- Weierstrass–Enneper parameterization
In
mathematics , the Weierstrass–Enneper parameterization ofminimal surface s is a classical piece ofdifferential geometry .Alfred Enneper andKarl Weierstrass studied minimal surfaces as far back as1863 .Let ƒ and "g" be functions on either the entire complex plane or the unit disk, where "g" is meromorphic and ƒ 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 ƒ"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:
:egin{align} x_k(zeta) &{}= Re left{ int_{0}^{zeta} varphi_{k}(z) , dz ight} + c_k , qquad k=1,2,3 \ varphi_1 &{}= f(1-g^2)/2 \ varphi_2 &{}= old{i} f(1+g^2)/2 \ varphi_3 &{}= fgend{align}
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 ƒ("z") = 1, "g"("z") = "z".References
Wikimedia Foundation. 2010.