# Weierstrass–Enneper parameterization

Weierstrass–Enneper parameterization

In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.

Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.

Let &fnof; and "g" be functions on either the entire complex plane or the unit disk, where "g" is meromorphic and &fnof; is analytic, such that wherever "g" has a pole of order "m", "f" has a zero of order 2"m" (or equivalently, such that the product &fnof;"g"2 is holomorphic), and let "c"1, "c"2, "c"3 be constants. Then the surface with coordinates ("x"1,"x"2,"x"3) is minimal, where the "x""k" are defined using the real part of a complex integral, as follows:

:

The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O. "Minimal surfaces", vol. I, p. 108. Springer 1992. ISBN 3540531696]

For example, Enneper's surface has &fnof;("z") = 1, "g"("z") = "z".

References

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