Enneper's Minimal Surface

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EnnepersMinimalSurface

A self-intersecting minimal surface which can be generated using the Enneper-Weierstrass parameterization with

f(z) = 1

(1)

g(z) = zeta.

(2)

Letting z=re^(iphi) and taking the real part give

x = R[re^(iphi)-1/3r^3e^(3iphi)]

(3)

= rcosphi-1/3r^3cos(3phi)

(4)

y = R[ire^(iphi)+1/3ir^3e^(3iphi)]

(5)

= -1/3r[3sinphi+r^2sin(3phi)]

(6)

z = R[r^2e^(2iphi)]

(7)

= r^2cos(2phi),

(8)

where r in [0,1] and phi in [-pi,pi). Eliminating r and phi then gives the implicit form

[画像: ((y^2-x^2)/(2z)+2/9z^2+2/3)^3 -6[((y^2-x^2))/(4z)-1/4(x^2+y^2+8/9z^2)+2/9]^2=0, ]

(9)

so Enneper's minimal surface is algebraic of order 9.

The coefficients of the first fundamental form are

E = -2cos(2phi)

(10)

F = 4rcosphisinphi

(11)

G = 2r^2cos(2phi),

(12)

the second fundamental form coefficients are

e = (1+r^2)^2

(13)

f =

(14)

g = r^2(1+r^2)^2,

(15)

and the Gaussian and mean curvatures are

K = [画像:-4/((1+r^2)^4)]

(16)

H = 0.

(17)

Letting z=u+iv gives the figure above, with parametrization

x = u-1/3u^3+uv^2

(18)

y = -v-u^2v+1/3v^3

(19)

z = u^2-v^2

(20)

(do Carmo 1986, Gray 1997). In this parameterization, the coefficients of the first fundamental form are

E = (1+u^2+v^2)^2

(21)

F =

(22)

G = (1+u^2+v^2)^2,

(23)

the second fundamental form coefficients are

e = -2

(24)

f =

(25)

g = 2,

(26)

the area element is

dA=(1+u^2+v^2)du ^ dv,

(27)

and the Gaussian and mean curvatures are

K = [画像:-4/((1+u^2+v^2)^4)]

(28)

H = 0.

(29)

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References

Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40, 1990.do Carmo, M. P. "Enneper's Surface." §3.5C in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.Enneper, A. "Analytisch-geometrische Untersuchungen." Z. Math. Phys. 9, 96-125, 1864.GRAPE. "Enneper's Surfaces." https://archive.ins.uni-bonn.de/numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/enneper.html.Gray, A. "Examples of Minimal Surfaces," "The Associated Family of Enneper's Surface," and "Enneper's Surface of Degree n." §30.2 and 31.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 358, 684-685, and 726-732, 1997.JavaView. "Classic Surfaces from Differential Geometry: Enneper." http://www.javaview.de/demo/surface/common/PaSurface_Enneper.html.Maeder, R. The Mathematica Programmer. San Diego, CA: Academic Press, pp. 150-151, 1994.Nordstrand, T. "Enneper's Minimal Surface." http://jalape.no/math/enntxt.htm.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, p. 65, 87, and 143, 1986. Wolfram Research, Inc. "Mathematica Version 2.0 Graphics Gallery." https://library.wolfram.com/infocenter/Demos/4664/.

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Enneper's Minimal Surface

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Weisstein, Eric W. "Enneper's Minimal Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EnnepersMinimalSurface.html

Enneper's Minimal Surface

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