The Catenary

Pin a chain at two points and let it hang. It does not settle into a parabola — it finds the catenary, y = a·cosh(x/a), the one curve that minimises its own potential energy. Give it more slack and it sags deeper; pull it taut and the dip vanishes. Drag the pins to move the ends.

The pins

Span 2.00

Height offset 0.00

The chain

Slack (length / span) 1.35×

Compare & prove

Overlay the best-fit parabola Show the relaxing bead chain

Read-outs

shape parameter a—

lowest point (sag)—

chain length L—

PE — catenary—

PE — parabola—

—

Presets

the catenary — a·cosh(x/a)

the best-fit parabola (the impostor)

Galileo guessed a hanging chain was a parabola; in 1691 Leibniz, Huygens and Johann Bernoulli proved it is the catenary — from catena, “chain.” Inverted, it is the line of pure compression: the shape of a self-standing arch. · its 2-D cousin, the Soap Film → · ↗ The First Integral — the cosh’s conserved a, shared with two other curves by one law.