Visualizing elliptic curves
Some aspects of elliptic curves make more sense when you think of elliptic curves in projective spaces instead of affine spaces. I was reminded of this when I recently came across the following incredibly cool graph by Rupert Millard that shows the parabola y = x2 and the cubic y = x3 in projective space. (If I could give some sort of award for the coolest graphic on the Internet, there's a good chance that I would give it to this one. But I think that cubics in projective spaces are inherently cool and interesting, so there'd be lots of bias affecting that particular award.)
Creating similar graphs for a few elliptic curves sounded like an interesting project, but after a bit of thought it seems that it take more than a few minutes. I haven't had a chance to look at Mr. Millard's source code yet, but I'd guess that making the graphs of the parabola and the cubic was relatively easy because it's easy to parametrize these curves using rational functions. It's also easy to parameterize some elliptic curves with rational functions. You can parametrize the curve y2 = x3 by
x(t) = t2
y(t) = t3
and you can parametrize the curve y2 = x3 + x2 by
x(t) = t2 – 1
y(t) = t3 – t
But there's no rational parametrization possible for most elliptic curves.
It's possible to parameterize elliptic curves using the Weierstrass ℘-function, of course, by
x(z) = ℘(z)
y(z) = ℘′(z)
but the ℘-function is definitely not a rational function.
With any luck, I'll have some free time soon so that I can look more closely at how hard it would be to modify Mr. Millard's code to graph elliptic curves. Maybe it's not as hard as my first impression said that it might be.