主题
Search

Enneper 最小曲面

DOWNLOAD Mathematica Notebook下载 Wolfram Notebook
EnnepersMinimalSurface

一个自相交的最小曲面，可以使用 Enneper-Weierstrass 参数化生成，其中

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

令 z=re^(iphi) 并取实部得到

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 则得到隐式形式

((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 阶代数曲面。

第一基本形式的系数为

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

第二基本形式系数为

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

高斯和平均曲率为

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

令 z=u+iv 得到上图，参数化为

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）。在此参数化中，第一基本形式的系数为

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

第二基本形式系数为

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

面积元素为

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

高斯和平均曲率为

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

另请参阅

Chen-Gackstatter 曲面，Enneper-Weierstrass 参数化

使用 Wolfram|Alpha 探索

参考文献

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. Wolfram Research, Inc. "Mathematica Version 2.0 Graphics Gallery." http://library.wolfram.com/infocenter/Demos/4664/.

请引用为

Weisstein, Eric W. “Enneper 最小曲面。” 来自 MathWorld——Wolfram Web 资源。 https://mathworld.net.cn/EnnepersMinimalSurface.html

主题分类