Rembs' Surfaces

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

$$x = a(U\cos u-U'\sin u)$$ (1)

$$y = -a(U\sin u+U'\cos u)$$ (2)

$$z = {v-aV'},$$ (3)

where

$$U \equiv {\cosh(u\sqrt{C}\,)\over\sqrt{C}}$$ (4)

$$V \equiv {\cos(v\sqrt{C+1}\,)\over\sqrt{C+1}}$$ (5)

$$a \equiv {2V\over (C+1)(U^2-V^2)}.$$ (6)

The value of $v$ is restricted to

$$\vert v\vert\leq v_0 \equiv {\pi\over 2\sqrt{C+1}}$$ (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

References

Fischer, G. (Ed.). Plate 88 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 84, 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. 39-40, 1986.

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