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Enneper-Weierstrass Parameterization

A parameterization of a minimal surface in terms of two functions f(z) and g(z) as

[x(r,phi); y(r,phi); z(r,phi)]=Rint[f(1-g^2); if(1+g^2); 2fg]dz,

where z=re^(iphi) and R[z] is the real part of z. Examples are given in the following table.

surfacef(z)g(z)
Bour's minimal surface1sqrt(z)
Enneper's minimal surface1z
Henneberg's minimal surface2(1-z^(-4))z
Scherk's second minimal surface4/(1-z^4)iz
trinoid(z^3-1)^(-2)z^2

See also

Bour's Minimal Surface, Enneper's Minimal Surface, Henneberg's Minimal Surface, Minimal Surface, Scherk's Minimal Surfaces, Trinoid

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References

Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40, 1990.do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 41, 1986.Gray, A. "Minimal Surfaces via the Weierstrass Representation." Ch. 32 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 735-760, 1997.Weierstrass, K. "Über die Flächen deren mittlere Krümmung überall gleich null ist." Monatsber. Berliner Akad., 612-625, 1866. Wolfram Research, Inc. "Weierstrass Surfaces." http://library.wolfram.com/infocenter/Demos/133/.

Referenced on Wolfram|Alpha

Enneper-Weierstrass Parameterization

Cite this as:

Weisstein, Eric W. "Enneper-Weierstrass Parameterization." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Enneper-WeierstrassParameterization.html

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