You may have heard of the fact that elliptic curves are groups. Perhaps you have little idea what an elliptic curve is, or in what sense we mean that it is a group. I'd like to discuss the group law for elliptic curves, as viewed from the degenerate case of the circle.

First, recall that an (affine) elliptic curve $\displaystyle C$ is a non-singuler curve of the form $\displaystyle f(x,y)=0$, where $\displaystyle f \in \mathbb{R}[x,y]$ is a third-degree polynomial. Notice that any line $\displaystyle \alpha x +\beta y = 0$ which intersects the curve in two points intersects it in a third ; indeed, substituting the equation of the line into the equation defining $\displaystyle C$, we obtain a third-degree equation for which two roots are given, by assumption; therefore, there exists a third real root.

We define a binary operation on $\displaystyle C$, as follows : for two points $\displaystyle P,Q$, we let $\displaystyle P*Q$ be the third point of intersection of $\displaystyle C$ with the line through $\displaystyle P$ and $\displaystyle Q$. This operation trivially satisfies $\displaystyle P*Q=Q*P$ and $\displaystyle P*(P*Q)=Q$. By choosing a point $\displaystyle O$ on the curve, which we define as the identity element of the group, we define another operation by $\displaystyle P+Q=O*(P*Q)$. Then it follows that $\displaystyle P+Q=Q+P$, and $\displaystyle P+O=P$. Thus $\displaystyle C$ will have been made into a group if we can prove that $\displaystyle +$ is associative. I'm not going to prove this; although it's not terribly difficult, it's certainly less obvious than proving that the other two axioms for a group are satisfied.

Now let's take a look at the circle group. What does it mean to add points on the circle? It means to multiply them as complex numbers, or to add their angles with the positive $\displaystyle x$ axis, or to compose the two plane rotations which they evidently describe. But there's a way to add points on the circle without any rotations. It's based on the theorem which states that two parallel lines which intersect the circle sandwich arcs of equal length. Using this theorem, we can "translate" an arc on the circle simply by drawing two parallel lines from its endpoints. So we pick a point, say $\displaystyle 1=(1,0)$, to be the identity of the group. Now to add two points $\displaystyle P$ and $\displaystyle Q$, we draw a line $\displaystyle L$ from $\displaystyle P$ to $\displaystyle Q$, and a second line $\displaystyle L'$ through $\displaystyle 1$ and parallel to $\displaystyle L$; the intersection of $\displaystyle L'$ with the circle is $\displaystyle P+Q$.

Now suppose that we were adding points on the circle as if it were an elliptic curve. Suppose that there were a vertical line $\displaystyle L$, very distant to the right of the circle. To add $\displaystyle P$ and $\displaystyle Q$, we would draw the line through $\displaystyle P$ and $\displaystyle Q$, which would intersect the line somewhere very far. Then, from this point we would draw another line through $\displaystyle 1$, which would intersect the circle in $\displaystyle P+Q$. Now the two lines which we have drawn are almost parallel, provided that $\displaystyle L$ is very distant. As $\displaystyle L$ becomes more and more distant, the point $\displaystyle P+Q$ thus approaches its traditional value.

It would be interesting to back all of this up with some good ol' equations. Maybe I'll do that later if I'm bored. I just wanted to share. (Cool)