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Concept
Weierstrass–Enneper parameterization
Résumé
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 f(z) = 1, g(z) = zm.
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 .
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.
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.
Source officielle
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