Geometric notion of addition for the real projective line

Draw a vertical line $\ell$ that meets both lines $a$ and $b$. (This is always possible: if one or both of the lines is vertical, use it.) Pick a point $p$ common to $a$ and $\ell$, and similarly a point $q$ common to $b$ and $\ell$. ($p$ is unique unless $a$ is vertical, in which case it is any point on $a$; the same goes for $q$ and $b$.) Construct a third point $r$ on $\ell$ whose $y$-coordinate is the sum of those of $p$ and $q$. The line passing through the origin and $r$ is $a+b$.