Comment by Kovah
11 hours ago
I'm really not into math and got really lost in the second half of "Adding points on a curve". Just don't understand what the author wants to tell me with the grouping and the role of the identity element, which is called infinity but is zero?
However, after looking at the next section and playing with the chart I immediately got the idea where the whole article is heading. Interesting to see how this works.
Just to toss on some info you might already know, the mention about grouping is related to group theory. [0] If a set satisfies those 3 axioms, there's some assertions you can build off that are common to all group theory sets, and having an identity element is one of them. It's weird that it's NOT zero, but in this case, infinity behaves LIKE zero. (Imagine going infinitely along the curve on the x-axis towards the open part of the curve, so therefore going infinitely up/down the y-axis. At somepoint, you're essentially have a vertical line between the original point, and your infinitely far away point, which points at the exact opposite side of the curve, which reflects back to the original point.) For natural numbers, zero is the identity, since X + 0 = X, in the same way P + infintelyfarawaypoint = P in this set.
To use a dumb analogy, it's polymorphism where your interface is something like regular old natural numbers: as long as your class behaves like natural numbers in some key ways, you can pass them to any add()/subtract()/multiply() functions relying on that behaviour.
[0] https://en.wikipedia.org/wiki/Group_(mathematics)#Definition
There is a slight bug on the interaction. When you set P=Q or for example you can't get the one P at the top and Q at the bottom. The lines disappear.
Basically you need the "infinite/zero" point to compensate for a situation when you have two points completely perpendicular to the x-axis. AKA it is not intersecting a third point. So it intersects this special "infinite" point.
And conceptually why you need this "infinite" point is that without it you can't add points together properly.
Say for counter argument instead of doing this "flip or mirror" across the x-axis (in the interaction it is the red dot appearing). And instead the red dot just appears on the same side as the two points being added on the curve - without the flipping.
If P1+P2 = Q instead of this Q' that is flipped. And P2+Q = P1
If you try and add P1+P2+Q you would get either Q+Q or P1+P1 depending on if you did (P1+P2)+Q or adding up P1+(P2+Q) which are not equal.
so you need this red dot flipping thing happening in the interaction. However, if you have this flipping that means P1+P2 = Q' which is the mirror flip of Q.
So Q'+Q need to equal this special infinite/zero point to ensure associativity works.
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