Catenary
Classic Cartesian curvesThe hyperbolic-cosine curve y = a·cosh(x/a) of a hanging chain — subtly different from a parabola.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Catenary?
Hang a uniform chain from two points and it settles into the catenary y = a·cosh(x/a). Galileo guessed the shape was a parabola; Joachim Jungius disproved that experimentally, and in 1691 Leibniz, Huygens, and Johann Bernoulli each derived the true equation in response to Jakob Bernoulli’s challenge. The name comes from the Latin catena, “chain”.
Flip it upside down and the pure-tension chain becomes a pure-compression arch — Robert Hooke’s insight “as hangs the flexible line, so but inverted will stand the rigid arch”. That is why the Gateway Arch in St. Louis and Gaudí’s hanging-chain models follow (weighted) catenaries rather than parabolas.
Key properties
- Arc length from the vertex: s = a·sinh(x/a); the sag-to-tension geometry is all in one constant a.
- The horizontal tension component is constant along the chain.
- Near the vertex it is approximated by the parabola y ≈ a + x²/(2a) — the source of Galileo’s confusion.
- Rotating it about the x-axis gives the catenoid, the only non-planar minimal surface of revolution.
- A square wheel rolls smoothly on a road made of inverted catenary humps.
Where you'll see it
Hanging cables and chains, the Gateway Arch, Gaudí’s architecture, suspension bridge design, and soap-film catenoids.