differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, a Richmond surface is a minimal surface first described by

Herbert William Richmond Herbert William Richmond (born on the 17 July 1863 in Tottenham, England) was a mathematician who studied the Cremona–Richmond configuration. One of his most popular works is an exact construction of the regular heptadecagon in 1893 (which was c ...

in 1904. It is a family of surfaces with one planar end and one

Enneper surface In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: \begin x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\ y &= \tfrac v \left(1 - \tfracv^2 + u^2\righ ...

-like self-intersecting end. It has

Weierstrass–Enneper parameterization In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863. Let f and g be functions on either ...

f(z)=1/z^2, g(z)=z^m

. This allows a parametrization based on a complex parameter as :

\end

The

associate family In differential geometry, the associate family (or Bonnet family) of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has the representation :x_k(\zeta) = \Re \left\ + ...

of the surface is just the surface rotated around the z-axis. Taking ''m'' = 2 a real parametric expression becomes:John Oprea, The Mathematics of Soap Films: Explorations With Maple, American Mathematical Soc., 2000 :

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

References

{{Minimal surfaces Minimal surfaces