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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=1/((zeta^3-1)^2)
(1)
g=zeta^2
(2)

(Dickson 1990). Explicitly, it is given by

x=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=-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=R[-2/3-2/(3(r^3e^(3itheta)-1))].
(5)

The coefficients of the first fundamental form are given by

E=((1+r^4)^2)/([1+r^6-2r^3cos(3phi)]^2)
(6)
F=0
(7)
G=(r^2(1+r^4)^2)/([1+r^6-2r^3cos(3phi)]^2),
(8)

and the coefficients of the second fundamental form by

e=(8r^4-4r(1+r^6)cos(3phi))/([1+r^6-2r^3cos(3phi)]^2)
(9)
f=(4r^2(r^6-1)sin(3phi))/([1+r^6-2r^3cos(3phi)]^2)
(10)
g=(4r^3(1+r^6)cos(3phi)-8r^6)/([1+r^6-2r^3cos(3phi)]^2).
(11)

The area element is

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=-(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." http://library.wolfram.com/infocenter/Demos/4664/.

Cite this as:

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

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