Math Art
Bour's Minimal Surface

Bour's Minimal Surface

Bour's Minimal Surface is a minimal surface, given by a parametrisation, self-intersecting.

Open Bour's Minimal Surface in the interactive viewer →

aperiodic def-parametric implemented minimal self-intersecting tradition-classical

About

One of the minimal surfaces that can be bent onto a surface of revolution without stretching. Bour classified these in 1862; the family is the standard source of examples where an intrinsic property -- what the surface measures like from inside -- and an extrinsic one -- how it sits in space -- come apart.

Formula

x(u, v)
u⁢cos⁡(v)−u2⁢cos⁡(2⁢v)2u \cos\left(v\right) - \frac{u^{2} \cos\left(2 v\right)}{2}
y(u, v)
−u⁢sin⁡(v)−u2⁢sin⁡(2⁢v)2-u \sin\left(v\right) - \frac{u^{2} \sin\left(2 v\right)}{2}
z(u, v)
43⁢u32⁢cos⁡(3⁢v2)\frac{4}{3} u^{\frac{3}{2}} \cos\left(\frac{3 v}{2}\right)

Properties

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

Sources

Chart as curated in tools/surfdb/charts.py, verified numerically against the curvature condition the record claims (measured over the chart by the validator and the charts self-test).