Klein Bottle - Parameterization

Parameterization

The "figure 8" immersion (Klein bagel) of the Klein bottle has a particularly simple parameterization. It is that of a "figure-8" torus with a 180 degree "Möbius" twist inserted:

$\begin{align} x & = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u\\ y & = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u\\ z & = \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v \end{align}$

In this immersion, the self-intersection circle is a geometric circle in the xy-plane. The positive constant r is the radius of this circle. The parameter u gives the angle in the xy-plane, and v specifies the position around the 8-shaped cross section. With the above parameterization the cross section is a 2:1 Lissajous curve.

The parameterization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a simplified version:

$\begin{align} x & = \frac{ \sqrt{2} f(u) \cos u \cos v (3\cos^{2}u - 1) - 2\cos 2u}{80\pi^{3}g(u)}-\frac{3\cos u -3}{4}\\ y & = -\frac{f(u)\sin v}{60\pi^{3}}\\ z & = -\frac{\sqrt{2}f(u)\sin u \,\cos v}{15\pi^{3}g(u)}+\frac{\sin u \cos^{2} u + \sin u}{4}-\frac{\sin u\,\cos u}{2} \end{align}$