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\begin{document}

$\R \bP^1$ is the space of lines through the origin in $\R^2$.
To coordinatize it, it is useful to view it as the quotient space of   $\R^2 \setminus \{(0,0)\}$
by the equivalence relation   $(x,y) \sim (u,v) \iff span(x,y) = span(u,v)$.
We write the quotient map $$\R^2 \setminus \{(0,0)\} \to \R \bP^1; \quad (X, Y) \mapsto [X, Y].$$ 
Observe that $[X, Y] = [\lambda X, \lambda Y]$ for any $\lambda  \in \R$, $\lambda \ne 0$.
We call $[X, Y]$ the  HOMOGENEOUS COORDINATES of the line $\ell$.

AFFINE CHARTS.  


 \begin{figure}[h]
\scalebox{0.4}{\includegraphics{ProjLine.pdf}} 
%\label{fig_shapesphere1}
\end{figure}

$\R \bP^1$ is the union of two large   open sets, called ``affine spaces'' , 
$U_x$ and $U_y$:
$$\R \bP ^1 = U_x \cup U_y.$$  
We give three equivalent descriptions of 
$U_x$.  \begin{itemize}  \item{}$U_x$ consists of all lines through the origin  which  intersect the line $Y=1$.
\item{}  $U_x$ consists of all lines through the origin which can be expressed by an
equation of the form 
$$X = M Y ,  M \in \R.$$
\item{}  $U_x$ consists of all lines EXCEPT the x-axis  $Y = 0$.
  \end{itemize}
 The intersection point of a  line  $\ell \in U_x$ with the line $Y=1$
 is $(M,1) = (x,1)$.  We call $x$ the ``affine coordinate'' of $\ell$ in the chart $U_x$. 
 The corresponding coordinate chart is then  
 $$\phi_x: U_x \to \R;   [X, Y] \mapsto x = X/Y = M.$$
 with inverse map
 $$\phi_x ^{-1}  : \R \to U_x;  \phi_x ^{-1} (x) = [x, 1].$$
 As $x \to \infty$ the line $[x,1]$ tends to the line $\ell_{\infty} = [1, 0]$, which is the X-axis.
 Thus, we can say $\R \bP ^1 = \R \cup \{ \infty \}$. 
 
 
{\bf Chart covering infinity} To cover the rest of $\R \bP^1$, which is to say to cover the   missing $\ell_{\infty}$ , we switch the roles of X and Y.
Thus let 
$$U_y = \{ [X, Y];  X \ne 0 \} =  \R \bP^1 \setminus \{ \ell_0 \} =\text{  lines intersecting the line} X =1 .$$
Here $\ell_0$ is the Y-axis, which is the locus $X = 0$.  
Any line $\ell \in U_y$ can be expressed in standard slope form as
$$Y = mX$$
and, since $Y \ne 0$ somewhere on $\ell$ we have    
$$\ell = [X, Y] = [1, Y/X] = [1, y]$$
allowing us to define the affine coordinate $y$
according to 
$$\phi_y: U_y \to \R;   [X, Y] \mapsto y = Y/X.$$
 with inverse map
 $$\phi_y ^{-1} : \R \to U_x;  \phi_y ^{-1} (y) = [1,y].$$
 Note : the slope $m$ is this affine coordinate:
 $$m = Y/X = y$$
 
 {\bf Coordinate overlap} or Transition function.
 $U_x \cap U_y =$   lines through $(0,0)$ except the Y-axis and the X-axis.  
Alternatively:
$$U_x \cap U_y = \R \bP^1 \setminus \{ \ell_{\infty}, \ell_0 \}$$
% where $\ell_0$ is the y-axis (which is given by $x =0$ in the $x$-affine chart. 
  The coordinate overlap map $\phi_y \circ \phi_x ^{-1}$ has domain $\R \setminus \{ 0 \}$
 and is given by 
 $$x \mapsto [x, 1] = [1, 1/x] = [1, y] \mapsto y = 1/x.$$
 SEE FIGURE.  Alternatively,    any line  $\ell \in U_x \cup U_y$ is expressed in homogeneious coordinates as  an  $ [X, Y]$ with  $X \ne 0$ and $Y \ne 0$
 so we can write $\ell$  as $[X/Y, 1]$ and also as $[1, Y/X]$ Setting $x = X/Y,  y = Y/X$
 we see that the overlap is $x \mapsto 1/ x$.
 
 \vskip .3cm 
 OTHER FIELDS.  All these considerations work with any field $\K$ replacing $\R$.  For our purposes, 
 $\C$ is a good field.  We have $\C \bP^1 = $ complex lines through the origin  in $\C^2$.
 Also $\C \bP^1  =  \C \cup \{ \infty \} = S^2 = $ Riemann sphere.
 
   

\end{document}


Labelling  the point of intersections yields coordinates on $U_x$ and $U_y$.
So we send a line in $U_x$ to the unique point $x \in \R$ such that 
$(x, 1) = \ell \cap \{y =1\}$.   