Math Art
Klein Bottle (Figure-8)

Klein Bottle (Figure-8)

Klein Bottle (Figure-8) is a topological surface, given by a parametrisation, immersed.

Open Klein Bottle (Figure-8) in the interactive viewer →

aperiodic def-parametric immersed implemented topological

Formula

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

Properties

Family
topological
Given by
a parametrisation
Curvature
no condition imposed
Periodicity
not periodic
Ends
0
Embedding
immersed
Fidelity
exact
Exactness
elementary

Definition

ring radius 2 fixed to the builder default. The u seam closes with v reversed (the figure-8 makes a half-turn per revolution)

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.33 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.09814).