Trinoid

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trinoid

A minimal surface discovered by L. P. M. Jorge and W. Meeks III in 1983 with Enneper-Weierstrass parameterization

f = [画像:1/((zeta^3-1)^2)]

(1)

g = zeta^2

(2)

(Dickson 1990). Explicitly, it is given by

x = [画像:R[(re^(itheta))/(3(1+re^(itheta)+r^2e^(2itheta)))-(4ln(re^(itheta)-1))/9+(2ln(1+re^(itheta)+r^2e^(2itheta)))/9]]

(3)

y = [画像:-1/9I[(-3re^(itheta)(1+re^(itheta)))/(r^3e^(3itheta)-1)+(4sqrt(3)(r^3e^(3itheta)-1)tan^(-1)((1+2re^(itheta))/(sqrt(3))))/(r^3e^(3itheta)-1)]]

(4)

z = [画像:R[-2/3-2/(3(r^3e^(3itheta)-1))].]

(5)

The coefficients of the first fundamental form are given by

E = [画像:((1+r^4)^2)/([1+r^6-2r^3cos(3phi)]^2)]

(6)

F =

(7)

G = [画像:(r^2(1+r^4)^2)/([1+r^6-2r^3cos(3phi)]^2),]

(8)

and the coefficients of the second fundamental form by

e = [画像:(8r^4-4r(1+r^6)cos(3phi))/([1+r^6-2r^3cos(3phi)]^2)]

(9)

f = [画像:(4r^2(r^6-1)sin(3phi))/([1+r^6-2r^3cos(3phi)]^2)]

(10)

g = [画像:(4r^3(1+r^6)cos(3phi)-8r^6)/([1+r^6-2r^3cos(3phi)]^2).]

(11)

The area element is

[画像: dA=(r(1+r^4)^2)/([1+r^6-2r^3cos(3phi)]^2)dr ^ dphi. ]

(12)

The Gaussian and mean curvatures are given by

K = [画像:-(16r^2[1+r^6-2r^3cos(3phi)]^2)/((1+r^4)^4)]

(13)

H = 0.

(14)

See also

Enneper-Weierstrass Parameterization, Minimal Surface

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References

Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40, 1990.Ogawa, A. "The Trinoid Revisited." Mathematica J. 2, 59-60, 1992. Wolfram Research, Inc. "Mathematica Version 2.0 Graphics Gallery." https://library.wolfram.com/infocenter/Demos/4664/.

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Cite this as:

Weisstein, Eric W. "Trinoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Trinoid.html

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