polycatenary

Polycatenary is a generalization of the catenary curve and describes the shape of a flexible cable, chain, or network of cables that hang under gravity when composed of multiple strands or segments connected at joints. In such systems, each segment may have its own density, tension, or end conditions, leading to a curve that is typically piecewise formed by individual catenaries joined together at the connections.

Mathematically, a polycatenary arises when the hanging system is modeled as several connected elements, each governed by the catenary equation y = a cosh((x − h)/a) + k for its segment, with parameters a, h, and k varying between segments. The segments are constrained by continuity of position and slope at the joints, and by equilibrium of tensile forces. As a result, the overall shape is determined by solving a set of coupled equations that express energy minimization and force balance, often requiring numerical methods for complex networks.

Special cases are instructive: if all segments share the same density and tension and meet in a single continuous chain, the polycatenary reduces to the standard catenary. If there are distinct segments with different properties, the curve becomes a concatenation of catenaries with matching conditions at the joints.

Applications appear in architecture and engineering, especially in the design of cable nets, photometric or decorative hanging structures, and polycatenary arches. The term is used in some literature to describe multi-strand or networked hanging systems, though many treatments simply refer to catenaries or catenary systems.