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Catalan's Surface

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CatalansSurface

A 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)=asin(2phi)-2aphi+1/2av^2cos(2phi)
(4)
y(r,phi)=-acos(2phi)-1/2av^2cos(2phi)
(5)
z(r,phi)=2avsinphi,
(6)

where

v=-r+1/r
(7)

(do Carmo 1986).

The first fundamental form has coefficients

E=2cosh^2(1/2v)(coshv-cosu)
(8)
F=0
(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=(sech^2(1/2v))/(sqrt(8(coshv-cosu)))
(14)
kappa_2=-(sech^2(1/2v))/(sqrt(8(coshv-cosu))),
(15)

the mean curvature is

H=0
(16)

and the Gaussian curvature is

K=(sech^4(1/2v))/(8(cosu-coshv)).
(17)

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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." Comptes Rendus 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-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Catalan.html.

Cite this as:

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

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