Sievert's Surface

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

$\displaystyle x$	$\textstyle =$	$\displaystyle r\cos\phi$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle r\sin\phi$	(2)
$\displaystyle z$	$\textstyle =$	$\displaystyle {\ln[\tan({\textstyle{1\over 2}}v)]+a(C+1)\cos v\over \sqrt{C}},$	(3)

where

$\displaystyle \phi$	$\textstyle \equiv$	$\displaystyle -{u\over\sqrt{C+1}}+\tan^{-1}(\tan u\sqrt{C+1}\,)$	(4)
$\displaystyle a$	$\textstyle \equiv$	$\displaystyle {2\over C+1-C\sin^2 v\cos^2 u}$	(5)
$\displaystyle r$	$\textstyle \equiv$	$\displaystyle {a\sqrt{(C+1)(1+C\sin^2 u)}\,\sin v\over\sqrt{C}},$	(6)

with $\vert u\vert<\pi/2$ and $0<v<\pi$ (Reckziegel 1986).

References

Fischer, G. (Ed.). Plate 87 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 83, 1986.

Reckziegel, H. ``Sievert's Surface.'' §3.4.4.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 38-39, 1986.

Sievert, H. Über die Zentralflächen der Enneperschen Flachen konstanten Krümmungsmaßes. Dissertation, Tübingen, 1886.