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  and  be functions on either the entire complex plane or the unit disk, where  is meromorphic and  is analytic, such that wherever  has a pole of order ,  has a zero of order  (or equivalently, such that the product  is holomorphic), and let  be constants. Then the surface with coordinates  is minimal, where the  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. For example, Enneper's surface has , . Parametric surface of complex variables
The Weierstrass-Enneper model defines  a minimal surface  () on a complex plane (). Let  (the complex plane as the  space), the Jacobian matrix of the surface can be written as a column of complex entries:

where  and  are holomorphic functions of . The Jacobian  represents the two orthogonal tangent vectors of the surface:

The surface normal is given by

The Jacobian  leads to a number of important properties: , , , . The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface. The  derivatives can be used to construct the first fundamental form matrix:

and the second fundamental form matrix

Finally, a point  on the complex plane maps to a point  on the minimal surface in  by

where  for all minimal surfaces throughout this paper except for Costa's minimal surface where . Embedded minimal surfaces and examples
The classical examples of embedded  complete minimal surfaces in  with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function :

where  is a constant. Helicatenoid 
Choosing the functions  and , a one parameter family of minimal surfaces is obtained. Choosing the parameters of the surface as :

At the extremes, the surface is a catenoid  or a helicoid . Otherwise,  represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the  axis in a helical fashion. Lines of curvature
One can rewrite each element of second fundamental matrix as a function of  and , for example

And consequently the second fundamental form matrix can be simplified as

One of its eigenvectors is  which represents the principal direction in the complex domain. Therefore, the two principal directions in the  space turn out to be

See also

 Associate family
 Bryant surface, found by an analogous parameterization in hyperbolic space

References

Differential geometry
Surfaces
Minimal surfaces