helicoid catenoid 001
Continuous deformation of part of Helicoid into Catenoid and vice versa.

The helicoid and catenoid are a family of minimal surfaces. Each surface in the family is isometric to any other, meaning, the length between any two points A and B of any curve on a surface, remain unchanged. This means that, if you are a 2-dimensional creature living on the surface, it is impossible to tell whether your world is helicoid or catenoid.

helicoid rotating 001
This shows a rotating helicoid, watch the bottom end.
helicoid patch ana
helicoid (anaglyph)
helicoid wire ana
helicoid (anaglyph wireframe)
  Px = bb*(cos(aa)*sinh(v)* sin(u) + sin(aa) * cosh(v) * cos(u))
  Py = bb*(-cos(aa)*sinh(v) * cos(u) + sin(aa) * cosh(v) * sin(u))
  Pz = bb * (cos(aa) * u + sin(aa) * v)

Related Surface of Catenoid

  1. Catenoid
  2. Helicoid-Catenoid
  3. Catenoid-Enneper
  4. Catenoid Fence