# Geometric interpretation of elliptic curve point addition in projective space

I am familiar with the idea of considering an elliptic curve in projective space, particularly thanks to this excellent Crypto SE post. What I find most satisfying about it is the fact that it unifies the usual elliptic curve points $(x,y)$ with the very distinct point $\mathcal{O}$.

In affine space we can interpret addition by drawing a chord between the two points (or tangent if $P_1=P_2$), finding its intersection with the rest of the curve, and reflecting across the $x$-axis. Unfortunately this falls apart for additions involving $\mathcal{O}$.

My question is whether there is an analagous interpretation of addition in projective space which also works for $\mathcal{O}$. I was hoping to discover one by treating chords between points as planes, but haven't been able to come up with anything.

To add $P$ and $Q$, you connect them with a line, and intersect that line with the curve to obtain $R$ as the third point of intersection. Then you connect $R$ to $\mathcal O$ with another line, and obtain $S$ as the third point of intersection. That point is the result of your addition.
In the regular affine picture following established conventions, $\mathcal O$ is $[0:1:0]$, the point at infinity in $y$ direction. So the second line in the description above would be vertical (constant $x$ coordinate) and $S$ will be the mirror image of $R$.
So how does this behave if one of the points of the addition is $\mathcal O$? Suppose w.l.o.g. that $Q=\mathcal O$. So the first line is the line defined by $P$ and $\mathcal O$. The third point on that line is $R$. Then the second line is defined by $R$ and $\mathcal O$, so it is the same as the first. And the third point of intersection is now again $P$, leading to the expected identity $P+\mathcal O=P$. Working as expected.