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MINIMAL HELICOID

Surfaces studied by Scherk in 1834.
Also called Scherk surfaces (see the other Scherk surfaces here). See Darboux p. 328. |

Cartesian parametrization: . |

The helicoids that are minimal surfaces have the above parametrization. They all are locally isometric to one another.

For a = 0, we get the
catenoid, and for , the right helicoid.

The above parametrization comes from the Weierstrass parametrization of a minimal surface:, taking , , . |

The minimal helicoids therefore vary depending only on their Bonnet angle ; and the animation above represents, in some way, a complex rotation of a right helicoid.

The coordinate line obtained for u = 0 in the above parametrization is a circular helix that is a geodesic of the minimal helicoid; the latter is therefore a Björling surface associated to a circular helix.
In the case of the catenoid, the helix becomes a circle and in the case of the right helicoid, it becomes the axis.
The coordinate lines for constant |

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© Robert FERRÉOL 2017