📂Geometry

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