catenarys

A catenary is the curve formed by a perfectly flexible, uniform chain or cable suspended by its endpoints and acted on by gravity. When the lowest point of the chain is at the origin, the curve is described by y = a cosh(x/a), where a > 0 is a scale parameter.

The parameter a equals the horizontal component of tension divided by the weight per unit length (a = H/w). The horizontal tension H is constant along the chain, and the tension at position x is T(x) = H cosh(x/a). The curve is symmetric about its lowest point and can be parameterized by x = a sinh t, y = a cosh t.

Compared with a parabola, the catenary is not a parabola, though near the vertex it is well approximated by a parabola: y ≈ a + x^2/(2a) for small x. The name derives from its association with hanging chains; the term comes from the Latin catena, meaning “chain.”

History and naming: the curve was studied in connection with hanging chains in the 17th century and is often attributed to analyses by Huygens. The mathematical form involves the hyperbolic cosine function, hence its common shorthand as a cosh curve.

Applications: catenaries and catenary arches are used in architecture and structural engineering because they distribute thrust efficiently. They appear in suspension systems, arches, and cables, and their properties help design stable, load-distributing structures. The Gateway Arch is frequently described as a weighted catenary arch.