Jorge-Meeks k-noid

Jorge-Meeks k-noid is a minimal surface, given by a Weierstrass representation, genus 0, immersed.

Open Jorge-Meeks k-noid in the interactive viewer →

aperiodic complete-finite-total-curvature def-weierstrass immersed implemented minimal tradition-classical

Formula

Gauss map g

z^{-1 + n}

Height differential dh

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

Properties

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

Definition

Extracted from the shipped row's source by AST (complete multi-line lambdas, phi triple (dh = phi3, g = phi3/(phi1 - i*phi2), a Weierstrass-representation identity)) 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) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 2.48e-16); dh: matches over 144 complex samples (worst 0).

L. P. Jorge and W. H. Meeks III, 'The topology of complete minimal surfaces of finite total Gaussian curvature', Topology 22 (1983).
L. P. Jorge, W. H. Meeks III, 'The topology of complete minimal surfaces of finite total Gaussian curvature', Topology 22 (1983) 203-221; M. Weber (mirror: minsurf ch222): 'A k-Noid is a minimal surface with k catenoidal ends.'