★★☆ Moderate

⛓️ Catenary

A uniformly dense chain hanging freely between two points takes the shape y = a·cosh(x/a) — the catenary. Drag the anchor points and watch the exact chain curve update in real time. Compare with a parabola.

Cable length L: 500 px Density (sag): 1.0

a = — px

Sag = — px

H-span = — px

T_mid = — (norm.)

y = a·cosh(x/a) │ T(x) = w·a·cosh(x/a) │ s = a·sinh(x/a)

Catenary (exact) Parabola (approx.) Tension vectors

The Catenary Curve

Galileo thought the hanging chain was a parabola — he was wrong. The true shape is the catenary, described by the hyperbolic cosine: y = a·cosh(x/a). The constant a = T₀/(wg), where T₀ is the horizontal component of tension and w is weight per unit length.

The catenary appears in suspension bridges, power lines, arch design (inverted), and even the shape of a soap film between two rings. Drag the yellow anchor points to explore how the curve responds to different spans and sag values.