The real projective line $\mathbb{RP}^1 = \mathbb{R} \cup {\infty}$ is usually identified with (or defined as) the set of lines passing through the origin in $\mathbb{R}^2$. Thus, the number $m\in \mathbb{R}$ corresponds to the unique line with finite slope $m$, and $\infty$ corresponds to the unique vertical line.
Algebraically, we can define the usual extensions of addition, multiplication, negatives and reciprocals from $\mathbb{R}$ to $\mathbb{RP}^1$, with a few cases left undefined. My goal is to find a good geometric interpretation of these operations.
Geometrically, given a line $a \in \mathbb{RP}^1$, the additive inverse $-a$ and reciprocal $\frac{1}{a}$ can be clearly constructed as respective reflections over the lines $0$ and $1$.
On a similar way, multiplication of projectively extended real numbers is just composition of the underlying linear relations, and can be computed geometrically by passing to $\mathbb{R}^3$. To construct $a\cdot b$, one first extends the line $a$ in XY perpendicularly to a plane, then does the same with the line $b$ in YZ, and finally projects the intersection of these two planes onto XZ to find the desired product:
This allows one to find, for example, a consistent meaning to the expression $0 \cdot \infty$ by including the elements of the two trivial Grassmannians $\bot \in \mathrm{Gr}(0, 2)$ and $\top \in \mathrm{Gr}(2, 2)$ among the possible outcomes of multiplication, as is pointed out here. In fact, the composition of any two functions or relations $\mathbb{R} \to \mathbb{R}$ can be performed in this way using their graphs in $\mathbb{R}^2$.
On the other hand, so far I haven't been able to find any geometric interpretation of addition in the projective line. So my question is:
How can we construct the sum of two lines in $\mathbb{RP}^1$ geometrically?
To be more specific, by "geometrically" I mean in the sense of classical compass-and-straightedge constructions and their obvious generalization to higher dimensions (for example, one is allowed to construct a plane passing through three nonaligned points, a sphere given a point and a radius, take intersections, etc.). Alternatively, since the three notions above are still valid if we replace lines with the graphs of arbitrary relations, it could be interesting to look for an answer that is generalizable in this sense (if it exists). In any case, an acceptable construction should be able to reproduce the addition of real numbers, and the identities $\infty + a = a + \infty = \infty$ for any $a \neq \infty$.
What I've tried
The first thing that came to my mind is to use an auxiliary vertical line at a nonzero distance from the origin:
Unfortunately this approach breaks down for vertical lines. The definition of addition for linear relations given the link above seems to suggest a three-dimensional construction like in the case of multiplication, but I've failed to find something that works.