Math Art
Richmond (generalized, g = z^k)

Richmond (generalized, g = z^k)

Richmond (generalized, g = z^k) is a minimal surface, given by a parametrisation, immersed.

Open Richmond (generalized, g = z^k) in the interactive viewer →

aperiodic def-parametric immersed implemented minimal tradition-classical

Formula

x(u, v)
−cos⁡(2⁢v)4⁢u2−u4⁢cos⁡(4⁢v)8\frac{-\cos\left(2 v\right)}{4 u^{2}} - \frac{u^{4} \cos\left(4 v\right)}{8}
y(u, v)
sin⁡(2⁢v)4⁢u2+u4⁢sin⁡(4⁢v)8\frac{\sin\left(2 v\right)}{4 u^{2}} + \frac{u^{4} \sin\left(4 v\right)}{8}
z(u, v)
u⁢cos⁡(v)u \cos\left(v\right)

Properties

Family
minimal
Given by
a parametrisation
Curvature
zero mean curvature
Periodicity
not periodic
Ends
0
Embedding
immersed
Fidelity
exact
Exactness
elementary

Sources

Weierstrass data reproduced numerically against the shipped implementation (math_art/minsurf/zoo.py, row RICHMOND_K) 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 0).