Math Art
Breather Surface

Breather Surface

Breather Surface is a surface of constant negative Gaussian curvature, given by a parametrisation, self-intersecting.

Open Breather Surface in the interactive viewer →

aperiodic constant-curvature def-parametric implemented k-const-negative self-intersecting tradition-classical

Formula

x(u, v)
−u+2⁢(1−0.42)⁢cosh⁡(0.4⁢u)⁢sinh⁡(0.4⁢u)0.4⁢((1−0.42)⁢cosh⁡(0.4⁢u)2+0.42⁢sin⁡(1−0.42⁢v)2)-u + \frac{2 \left(1 - 0.4^{2}\right) \cosh\left(0.4 u\right) \sinh\left(0.4 u\right)}{0.4 \left(\left(1 - 0.4^{2}\right) \cosh\left(0.4 u\right)^{2} + 0.4^{2} \sin\left(\sqrt{1 - 0.4^{2}} v\right)^{2}\right)}
y(u, v)
2⁢1−0.42⁢cosh⁡(0.4⁢u)⁢(1−0.42⁢cos⁡(1−0.42⁢v)⁢cos⁡(v)+sin⁡(1−0.42⁢v)⁢sin⁡(v))0.4⁢((1−0.42)⁢cosh⁡(0.4⁢u)2+0.42⁢sin⁡(1−0.42⁢v)2)\frac{2 \sqrt{1 - 0.4^{2}} \cosh\left(0.4 u\right) \left(\sqrt{1 - 0.4^{2}} \cos\left(\sqrt{1 - 0.4^{2}} v\right) \cos\left(v\right) + \sin\left(\sqrt{1 - 0.4^{2}} v\right) \sin\left(v\right)\right)}{0.4 \left(\left(1 - 0.4^{2}\right) \cosh\left(0.4 u\right)^{2} + 0.4^{2} \sin\left(\sqrt{1 - 0.4^{2}} v\right)^{2}\right)}
z(u, v)
2⁢1−0.42⁢cosh⁡(0.4⁢u)⁢(1−0.42⁢cos⁡(1−0.42⁢v)⁢sin⁡(v)−sin⁡(1−0.42⁢v)⁢cos⁡(v))0.4⁢((1−0.42)⁢cosh⁡(0.4⁢u)2+0.42⁢sin⁡(1−0.42⁢v)2)\frac{2 \sqrt{1 - 0.4^{2}} \cosh\left(0.4 u\right) \left(\sqrt{1 - 0.4^{2}} \cos\left(\sqrt{1 - 0.4^{2}} v\right) \sin\left(v\right) - \sin\left(\sqrt{1 - 0.4^{2}} v\right) \cos\left(v\right)\right)}{0.4 \left(\left(1 - 0.4^{2}\right) \cosh\left(0.4 u\right)^{2} + 0.4^{2} \sin\left(\sqrt{1 - 0.4^{2}} v\right)^{2}\right)}

Properties

Family
constant-curvature
Given by
a parametrisation
Curvature
constant negative Gaussian 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).