Math Art
Scherk-Enneper

Scherk-Enneper

Scherk-Enneper is a singly periodic minimal surface, given by a Weierstrass representation, immersed, with a rod group.

Open Scherk-Enneper in the interactive viewer →

def-weierstrass immersed implemented minimal minimal-periodic singly-periodic tradition-classical

Formula

Gauss map g
(−z−1+3⁢k+z−1+k⁢a2⁢k)−1+z2⁢k⁢a2⁢k\frac{\left(-z^{-1 + 3 k} + z^{-1 + k} a^{2 k}\right)}{-1 + z^{2 k} a^{2 k}}
Height differential dh
i⁢z−1+k⁢(z2⁢k−1a2⁢k)⁢(z2⁢k−a2⁢k)(−1+z2⁢k)3\frac{i z^{-1 + k} \left(z^{2 k} - \frac{1}{a^{2 k}}\right) \left(z^{2 k} - a^{2 k}\right)}{\left(-1 + z^{2 k}\right)^{3}}

Properties

Family
minimal-periodic
Given by
a Weierstrass representation
Curvature
zero mean curvature
Periodicity
singly periodic
Ends
0
Embedding
immersed
Symmetry kind
rod group
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 SP_SCHERK_ENNEPER) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 3.33e-16); dh: matches over 144 complex samples (worst 3.18e-14).