This visualization is incredible. In ten seconds, it gave me a stronger intuition for why cyclic groups are so important than when I took abstract algebra and cryptography in college. Thank you!
The animated elliptic curve
11–20 of 62 posts
Re: The animated elliptic curve
#12My broken record take on this stuff is that whatever value there is in visualizing elliptic curves, there's more intuition to build just by playing with the curve formulae directly (it's pretty basic math) and seeing how the operations work in code. I don't have the math background to back this up but my understanding of the group rules is that they're --- at least for the application we're using them in --- a little…
For what it’s worth, in the math context (not thinking about applications), the group law is extremely natural. Every variety has an associated group called the Picard group tells you something about geometry of the variety. But for elliptic curves, it turns out there is a bijection between the complex points on the curve and the elements of (the degree 0 subgroup of) its Picard group, so it inherits the group structure this way. This is the same group structure as the usual one defined explicitly. I might write more about this when I get home.
Re: The animated elliptic curve
#13My broken record take on this stuff is that whatever value there is in visualizing elliptic curves, there's more intuition to build just by playing with the curve formulae directly (it's pretty basic math) and seeing how the operations work in code. I don't have the math background to back this up but my understanding of the group rules is that they're --- at least for the application we're using them in --- a little…
For what it’s worth, in the math context (not thinking about applications), the group law is extremely natural. Every variety has an associated group called the Picard group tells you something about geometry of the variety. But for elliptic curves, it turns out there is a bijection between the complex points on the curve and the elements of (the degree 0 subgroup of) its Picard group, so it inherits the group struct…
Yes please.
Re: The animated elliptic curve
#14Beautiful
Re: The animated elliptic curve
#15Author here, let me know if there are any questions or comments!
Nice work!
As an algebraic geometer, I have a minor correction: The graphic "examples of elliptic curves" features the singular curve y^2 = x^3. This is not an elliptic curves, because by definition elliptic curves are smooth.
Re: The animated elliptic curve
#16Author here, let me know if there are any questions or comments!
It's great! Minor correction: "In real numbers there are two square roots for EVERY non-zero number. The same is true in Fp...." "...only half the non-zero members of Fp have square roots"
Re: The animated elliptic curve
#17Author here, let me know if there are any questions or comments!
> associative: addition of additions has the same result as adding the points individually
You should mention the generic rule: P+(Q+R) = (P+Q)+R, even if it's much more tricky to show than P+(P+P)=(P+P)+P.
Re: The animated elliptic curve
#18Author here, let me know if there are any questions or comments!
> associative: addition of additions has the same result as adding the points individually You should mention the generic rule: P+(Q+R) = (P+Q)+R, even if it's much more tricky to show than P+(P+P)=(P+P)+P.
Good idea, let me tweak that...
(pushed)
Re: The animated elliptic curve
#19Author here, let me know if there are any questions or comments!
Nice work! As an algebraic geometer, I have a minor correction: The graphic "examples of elliptic curves" features the singular curve y^2 = x^3. This is not an elliptic curves, because by definition elliptic curves are smooth.
Good spotting. I actually based that animation on the grid of sample curves at https://en.wikipedia.org/wiki/Elliptic_curve , which includes A=B=0 in the illustration but makes the point it's not a valid curve.
I didn't think anyone would notice/care, but I'll tweak it to skip over that example.
Re: The animated elliptic curve
#20Author here, let me know if there are any questions or comments!
It's great! Minor correction: "In real numbers there are two square roots for EVERY non-zero number. The same is true in Fp...." "...only half the non-zero members of Fp have square roots"
Took me a few re-reads to see what you mean. Will fix!