Catenaria

Catenaria, or catenary, is the plane curve that describes the shape of a uniform, flexible chain or cable suspended from its ends and acted upon only by gravity. It is the natural shape of a hanging chain in a uniform gravitational field. In a Cartesian coordinate system with the lowest point at the origin, the curve can be described by y = a (cosh(x/a) − 1), where a > 0 is a constant. Equivalently, choosing coordinates so that the horizontal tangent occurs at the endpoints yields y = a cosh(x/a) up to a vertical shift. The parameter a relates to the horizontal component of tension T0 and the weight per unit length w by a = T0/w. The curve minimizes the potential energy for the given endpoints and length.

Properties: The slope is dy/dx = sinh(x/a) and the curvature is κ = (1/a) cosh(x/a). The tension along the chain has a constant horizontal component T0; the vertical component increases with x. The catenary differs from a parabola; for small sag or shallow spans a parabola is a common approximation, but the catenary exactly describes the physics of a uniform chain.

Applications: In architecture and civil engineering the inverted catenary is used for arches and suspension cables because it yields favorable thrust properties. A catenary arch, formed by inverting the curve, distributes loads with minimal horizontal thrust. Real-world cables, such as electrical transmission lines and suspension bridge cables, are well described by the catenary, though practical designs often employ parabolic approximations or numerical models. The term catenary derives from the Latin catena, chain; the curve was studied in the 17th century, and the name was popularized by Leibniz.