k-Noid with Enneper Ends

k-Noid with Enneper Ends is a minimal surface, given by a Weierstrass representation, immersed.

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aperiodic def-weierstrass immersed implemented minimal tradition-classical

Formula

Gauss map g

\frac{z^{-1 + k} \left(R^{k} - z^{k}\right)}{-1 + z^{k} R^{k}}

Height differential dh

\frac{1 - \frac{z^{k} + z^{-k}}{R^{k} + R^{-k}}}{z \left(-2 + z^{k} + z^{-k}\right)^{2}}

Properties

Family: minimal
Given by: a Weierstrass representation
Curvature: zero mean curvature
Periodicity: not periodic
Ends: 0
Embedding: immersed
Fidelity: exact
Exactness: numerical-integral

Definition

Extracted from the shipped row's source by AST (complete multi-line lambdas, g/dh lambdas) and reproduced numerically against the shipped callables over the complex plane, exactly -- no scalar slack.

Sources

Weierstrass data reproduced numerically against the shipped implementation (math_art/minsurf/zoo.py, row KNOID_ENN_ENDS) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 0); dh: matches over 144 complex samples (worst 7.83e-13).

B. Riemann (1867) and F. J. Lopez, A. Ros, J. Differential Geom. 33 (1991) for the (never embedded) finite Riemann family; the symmetrized member follows M. Weber, minimalsurfaces.blog, "Symmetrized Finite Riemann".
H. Karcher, "Construction of minimal surfaces" (1989) for the symmetrization method (double Enneper, k-noid families);
L. P. Jorge, W. H. Meeks III, Topology 22 (1983) for the k-noids; data after M. Weber, minimalsurfaces.blog ("Symmetrized Double Enneper", "k-Noids with Enneper Ends", "Antiprismatic k-Noids").
L. Henneberg (1875); R. Kusner, Bull. Amer. Math. Soc. 17 (1987) 291-295; F. J. Lopez, Duke Math. J. 71 (1993) 23-30 -- full citations in the weierstrass symtail engine block.