Coordinates of a Möbius Strip in Three-Dimensional Space
Definition1
A surface represented by the coordinate chart mapping $\mathbf{x}$ is called a Möbius band.
$$ \mathbf{x}(\theta, v) = (\cos \theta, \sin \theta, 0) + v(\textstyle \sin\frac{\theta}{2}\cos\theta, \sin\frac{\theta}{2}\sin\theta, \cos\frac{\theta}{2}) $$ $$ \textstyle -\pi \lt \theta \lt \pi, \quad -\frac{1}{2} \lt v \lt \frac{1}{2} $$
Properties
- The Möbius band is not an orientable surface
Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p87 ↩︎