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 Incoming links: Catenary

Catenoid Created 2026-09-24 Updated 2026-10-03

A catenoid is the surface of revolution obtained by rotating a catenary about its directrix. One conformal parametrization is

(u, v) \mapsto (cosh u cos v, cosh u sin v, u) .

Enforce the fixed arc length with a multiplier. Apart from an irrelevant common factor, the augmented integrand is

f = (g z + λ) 1 + z^{'2} .

\frac{g z + λ}{1 + z ^{'2}} = constant .

After absorbing constants into

z_{0}

and a positive scale

B

, integration yields the catenary

z - z_{0} = - B cosh (x / B) .

The endpoint conditions and prescribed length give

z_{0} = B cosh (a / B), L = 2 B sinh (a / B) .

The second equation determines

B > 0

implicitly because

L > 2 a

, and the first then determines

z_{0}