The group law on a circle
It's fairly well known that you can define way to add points on an elliptic curve – defining a so-called "group law" for the points. It turns out that it's not too hard to do the same thing for a circle. This shouldn't be too surprising: an elliptic curve is roughly the product of two circles, so we should expect the same structure on just a single circle. And as I previously mentioned, all quadratics are really just circles if we define our coordinates correctly, so we should even expect this idea to work on other conics than circles.
With a circle, it's easy to find an appropriate group law: given points P = eia and Q = eib, just define P + Q = ei(a+b). That corresponds to adding the angles to the points. But if we think a bit more about what this means geometrically, it's easy to apply the same idea to adding points on parabolas or hyperbolas.
The slope of the line through P and Q is just
m = (cos b – cos a) / (sin b – sin a)
Now let's draw a line through (1,0) with a slope of m. This line is given by
y = m (x -1)
Substituting that into the equation of the circle
x2 + y2 = 1
we find that
x2 + m2(x – 1)2 = 1
(1 + m2) x2 – 2 m2 x + m2 = 0
One solution to this is obvious: x = 1.
Once we see that, we can factor out (x – 1) to get the other root: x = (m2 – 1) / (m2 + 1).
And if we then substitute the definition of m in terms of a and b and simplify things a bit we find that this other root actually x = cos(a + b) so that the sum of the two points is (cos(a + b), sin(a + b)).
This leads to the following geometric way of thinking about the group law on a circle
- Draw a line through the two points P and Q
- Draw the line with the same slope that passes through the point (1,0)
- Call the second point where that line intersects the circle the point P + Q.
And just like the point at infinity is the additive identity in an elliptic curve group, the point O = (1,0) is the additive identity in this case. (And just like with elliptic curves, there's nothing special about this point. Any other one would have worked just as well.)
What does this look like?
Here's a picture that shows what adding the points (3/5,4/5) and (-4/5,-3/5) to get (0,-1) looks like:
Tomorrow: a picture of what doing this with points on a parabola looks like.