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Rembs' Surfaces

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

$\displaystyle x$ $\textstyle =$ $\displaystyle a(U\cos u-U'\sin u)$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle -a(U\sin u+U'\cos u)$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle {v-aV'},$ (3)

$\displaystyle U$ $\textstyle \equiv$ $\displaystyle {\cosh(u\sqrt{C}\,)\over\sqrt{C}}$ (4)
$\displaystyle V$ $\textstyle \equiv$ $\displaystyle {\cos(v\sqrt{C+1}\,)\over\sqrt{C+1}}$ (5)
$\displaystyle a$ $\textstyle \equiv$ $\displaystyle {2V\over (C+1)(U^2-V^2)}.$ (6)

The value of $v$ is restricted to
\vert v\vert\leq v_0 \equiv {\pi\over 2\sqrt{C+1}}
\end{displaymath} (7)

(Reckziegel 1986), and the values $v=\pm v_0$ correspond to the ends of the cleft in the surface.

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


Fischer, G. (Ed.). Plate 88 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 84, 1986.

Reckziegel, H. ``Sievert's Surface.'' § in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 39-40, 1986.

Rembs, E. ``Enneper'sche Flächen konstanter positiver Krümmung und Hazzidakissche Transformationen.'' Jahrber. DMV 39, 278-283, 1930.

© 1996-9 Eric W. Weisstein