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The Enneper surfaces are a three-parameter family of surfaces with constant negative curvature (and nonconstant Mean Curvature). In general, they are described by Elliptic Functions. However, special cases which can be specified parametrically using Elementary Functions include the Kuen Surface, Rembs' Surfaces, and Sievert's Surface.
References
Enneper, A. ``Analytisch-geometrische Untersuchungen.'' Nachr. Königl. Gesell. Wissensch. Georg-Augustus-Univ. Göttingen
12, 258-277, 1868.
Fischer, G. (Ed.). Plate 92 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.
Braunschweig, Germany: Vieweg, p. 88, 1986.
Reckziegel, H. ``Enneper's Surfaces.'' §3.4.4 in
Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, pp. 37-39, 1986.