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.

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.

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

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.