[aerovistae]

Great guide but I run into a question almost immediately when it says "This line always intersects the elliptic curve at a 3rd point" and then subsequently "This line always intersects the elliptic curve at a 2nd point."

Both of those statements seem false, since it's possible to pick points such that you get a vertical line near the left, like by putting the point on the X-axis (0 on the Y-Axis) in the second interactive example. Sure enough, doing so crashes the page.

Is there just, like, limitations to where you can place the points? As in, you can place them anywhere so long as you're not creating a vertical line?

[betterunix2]

You are thinking in terms of affine coordinates i.e. (x,y) coordinates. The math is better explained (but not as easily to visualize) in projective coordinates. The conversions work as follows:

(x,y) --> (x : y : 1) (x : y : z) --> (x/z, y/z)

In that case, the "vertical line" will intersect the elliptic curve at (0 : 1 : 0), which is the identity element of the group.

Tangent lines are considered to intersect the curve "twice," (EDIT: "twice" at the point where the line is tangent and one more time at a third point) which is analogous to a polynomial equation having a double root. There is also a triple intersection case (EDIT: "triple" at a single point, all lines intersect the curve three times), which is possible at (0 : 1 : 0).

I'll leave it to the "reader" to work out the group law in projective coordinates based on the conversions given above. One neat trick: you can avoid inversions in the field by using projective coordinates and cleverly using the Z coordinate; this is a common optimization used in practice.

[desertrider12]

Any time the line is tangent to the curve, you treat that one point as a double intersection. You can show this by setting the line equation equal to the elliptic curve and solving for the x coordinates of the 3 intersections. The double intersection's x coordinate is a double root, just like x=0 in x^2 = 0.

A vertical line is perfectly valid (could be tangent or not), and in this case the "sum" is the "infinity point", which isn't any specific location in the grid but is a useful idea to make the math work. The infinity point is the additive identity: P + inf = P. For the amount of detail in this article, it should have mentioned this.

[royalfork]

I should really fix the browser crash, but I always thought it fitting that graphing infinity crashes the browser.

[stablemap]

We should really work in the projective plane, so as to have a point “at infinity” which resolves this issue and serves as the identity element for the group.