Kuen Surface
KuenSurface
The Kuen surface is a special case of Enneper's negative curvature surfaces which can be given parametrically by
x = [画像:(2(cosu+usinu)sinv)/(1+u^2sin^2v)]
(1)
y = [画像:(2(sinu-ucosu)sinv)/(1+u^2sin^2v)]
(3)
z = [画像:ln[tan(1/2v)]+(2cosv)/(1+u^2sin^2v)]
![画像:ln[tan(1/2v)]+(2cosv)/(1+u^2sin^2v)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/Inline15.svg){kind=link}
(5)
for v in [0,pi), u in [0,2pi) (Reckziegel et al. 1986; Gray et al. 2006, p. 484).
The Kuen surface appears on the cover of volume 2, number 1 of La Gaceta de la Real Sociedad Matemática Española (1999).
The coefficients of the first fundamental form are
![画像:(16u^2sin^2v)/([2+u^2-u^2cos^2(2v)]^2)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/Inline20.svg){kind=link}
F =
(7)
![画像:csc^2v-(16u^2sin^2v)/([2+u^2-u^2cos^2(2v)]^2),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/Inline26.svg){kind=link}
the second fundamental form coefficients are
![画像:-(4u[2-u^2+u^2cos^2(2v)]sinv)/([2+u^2-u^2cos^2(2v)]^2)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/Inline29.svg){kind=link}
f =
(10)
![画像:(4u[2-u^2+u^2cos^2(2v)]cscv)/([2+u^2-u^2cos^2(2v)]^2),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/Inline35.svg){kind=link}
and the surface area element is
![画像: dS=(4u[2-u^2+u^2cos(2v)])/([2+u^2-u^2cos^2(2v)]^2).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/NumberedEquation1.svg){kind=link}
The Gaussian and mean curvatures are
K = -1
(13)
![画像:-(cscv)/(4u)+1/4usinv[1+8/(2-u^2+u^2cos(2v))],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/Inline41.svg){kind=link}
so the Kuen surface has constant negative Gaussian curvature, and the principal curvatures are
kappa_1 = [画像:(4usinv)/(2-u^2+u^2cos(2v))]
(15)
kappa_2 = [画像:-([2-u^2+u^2cos(2v)]cscv)/(4u)]
![画像:-([2-u^2+u^2cos(2v)]cscv)/(4u)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/KuenSurface/Inline47.svg){kind=link}
(16)
(Gray 1997, p. 496).
See also
Enneper's Negative Curvature SurfacesExplore with Wolfram|Alpha
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References
Cover of La Gaceta de la Real Sociedad Matemática Española 2, 1999.Fischer, G. (Ed.). Plate 86 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 82, 1986.Gray, A.; Abbena, E.; and Salamon, S. "Kuen's Surface as Bianchi Transform of the Pseudosphere." §21.7 In Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. Boca Raton, FL: CRC Press, pp. 484 and 704-706, 2006.JavaView. "Classic Surfaces from Differential Geometry: Kuen." http://www.javaview.de/demo/surface/common/PaSurface_Kuen.html.Kuen, T. "Ueber Flächen von constantem Krümmungsmass." Sitzungsber. d. königl. Bayer. Akad. Wiss. Math.-phys. Classe, Heft II, 193-206, 1884.Nordstrand, T. "Kuen's Surface." http://jalape.no/math/kuentxt.htm.Reckziegel, H. "Kuen's Surface." §3.4.4.2 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 38, 1986.Referenced on Wolfram|Alpha
Kuen SurfaceCite this as:
Weisstein, Eric W. "Kuen Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KuenSurface.html