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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:

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  • Enneper-Weierstraß-Konstruktion (de)
  • Параметризация Вейерштрасса — Эннепера (ru)
  • Weierstrass–Enneper parameterization (en)
  • 魏尔斯特拉斯-恩内佩尔曲面 (zh)
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  • Die Weierstraß-Darstellung, manchmal auch Enneper-Weierstraß- oder Weierstraß-Enneper-Konstruktion, ist eine nach Karl Weierstraß bzw. Alfred Enneper benannte Darstellung von Minimalflächen. Letztere sind reguläre Flächen im reellen Vektorraum , die in der Natur als Seifenhautflächen vorkommen, und daher "reelle" Gebilde. Es mag daher verwundern, dass bei deren Beschreibung holomorphe Funktionen zu Tage treten, wie das bei der hier zu besprechenden Darstellung der Fall ist. (de)
  • Параметризация Вейерштрасса — Эннепера минимальных поверхностей — классический раздел дифференциальной геометрии. Альфред Эннепер и Карл Вейерштрасс изучали минимальные поверхности ещё в 1863 году. (ru)
  • 在微分几何中，魏尔斯特拉斯-恩内佩尔参数化（WE曲面、魏恩曲面、Weierstrauss-Enneper surfaces）是二维极小曲面的参数化。 它以恩内佩尔（Enneper）和魏尔斯特拉斯的名字命名。他们在1863年发现了这个参数化。 设 f 是解析函数、g 是亚纯函数、fg2 是 全纯函数、c1, c2, c3 是常数。若(x1,x2,x3)是曲面M的坐标以及 则M是极小流形。逆命题也是事实：若曲面M有上面的参数化，则M是极小的。 比方说，恩内佩尔曲面具有 。 使用魏爾斯特拉斯橢圓函數。 * (zh)
  • 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: (en)
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  • http://commons.wikimedia.org/wiki/Special:FilePath/Helically_Rotated_Catenary_(Helicatenoid).jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Lines_of_curvature_make_a_quadrangulation_of_the_domain.jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/The_fundamental_domain_(C)_and_the_3D_surfaces._The_continuous_surfaces_are_made_of_copies_of_the_fundamental_patch_(R3).jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/Weierstrass_parameterization_facilities_fabrication_of_periodic_minimal_surfaces.jpg
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  • Die Weierstraß-Darstellung, manchmal auch Enneper-Weierstraß- oder Weierstraß-Enneper-Konstruktion, ist eine nach Karl Weierstraß bzw. Alfred Enneper benannte Darstellung von Minimalflächen. Letztere sind reguläre Flächen im reellen Vektorraum , die in der Natur als Seifenhautflächen vorkommen, und daher "reelle" Gebilde. Es mag daher verwundern, dass bei deren Beschreibung holomorphe Funktionen zu Tage treten, wie das bei der hier zu besprechenden Darstellung der Fall ist. (de)
  • 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. (en)
  • Параметризация Вейерштрасса — Эннепера минимальных поверхностей — классический раздел дифференциальной геометрии. Альфред Эннепер и Карл Вейерштрасс изучали минимальные поверхности ещё в 1863 году. (ru)
  • 在微分几何中，魏尔斯特拉斯-恩内佩尔参数化（WE曲面、魏恩曲面、Weierstrauss-Enneper surfaces）是二维极小曲面的参数化。 它以恩内佩尔（Enneper）和魏尔斯特拉斯的名字命名。他们在1863年发现了这个参数化。 设 f 是解析函数、g 是亚纯函数、fg2 是 全纯函数、c1, c2, c3 是常数。若(x1,x2,x3)是曲面M的坐标以及 则M是极小流形。逆命题也是事实：若曲面M有上面的参数化，则M是极小的。 比方说，恩内佩尔曲面具有 。 使用魏爾斯特拉斯橢圓函數。 * (zh)
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