Catenoid-Helicoid Associate Family

Catenoid-Helicoid Associate Family is a minimal surface, given by a parametrisation, genus 0, embedded or immersed depending on parameters, with a continuous symmetry group.

Open Catenoid-Helicoid Associate Family in the interactive viewer →

aperiodic def-parametric implemented minimal tradition-classical

Formula

x(u, v)

\frac{\sqrt{2}}{2} \left(\cosh\left(u\right) \cos\left(v\right) + \sinh\left(u\right) \sin\left(v\right)\right)

y(u, v)

\frac{\sqrt{2}}{2} \left(\cosh\left(u\right) \sin\left(v\right) - \sinh\left(u\right) \cos\left(v\right)\right)

z(u, v)

\frac{\sqrt{2}}{2} \left(u + v\right)

Properties

Family: minimal
Given by: a parametrisation
Curvature: zero mean curvature
Periodicity: not periodic
Genus: 0
Ends: 0
Embedding: embedded or immersed depending on parameters
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).

O. Bonnet, 'Memoire sur la theorie generale des surfaces', J. Ecole Polytechnique 19 (1853) -- the associate family.
H. A. Schwarz, Gesammelte Mathematische Abhandlungen (1890).