived from the Latin word for "chain." In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola (MacTutor Archive). The curve is also called the alysoid and chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli.

Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690 (MacTutor Archive). If you roll a parabola along a straight line, its focus traces out a catenary. As proved by Euler in 1744, the catenary is also the curve which, when rotated, gives the surface of minimum surface area (the catenoid) for the given bounding circle.

The parametric equations for the catenary are given by

(1) (2) (3)

where corresponds to the vertex and is a parameter that determines how quickly the catenary "opens up." Catenaries for values of ranging from 0.05 to 1.00 by steps of 0.05 are illustrated above.

The arc length, curvature, and tangential angle for are given by

(4) (5) (6)

The slope is proportional to the arc length as measured from the center of symmetry.

The Cesàro equation is

(7)

The St. Louis Arch closely approximates an inverted catenary, but it has a nonzero thickness and varying cross sectional area (thicker at the base; thinner at the apex). The centroid has half-length of feet at the base, height of 625.0925 feet, top cross sectional area 125.1406 square feet, and bottom cross sectional area 1262.6651 square feet.

The catenary also gives the shape of the road (roulette) over which a regular polygonal "wheel" can travel smoothly. For a regular -gon, the Cartesian equation of the corresponding catenary is

(8)

where