Bour's Minimal Surface
BoursMinimalSurface
Gray (1997) defines Bour's minimal curve over complex z by
x^' = [画像:(z^(m-1))/(m-1)-(z^(m+1))/(m+1)]
(1)
y^' = [画像:i((z^(m-1))/(m-1)+(z^(m+1))/(m+1))]
(2)
z^' = [画像:(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 =
(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 = [画像:-1/(r(r+1)^4)]
(16)
H = 0.
(17)
See also
Cross-Cap, Enneper-Weierstrass Parameterization, Minimal SurfaceExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Bour's Minimal SurfaceCite this as:
Weisstein, Eric W. "Bour's Minimal Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BoursMinimalSurface.html