Catenoid

Catenoid is a minimal surface, given by a parametrisation, genus 0, 2 ends, embedded, with a continuous symmetry group.

Open Catenoid in the interactive viewer →

aperiodic complete-finite-total-curvature def-parametric embedded implemented minimal tradition-classical tradition-physical

About

Hang a chain between two points and it settles into a catenary. Spin that curve about the axis between them and you get the catenoid -- the shape a soap film takes when it spans two parallel rings. It was the first minimal surface found after the plane, and apart from the plane it is the only one that is also a surface of revolution. Pull the rings too far apart and the film has no stable shape left to take, and it snaps.

Formula

x(u, v)

\cosh\left(v\right) \cos\left(u\right)

y(u, v)

\cosh\left(v\right) \sin\left(u\right)

z(u, v)

v

Properties

Family: minimal
Given by: a parametrisation
Curvature: zero mean curvature
Periodicity: not periodic
Genus: 0
Ends: 2
Embedding: embedded
Orientable: yes
Symmetry kind: continuous symmetry group
Fidelity: exact
Exactness: elementary

Sources

Chart as curated in tools/surfdb/charts.py, verified numerically against the curvature condition the record claims (measured over the chart by the validator and the charts self-test).

L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (1744) -- the catenary of revolution as the minimal surface of revolution.
J. B. Meusnier, 'Memoire sur la courbure des surfaces', Memoires des savans etrangers 10 (1785, read 1776) -- identified the catenoid and helicoid as minimal.
M. Weber, minimalsurfaces.blog (mirror: minsurf ch189, ch202, ch222): total curvature -4pi; 'the catenoid is a 2-Noid (in fact, the only one)'.