Richmond surface explained

In differential geometry, a Richmond surface is a minimal surface first described by Herbert William Richmond in 1904.^[1] It is a family of surfaces with one planar end and one Enneper surface-like self-intersecting end.

f(z) =1 /z ^{2,} g(z) = z ^{m} . This allows a parametrization based on a complex parameter as\begin{align} X(z) & = \Re[( -1 / 2 z) - z ^{2m+1} /( 4 m +2 )]\\ Y(z) & = \Re[( - i/ 2 z) + i z ^{2m+1} /( 4 m +2 )]\\ Z(z) & = \Re[z ^{m} / m] \end{align} The associate family of the surface is just the surface rotated around the z-axis.

Taking m = 2 a real parametric expression becomes:^[2]

\begin{align} X(u,v) & = ( 1 / 3 )u ^{3} - uv ^{2} + (u)/(u ^{2+v} ^{2}) \\ Y(u,v) & = - u ^{2v} + ( 1 / 3 )v ^{3} - (v)/(u ^{2+v} ^{2}) \\ Z(u,v) & = 2 u \end{align}

Notes and References

1. [Jesse Douglas]
2. John Oprea, The Mathematics of Soap Films: Explorations With Maple, American Mathematical Soc., 2000