catenarias

A catenary is the curve formed by a flexible, uniform chain or cable suspended by its ends and acted on by gravity. The term derives from Latin catena, chain, and the curve is closely associated with the study of hanging cables and wires. In its standard form, the curve can be described by the equation y = a cosh((x - x0)/a) + y0, where cosh is the hyperbolic cosine and a is a positive constant related to the horizontal component of tension and the weight per unit length of the chain. A common equivalent description uses the lowest point at the origin, giving y = a cosh(x/a). The slope satisfies dy/dx = sinh((x - x0)/a).

For a uniform chain under gravity with negligible bending stiffness, the catenary is the exact shape. In contrast, under a uniform transverse load or small sag compared with the span, the profile can be approximated by a parabola y ≈ (w/H) x^2, where w is weight per unit length and H is the horizontal tension. The catenary’s curvature is not constant, unlike a parabola.

Applications of catenaries are widespread. They are fundamental in the design of overhead electrical lines for railways and power systems, where sag and tension determine clearance and performance. The curve also appears in architectural cables, suspension structures, and decorative arches. The mathematical form is related to the catenoid, a minimal surface obtained by rotating a catenary around its axis.

Historically, the curve was studied in the 17th century, with notable contributions by Christiaan Huygens. The name catenary reflects its origin in the study of chains and hanging cables.