Catenoid

A catenary of revolution. The catenoid and plane are the only surfaces of revolution which are also minimal surfaces. The catenoid can be given by the parametric equations

			(1)
			(2)
			(3)

where .

The line element is

(4)

The first fundamental form has coefficients

			(5)
			(6)
			(7)

and the second fundamental form has coefficients

			(8)
			(9)
			(10)

The principal curvatures are

			(11)
			(12)

The mean curvature of the catenoid is

(13)

and the Gaussian curvature is

(14)

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

			(15)
			(16)
			(17)

where corresponds to a helicoid and to a catenoid.

This deformation is illustrated on the cover of issue 2, volume 2 of The Mathematica Journal.