|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Weierstrass–Enneper parameterization''' of [[minimal surface]]s is a classical piece of [[differential geometry]].
| | They call me Emilia. For years I've been operating as a payroll clerk. The favorite pastime for my kids and me is to perform baseball but I haven't made a dime with it. Years ago we moved to North Dakota.<br><br>Here is my homepage ... [http://www.youporntime.com/blog/12800 http://www.youporntime.com] |
| | |
| [[Alfred Enneper]] and [[Karl Weierstrass]] studied minimal surfaces as far back as 1863.
| |
| | |
| Let ƒ and ''g'' be functions on either the entire complex plane or the unit disk, where ''g'' is [[meromorphic function|meromorphic]] and ƒ is [[analytic function|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''<sup>2</sup> is holomorphic), and let ''c''<sub>1</sub>, ''c''<sub>2</sub>, ''c''<sub>3</sub> be constants. Then the surface with coordinates (''x''<sub>1</sub>,''x''<sub>2</sub>,''x''<sub>3</sub>) is minimal, where the ''x''<sub>''k''</sub> are defined using the real part of a complex integral, as follows:
| |
| | |
| :<math>\begin{align}
| |
| x_k(\zeta) &{}= \Re \left\{ \int_{0}^{\zeta} \varphi_{k}(z) \, dz \right\} + c_k , \qquad k=1,2,3 \\
| |
| \varphi_1 &{}= f(1-g^2)/2 \\
| |
| \varphi_2 &{}= \bold{i} f(1+g^2)/2 \\
| |
| \varphi_3 &{}= fg
| |
| \end{align}</math>
| |
| | |
| The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.<ref name="DHWK">Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O. ''Minimal surfaces'', vol. I, p. 108. Springer 1992. ISBN 3-540-53169-6</ref>
| |
| | |
| For example, [[Enneper's surface]] has ƒ(''z'') = 1, ''g''(''z'') = ''z''.
| |
| | |
| ==See also==
| |
| | |
| * [[Associate family]]
| |
| * [[Bryant surface]], found by an analogous parameterization in [[hyperbolic space]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{DEFAULTSORT:Weierstrass-Enneper parameterization}}
| |
| [[Category:Differential geometry]]
| |
| [[Category:Surfaces]]
| |
| [[Category:Minimal surfaces]]
| |
| | |
| | |
| {{differential-geometry-stub}}
| |
They call me Emilia. For years I've been operating as a payroll clerk. The favorite pastime for my kids and me is to perform baseball but I haven't made a dime with it. Years ago we moved to North Dakota.
Here is my homepage ... http://www.youporntime.com