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Bour's Minimal Surface


BoursMinimalSurface

Gray (1997) defines Bour's minimal curve over complex z by

x^'=(z^(m-1))/(m-1)-(z^(m+1))/(m+1)
(1)
y^'=i((z^(m-1))/(m-1)+(z^(m+1))/(m+1))
(2)
z^'=(2z^m)/m,
(3)

and then derives a family of minimal surfaces.

The order-three Bour surface resembles a cross-cap and is given using Enneper-Weierstrass parameterization by

f=1
(4)
g=sqrt(z)
(5)

or explicitly by the parametric equations

x=rcostheta-1/2r^2cos(2theta)
(6)
y=-rsintheta-1/2r^2sin(2theta),
(7)
z=4/3r^(3/2)cos(3/2theta)
(8)

(Maeder 1997). It is an algebraic surface of order 16.

The coefficients of the first fundamental form are given by

E=1+r^2
(9)
F=0
(10)
G=r^2(r^2+1),
(11)

and the coefficients of the second fundamental form by

e=-r^(-1/2)cos(3/2phi)
(12)
f=sqrt(r)sin(3/2phi)
(13)
g=r^(3/2)cos(3/2phi).
(14)

The area element is

 dA=r(r+1)^2dr ^ dphi.
(15)

The Gaussian and mean curvatures are given by

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

See also

Cross-Cap, Enneper-Weierstrass Parameterization, Minimal Surface

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 732-733, 1997.Maeder, R. Programming in Mathematica, 3rd ed. Reading, MA: Addison-Wesley, pp. 29-30, 1997.

Cite this as:

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

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