This is an example in lecture for a chart:
\begin{split} M &= \{\text{affine line in } \mathbb R^2\}\\ U &= \{\ell \in M \ |\ \ell \text{ not vertical}\}\\ V &= \{\ell \in M \ |\ \ell \text{ not horizontal} \} \end{split}
\begin{split} \phi &: U \rightarrow \mathbb R^2, \ell \mapsto (m,b) \text{ with }\ell \text{ given by } \ell: y= mx+b \\ \psi &: V \rightarrow \mathbb R^2, \ell \mapsto (n,c) \text{ with }\ell \text{ given by } \ell: x= ny+c \end{split}
So: $(\psi \circ \phi^{-1})(m,b) = ({1\over m} , {-b \over m})$
Could someone explain why here we obtain a 2-dimensional surface defined as an open mobius strip.
based on :The set of lines in $\mathbb{R}^2$ is a Möbius band? i understand that a set of affine lines in $R^2$ makes the projective space, which can be formed by gluing a mobius strip and a circle, or two mobius strips, however maybe could i get further explanation following this specific example?