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Enneper 最小曲面

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EnnepersMinimalSurface

一個自相交的最小曲面，可以使用 Enneper-Weierstrass 引數化生成，其中

f(z)=1
(1)
g(z)=zeta.
(2)

令 z=re^(iphi) 並取實部得到

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)

其中 r in [0,1] 且 phi in [-pi,pi)。消除 r 和 phi 則得到隱式形式

((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)

因此 Enneper 最小曲面是 9 階代數曲面。

第一基本形式的係數為

E=-2cos(2phi)
(10)
F=4rcosphisinphi
(11)
G=2r^2cos(2phi),
(12)

第二基本形式係數為

e=(1+r^2)^2
(13)
f=0
(14)
g=r^2(1+r^2)^2,
(15)

高斯和平均曲率為

K=-4/((1+r^2)^4)
(16)
H=0.
(17)

令 z=u+iv 得到上圖，引數化為

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）。在此引數化中，第一基本形式的係數為

E=(1+u^2+v^2)^2
(21)
F=0
(22)
G=(1+u^2+v^2)^2,
(23)

第二基本形式係數為

e=-2
(24)
f=0
(25)
g=2,
(26)

面積元素為

dA=(1+u^2+v^2)du ^ dv,
(27)

高斯和平均曲率為

K=-4/((1+u^2+v^2)^4)
(28)
H=0.
(29)

另請參閱

Chen-Gackstatter 曲面，Enneper-Weierstrass 引數化

使用 探索

參考文獻

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." http://www-sfb256.iam.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-sfb288.math.tu-berlin.de/vgp/javaview/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.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, p. 65, 87, and 143, 1986. , Inc. "Mathematica Version 2.0 Graphics Gallery." http://library.wolfram.com/infocenter/Demos/4664/.

請引用為

Weisstein, Eric W. “Enneper 最小曲面。” 來自 —— 資源。 https://mathworld.tw/EnnepersMinimalSurface.html

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