Kuen Surface

A special case of Enneper's Negative Curvature Surfaces which can be given parametrically by

$$x = {2(\cos u+u\sin u)\sin v\over 1+u^2\sin^2 v}$$ (1)

$$= {2\sqrt{1+u^2}\,\cos(u-\tan^{-1} u)\sin v\over 1+u^2\sin^2 v}$$ (2)

$$y = {2(\sin u-u\cos u)\sin v\over 1+u^2\sin^2 v}$$ (3)

$$= {2\sqrt{1+u^2}\,\sin(u-\tan^{-1} u)\sin v\over 1+u^2\sin^2 v}$$ (4)

$$z = \ln[\tan({\textstyle{1\over 2}}v)]+{2\cos v\over 1+u^2\sin^2 v}$$ (5)

for $v\in [0,\pi)$, $u\in [0,2\pi)$ (Reckziegel et al. 1986). The Kuen surface has constant Negative Gaussian Curvature of $K=-1$. The Principal Curvatures are given by

$$\kappa_1 = -{u\cos({\textstyle{1\over 2}}v)[-2-u^2+u^2\cos(2v)]^4\sin({\textstyle{1\over 2}}v)\over 2[2-u^2+u^2\cos(2v)](1+u^2\sin^2 v)^4}$$ (6)

$$\kappa_2 = {[-2-u^2+u^2\cos(2v)]^4[2-u^2+u^2\cos(2v)]\csc(v)\over 64u(1+u^2\sin^2 v)^4}.$$ (7)

See also Enneper's Negative Curvature Surfaces, Rembs' Surfaces, Sievert's Surface

References

Fischer, G. (Ed.). Plate 86 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 82, 1986.

Gray, A. ``Kuen's Surface.'' §19.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 384-386, 1993.

Kuen, T. ``Ueber Flächen von constantem Krümmungsmaass.'' Sitzungsber. d. königl. Bayer. Akad. Wiss. Math.-phys. Classe, Heft II, 193-206, 1884.

Nordstrand, T. ``Kuen's Surface.'' http://www.uib.no/people/nfytn/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.