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Help!

Is there any function $\varphi : \mathbb{R^2} \rightarrow \mathbb{A^2}$ That can associate to every line in $\mathbb{R^2}$ a line in $\mathbb{P^2}$?.

$\mathbb{A^2} = \lbrace \left[x:y:1\right] : x,y \in \mathbb{R} \rbrace$

I am trying to find a way to show that two parallel lines in $\mathbb{R^2}$ that can be associated to two projective lines that intersect at a point of infinity.

Thanks.

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    Yes the map $(x,y) \mapsto [x:y:1]$ does this (as discussed in comments http://math.stackexchange.com/questions/2121326/projective-line) – 2017-01-31
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    Since every two lines in $\mathbb{P}^{2}$ have a nontrivial intersection, there is nothing really to show; when two parallel lines of $\mathbb{R}^{2}$ are embedded in $\mathbb{P}^{2}$, their intersection will *have* to be on the line at infinity. – 2017-01-31
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    That means that if I have any straight line through that function that straight can be "behaved" as a projective line ?. Thanks for the patience – 2017-01-31
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    Any line in the projective plane has a point of infinity when I say that a line in R2 behaves as a projective line the point of infinity is not contained in this line is true right? – 2017-01-31

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The parallel lines $y = mx + b$ and $y = mx + c$ correspond to $y = mx + bz$ and $y = mx + cz$ in projective space with coordinates $[x:y:z]$. In the chart $z=1$, you get the original equations. These equations have the common solution $[1:m:0]$, which is a "point at infinity" (since $z=0$).