Catalan Minimal Surface
CatalanMinimalSurface
Catalan's minimal surface is minimal surface given by the parametric equations
x(u,v) = u-sinucoshv
(1)
y(u,v) = 1-cosucoshv
(2)
z(u,v) = 4sin(1/2u)sinh(1/2v)
(3)
(Gray 1997), or
x(r,phi) = a[sin(2phi)-aphi+1/2v^2cos(2phi)]
(4)
y(r,phi) = -a[cos(2phi)+1/2v^2cos(2phi)]
(5)
z(r,phi) = 2avsinphi,
(6)
where
| [画像: v=-r+1/r ] |
(7)
|
(do Carmo 1986).
The first fundamental form has coefficients
E = 2cosh^2(1/2v)(coshv-cosu)
(8)
F =
(9)
G = 2cosh^2(1/2v)(coshv-cosu),
(10)
and the second fundamental form has coefficients
e = -cosh(1/2v)sin(1/2u)
(11)
f = cos(1/2u)sinh(1/2v)
(12)
g = cosh(1/2v)sin(1/2u).
(13)
The principal curvatures are
kappa_1 = [画像:(sech^2(1/2v))/(sqrt(8(coshv-cosu)))]
(14)
kappa_2 = [画像:-(sech^2(1/2v))/(sqrt(8(coshv-cosu))),]
(15)
the mean curvature is
| H=0 |
(16)
|
and the Gaussian curvature is
See also
Catalan Ruled SurfaceExplore with Wolfram|Alpha
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References
Catalan, E. "Mémoire sur les surfaces dont les rayons de courbures en chaque point, sont égaux et les signes contraires." C. R. Acad. Sci. Paris 41, 1019-1023, 1855.do Carmo, M. P. "Catalan's Surface" §3.5D in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 45-46, 1986.Fischer, G. (Ed.). Plates 94-95 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 90-91, 1986.Gray, A. "Catalan's Minimal Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 692-693, 1997.JavaView. "Classic Surfaces from Differential Geometry: Catalan Surface." http://www.javaview.de/demo/surface/common/PaSurface_Catalan.html.Referenced on Wolfram|Alpha
Catalan Minimal SurfaceCite this as:
Weisstein, Eric W. "Catalan Minimal Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CatalanMinimalSurface.html