Richmond surface

File:Richmond Surface.png

Richmond surface for m=2.

In differential geometry, a Richmond surface is a minimal surface first described by Herbert William Richmond in 1904. It is a family of surfaces with one planar end and one Enneper surface-like self-intersecting end. It has Weierstrass–Enneper parameterization $f(z)=1/z^2,g(z)=z^m$. This allows a parametrization based on a complex parameter as

$\begin{array}{rl}X(z)& =\mathrm{\Re }[(-1/2z)-z^{2m+1}/(4m+2)]\\ Y(z)& =\mathrm{\Re }[(-i/2z)+i{z}^{2m+1}/(4m+2)]\\ Z(z)& =\mathrm{\Re }[z^m/m]\end{array}$

The associate family of the surface is just the surface rotated around the z-axis. Taking m = 2 a real parametric expression becomes:

$\begin{array}{rl}X(u,v)& =(1/3)u^3-u{v}^2+\frac{u}{u^2+v^2}\\ Y(u,v)& =-u^{2}v+(1/3)v^3-\frac{v}{u^2+v^2}\\ Z(u,v)& =2u\end{array}$