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Function Repository Resource:

 EnneperWeierstrass

Source Notebook

Compute the Enneper-Weierstrass parametrization

Contributed by: Wolfram Staff (original content by Eric W. Weisstein)

ResourceFunction["EnneperWeierstrass"][f,g,{z,r ,ϕ}]

compute the Enneper-Weierstrass parametrization for curves f and g, where z=rei ϕ.

Details and Options

ResourceFunction["EnneperWeierstrass"] gives a parametrization of a minimal surface defined in terms of two complex analytic functions f and g.

Examples

Basic Examples (2) 

A Bour's minimal surface:

In[1]:=
bours = ResourceFunction["EnneperWeierstrass"][1, Sqrt[ z], {z, r, \[Phi]}]
Out[1]=
In[2]:=
ParametricPlot3D[Evaluate[bours], {r, 0, 1}, {\[Phi], 0, 4 \[Pi]}, PlotPoints -> 80]
Out[2]=

A trinoid:

In[3]:=
trinoid = FullSimplify[ ResourceFunction["EnneperWeierstrass"][1/(z^3 - 1)^2, z^2, {z, r, \[Phi]}], r > 0 && \[Phi] > 0]
Out[3]=
In[4]:=
ParametricPlot3D[ Evaluate[trinoid], {r, 0., .999}, {\[Phi], 0., 4 \[Pi]}, PlotPoints -> 80, MaxRecursion -> 3, BoxRatios -> {1, 1, .35}]
Out[4]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 26 February 2020

Source Metadata

  • Citation:
    • Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

License Information