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

(1)

In Cartesian coordinates, it is

(2)

It can be given in parametric form by

			(3)
			(4)
			(5)

which has an obvious generalization to the elliptic helicoid. Writing instead of gives a cone instead of a helicoid.

The first fundamental form coefficients of the helicoid are given by

			(6)
			(7)
			(8)

and the second fundamental form coefficients are

			(9)
			(10)
			(11)

giving area element

(12)

Integrating over and then gives

			(13)
			(14)

The Gaussian curvature is given by

(15)

and the mean curvature is

(16)

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

			(17)
			(18)

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

			(19)
			(20)
			(21)

where corresponds to a helicoid and to a catenoid.

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

Helicoids occur in architectural forms such as helical staircases and ramps, as well as in screw-type machine-building products (Bock Hyeng et al. 2025).