In differential geometry, a k-noid is a minimal surface with k catenoid openings. In particular, the 3-noid is often called trinoid. The first k-noid minimal surfaces were described by Jorge and Meeks in 1983.[1]
The term k-noid and trinoid is also sometimes used for constant mean curvature surfaces, especially branched versions of the unduloid ("triunduloids").[2]
f(z)=1/(zk-1)2,g(z)=zk-1
\begin{align} X(z)=
| 1 | |
| 2 |
\Re\{(
| -1 | |
| kz(zk-1) |
)[
| k-1) | |
| &(k-1)(z | |
| 2F |
| k)\\ & | |
| 1(1,-1/k;(k-1)/k;z |
{}-(k-1)z2(z
| k-1) | |
| 2F |
| k) | |
| 1(1,1/k;1+1/k;z |
\\ &{}-kzk+k+z2-1]\} \end{align}
\begin{align} Y(z)=
| 1 | |
| 2 |
\Re\{(
| i | |
| kz(zk-1) |
)[
| k-1) | |
| &(k-1)(z | |
| 2F |
| k) | |
| 1(1,-1/k;(k-1)/k;z |
\\ &{}+(k-1)z2(z
| k-1) | |
| 2F |
| k)\\ & | |
| 1(1,1/k;1+1/k;z |
{}-kzk+k-z2-1)]\} \end{align}
Z(z)=\Re\left\{
| 1 | |
| k-kzk |
\right\}
where
2F1(a,b;c;z)
\Re\{z\}
z
It is also possible to create k-noids with openings in different directions and sizes,[4] k-noids corresponding to the platonic solids and k-noids with handles.[5]