Helicoid

The (circular) helicoid is the minimal surface having a (circular) helix as its boundary. It is the only ruled minimal surface other than the plane (Catalan 1842, do Carmo 1986). For many years, the helicoid remained the only known example of a complete embedded minimal surface of finite topology with infinite curvature. However, in 1992 a second example, known as Hoffman's minimal surface and consisting of a helicoid with a hole, was discovered (Sci. News 1992). The helicoid is the only non-rotary surface which can glide along itself (Steinhaus 1999, p. 231).

The equation of a helicoid in cylindrical coordinates is

In Cartesian coordinates, it is

It can be given in parametric form by

which has an obvious generalization to the elliptic helicoid. Writing z=-cu instead of z=cv gives a cone instead of a helicoid.

The first fundamental form coefficients of the helicoid are given by

and the second fundamental form coefficients are

giving area element

Integrating over v in [0,theta] and u in [0,r] then gives

The Gaussian curvature is given by

and the mean curvature is

making the helicoid a minimal surface. The Gaussian curvature can be given implicitly by

The helicoid can be continuously deformed into a catenoid by the transformation

where alpha=0 corresponds to a helicoid and alpha=pi/2 to a catenoid.

If a twisted curve C (i.e., one with torsion tau!=0) rotates about a fixed axis A and, at the same time, is displaced parallel to A such that the speed of displacement is always proportional to the angular velocity of rotation, then C generates a generalized helicoid.

See also

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References

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Cite this as:

Weisstein, Eric W. "Helicoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Helicoid.html

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