# 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.

## 1 Answer

Let me try to formulate the group operation in purely projective terms, without any use of coordinates and without a designated line at infinity.

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.

treating chords between points as planes

Note that the vocabulary I chose above was purely 2d: points, lines and a curve. There is some value in considering a 3d embedding when you first learn about projective plane, homogeneous coordinates and so on. But in my experience that view can become distracting in the long run for most operations. Yes, you can view some coordinate vector as the normal vector of a plane which intersects the drawing plane in the line you are talking about, but that usually doesn't really matter. All that matters is that two points define a line, even if one or both of them are at infinity. Two lines intersect in a point, even if they are parallel or one of them is at infinity. And a line intersects an elliptic curve in three points, or perhaps one if the field is not algebraically closed. If you have a decent intuition for these concepts, imagining this embedded in 3d will add little value but complicate vocabulary.