Henneberg's Minimal Surface

A minimal surface and double algebraic surface of 15th order and fifth class which can be given by parametric equations

The coefficients of the first fundamental form of this parameterization are given by

and the coefficients of the second fundamental form are

giving area element

and Gaussian and mean curvatures are

A slightly different version of this surface (also algebraic of order 15 but with slightly different coefficients) can also be obtained from the Enneper-Weierstrass parameterization with

which gives a parameterization of the form

The coefficients of the first fundamental form of this parameterization are given by

and the coefficients of the second fundamental form are

giving area element

and Gaussian and mean curvatures are

Henneberg's minimal surface is a nonorientable surface defined over the unit disk. It is an immersion of the real projective plane that has been multiply punctured (once at the origin and four times at each of the roots of the metric). Consequently, it is not a complete surface. The total curvature is .