Kusner Projective Plane (p planar ends)

Kusner Projective Plane (p planar ends) is a minimal surface, given by a Weierstrass representation, immersed.

Open Kusner Projective Plane (p planar ends) in the interactive viewer →

aperiodic def-weierstrass immersed implemented minimal non-orientable tradition-classical

Formula

Gauss map g

\frac{z^{-1 + p} \left(z^{p} - s\right)}{1 + s z^{p}}

Height differential dh

\frac{i z^{-1 + p} \left(1 + s z^{p}\right) \left(z^{p} - s\right)}{\left(-1 + z^{2 p} + \frac{2 s z^{p}}{-1 + p}\right)^{2}}

Properties

Family: minimal
Given by: a Weierstrass representation
Curvature: zero mean curvature
Periodicity: not periodic
Ends: 0
Embedding: immersed
Orientable: no
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 KUSNER_RP2) 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 3.71e-14).

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.
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").