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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?

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    It looks me strange that $(\psi \circ \phi^{-1})(m,b) = ({1\over m} , {-b \over m})$ doesn't involve parameters $n,c$ as well. – 2017-01-15
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    There is a typo, the domain of $\psi$ is $V$. What is $U\cap V$? Can you see how the change of variables acts on different components? – 2017-01-15
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    $U \cap V$ is the set of affine lines which are neither vertical or horizontal. and the map takes the slope and constant to the $R^2$ plane. but i dont quite see the change of variable, do you mind elaborating? – 2017-01-15
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    I'm trying to guide you to the answer without telling it to you. Think about gluing strips together to get a Mobius band and try to recognize it in the example. – 2017-01-16

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