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MINIMAL HELICOID

Surfaces studied by Scherk in 1834.
Also called Scherk surfaces (see the other Scherk surfaces here), or helcats.
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