Helicoid

Helicoid is a singly periodic minimal surface, given by a parametrisation, genus 0, 1 end, embedded, with a continuous symmetry group.

Open Helicoid in the interactive viewer →

chiral complete-infinite-total-curvature def-parametric embedded implemented minimal singly-periodic tradition-classical

About

The surface swept by a horizontal line as it rises and rotates at a constant rate -- a spiral ramp, continued forever in both directions. It is the only ruled minimal surface other than the plane, meaning it can be built entirely out of straight lines. It is also the catenoid's twin: each can be bent into the other without any stretching, through a family of minimal surfaces that are all locally the same surface in different poses.

Formula

x(u, v)

v \cos\left(u\right)

y(u, v)

v \sin\left(u\right)

z(u, v)

u

Properties

Family: minimal
Given by: a parametrisation
Curvature: zero mean curvature
Periodicity: singly periodic
Genus: 0
Ends: 1
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).

J. B. Meusnier, 'Memoire sur la courbure des surfaces', Memoires des savans etrangers 10 (1785, read 1776).
E. Catalan, 'Sur les surfaces reglees dont l'aire est un minimum', J. Math. Pures Appl. 7 (1842) -- the helicoid is the only ruled minimal surface besides the plane.