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 v are catenoidal helices; they are generatrices of the helicoid. In the case of the catenoid, they become catenaries, and in the case of the right helicoid, lines perpendicular to the axis.

 

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© Robert FERRÉOL  2017