Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation
Dually, the tangent plane at the point with given parameters is where
Its coefficients satisfy the implicit degree-6 polynomial equation
It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization for integer k>1.[3] It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in for n up to 7.[7]
See also [8][9] for higher order algebraic Enneper surfaces.
References
^J.C.C. Nitsche, "Vorlesungen über Minimalflächen", Springer (1975)
^Francisco J. López, Francisco Martín, Complete minimal surfaces in R3
^ a bUlrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. ISBN 978-3-642-11697-1.
^R. Osserman, A survey of Minimal Surfaces. Vol. 1, Cambridge Univ. Press, New York (1989).
^Cosín, C., Monterde, Bézier surfaces of minimal area. In Computational Science — ICCS 2002, eds. J., Sloot, Peter, Hoekstra, Alfons, Tan, C., Dongarra, Jack. Lecture Notes in Computer Science 2330, Springer Berlin / Heidelberg, 2002. pp. 72-81 ISBN 978-3-540-43593-8
^Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569
^E. Güler, Family of Enneper minimal surfaces. Mathematics. 2018; 6(12):281. https://doi.org/10.3390/math6120281
^E. Güler, The algebraic surfaces of the Enneper family of maximal surfaces in three dimensional Minkowski space. Axioms. 2022; 11(1):4. https://doi.org/10.3390/axioms11010004