Richmond

Richmond is a minimal surface, given by a parametrisation, genus 0, 2 ends, self-intersecting.

Open Richmond in the interactive viewer →

aperiodic complete-finite-total-curvature def-parametric implemented minimal self-intersecting tradition-classical

About

A family of minimal surfaces with two ends, one planar and one of higher order, obtained by the simplest Weierstrass data after Enneper's. Increasing the order winds the higher end round more times and the surface acquires more symmetry.

Formula

x(u, v)

\frac{-\cos\left(v\right)}{2 u} - \frac{u^{3} \cos\left(3 v\right)}{6}

y(u, v)

\frac{\sin\left(v\right)}{2 u} + \frac{u^{3} \sin\left(3 v\right)}{6}

z(u, v)

u \cos\left(v\right)

Properties

Family: minimal
Given by: a parametrisation
Curvature: zero mean curvature
Periodicity: not periodic
Genus: 0
Ends: 2
Embedding: self-intersecting
Orientable: yes
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).

H. W. Richmond, 'On the simplest algebraic minimal curves, and the derived real minimal surfaces', Proc. Cambridge Phil. Soc. 19 (1904).