File:Klein Bottle Parametrized Symmetric Half.png

English: A symmetric half of a Klein bottle. When joined along the border with its other half which is the mirror image of this one, with the mirror along the plane of the border of this half, then a Klein bottle is produced. The symmetric half is (homeomorphic to) a Möbius strip.

The parametrization for the figure is ${\displaystyle {\begin{aligned}{\begin{bmatrix}x\\y\\z\end{bmatrix}}=&{\begin{bmatrix}{3 \over 4}\cos t-{1 \over 4}\sin t-{1 \over 2{\sqrt {2}}}\sin ^{2}t\\{3 \over 4}\sin t-{1 \over 4}\cos t-{1 \over 2{\sqrt {2}}}\sin ^{2}t\\0\end{bmatrix}}\\&+{\rho +\sin(t-\psi )-{1 \over 2}\sin 2(t-\psi )+{1 \over 3}\sin 3(t-\psi )-{1 \over 4}\sin 4(t-\psi )+{1 \over 10}\sin 5(t-\psi ) \over 10}\\&\qquad \qquad \cdot {\Biggl (}(\sin \phi ){\begin{bmatrix}0\\0\\1\end{bmatrix}}+{(-1)^{{\bigl \lfloor }{t-2.356194 \over 2\pi }{\bigr \rfloor }}\cos \phi \over {\sqrt {{5 \over 8}+{1 \over 4}\sin ^{2}2t+{3 \over 8}\sin 2t+{1 \over 2{\sqrt {2}}}(\sin 2t)(\sin t-\cos t)}}}{\begin{bmatrix}-{3 \over 4}\cos t-{1 \over 4}\sin t+{1 \over 2{\sqrt {2}}}\sin 2t\\-{3 \over 4}\sin t-{1 \over 4}\cos t-{1 \over 2{\sqrt {2}}}\sin 2t\\0\end{bmatrix}}{\Biggr )}\end{aligned}}}$

where ${\displaystyle 0\leq t<2\pi }$, ${\displaystyle \pi \leq \phi <2\pi }$, ${\displaystyle \psi =5.497786=\pi +2.356194}$ is a constant, and ${\displaystyle \rho =2.4}$ is slightly variable (e.g., it could be set to 2.3 instead). The constant 2.356194... is the solution to the equations ${\displaystyle -{\tfrac {3}{4}}\cos t-{\tfrac {1}{4}}\sin t+{\tfrac {1}{2{\sqrt {2}}}}\sin 2t=0=-{\tfrac {3}{4}}\sin t-{\tfrac {1}{4}}\cos t-{\tfrac {1}{2{\sqrt {2}}}}\sin 2t}$.

Graphed using Grapher.